From e8bd56ea08cd43b7b5e319ae9eae92ca014eea37 Mon Sep 17 00:00:00 2001 From: krahets Date: Thu, 6 Jul 2023 00:08:29 +0800 Subject: [PATCH] deploy --- 404.html | 16 + chapter_appendix/contribution/index.html | 16 + chapter_appendix/installation/index.html | 22 +- chapter_array_and_linkedlist/array/index.html | 16 + chapter_array_and_linkedlist/index.html | 16 + .../linked_list/index.html | 16 + chapter_array_and_linkedlist/list/index.html | 16 + .../summary/index.html | 16 + .../backtracking_algorithm/index.html | 16 + chapter_backtracking/index.html | 16 + .../n_queens_problem/index.html | 16 + .../permutations_problem/index.html | 16 + .../subset_sum_problem/index.html | 16 + chapter_backtracking/summary/index.html | 16 + chapter_computational_complexity/index.html | 16 + .../performance_evaluation/index.html | 16 + .../space_complexity/index.html | 16 + .../summary/index.html | 16 + .../time_complexity/index.html | 16 + .../basic_data_types/index.html | 16 + .../character_encoding/index.html | 16 + .../index.html | 16 + chapter_data_structure/index.html | 16 + .../number_encoding/index.html | 16 + chapter_data_structure/summary/index.html | 16 + .../dp_problem_features/index.html | 22 +- chapter_dynamic_programming/index.html | 16 + .../intro_to_dynamic_programming/index.html | 16 + .../knapsack_problem.assets/knapsack_dfs.png | Bin 0 -> 105862 bytes .../knapsack_dfs_mem.png | Bin 0 -> 105434 bytes .../knapsack_dp_comp_step1.png | Bin 0 -> 64074 bytes .../knapsack_dp_comp_step2.png | Bin 0 -> 68113 bytes .../knapsack_dp_comp_step3.png | Bin 0 -> 69230 bytes .../knapsack_dp_comp_step4.png | Bin 0 -> 68402 bytes .../knapsack_dp_comp_step5.png | Bin 0 -> 66215 bytes .../knapsack_dp_comp_step6.png | Bin 0 -> 67079 bytes .../knapsack_dp_step1.png | Bin 0 -> 62880 bytes .../knapsack_dp_step10.png | Bin 0 -> 66144 bytes .../knapsack_dp_step11.png | Bin 0 -> 66009 bytes .../knapsack_dp_step12.png | Bin 0 -> 70384 bytes .../knapsack_dp_step13.png | Bin 0 -> 70695 bytes .../knapsack_dp_step14.png | Bin 0 -> 59214 bytes .../knapsack_dp_step2.png | Bin 0 -> 67470 bytes .../knapsack_dp_step3.png | Bin 0 -> 67887 bytes .../knapsack_dp_step4.png | Bin 0 -> 67682 bytes .../knapsack_dp_step5.png | Bin 0 -> 67313 bytes .../knapsack_dp_step6.png | Bin 0 -> 65647 bytes .../knapsack_dp_step7.png | Bin 0 -> 67878 bytes .../knapsack_dp_step8.png | Bin 0 -> 68890 bytes .../knapsack_dp_step9.png | Bin 0 -> 69028 bytes .../knapsack_example.png | Bin 0 -> 70230 bytes .../knapsack_problem/index.html | 2765 +++++++++++++++++ chapter_graph/graph/index.html | 16 + chapter_graph/graph_operations/index.html | 24 +- chapter_graph/graph_traversal/index.html | 16 + chapter_graph/index.html | 16 + chapter_graph/summary/index.html | 16 + chapter_hashing/hash_algorithm/index.html | 16 + chapter_hashing/hash_collision/index.html | 16 + chapter_hashing/hash_map/index.html | 24 +- chapter_hashing/index.html | 16 + chapter_hashing/summary/index.html | 16 + chapter_heap/build_heap/index.html | 16 + chapter_heap/heap/index.html | 16 + chapter_heap/index.html | 16 + chapter_heap/summary/index.html | 16 + chapter_heap/top_k/index.html | 16 + .../algorithms_are_everywhere/index.html | 18 +- chapter_introduction/index.html | 16 + chapter_introduction/summary/index.html | 16 + chapter_introduction/what_is_dsa/index.html | 16 + chapter_preface/about_the_book/index.html | 16 + chapter_preface/index.html | 16 + chapter_preface/suggestions/index.html | 16 + chapter_preface/summary/index.html | 16 + chapter_reference/index.html | 16 + chapter_searching/binary_search/index.html | 16 + .../binary_search_edge/index.html | 16 + chapter_searching/index.html | 16 + .../replace_linear_by_hashing/index.html | 16 + .../searching_algorithm_revisited/index.html | 16 + chapter_searching/summary/index.html | 16 + chapter_sorting/bubble_sort/index.html | 16 + chapter_sorting/bucket_sort/index.html | 18 +- chapter_sorting/counting_sort/index.html | 16 + chapter_sorting/heap_sort/index.html | 16 + chapter_sorting/index.html | 16 + chapter_sorting/insertion_sort/index.html | 16 + chapter_sorting/merge_sort/index.html | 16 + chapter_sorting/quick_sort/index.html | 16 + chapter_sorting/radix_sort/index.html | 16 + chapter_sorting/selection_sort/index.html | 16 + chapter_sorting/sorting_algorithm/index.html | 16 + chapter_sorting/summary/index.html | 16 + chapter_stack_and_queue/deque/index.html | 16 + chapter_stack_and_queue/index.html | 16 + chapter_stack_and_queue/queue/index.html | 16 + chapter_stack_and_queue/stack/index.html | 16 + chapter_stack_and_queue/summary/index.html | 16 + .../array_representation_of_tree/index.html | 16 + chapter_tree/avl_tree/index.html | 16 + chapter_tree/binary_search_tree/index.html | 16 + chapter_tree/binary_tree/index.html | 16 + chapter_tree/binary_tree_traversal/index.html | 16 + chapter_tree/index.html | 16 + chapter_tree/summary/index.html | 16 + index.html | 16 + search/search_index.json | 2 +- sitemap.xml | 169 +- sitemap.xml.gz | Bin 812 -> 823 bytes 110 files changed, 4197 insertions(+), 99 deletions(-) create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs_mem.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step1.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step2.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step3.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step4.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step5.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step6.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step1.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step10.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step11.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step12.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step13.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step14.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step2.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step3.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step4.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step5.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step6.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step7.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step8.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_step9.png create mode 100644 chapter_dynamic_programming/knapsack_problem.assets/knapsack_example.png create mode 100644 chapter_dynamic_programming/knapsack_problem/index.html diff --git a/404.html b/404.html index 490dd5645..06046398a 100644 --- a/404.html +++ b/404.html @@ -1889,6 +1889,8 @@ + + @@ -1939,6 +1941,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_appendix/contribution/index.html b/chapter_appendix/contribution/index.html index c8cf7945c..2a3190702 100644 --- a/chapter_appendix/contribution/index.html +++ b/chapter_appendix/contribution/index.html @@ -1900,6 +1900,8 @@ + + @@ -1950,6 +1952,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_appendix/installation/index.html b/chapter_appendix/installation/index.html index 0322224c5..5b63df84a 100644 --- a/chapter_appendix/installation/index.html +++ b/chapter_appendix/installation/index.html @@ -15,7 +15,7 @@ - + @@ -1900,6 +1900,8 @@ + + @@ -1950,6 +1952,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -2405,7 +2421,7 @@ diff --git a/chapter_array_and_linkedlist/index.html b/chapter_array_and_linkedlist/index.html index fd879af16..73c47f738 100644 --- a/chapter_array_and_linkedlist/index.html +++ b/chapter_array_and_linkedlist/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_array_and_linkedlist/linked_list/index.html b/chapter_array_and_linkedlist/linked_list/index.html index 07927c550..ceaafe1bc 100644 --- a/chapter_array_and_linkedlist/linked_list/index.html +++ b/chapter_array_and_linkedlist/linked_list/index.html @@ -1964,6 +1964,8 @@ + + @@ -2014,6 +2016,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_array_and_linkedlist/list/index.html b/chapter_array_and_linkedlist/list/index.html index 0677c4a91..177047d27 100644 --- a/chapter_array_and_linkedlist/list/index.html +++ b/chapter_array_and_linkedlist/list/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_array_and_linkedlist/summary/index.html b/chapter_array_and_linkedlist/summary/index.html index 2bac34bf0..26a79aebd 100644 --- a/chapter_array_and_linkedlist/summary/index.html +++ b/chapter_array_and_linkedlist/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/backtracking_algorithm/index.html b/chapter_backtracking/backtracking_algorithm/index.html index c810bdd46..459ba2b42 100644 --- a/chapter_backtracking/backtracking_algorithm/index.html +++ b/chapter_backtracking/backtracking_algorithm/index.html @@ -1978,6 +1978,8 @@ + + @@ -2028,6 +2030,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/index.html b/chapter_backtracking/index.html index c960b8288..ee53db9d8 100644 --- a/chapter_backtracking/index.html +++ b/chapter_backtracking/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/n_queens_problem/index.html b/chapter_backtracking/n_queens_problem/index.html index 7ce6f833b..53a8e86c8 100644 --- a/chapter_backtracking/n_queens_problem/index.html +++ b/chapter_backtracking/n_queens_problem/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/permutations_problem/index.html b/chapter_backtracking/permutations_problem/index.html index 4dbbfd620..8ca5a3dfa 100644 --- a/chapter_backtracking/permutations_problem/index.html +++ b/chapter_backtracking/permutations_problem/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/subset_sum_problem/index.html b/chapter_backtracking/subset_sum_problem/index.html index ad3b08d21..73091cdc6 100644 --- a/chapter_backtracking/subset_sum_problem/index.html +++ b/chapter_backtracking/subset_sum_problem/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_backtracking/summary/index.html b/chapter_backtracking/summary/index.html index 41f6fb553..8fbb2476f 100644 --- a/chapter_backtracking/summary/index.html +++ b/chapter_backtracking/summary/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_computational_complexity/index.html b/chapter_computational_complexity/index.html index c31cdd8c1..12bd5c365 100644 --- a/chapter_computational_complexity/index.html +++ b/chapter_computational_complexity/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_computational_complexity/performance_evaluation/index.html b/chapter_computational_complexity/performance_evaluation/index.html index 82ee791d8..fcff461ac 100644 --- a/chapter_computational_complexity/performance_evaluation/index.html +++ b/chapter_computational_complexity/performance_evaluation/index.html @@ -1977,6 +1977,8 @@ + + @@ -2027,6 +2029,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_computational_complexity/space_complexity/index.html b/chapter_computational_complexity/space_complexity/index.html index a1b2f1adb..e59d3330f 100644 --- a/chapter_computational_complexity/space_complexity/index.html +++ b/chapter_computational_complexity/space_complexity/index.html @@ -2005,6 +2005,8 @@ + + @@ -2055,6 +2057,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_computational_complexity/summary/index.html b/chapter_computational_complexity/summary/index.html index ccdac2d57..f3f092ea4 100644 --- a/chapter_computational_complexity/summary/index.html +++ b/chapter_computational_complexity/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_computational_complexity/time_complexity/index.html b/chapter_computational_complexity/time_complexity/index.html index 7e7ab9453..3478e8cf8 100644 --- a/chapter_computational_complexity/time_complexity/index.html +++ b/chapter_computational_complexity/time_complexity/index.html @@ -2053,6 +2053,8 @@ + + @@ -2103,6 +2105,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/basic_data_types/index.html b/chapter_data_structure/basic_data_types/index.html index 34cca17b2..3e2d96bce 100644 --- a/chapter_data_structure/basic_data_types/index.html +++ b/chapter_data_structure/basic_data_types/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/character_encoding/index.html b/chapter_data_structure/character_encoding/index.html index c7e37a2a2..a7b02b309 100644 --- a/chapter_data_structure/character_encoding/index.html +++ b/chapter_data_structure/character_encoding/index.html @@ -1971,6 +1971,8 @@ + + @@ -2021,6 +2023,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/classification_of_data_structure/index.html b/chapter_data_structure/classification_of_data_structure/index.html index 0632449a2..83a1d8532 100644 --- a/chapter_data_structure/classification_of_data_structure/index.html +++ b/chapter_data_structure/classification_of_data_structure/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/index.html b/chapter_data_structure/index.html index 476997b84..8bd16704a 100644 --- a/chapter_data_structure/index.html +++ b/chapter_data_structure/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/number_encoding/index.html b/chapter_data_structure/number_encoding/index.html index b2ccda969..74dc9684b 100644 --- a/chapter_data_structure/number_encoding/index.html +++ b/chapter_data_structure/number_encoding/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_data_structure/summary/index.html b/chapter_data_structure/summary/index.html index 0f2bda460..c4ea2369a 100644 --- a/chapter_data_structure/summary/index.html +++ b/chapter_data_structure/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_dynamic_programming/dp_problem_features/index.html b/chapter_dynamic_programming/dp_problem_features/index.html index 072aa0ed6..3f0eb6269 100644 --- a/chapter_dynamic_programming/dp_problem_features/index.html +++ b/chapter_dynamic_programming/dp_problem_features/index.html @@ -18,7 +18,7 @@ - + @@ -1902,6 +1902,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -2650,13 +2666,13 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] -
  • + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_dynamic_programming/intro_to_dynamic_programming/index.html b/chapter_dynamic_programming/intro_to_dynamic_programming/index.html index a6fd92bfb..cabeac839 100644 --- a/chapter_dynamic_programming/intro_to_dynamic_programming/index.html +++ b/chapter_dynamic_programming/intro_to_dynamic_programming/index.html @@ -1902,6 +1902,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs.png b/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs.png new file mode 100644 index 0000000000000000000000000000000000000000..fa1e10926d5bb728c368c9073fe8dea3ce4b95cc GIT binary patch literal 105862 zcmd43by!qw_XawHql7flAPv$;cY{cGcXxLqBHc*0ba$t858W-@o#GjM_5FV5`>yl< z88+A6%+6;$EAF+{y$P0;{(u070|x*A5I%|t%L4!}A)8QO7%0e!pFkW4Spgm7KL`TK zM)CF_e|$Dl{b(#D1)zd#!vJ0aQ2{TW4}tsufH=QTem+GUi2LuhJn-G0V<6WIGy{Ns z9}fViL)On9UC8hAIsdF*WCH(l#Z0KbN52GQzWBQh1%@0A@DUd9gsfm~#nc=C0L1R+ zU*K12L|n*o13n51D7pd<)1Vvhdy?{OMg=%_#E=DuVPKI3QG-$~U%c96VBP$9pyuKH zJ#T(Kt>QpyER2Uq*Q1x!*wP-67>+20ABL1P8)^{6Xc=#;?ZYpnLK~M1hp`a~&!4+y zN-B^~tCzKk&($xbulT{jRW-rBFhT#d@He6LOSF8Fl9H;etwlgU80lk+W72STX62Y$ zsj}A_B~NdZ5BaZiAb$ctf+D%Fl4x0^{akJ67#Q(!aX)r;DEeueg_6WS8T3U!?#{-_ zy8YvaEk(VXgs9=CdPPPQWZ-`HO?mP;jaS_&k z3ko8-T{|f0S1Ff9Hsb8&P`qtKzVkr@oR04<4DR68FgxbFZ;qEO_A!eqlf0!~aV=Y9 z&tm&&MQFdb?6J>(^S_cu_5;Cot0ahNo!jBe8CL9RUx)3~+JX_6qye$kvt?RKusxyo zPTM+A!r89#i@hT-s^hS>d@iOFv-sqyJY;D9GnL3dlC{aNyo+M&Tr2)(R891q_+=d%U7$6hSGUF#4bi` zyqV5KXWC#lejM_Ab*)<|euMGPQ~I?L>4qoQDwmBgE=vGu_*>>bYe7-tmU0lgOcltL z6;Qlnpnm&)r+Vzvf(~5gQtz1L{vyrXV)+RDAGz&9`Jq!Q&nX66?2MT-`3&hz=1J|6 zf#w?QHbuj*#se&yod@cy7FRq@S{0-@jP|DrVZ?OEm6Vj;SYB+8I1a9OUW~L=f3#QC z(2$^g7nVZnFND^?WIRmRR#tXADZ}^kuzGBJIEl$_qZ_PWu2CNm`2BkP13|Rq@ls>f z04I7Q#GoSip62L)B%je5A4k|$H5=`Xh$(4DL@>2JkIHbj{RIJe0NbF0pYBJk+Itd@XOG`_$3U8G7<A{>0*MI?49HC_NOM;c6PJFH$tX#?T_bR z!wk>!ftk(4{psTNvv8Z+3E3Q@p?EdYc*&(GPuL|Q*14NipH+NkEcTsL2fBu}Pl&Mr zNVlGY*5Y**GDtukHbu3OZG#wBp)Ii_h`XjaE19a)oX)? zG!~2atxXAxgVRoE0@A=WwY*%%X!_R6n506ju$QLeHZ5nOIyw0As1$qtu>DI;@h(L* zoh5Hb{+M~Icn38Ht!4eU);gijE9d z9UU=V|JePWIzj&XZ15sTR_As;K0dyxDeXi%kvc6djY03ziDgL_Nk zRTCeGQD{9E%<|O$SwnFl7<6=WbpVe`+(9s+nP!DzvjS*0vj_DcJeYvo>k(u(DsQLq zoqd44R-x5#BjgXJtgpWT+HYD<++y7>Zoagd!8FfAFx$o4 zm7U`=iq1AhIE0J~e29Eo%bewgh{fO=-2s1`#%`x?YfG;MXyEa%Ia?1Ta7PmAYA6p} zp=P#A?SB_=eN@hsBSaL<8?cF8@G`PXHUA&qf1l)Q@v{_fW}@*r!WZIpV|$^bSF-(3 zS|Db;N_Y8fGFt=A0tQy2(^ zb;Vf>cV+tZgMtsXmGZc+e%0G!mvU-(-XsH`yC+jW19mE*7q%T@raTe#r6)zVfdvXn zVFtO42W8Evr7X(Hh?|&nn!6KVm@=*AnLD#)*Vm~&Pj@@%9(3tDc%L@Ds%Uox9o2R) z?cP8xd8IYIPNX%Jb+oh^wXdc?*YbpClY%f)BtLWPw{1|zBtFS@B9!^7fw4bBp> z12K%&led_$kFsxu={#;=N`4eocR2HznEgR-dn_t*uBj7K;RVzrGXerE)~Mm zpTwM)+{gmVh>WKO=hBH(m!#3WFBY+Q&DfQugpBRiK;;+EmH^|WG>X@8s$V(N+@ZEAChK!Xfp$7?vxj*4+>{*-Z$@=kf zv(Md-j=O<7t0V_kO&SE^Bs#@$Hkmi+6Dq(mh0kn8@*VeX=>tAsd8ptx?qSzi8Dzar zOaWYz;W8PANqy?!HW>yPBXG1%4O<5UB%yr2lx3%|I`$ajxyh_)4B4JD5)prhwsre- zA<~SxpgE~pmn5tE{M(LmD=Bcy=Ap9{|1#vDe1YxUQ^Tr2wbVSDfHAhww2LHg?x(IrVB|se)X#h%(WlBy(ak@-*-D z08AP-yu?Zk)V#4Ph|xLMi>3Rfu%$R71V{W3>gN2SEN$5+X^yXM{F%*^q9PuFC1&|wOi4-vjLnt4X%eEQ3QQ%I?5H1i zM_ncNH8VY`dAanG4ik!w#D$5gxb?S(#kY5oo$xl)gve*(G35>f5?R zU8vAm?cXY&X#>NsF!Sgf6;wcS)7zUeJ4N5q(HkUMbA0E+dk(V-Az73I|6Zl;Kjs%_NS}z$%hVgg>C~6m+Um)oUuu-NprOXo|f~x zk?z^oL3EA3=uu1%>YBsQgO_g!6Xv~TjYS(4SD3nQD}^U|b?CeXBNSDu(-O7+{|pj*f>6$pXQ7cf+;lwqsJ&*vxzw(E(WfHUblyN z)i9GzFB6+IK?6L{N_xRszoy+;{4WG66zsUs>dsR6Y2PoZMWegU3bbUYd8ad{=u|`~ z%OL{E?g+zS3A;{Xj2Ia1XNt_Wl_#5Q+JDxGZ2W5ZjRDu3CK(V(NHSO3Wp0w{8>nMc`Y!rbp{y5`0lDaPoLD zhIHi8(QB=&?~R*N;|ZQ{)JEPm+1x&JE^Tj4_?sX{g@8=QO;zVhP6a+ffJU|2(9VRL zf#$besJ?Qr^gmAO7p(4!Zru2!jO9xe*WRWpk0{X~ZMJuq7nVrziEfFXgghG)T=;3~ znKI0w=FBQ_zjdb)ajIgmWnt#q-Xp*Xo{+^-$Q)A!NOo=v=Qy3Nb!G}R)RV3-boh^u zX}w}E@QAGpDd&2GBzkBT;nW(J{kiZ6gu#1XbQNt+Eq5oBvoO}yYC}-|z*$8i{-J}o zl9w%3mGpU10F7l$PE5j;?<`vk0mp=q$Hg_mDW{H_wD(f z8fRDhaUD6xKpfr$;YE!~Erneu3?F%bhr&bI$E@Vyg6AL;RR$X6zL5MucA=%GU^I?D z<6E0tbv^?jCkRk7UydS94p8-2X2{0(+dLHRJEJ#w!k z!SD1y3~<$tz9_s^pS)04t6tV(gBajhWXE1};7fKC&a7~j0 zvU6ASzfsfs+&49RsNk~d#kP!=xVy)$Yi#40i1LeDT+~Iw<<*&MB=kuHMK!|M_Bx$v zg>UDI<0n_E2S-B&o#Yci?-$nZ*n3SmSa`EH)r=p~muy4U#Oi0k{MQ%kCakW1zpP-I4n6joPXrP2FBmq~ojDUIl z_7|~d9;z*_mGqs_Bp+1O1~K<|9L48w_qK^HM5$>0SQn*u2Ziww*wCi=eFw(Z?5~QI zza60g?CEu{YtR=(ChUzklA6VZ+1({=@cUnkl9;fV zoCX-8Vv?O=YT|<1@S;8aciSm8*&#^?LnHV{t(2&(LQ)5`CaH4GE0$=m-$-qmkLpw@ z)kbHpNIvK3WmCf0G~t`r&j^WLqoGN{Q;TsXII*F!-OBe_zLx-RE^L2n!^LkyYM=0 z+&MIgxI%YP&Y_$UV0a0Kx;bpG5HP5^#JXB0wK+nFlJ^2u-FEPCEkgXu z#xNIBtUl@GrGHlrORUi=?suZVPqI$F1zqehU`wUj_%d_Gvcy*|O=vd(JG3Rr*<|QG zILmVrf5fwhs=h&ez1fNvR)h5gp}L0?7iUb>=%&F7wnfw_F9nhHj1`vE;S1CLqDv{_ zv&^64&lU{W(_@i$@C0O|2x#_?D^7&_XQBYWDIEJLS;7hHraf}YYU=>pP!|b=t2ggo zkuH*UxsFxg(8i>!7v_0m=MD#Lv8}k0Txti82@ZqQ3+};^8)ALn%e3%rjBxr&n{yLI z>k`khrhO_5TR5kq2q)OCe2+Sen#;Cc-kDMZvdJ5M8vPK=->yvT1=P%Nt(xy-CU4QF zXIUE303X9qbZ^nW@s+n;s~dNH-%qZVC^|@Q5pEjP3XP#oAoext;2vwH%3U&<5}LL@gC= zFDYf|OTO7%QhmC+@%eRwtu1kdeE~nFsvncuDSZR1j_aXhIJp0^5`i*XIU}i$+%Oh_ zPw@+^z_Y9LHo2Ynk1#V=VZ{J94C?6^}r zId$n`-423!mR4x}e>g-KDd~MCfIT&7`}OqD`ch=8%FRGF*k#EMI-&tyjQlOnFu1LMB9%AH4!-6c2Ry9WzSnUYYT)zFW3ka=!VC zR8LWZ=n}d?6wEcuxre2>`REk71fV*ZoTHoV;(r`6{|O4e=)#mjPfw2k(bT%vd%5JG z4SSD)2f9vQvpa*bL&2xEr6wQK8O2)RWr{+)Kb34kE!er)Pfp3zI}*LuAgsym1SF4^ zABI5bSpwgENQpXX6x3w)OL(TL2lzuvg;lzWmy2v~(WJnM~_LG}wHf$Xu1)I@*Y;}=JAI-_B8Wt8-A>+Ge*vMlk(oEcMn zLAGCh0TjKSuPp|I@+uL7971^tP!#@qe@rOUj=ghIC8YnoeFKQpcXMO>N}|J?ZWx_mxL7c6vP=k$?xWG$|%hf)mBYj9^@_}(BX48=}Q{PZ9P(6 zJe1HiMgIxxhA_a-v@BLuR<&Ld60h~HEg|ihk&%(ZzHr=x1R{1tso&!ClZBL1thCSf zVhpbO)YUZv(cV?%!+;w*+FMc|+3H%n&$``WI(c3y-~C0V8<5#RGs%tnjSj3q@z$zM z#y9)-d2LK(dAOaUv-yp+!J*)ekmSW}*&=Xk)*dmYMV3l$A_;0|MgFH3UzFl$E^A}*zHPfGRxM5Vp zKgE25`w~5y30`Mvg#J@a;Lk&e0*uyBl->V^cX1tM23#+Rv0*AM6&c?M8U0ZsKlo?W zBSKQ2epWC)XHbko>#DQJC0%e~*=|JX4<2$c+S-FK9^|Jxf8GlO!Y)N)7vCz&S{frWqfHY&lP3c=je zy1=|ZytGLXLuq@bF6HBc>9Sn-o>XaI!NfmVuOW_KECE51vct5>@5#t)hY7=Fe>&Ll zgPKiuKI-m_Z4vo7;_<8P>Qvb#>@R8}g!(ALPY3>Z#jg0f zwIqfB^*WEH!MhhhAyD!_-EIg@ymi0cBI)u$>7nT6$^%;w@YiJhq*0#Y7-;b}*-ks9<<|!;=VR;om}45Pt}2)N98OUUqbiXGu)Nccj8PjN-bq zsr+mz(sypt>qR(bh~~F;KDJ`Axhd3Xj?x%T@f?^UQqbbPARoCX(!un3YM9>%=NZHQ zx5`Hxl?~A?h0D&ux!>wN3zCfy5)X9Ew~kQwwDGv%*YjeNG|0icX9isZ$%;%sOJVsS zqy74jV$AcpC!g+Te~8>xz}9g^7r#W#RwWWa&vJ5H3ZkVZ#4Bv2XkU%8^LWis#=~x} zCVeweV@C~csd>sZ;q_kFiWFH0_|x8k*oBxJl)azyICQ@LvnlbfqF`yWK7J3z)T5;B zr#20*$(E)Lq4KOZk;+_cU27KRJsaPI=ntMUEBpi15R}ziyZ2siF2rxNUD_djd|ZP@ zTFZFsJZ70jn9h0By@B>&{WaFutfeHUxTjQMS5>8edu`*o>mL|${yNEGL0`7#bv_+u zw1$RO9zvS3vZ`^kg>$J|Q-=<>Rv-;UbO1}KGuv; zlW<;-N8ht|4qOAN6V&zPH5Ief_qrnj7;aUR-?Ug)Ky>o|iRf>LGv9YmUREe7KHKgt zG*pR5x0Vb3^3`B(?woz(G=YcVa@U~&`d00jj)&L90ZTZ`gT*!d;v1GnNTWWi2n(Is zE7dh@Y?znD?lfm9Ra2tw@nUPt=UOvWVk0A&({Uk*WXy5jWvr6NrM%(5c3H1^u-FDy z9f!``=7}tvP2drC?{8Tvh~FMH3LrhYlP&+uI^dT_?(hzU%YC;JU~4DXE0WYNri$(I zc78pZ?ReExZ%5!rnhX_7(|$s3eq!t`o5xCy1ZpEIGs328>lq2P=!#=+0#=!2v>SyX zYoGyDUl=ySmyU&Hi)AucX~6ab_1~H~ag-&*nqrTF3%H-HDM}dhCB81W%i!umvuw%z zwtLD`QB{Ezj9oq>hrG+}xKkt}?!~r$#$tdZ-H5nIJ_28)1$E z4{E0l=fzNbW>aNvE_0Gs&T)UTROd5f}d^uA55A z+j`Nwm6Y8G;%0^^*bTHS&R#aYtRH=(8XdT$V#;Fno&SY(K@c5acu37m%sy{^xu|@SUiCpyOP{2#(d%x`EIGYGG$cImIpXslj zIacF>xOJ9%tx&TB`_FX#j#p+XY@Tx!M_^^ih6{=hPUI2@N4#}hXZF=GrwyCVvydn< z#J9aSevp1 zGHox7wcQ>joIF7V98R2+E_j?2=_puY3iLyV!t@^)>aQwd?(qxib}hMzo1>uF{5MvE7NpPT;IqtcUH0Lm80 zS=>{$xlV~&Bl4q5>lJ+%+FPafoH5gbhm)1VvknxRrWP>Gja6PZ(^-E2dk}vW1bIr= zQDmP#V{QHz)Q(U4Oxe9u?}W^6K94ADNkp=phoLah#7xJuK+O@`S@te+?4BMD1qxN+ zXka_p_vX5F!7A7?U~jEXA=hfTG*oVk-~Y_$kzut1fBxuC(I@k*`q{PoULa5oioNh{ zAlz?S1GWHwZBK=*b-#4qFL{@HSZk%?fYE>{q5a@Ji3>+Ph+eZvZ*MAVu{C&${3ti; zuVf*8H`p~&&v#TYzY#qLQ1jdB(zz&-a@Ac!GfUYh!J+!3|9tNl9G}%w&;R$`3&q#O zmZLFe+Y`T{cQ!5FIO(n(%cv|Z-`q^&|0x0RgXBXYNu)Us{a58@;1rY4(Y&ZaZnOS> z8Daf{6O3Z_=zlVme=!%N)&YV%XPE17QO;jk|7Rzt!u!R)Qu%iSTL2;+lDE}V|0H35 zrS^M2uw?c3>4wC9ow@!@l?wwZCOknuJjkj3$-VeF(IE4jGc z!-kb%O(xV&w*Jl9yC8K9lUpSIX-H1cFAJ3*)l9=i!6rnV!8|=WG*C`cB$>QXK=|jO zqMjh}Ve%Jt@t5Z>kVzTzz}+F)A-XO4_c8nq7L(8U-+4q+!k-};vO~rU{|i`f4>>$Z zVr^fcoNkbQaOs{xV1z!MogP*^i}-I&_v$(2$qXzl`kgK$bU-4DaYXV~xLm^+4ETz{ zOF|&~y9qmAP|He>uB8DFj4>c@8BGj%g0HQZ?GH z{1?P5zQw<@NXTfSM4!jDt5-zzJT|Z!6c*84 zz31flx2q2>_=|28+qyd!yFaOMs%2Z`mNmYnEXP^hgc^zv2Q+fjay$+#B%t%qgbVlB zx=G}a#(dUKM2}4LOxf$h@?C!DcZ7b9a?gt&G9>O1$-eo`*-H{7g!QtsFgJHToU622 ztUFq)C*bvDzZI^kuKG|D!5-z*A#Ina6`Q6I8(Yg3X?3|fzS$Q^z-GM!Y025!%Xc#4 zm1ODvnMr<-don~lWzdwo|Cuc_Pa1a-)`-7JE|gNVu#tT+JiCheOJLhz8G7#!twF8hpAOm60Li!||}RiJP|dXrr}V zH;XAm{)nUa;dw~A9#G-o;Skze4w+#2@nSuF4Io(6(Za%l7VSCj9{&D=&vvZ?%pCG>%>_cq zg8+^597XBphwX=fx@^yOLdk_t-HtXGRIOUVzaSAr&>0C`ZX~>soSaXDaG(;gw%fv} z5nEeYTJW%krRF!4@smaW!-yP^gxFn;>Nx+eONv4+IiDUh_{S6aZkzmi z`YFeCau!b1mNosB*)(c02~v;CGIOVo5%fOm!BiIPwk;7ve9Jut7tDC_Ab72P^|kts z+2s&G%PsSd`_BEIro6~ez-WvBYDr}~BEs66) za1?keUtC|!@Os=0~c$z8hhSOsAz%+p!&i9uW}Tq+D~|27zl zkTdb~)l4KtU-1Y4OPqWawr=w5L-_Myb)yXoP!!2;D?=;Woj5VI2%(N(6;)K2{Fvh9 z>r~NbCYc%in5tDPv(Oh!q7FaA&4&b0x}L1G!fWNV-AZUwLy(@SK^+1vD}xeCyq@KE z%^or1;B`ZvCaR3zBI0L*VbBi>pUL3S>abkyFIlCVW)%DRMVk!+(xKK>*5O@^Zi87{ zX^yrZMTTl%<{BT$8O*BfrFQ+%A1yb)1Nm< zcwksyXHNQCH6}!W0TV@_=1ohywXpq!p$ogle0x_2M+}be#rBpXjA=V+*v*Uu*_s(o z6O$5<%kGGr4EY;{Ah+)FEbu-0ogrM4i46($C!!ih;OuVg~%pt)dYrEs$Mj7!e`^V!A$N&%| z4L*d6X4&iX0QG$MAbg(y^-({&4g@xiABT8lB_UB7~z2h8vf%$WHJw zGc$us(lrgRWl-&H@DYS+Yh1!YQg-zIt_8eTt^07=h1AFueDr?cb*o86fc-zkyInOX z&!DO^k>p2jN%dESCIqeaXns#bguiy4*3%_jjA(f&+#&HtF!T$!w$=?R*B^i=^{u~+ zq65ovo5%o)w&W5dxI-kw&4h%Fgvj&UGn>IXez4z<_#GopKn!PAD=pX*IIpfWy{@L) z?kIWhcS%;V8(7Z1<3Z@opK_NQ>_`&8>xsJVI#o)=%JWUInA)yHn`(!L@w6JORtts- zX0lkTWP9#&CnSDGkY+(fJfvAb1WI(xG~rQ^Qy{AU$R3i_uYSv{r>-Rv1QuaTIu5UW zB_k&ntr4Ycaq5`nWTQP$vcOHkYW+!v`IVkWo~N9ehBP;@ZGUdX=a(1xR^8VEaF=JL znf&pJC;NCy(&Ph#11a@L$DPSE#yxqXm43S=P6;(2~(Re(C5UClQCMq^YtGynaQ8V243#9=**YhSC_0}RU#uDF?2mo8`w+8XKoV(i# z)3!!@2FMafW8}>QVGCY!ya^>1A?$%!v)x!}T&CwlSh4YWSUmo5k#^$QH>>GH!KQUU z#$RlcjpF}E2Jv4vZ6{q8kcIzsh} zJl&RWBT_gDFIH{olAw4*eADx8Ry@iT8sEc7pL9s0C~W*7<)76k`9j_ab+J=zJdzAT z3nUkc+qmA49?zCa=e~oG1<89Ue3O>OX)062WgYAXCUPa*3afa2JzhXq<$POl3QMgU z(Aw^MWON^1TpVBJlBTmR5Ss;gnHb<#dHPcOlrUqQ+>35fOwOx90GC0uTnmgLv)?UZ ze9L?|qR!&|+Oa=@I1SN{)>;*qaWX>Q|RRouZE69K<`{A zuw#Ks+qd9@W64({0idKi1vxB@PFpG`Ik8%0$YQY`m~5Rw1}B=MEoN_|k)Rr?3@&xQaeAKWX{^0?O;Ry?m4+nD zeLN>7WG$#K23CCS?W{79PSVh`ulU*nU7-e;8v z`)i5P&;%SesHTtzA)1D?E@`?$iEThQ{ImMceOVjpeuj)4R&$5@%e|E$*b4*)2yK?{ z=jCJ|62UfdZiv>6gwDNupte(kvxf4IwK$H=aaX?TPnpzI*iYvP?U&0_h|y<|9ZeX=a^+yf+H1bBLqXK^X_O4G~(%pX^`8`l&I}w_;~*cxsX=Me;rKb+YFFO z2;%>On#pw$&bFef`ueJ`?elw5<$U8WQ2WlcW+A&{tU?69+1R6zM0(4nw~~us&))iJ z4|+v*E0{IAz(87DmJpUZ?A5{yf>nrMycs+-t7dKeaOCBYjeTFU-!UN@^>`d?HNi zvsK|pWD6l0^tOkzuUaFzgk`jkhVb*DW=_&ZjA5P-u6}&m3v=DvU!Y~49LpF-j(g$xofPv-TgcmbNwKU7(z>qsoL~nX!r^6+`hXLZ?jkX&UI{?{ zhJpeF7RbOdh<}#hOaV==e3`NO2u*P!Zg@IG-#!eEp075MNn*^Ek%+HidqKD)OIgDo zq)j7{rPMhbvB5GPDxvfnaggxayjk4K&ya@CNGwz7Ie-qDhN3{c*d(lc@Vh!>J}(p7 zd$TtA3&LFu)gfsCllg9vOw3P_t)XXhs8dyXCs*32Q_mmQ^d}j5sN5^FnVUFEuFq; zmstClSshOwJ5Nfz-Ws2msYUgn8vN1IW~n&UZ?yPhi-_@3v>EF41gSP7oJ$BGBhvCQ zK;7mwf#Kuzu2m(S&rpaA4{M6Vm+tGXO@nu%Y&!XaQO3b2^}nude#IMqm3I;*w1Wy7 z>qhfXm{5t>q$|fM53OPY>Vas>N%B?pzI#BT5K4#Y7oot-Y^PRpWCCVhY0Y$Y#m(|g zc$s*jI{nAu&4AbB$;R#w|nSDlZk@2s2F3TG#Jq5V;djhRrV0AuR&`) zYc62I_(rv21m$W4cIx>uhD%^IM~vL%`9?osjegl^Xonnk)jX#UfAkgnWD@!g+K5)l zHE_k)5t&q5Y9)3fo}DgN(jF@9{!9V$bB04&?qX2x&>k7mXnN0n1AMyvHDxicmQ9#@ zZdUi@epoU4Ru$a)41$(6&p22A#=Ua(9)dKcU-m(lB7kNatQE7ia!eg-ymS4_Hq>B& z%IY$k-4@N7!CC1_x5XvLNhwc1LjTgCL^gD5(_xY$;aZ79ga)=mk#p%TUdj9x zeDW$%dGJG9p?)3?qP!%c6wNmMz0$|MjRdoK?#-Iad@NI#k6poZ}4pg~@{ zdRGi9DJQ-_yXc%E^GX*tzxHlhnPj1v1RwoueS>e8luGOU+63bS& zahsiA_aZYASk9JBD-na+L5cL4DK8e>eUn}v`g=v!)1RA-%=6T)|4QqS?<|c!fj!>YncB!p1U+vCW0~X8o7xW#qE4# z6VBL8pHDXt7Tqumi*bdN%xRt09K*n{-J_EeC<*O{Ag#9)YqN{C4JsO_6+klA2!usk ze$zD^8MMCeUSNW00S9=ZKfwse*W)9VzM}eBqpTSX61%npY+`u|@}#;c=s@`M%S|rg zT9V(4vN5+h98t220)y4u%~1Aku2|x;5+JF`eIlS;p4WY@fnG=bSmj77=QBNrLVwQs z*y_(ByB;TvkSXcx@4AueJ1m~^vb#8Iz6x7iJcRYMgU`Y2c7+gsCm#8E16J{Bv3 zRokp-A6ywHPqsf3!!FUhVlITd&4FNXot*OCA}1Ehg=#@9?U#dEvg-K?&!#D9yWm#zw8D81N5m>Ap^Sv+moUm|1|~Bu9aEL^hvVikx=Q91 z9&kmxfxM`o$7bs~*GUW0W$bVYxpE>VSDyEYTGJzc;3@&Vn~i630GaRPprhtj@|vB`s}TNxunFq-lM zz~Y4{F01%x28TYD1dK;{u&<3QZ_G1Tdv!9OxPFBAhHQC7GtImg=#r^SS)q`jWmR}>x&^nFI9^W%32zcg=Im-nWW$e?>mGZnUJ6B z!nzJ#3UZu=mjbVEWvQr<&Yp+(7q#w71x{-CiTAaQphY@ZiD$$Br1$#rzlZXlpmr4P zS)rpgs+D|Uh_;AjMHs;vsdKtyta?{dnWgtFt=c%9A8*rbLRsM@8jt`O_V&`e5+!nC zQ>LSvp)Hz@OefHjwh~3ieFZ@NW!a@vA}Iv;rX)$(6_!ChqFkstUPG(ICe$logubv} z*a%W}c`S;KQ;g5xSwL4H+9UznOL~0#QKVEn+zGVW`&>rSRoi*Yz&Wl@De98Nwsx7t zHgVOR)HCN9_K`ouf|rIF{AOUu|42{+do(4OemKY|pFxdwU`1=#;FHz7Fiku_7Toqg z2mh8hr4j#HyILX^@KE{?k!=ut{GuAwVXz7HQ_`hTn8zUe-1c7P{?V~-?yUdUg8M24 zFE@JFwYHp6k6?ar)J%!muJ^5yo?FSeaV3VC;Fh1`P1ik$FTU2lDGKS0vYc-1vca zRZS~;_p1Snb!tl0ZAHoz8u~~^DVeCq#a6ZbmN-{;_E3q6wgh82)Eklnx*;iJqkgvymj!p)1xn z+^oDWm$1@x8GC~Gn^4!#!o&6K6{E_@(7GPiU3c7j^uy*qhclDb@nK&tr6N?N2!Aj%jX6UOA?j~p^6FKe^4yJ!*V!i{lIsHsW9_DiuFwi9OS zO2YwETvS12%^nRDOnqrR0V6B13!HDwc(^S4vtH_8iCt?epkYg6WDgFHKnqz#>&dw^ z{&}?j$Bc3=7kXa5IZbly96ClTNUhoI;++x-q|jf6oN1(gr*QM~L|s8g+cC6>-gWlr zZemv!#25we!Lh{Ryj9Sjpz+MPv-r-bcP;b8*c7iB}*b?UN)?}r^ZCKniE-&41o3A}HKuC+adEPF_#^)$9?`N`wqOK~|Xl^MEL zf&?=BUs6U`n*>v8D_=et6kbaQ1+cl=EOCB3hk)Q1=q!CmgXfYGh(mRhatyB$qK(gQ zJ&q@Rc0TTg(M4Um8}`*+rFBJVz1;|v+}#4@=Nc;V0VO0&@T%uuiZC+aR4|Z=nn`>S z_xut4;r1eu%?*B`_@lUc&9cuemk2;F!sE}mih|QD%dWk<10nckdTCUWn0bcTQ=%Yz*dYcV zmpPTR_kBilkwDSQc@wtSjnnyX*r3QY=(n!E^b2vWCs!n#%bdBerx)J$d-*65%CB)g zP)aPP*ny&Q8rPaa4Svl=Z zVMYj~SSRgEZ{SF>*tnAp0jLO;KV=K_cRa;>g;?R)gLZx-$)=SndGZPj&r(Hvnld`u z=^;<_4%Z|dZ?D*@ijLQGO0P?UZYLCXl-&rw+2Sfm znKsSxXYA{v`qk?qohN$~kINZ1hl_y9x8FR@Oe$TcEFbyIL;7;m;u8}9bw;mzZ(p(A zlzV*@pytNMsQyuU@U;^R7$-~6jCsX`dH**1L>6ieTKHKvyrEiiD*u2!;&u zy}q}xXU9&fxwpPhZ@I~{n9M8Zzc(*PPk<05!5*NfpgGt1hL<{Xj?(iPMK>OALJ?TW zSn=LSaclf~L$1$D=233hnd;`z0(m>zlgEI;&HyG!F8%eD)7LCNPSO#_cpT^%)czBc@<>qf-YdeNR@yL3Ayqi8lGzn6A%h{}f zTv8r(9`to$WAFR7sD6nR`yXR{ue^M;fx`f5O|ef&vaTLOa=PVoPLy|!l@Y9M4#N#c z(oU@W`78eqRc{%VN3eA5q6q|t;1Jwhf?IHcySux)yL)i=;O-6qg1fuByPrw2_kPdE z#V>dmX1b@VrPjI&j&+5P6ywo#VlT)a4*wu|bunF@upPqYI~OS`u$~PDF^W7OMi{`3 zcp%vjivDmmKr6uEygE>KTz8j%Ng|z8DK*{w7uBl7pv~a=KDU9>*HHu5L;uX(=R?ueHDmD zujKmP#{5}|04zj{7{@N0#P81m9CqjTN9YO6fLF9(f>)*A^j?n^IeHdA#&?af6`uNr z68uwJQQPKhsdCznUFo&H>DYh_G^R>di%awYRYrs8C4+KJxh>8}OEg~Nb8RZDWC#~a z9o7cOfHJfEO;Vq4|m?@iYW;)QEY z9q{1O;IC>aSlrhBM$b;DR zsbVOuUBB5;LI-Z)c{hXugh~npU(t|-AqY%8>Bw%5HiZL{+8Y;4)s4=eI}sJnh(+JS z+V8eru5zL&H-m-V(L0C-lK>|779CKa-aEy^a;MG<;Z?HAP*3P>OkFZTmPE8GRS zM%JsWTMAa++V(5q2t+xH2-V>!Nmpi-c-}wJ2Ol0{SL*0NGf$H&t2F`Wilk4A2B7{_ zhT5pa6 zQH9K85F3es+?pit-6N)1?H>Y#;yVavgtF8Xx)$xk;CqAuKhi-mNZwvmY~d60hX*F6 zyogY(4s**YnX0Q1;>`%I*RLC7DwKaxvp3wJT3936L*n4v64IFdt^C7X66Y+WexA~B zCa~?Xe^ZXd5VQb;ES>7G^0DXNV? zv?~s@_Sy~y(9NG-^b}B#L}9=pJavSfSIMLbmV+G27`9+Tzg&J zjH=tSt#83Z4F!;815#}CwmzeY!Cy#ySl|=UE+Yzhv=}9DCIXw){aq>l0vH$*LHuRt zfuvrr$=}>;#A`yymZ>NRcgl$UxA~bFHgP&N?i)}UuiBd{MYHg|KvMa?c}!wa5PA9! zy2n<%f)-gK9p)(2E{zLMK1%4#$_T%l_BklWY^2eZ12wNRMCV9bxE z`zgcdL&9-}kucHm5;qwx@CFj&feP0wn!|~Wj6mi<8l=jz3gOw$1pgT2Yb2qLwem0m&$28~27PfzPn=02E7lQX4u>OQ1rxt8Pq5~% zWf?KyAcyaHKP`+Q^#P>Cc0NEewaL<9{^DcHxR;yaSp}V#8xuf zZxcZ7restg7?VKebUKz7pd2E>tmE@Uq8Z&OJ4P*Xa+A^-{OMDj0sX@XiCb%snlhle zBBlYPXshNi-|%vF1wmTp@qwQGPm2IHNmaM_p`+u#lx&c(YpuLL%fu(74a;3Em5dTx zHRoj@c^GXyVvg!YW8Gy-)=;rZ!mR!`LC-gZya66wA9%VNQd`FqDxyG8XojE)f6cegC7m{{}Xav$57Lk%#Jrit@3BuO)PIvp4v775lr`UzrbL z79gbZ-~om$<~JV%x-vvkH`NaEx-kzIb}9%wt+EyDREfVI#0Og?<`cbZ$Spa7$gXrc70kjly>S_?BUxe5~|UqE+6LRHN#(Y z0ttXzNt6@}R;U{NYg}@{9)Ebh%x8DmTI0Fb|7n@%fxYDuNeJHuBppjF*2ho26NAe1 zL`{Rr7xmLSRG2rf6fo4+Wxg(9o4DRpx(EvuqD9j=NrWJp0(4dg;0Wyc zx}+Lpn*!gC1h$|Wlv2RU94SG(0B^dKEU9(liNT@%V{HZ!(0g$y~}je4T`+9G*Acf`Qgg{ToL$+P!~Ob zjLRofw{TKVDtIDp8|#$1w@Ko8>g7O$moY1wLcVaKxCE_xK;+Nj_)rjN;*89~qc|jp z2VqwrySeUpuUSeHFS`ox%0g7&A3rco4ztFXZsW%Srt!>)w(alY;{MS8K{)`%m4_0h zB)lasH{O(Ef;z&HHDDZ=)+I}XGXtdfOlo0S5B7c2Z+`9IM+4oI#tf? z093?rsw*3yD2#Y}o-p{I6b}#7C|qX@suq?hJm9IzXI^sX(<}H*i>yTZLi*TbESQwb#vCI(p z8$gIKIjF2*8@Q6QeZ6CGJV`{e+$aJLkk^t`pn-AEA&zg{<#Q4?)ONYe>8^`DWlZ$r z5Pyf0la6k@qv3JGs43)hFqYn3HLC?gIq3$}w>-F^t!;C+aIO%r89Z+|aXW?gm$h*} zF5WIK!oiLlb=561S64gonMbd-h-3agF38p&zO?_CMr1Jxl5%fPe(6gGWEy2Oaj1io zA%bCqG^pj}0s=-;VUD_;1NXM^0XLRvjkK+8T+TD(N15_kljjhCbxo8PjHcCDzBqBU z)ovj2yarZk0*o3o-K#T8qIfbARuh1Zdif zbe(l-ej&oVkeXUd$5gx6tGUePR3M-EZ1vvlach|`%I7Y?VU@Hl9yF_e7*t^HR9RtN zA|33uZzyy{7&`^!rASt4&yOl{MHFi%Mba1aRDfe`;o9C@?>bp0BqmFW`u8P#O5?L* z5tBqovx}-TnEZwB21BTN8PWYJStWjd?oV3my4-LQH`TTsv`(8Kiw?}D4pAW7l92nE z!C#X*ie8w;PFDJ$i=5iRJ7xK4tlHBxqEhAwY=LhF?zunezEIV0KJ`A+_K>J{aal#An< z`C=VM|GpQ6Rc_lUcIBZM0ZZJajmI>88;1nUx!~Ti#hSZ-64`P=+Dw_~R(yO#f21?x z=~f2r&x8iJT4#tIGc=>M7Plqkb`~bT&}1_L^Zx-eXhc`Tok~&V$pIMjMikVcM;y|E zqlIx?0=a#7?~#0py6Y^k@!*4~shJ@r8L!MXP9sKpo;_IKac!au{l8h-ErTzt{C@eE z>*@M~8``Uw_C3%{qF`WdaHQP=CRBczLRVZc34#!A0U%lv4nnu0N*xl!KAwCIe;@JB z-1EQhn9M&i>+9wcrT_cc-(Lb^$ z?oQ;r8#VjxazLHMnzmy%v8`HP)1Zt`AxMa@dQ8p3$*MK_CtE1d=xFTIljv;qoW5dg zNy(i5M|F4{6K9gpQNrqjq2Vyi?5te!8nlUsd&p6Q3^lZq*b$}NKTdW*62BLKEqnS5K< zy(pQ>6H)JYAjH6nAZSdFi_`Jw`~|GwVVqW(1EE*!?#cy*gWRZW?$w=f%eW6B#jY+c zYBoDL_gmNat$5o`*Y3}5`^o?DAYxvC_lIPeX$X*SJ&}yh@ir2_k1*K>r=OTxE!LNA zM$Xs5z=$E1jsWNn!utBy#28vfYP<9yvqP2Dz9B9zE<^Lh*2aU9QuWS-q^}Oi;t|=F zYV90ynhEl%hlJY@CGUr1H0q80Vlj#`^c(MQojQJdr_0A#o-9)fh{e_+%rnMctE_*U znL-zi3=QYi<$yKq&EE?~Ijy8+F`3BpX}?9sq(8HSeIh(x<(7Sz6p>i91Q)sq6@jLQ zZ`eF-#9nu8y~qs@r)?e4+yWcrbYkwfhSx-3&otg2bE1sNNJ`?gGdMn5bsXnazRTd` zY->1-mp=O9R@G}jOx}Fj;CF3I7FUBX1vDS1!75=W6f70C)8LMH$x*3FPZ0_7ot|WF z&z+L&r>jqHROY*X9Xy0F^q^nGvcLP2LSHJmzrT{&S-hPvK~z=3(kv*Hmf8(O3>N>q$oRw^Q8HInmL5w~ zTT|l?t#*F`FRUgp<^xLed}p+eZh($vfPkR`DiL1(e7BorYqa)*Vle@g9W{YCJaaUZ zHK6w7nK#4{omlTS`ki5Kv4eZEYODOIC^O7^yg%W5rt^VcJ&RvT>Uuwo@Op1Da;u#Z zIdq2mj$E<#S1PB4%8P0q$U%jrMt#X$$9ju^cJ^albkxfL@12mHirw@4r!mfPbZz=) zKfh*s&)Bmn+dlP3D&>}517HC_)+;ZA`2?I^FUtpI480~=s?8xU)h4T`CR%#zmX}-i zedjd%{i7PZKN6R|0lzUjr&;&DT2md~YSF{QzU=05TSPf3lgHtys%FzLuz-EB`MLD| zym+bBXx_~3PBTRR8t^cGk!R=NZOray_L>KbPypiOiRGNFX1h}RP21_zaGJJN=LJXm z-(CQSdf;Hi3&iZyMbT^-L4)wT%l8y;@tlUfux4>~b~WMQnz#03Qg*uN7XVCCwK~I> zh=#^7nitK~Lfxv$@EEPa%*D}lI~8GNw@+C!fypMD^F#`yQ|~(v5RQiPO;^>zOMNgxodwY@tx2KpgXbj&SQuGT2ml=bpeyE4SNG8iLbc)&jcp|C!B`GCXU@99k)8&vTx90%Cq0ir8J`LXIc1@!Kb>+9n_v{d##> zG~xbR6T@qz#+@~fk0HfPx({J{Kcb+jzaj839!I2yOjow!dB$bfw1Lcy&y~ zKjrgSC$j>`NBREf;bvW28N4mo*(dGMs0F*XHs?zXW;%a-zmv`K`>}?ju42M0qfR{Xoy-_2}LR z{$0>~;xfg0CiCO|(`XMZobuOlwO#o5KX32HRjcwn#tJuld5S>i<+Zuw^yBJRAO zxUsR=jeF9s`!n#|%xn%fu#HwO!V@~HY6D#Z8^>F?4@8x;RJ@+cDqnXZ@rqWMHFN5f z{%P>}w*GX~R)@pz7+~r@v90xt|MnF2BSjKffSKXfX7;fO{*lcWVG#ZPvbQ8}T8*Zv zho$41)YPh(dYpyxxXl}JgPRFbn;z;BT`#Z4toJ~1e?02`uH56w0gDB+l=yhgwox!^ z2Fc^OSvpnZQKRktq0UEjXUkyFA^$THOnDpCEfaiRUgs%dV$ZV$Tgd&a;GrQ}I=Yst zk|*OVT|T?+GW+BoRINWV=&XWS!xHe1+;CY=ZKenCHj^{F&zfDg$pSRevd*JN#t5>G z@LqYnJrHoG<-52PRoU75Sl5xwJkDl3)uM6D7aOC$%}>6rwbQh|@_gRuw=pbI%c~fZ z=JB29+tq%LT!p$-zn*TrEFI5ai*+5=7}8;IF%Wq_k#;TYmPeaza&tVj_U;|`;p70- zcC9(1!A+$aeu z=})YX0QGLVi&@QWsJ`=G#oO868O3l7svVNTat}d%Onb|DzYP;8`O8yTL_f;HU zD2-U~{9OCtV}eiu?$q!CDRi2_A#2Cs!_3Q35-kpyj02DjH81CII-z?JzjCT}|^Tj3Q=jORb(t9ZASH0=iFYinxF z&mK`P`v56#?{^rZ@iE?K3nOe{m|dylnA@|VvYy-t*yBtu%F{H;fg8Vg4Sqe7X=A z+j_zdHomaqU{t$1H}<0)`3;Fa-V2FqZbPr?#}I+cn#Oy0$y?;Txr0JnLVh%%{QSGO zxw5mr79SR_u+Of+v&f4*fz`|7=i3PHtAhr#sLn7JLrrSc^3^y;smB!Y3LMs&$FwjK zy7w*>Ha&qw(aFBc^s{_F&MMg#_m3!@W{Z`x zcwQWWhNny=v&lPu7StOqZ(33{cGeztktR{ccEKELFL)U|LvGW#>}?yLXboVM*yuNy z4=6;55u+p(HrGFPBYa8bKJ!Cxsw}qpN%}?h)%f~-#dI|+_d9@J7a6ZBHM#zxevc6g zpe$@2H|S_UPfY%00bYgh3(|sijvdXnVoS!Zbq`@T7A%sBoZrJ;a z_EDxwo9XFN|Gc)j+Vi(dNmkT!N2e`z7kRWQEmeekVT*q-9hE8;I!+MVn z==Rl@$Ne+PI8@rG7OL7=-fP$fZI2UIm7Jp_=BAZ2kTJ6`EXUlA51BMa3mS@w0Lgey zP1|ETl5M;eR&KP?D{z3v{5wOu>!7l=K;<0w1Ae2)T3hqY80YF|jjztH;bgT^5L782 zxGX|k$Z8Ip5>)}p5)W__;B&o=Pr9%K-&*m!UtaNCz+WB0~=U5Wo7 zjitzNR77=FVHH6+{;N^?&a;O%siWVXJxBvROUDP*t(_SgshSVP&tgFA zek2&RF+c~UEJE?_|9s-EW|%^K^|!3rNmS2+Ush+?*Sp2#sN0-X7K>vwG@E`JhcZBa zKFuOwUDrc?6_oOQx0f0-MP@^f%-$O7SGiw_3rf^M5|yB>EAjegm9rnLZ-n0CP2 zI0&Y@9Ha%;&b

    +%>$x-+?{6nMc~T7Vo^}oUHf$;{L-i)Ri!@C{yWWmyWY>^X@*> zIRJD~YI{A({;StH)YwUl-5g%cLmMVjy43L4SZGw=Ek=6Y-Xd8RdE1Gd$GO<}Vq3BN z>Z=M%sA>8tLAbb3?`cuA6}t-}74!XQ#ZU6)*SnX)BKnB7yY7!No%ww@nx6BTD+L=P zwJOOid&a!xiyiGx1H82F*6*t=%0nIZ+;R0viXtT_uTo>U?1WUTbPj%XCYMh(_JHv! zvn2uQ1{xGghpGKbIp@>wrC`~h=^&|AtNz=Ok&#ycP7$%|9%H%_juw>nO&Bt*07_wA z-q%a{RSLQ}W#`AWl6Kma#V!DzlHzQz(se0erZ=xw!y=>^r@a-VzZmul05$wh^EUlP zaofjs5~=WyUz9x#l`j%6h;qGr@7>!`Sl!Rg`iNm<7atJEb+`Vm&WX>Uj9^v-&hV4P zdO-)Bv7Yf${$~CB3Y7Z(tr>JGtB~jP6{)0|bDk9r{X)|$6KP61z*9v{_hji;kqD`$ z1t5mBpv+YB|`K{OE4ZFu7uVXp=r7rVuRkw(v#Y2qwtn}zY-sjUwt_wQ+(g|M%V1X zaJNgBZL(!7^FA!NK#gA3j`tp8Z~g*`b9&x%TS_Z-cEIphG;R|-s@~9m(&Z?t&cTgE z3koXeKDnDT-7Wi)kd^|Hjz<7P!E}Jdt?_hgJ;U<^g^M@)esN#W`K&qe+i?Az$3FmO zU+bVF%N7nlzQp^j2rW8tDb1UnYhyrG+0S4B0Ih6N{Df`i*SXrDaAlv=eu$G!S~&TU z<#lO#85CD4-Oh32^?)&5G5ym%Q7BFd+v}+-&NstR?QU1X`yF6&&>+7)p4}yMKtO-l zN|RJt`!&p~eMa3oxoJSw(7)F|=uftMu#RFYTd%DplevLt;+*iG^AK(!e9KhgpvPav zag&mAQc{P`o!Gn6Q-0v~r~U|fi3S3)pEccSnAmd|)RspKPh0ZyO7k;SQe%-4U^~ja z-4;e8n(oq{{nuRHs={+}?WpbXsS-vdz?BU3G*l`RrP$R!t55*KGNw$xWXt!U#($a}A zHyMNs4}eMZgi<`zk0;q5jC2O1si%#+42zbx6fRAY3FEp?tH3Y)GmG9}`Z}1!Z3$en zu%^rr8zv=eihRm?Qch$PjQ4?7ZnF$WqcvM=zCO^dV;AN8CXV$Yb*%0AwdH2=0Xi8a zMXc1MyQi#8W9O4N2}4W$+isG$RdZj}Johs!R{h0Rc&m{`orH%wnoUm~wqhp>+*fdU zGO1s5lCU&V!+-U$4RtAyQMb zLACNGU6XY%+}^(C6++p?o2O`Q*3y`i9jazU>)Xw`=V~cE3g*4(ztLeU$|W>@2r1iU zd9*Sdw%$lk*4ssNJUufh^*-z?jLv0pU!1EcT1}rS56+GDS>F%6CVMa12FE-3Y42`M z)!>QK#_(*Pibn(UebaEYNl$2$&bynUY$drvTI&!R5!VmB5#2n!Gihh9`{rv`Wf*}NTA z^H2j(<%1+Lnc0r)dq$=nOwTLd{4V>y!c9hTFDHLpqI{c6WR=GCy5ioQy(61C^8Ryt z3m|p~W27YIJPcA6iK|=akQ?huhUz{6KnZNCU%~9oBPvB&^yz517PTp=AI4 zwBb1HBX*TmG%LA%;iuH{aPxHP{rKZADaQNKX;E^0m8grV^!qzyB(zzyA5(c${I{vD zndB)WD#%-|&R-ew-ZqA8 zSFi5}JrEdUYQTw==E&sxWl}{A?G$N9Mf>|(Ji~y1Wa-*&79Hapfmcu97q(g6cg!hV z+5IV|ZDj$A;=?!+WD|{79`S88ZN4|xeP=U%Q}4IY(X!2xXomsNN^9qSNVWU({VYxk z7@YHOT{|eJs~fKMc-Im3-+IGPRm`P%!cR`r=KSE7Rklj^w7oL{H?P8nEZZPDu|vF< zZ+D@towm`|9>+9ZM_Tn7*T>q!U?={C0!Rm_U@S@v!^i6_@ckj|^p1rQF8H;S0B>m7 z%^brEGhk&m&Sw7)Mk-pmtnb%Mr~O+aaQ@SuYh4DMbkCov?za_Q zhMz5!NRipizW+NdUj5fyh(LGcz^QdU5b~rr8E3iGT#0z&Wn|`c@NoFsT3Jo@1Og{I zYqW4-1$A`~{b@%S5P%)BZ+;yScRMvh^D5FB`CNQ4qqiq8{L|iuZlEsd1NRjfZ&jJf z1G+q=-g2T~iTe%G77Sgcv2h*@5h_|PO{=~3dVPWOTZKB>Z>gNo7&({d=ht>=6?6}F zn}s^JdTdEt?%}pYNy)GGZyZ=W9YXtaXLmu`&Z|Vx8<~<)>;l0x)HWT)U?sE$(XX9X z=ebt>vU%8n~=gGc~zyFGpS`)W; zqOy*F(GM7K$CI|uQF=*wUm7DFH$G~=Z+R>|CCJ^!=3YgMO%)^?vU5I}^`Y)+4^vLL zCj<5HLCRgQZdD4Y*+wVCwS3L_#iZ-HRyq2Fg<`Tu;vRL{Y<+8eEZw9vFcd7?0XKT8 zC6E}WJ7S{7Xp2s0l>tIa7vWZ0#SBTs#V}AWUaNN;mfJ?iiclo{&cXP%6b-31Qj@Bk#nEIhLI`Lng$kFi{KwVYdOerT8+{)1u<&ZNs|6j5O$p|+Gkvy>1vSe zI7POUwxU7_~>2S4= zN<|N{7lGkRk$h7w5>1$1^K9Ci z)_>I0qF^ELYOx9tVyv~GDoZ29wy)(LtUbKH=1ns1X^@Y3^r$LSHgl02kow58{s>ow zB({3Qd&%)!qPcwy=M@oe+rwUmSWdO}T;}ovIfJ&DlM9vfAhLPgDY5blAj@n*LWtmO zXe?7Z>_B8G}A4VaA5SX?6iPTs5Y{UTwPr7x@H8 zr;mI=*GCQdE}fgv((gU)&q{eTFB3ls9A{T~K;X654hj#yddx4hQs0W})n#|O4zR9j zytiFuNokyEv7dZ3-?3Pcv=8rc+6zg+!)?Ayc^g`(jKZZ)6mgDd_UA2QBTQ|-KlvgL z|D}N%bR`A5P$DxfyofZ#E6@iK5sV+73~Hx44#MiftkeG7S_Hhw;0pOI5%2KdWP{26-}lIaNz*^N_z@`5_bfm5c2 zk`m(u)_gw?4Q;1UzL9<#L`+`pL3oSZth}`1nrCzp^_S@l=~kAGPd5pjPs0$D6T(Qi zTW#|#BdXJ0loiSf!usJBZQE0Qao_y&@PzPiR?X9xe((W|X&=SvW|57qK=6g`F)Rji zgVmqwnkS*CpWsXfMLB%xoKD~B-|lwgrVB#AD(hCNHooq&EJAQfS>ma(P6-RkbRHvA zunus>W_HbOtM5mo>R+iHC8^KkEJiIbq5_T02LDEygxKLY7LYeq+w|jf?DMO1lpd zolq%6mpRweRrSZL`7{h~)I1ThH42wdYnoo(wOukz>}aVn<`50zH&fFr>o-bn%jIhD z9OjC?u!q)1e)PuW=F8t0diEaX_HCARyhUj+O^5JP)#uI~@(Nl=qn=F^nx9xtE^^z{ zowJf5@jwZjqKB(p&ML)&qqBPwch7s04@f7)D{(~}Y*0LCInsNGZ9M8pi z-5r83j-*5K#G<*md~w!>MaYu3{uQQCOP+$=NyU=q)e|Y!HN4j zi2gSkvGU{f0qiOv@Mk#*zu`jZMqc$Id--BVxU(dZGUTD5v?u|+wbR3g9GvMJe)a*6B zf&cJ$KGU>RZ%8&UlrKz_Om~q_+|rYxZ(ta;Za3|PU_(hJvbU7@H9aTITOUmbo{(^$ zbB&H^ea7&s{O1O2kqL=0D|hfsOCyU0&-qmr>n zc`Nd8MY$9PWR{Z=n@quXtws;kdgFG-0X+H=MKfe z2qP)P-nTMqsNkLIn|i!vZ>qOAs;e1{r|~o&qQ>TKIZ5mMbOwLeN(W+V@yNPZ(_e8-O6zO3*O`n zI`>75SiI)fJu6467g6obr}y>u@mn3r_tMS>YK~2c9co%}Ata8#Fl6Jy?lx-&EU8|z zb?@w^y8Nbw3LLNR5|dWa4M)bA->9&^qZV!9wBcxeW~icX(X8yF446D@dj$&?Pm09D z%RrBOnnPgyA$h{%Kdt5BT>2W%TiqxI8D_5Mz*3AnB~1;+<_W9FKY`=}6+z!BIiWWO zcW38tQRUV9z_eSo+3JOEPlgBNF+YzL;W2WfU?1({b~lL^8F(mrREi%AH*Jjp?_qNyhTTr zbs%{)o3w8T^@P1~eRFMOR-Q)>)O}kiV!_#rNpW%4krUIzJ(^}IfL2(s6%mx+Xtcq} zO@TlBoV?qHl)St|b)GVw)h>y{Cz72u*rR=={5Vtm(l;EOS9w&F73#92soA*{@*8nA z3~4wD zpKAj#sS6Ic{Aw4WGXZ=;-{j}QhGug=Gc;h*mxuacP@VcRzX`zDK9Gm~`_F$b>VrtG0`VP?Dqm|>ttt-$^6?>VK>d`c6NIoKl~GLj ze=epgl$~GUtt~lB-wRT#JrS1WFel>0vi7lIw*}8DCYABjo3*~tYA)L zsOa~p4IkLX)L3sQzI&{xq-G@+U4GVciuj~4&@ti`N=|U72t2l&hLXFj*>Fs9bs93I<$ohYULflthraZQ2Uj+<$)RNp_GZi|TNDCi9`yed~%1pefgPGAJa zBA})&P@(oE@lJz;gydyE$z!jvsfo=B(+wZOF1Rbz)BGsyI9i*>spHb)Ol`A#uOfG< zwm_z{54SHvF@prC_|>P0+snZb#E;@mMc`b_rUhnP)a2g1LE~I5Gy3=D_=pg5Ns1r7 z+b)J)#r4Nk4Q36!fqsC zL=4fu)__oP0!LPW42CJZ0hD?!Rjw2s@0nw=+2H%V%I$ua&fz#D43Eun0+boV0Lq6( z%Ki;fe*!}^ltzcgqdx-ALWgH7iSy)D8n-*k6{6@C3_9&Xi%LEkjT(gaTt9gXHCW;e zP1e`&PJ>#ic3@ml{0VB16F-EY9zyx_^|zmHj{$y_yt;{@;col=8mILdmjrBbE};5X zX|!5Rpi;K*E0!;+**{q4#ire`Eco;3ko^S)tl^DR#URcYxonkLHF zepxTo9rhyd3i3`6>vV4&rnB2eP^6C*@Kr)#OeGmiIh~)IpPKZ`(nA#ISreduU+`$6Kn0@a zzlco@R>oK+Le1>Nof~kuJc`0F)NDi1^QQ<3k`5b)g|DQIoRJB85LgKa^lB5s#T?<|0K7 zG27{vW+*5sLc&FKS1r^afgBm+LGd|ILD&JA&J7&D$`TlO&=Z$Se6Y&ZyY%)RKS-Fp zv5d5`+Zh1U(gASD40;-Vs#`TfYJR<=Wf2AiV#?pChyjVs;V)$uh>R( z!kB26>6`$F^@)tU0OGLed=3-ZaLw!a6oB4IH;>Gyc4cBPnV5w*mO`{dq8Khr?mqXoxsWem7+67uEYQj>X>{$e*Bmeg3@T_~1E4J4|2z zR;_1cK4DHOmDv<1=-VM`9!pu9&@Eplg!oadX_6nDMwPaM=G%Xs@i#AojAQ@Yyd$wM zuh4=UCBjKo90F7-KBIfSW(m<->FXYbJ}I(*RvA~2cG06gD{6$=NQ_TkY9`Kk8)`$7h%a*I1_ zpVJ1f#f8rMuyQCrz6DfWT^$BCcL(?Z7d-q_8?Z)dXdH~bnxNFOF>6hUR#%9jK0q1RM8R5SplP+%v z75J*CO$Y5?oed5+^534e@`w5jXuAkSPh-~8>kQ0%ks^{lMbj2~TBs=7i@)N%2+j6O zt8;R;hP9N452cbum%pa*(pD4$jOEx49-p6tS2&=*_+z?5{F?nH6p^~VVFZ>nYy!Ac zYM~C5ETj5W-JcPMbm4c*6Epy&wxU>wt`MRnc=(vom^)^5drpG7NG$5R@xwafCzekf=bcBU{ z4{wP!HakC}q@|>!qC9(Fx)1wUWb5E3s?GTm2*LQDC!cKd#`G1!5@Yhc>7lt=`Jy$3iwQTGc(DJ2ZuepusgAIOJsl^y?1 zntnnc_hT2jn(HX!9y#tKvaal0Y3O*9)Uk=265=uuel%&OeLpJ-_e!PTNpIn)sjQP( z^VXd^v@|qlZ}PGf+gs20-PK)H7k$3c6rB;p8Y+RH2P+_7reCYn!h7)JOR0J+*VEm!>nIJ!RDNwnG> z7W%@`U*Is)zn-<0EB|+x;(%S~J}$M2JnKFB#T!Rh`)5D*!Ut@FTt*vy&Ap83)q^n1 zWS5|Qwf3H9Fnso@xf@)w!sg|2u$7F>V5<1$EsAoOy`ckxZIK9X!C)lyE7j=q=T}SO zf=Mn9LH+PR*3a^6KiH2ZY}O;AdDhJp{=1|23b7!Qdl3#HO`Jye1K@Vj1}b z94G4H?fJa5MC?G!R>LNU%dVd8>8dYkE{x=U?sf27xUTB0*t$NmQ-#|e*iF9RAw}!0 zXpwog7D^uf8cm=Dg7OsK#?D|ol)jE{fvm8ast`yKq;?3S_~YVwx89yjJ_I&=nWg1m z`dPR;!jDl4iEJ>o*SCYbXg1I(3HPP`eyDwwrr)3ZZ181NiW&aC1~T|QJe!4*X)4ht z!MKg#7p$ATdp7oqR1OcrUHk*I*>#v}49BvruCA4??PMnF_Su)}o$G6}i|@1N3Gk5) z?e@2H@5Prz){66Y$Io*J$O*2i6OHBh zt1?LO8G}ju_->U(Gh>#dfJ8vgrNxZb8MILax7?E9+DQaxU&8}zbV6=mp+w;GiJ4zV zMd#bz^du8f7X_TP4r7WyG}v@HA}fTr+K8v^p6;`cct;UqA+@Oq$nFPBXw=MFzp)Z|1J7} z{YS@gx^?iz3?;(!YmPuL|5Fs!SIRb->>fC2G)~-U$c7&(bU-0uy3yVG>z;zWB5xD| zoAu7O^WAHsmIsG)w$+#2_2#9;$$MVU)~7|ovpg$=@%=Heqhr0?D#Qz1P?ju-M~!Cx zyPse*U<>!Q<7U2yT|YZ}%m>ZNALTIPAzD?-j8B&&#%0#^dSgFwLH;;c5I9|!qBG*?<;f9-m22=omkpp-DLz%MrusQ0MFaiT@Iq6!*Sak=ooVP{%<11KZg9$)I$_w*kozX^_f%ZNm92ieC0oOzs{Pj#oi7pnCM`~j}Lg@l~%UgMhi8=m*F zs)B-+hTK(w$gq)=_47siob%c-9|e|lj6B)w>va~HVm5Lx6ljwWhR0?jWV;(qb`yPF;X3p^4jj3i`|x8$ISa5hrHWK9WnSC$-+{^i&nmaZN07Hkg=OPNy^Qum{?8ug>U z1$`hlE(h>LU7!Civ|cMWa{5Fog_yF+kyNpMSWNN{)e-~@MfcMZ4myx&*% z{-UUYnx5{{`<%V@61relT0*yQJz@p?hLBzQ!kX08^M`H}T7I8-<4Qx{-86H>tHL=t zx=9k=8kV+eOx@j&Im4>M$u7i#i+?wXdWzu*h6sjBat1H!}W)RB7B*lb*42zR?r99$eKOfOk5EMT4v$Z&WWrinv~9 zJ+}!5Uu}CZRj_)8?xB^V?_m&d$em<#YKLgBSZ}{3^apeCtUv?FBy9JVQ!ka-xnG0M z&GDi+e(7HDd2knZLUAq@bt=yHI?tu~9N;%E4WlR&MY>#eZ)^?f92@I8>nH2UefzB8 z^T+Z-@|97<#j^8wPC&iqm08WzI>6dts94`b{!JDjp1hQx;>t+NDe#l=oIUWFlx7Bx z8W;2sBeGf{FxeA!j^xyR8vVgv6AAn#x$41oqBq$GFdzoXW-xqcVeVFA)9zI)Gg0n3 z1c{cgVLvtf^(N!wZe{WTZ;k1_3yP~nhCf`}WqZr9?QhsJA$a~`_PNg^$ln{CofAa0 zo6{{$saOn&+of7bAu5JP33xPcPXw7hi%yMPhO0br*A&Tz{RrgsL_f$oqaAKUG4r>fXGH^OG|WY;KO$k6M;zmpj(S}Tyu4b5 zBI08h?C86^I!T99p9n9KN-HMq{>JfO89~0LgLab3uwsmDMUBzqij>X zR`IWC*)>%50Fop{hdc$F`!_X=nD6yB{YDI2uK#@lc+jT}(|d5xWO0FU(Z~Kbfp}uC}UbQlkjVu<9n}wF-bnF;c}>9DM>uTgW&O zGY_C@eEAOc2oOUUI9{Qe+IwLlieT}*h?&i0!gNM_%J+|ZJGBD@ zV;}Z&!bl?t;zFE9?y)H^+6LI@?4IIbEPp(P!qoH~c@B}6)T7_9hL$@S|z7`Z8a1&%+#Bmw8; z0?@&lDzm7#^|=}0K*T7HZo#W2{K*SF`y6L-!Dy*k$7>ihRud)pz*`ac8EZruQ7`ko zdVAl8!FX7bq#f!Lk{#8g@zG4HD>qlS`ZIy)>k47|FMe23-DSF2&u+*oWBSF<`^wwy zt`$BNZ+9cQ=AtGJok=3>vX-X7t-P)^A@eZxF!miU^=%;*Cg$Dw24R*5AD*U-{h4BR z*s7{L&)V)N+1bx>5e=rYDF={;^yc!-1$v3*k7O+n4UFUE^ri`X&x6QB%)DF}5)1@L z6lgY(VNrbA%|;1#wZKBV$-ZrXy?bTx=UXMbnt2gP0B)al09Uc?C!fO^e$421B&T>5 zu1(_2hdt6vW^UJnwc*i5%{#IdW@dGaDYvgK-GR9sOQGp-%$Cn5Bqa%nQG>QCyob)l(GabqybWqjTbC~v-R7?Uj4+Y(JoBFv(P;!|^$>-ePS z<(Mq=$T9P;#)#g?fGd&1yT!;|Ai*vx5&Z;L!iionF}a?<|9;E!S=he|HP;xkD& zrmqhTzfzd!o9$c|TJ4NyULS<{NH(VC{@%e>ygCa$19iY;9-QtI7!v`}_$n^Z+Gcte z-a=nbJozbyV#xQl<_E!#^Z0>5gBS+Oqqs=ay@vK1rpPaVTt?0f8&mgA9s936V6$Dw z?8Xv0+8brEI0D}0>LE+Am2>Iz`P~4a1lal{Ur7(=zHLs%$)Qv*x2MOD6MC}yM37Hn z9aro5KJ0`W`Lu@A1w%|`eyL{m-mV9sr`rmyJZzx*S-7fZhe^Ubg*Qba!Gavt9v(Gb zUH6?jvv-V5YO^QhjYI>d)^UeeO;~IH{KGxH@6R*uV%{S}(w&AS#Zl>)+~p--Nx_%F zmY&kV8G5{NiErnOizB~u_8{wL<%KM9g)g1?oOy``eL4YVmz^x&?v_&ta(@8=;vOvp zK|#sEA?|d5Y9Yp)n<>^}niH^YNQL%AiX$NjLIVoA!4TB5u_&cuA>+(;Q+)k9 zqnZ@og-2NKrdq4&*eQyxj`vncrQ|;ERj28kW^O|DZq4ULdpybeZtY3OYTwF|sXY2* zL82ti2PX%y>Dcx&EVmhi7GG1HcO$t|4Xg>50eJtB{z_uaLq8MP%%;kS$|cV@VQNv6 zZ553KoG-v>tO$DeqoDlX#juOLY;R_5lRP7R7}G}-$JnsM$O;Y#-C;dZR6j6fXd11H$JnnvgeE32@&&+1 z2?$;2#-I@nsUvdz%B-bzNdnE~#R>f;#l{~+?rcCmC0|D6g=m!S{E}`}VqWaiBXTi{tk~1 z<9=v{C_h$aB0_>KY%FbU>@oz57Z05GCR@u6lxHR#Us&Jvw$A<3f7D=e{S3ab=3m{D zjcf2Z3t=ERse3|CStC4v|6DT8mH2r5!3u{arQjz-oKEaRbqj03g8o$JqR6p*Aqp}N z$S;-rSPa^BX-ZE0migD9jC{_!JRf$+%%}A)zk|I3Z4x-h%W&bxVr*wFK&RW(s3DcZ zK7ePYQDMU#4uw^H^TGGuvHe~n zv3t@cb{0v`oy@Oay7<;-$-na<)*ul@gK>G~%-WH?jVklQ!oUR*%qp$b(I~eibn+Vm z^G?(E`<-@s9Piw`s*M9WpMl^COhNVI*^!iV1;KWVd2e#m*Z~Cj`sDnn8Wo3d$IqIH zz?3zhk{n~ol@_E|imMr_A#%bFrbT-fnEXdnQ>S+1QYZBnGwlxPu`lE@mO?kmc?2%D&2R??>_}ekzWpGL@j7EK9U6Q%| zalTo0j6K)ONtF)c#(`PCsgD(h^uA^sW2!fM1plkhChj7O*1na>g|Qo1dZs)S(IVCL zUM{UyoKIXfk7Y<%8Tp$DGKfDS8OE&RkUNO!uG(~XITdWUMvavj)@y3*GBa#zvicA0 z1xoq^l#V!WGJr89X=zdSEi++*#sqEUes&5bzU*7(w{=C{bN4Q}rPkPHbYMH&HTm}M zo_>LvR^gsQrmVr6A*RSwvy7xRLK=|<;pDgDgguzA@r(v~i^LqL($Rv?=RIs>^q+w{ z?N5%;4BTm++hy-}ni0oB8DN`7%}~XX2$@ptGf1bX0Ymr4V94pjMo&;!3^9kJ(;940 zY?gf|+Rd(0bF?Z;6yC)|E%@CL(eHQfcu$LKYVztzN{si~Gir2NS{@IzM|HEfs7-eg z`dbidAMrUSLS5-8@hDjg>E7>xg2J0h`J}N3gdRtZS#>RKU6e4Ai}S0FMB2Ntdt_#A zh|Cw{Y?LXuR(|f`!XaASRY5px=jFFTSKg#4lGfHToGzS)!cCsi(8!_us0jCANo4YE z#Sm}~huVfl^E6GI)~{wz^g%f=#%1lcIc1{Dn0a)9!Xsf~n3uq}^xFD0PX*Qo_9G&g zwYD27k$>ykMmt2d9d`B6XQ+U0{;mfZF5ei`WqD2Of0^dt(AuiLE&H)Ox-=@3#}cVm z3SzpFYCih#*;o{(J2iJ=>|F>SE;GZvfHlgKpfNrWM50nNGe}2M+qDlZP?Np3`l`kE zn;Mwh?_xmanzuInO)%&18+%y26t5sjvbVjL{2QPotyB^}XX~ee5?d_uRUyyWEPpvU zxzFYzx+lStRN6~R1F^)h#Hr4)R=<8f&8VpR-4m+nKgRiKTwj>u{5TEfZ?|*zER#G5 zi^KCfqweh(?Sa2-VrJkTuYJttwz_c2S&AmcCsV;lh6>1KbAO(&@DXpEHwj3Mu7n9w z;lMHM{dJfaYob*7XZan!$IK!*fC9c;gW+R0(?a4Fuc>cA{p+@FKi$K&I;T0`Z~`Q`Lm zE7D>+(?hZ)S31=mx5!>oaiWahP$f$(T``iW@E|5q^-oaMy30T3GF7}3$l&J!b~h{v9TOXB7}lC;j5q#F$C{8^JS&dIa4R7 zL}F#^H9s-`2uul3G7j^9Ali|DlpB*u9OEKKSRYv3*2Y)RId;B;vA-oSYujg)-ZwD5 z?3^8pZ?sbiK45VmU}I^(HticuG+wg5h!`aGAxgM0PN4R+N}z@qIF0)+f)%I3Pfu+1 z;DPe9Q(3q~pQuV%%Kn9E@*8zqp&Tu23o2va$dFtiF}CnJW3JMh3g)bDNj$KiRAWa{ zenIGS3x0+FIk}$Si>pRcX>6MEl`$e4icB(f-u{@CxxC{@_*KT5+;8w!uJk|B0_3=P zD{#aPABv?fHiyUj7M4hdd(`e@h_d+lR8k?bdiuj|L!GPag9e-&A9-A3slt!+{(goV zys;nrj@MHU&ZswdPto6{zJ2>{f+an0?xe#0MdHD|ef0wUb#``j^;!G14ocuCO}hlg z`Gek5#L{yJ*-u$Vm0(f}z$P#Nf6(BsKPeS=x=)bJp;fq8mJkEAjnsGO@pVwGfkCCx ztzt;$)=IM=O+qNXG42>sMX%udB)8{PtCmCfIg=#O6FVSI+k?C3xPgZ&UmN?}@4^!=LN#tI`G2U=ZxeH&Uf!)iJtzO=iinW6y0F%- zA}@1&-^AAOoIYM-d&O;TeiQQf1;tiYvebMC{<@N{b*=i6L@OJLX7lo=gTd0?zRRR` zPbEM7+f*`C{I~CU%H(u~cAy&Bg#VL)LPf|d*K~y>A|*zNx|j7EUS}s%cU$jt*gP3I zisH{4uw^8fC7PppaJrHVczvl1q6M{N3HH|ZeJ=HXa~BMA;bJz47AEc@817qeS*$DDFh$`-%hgp~ z*WJFc7THH%CVeCXRgW)m%`Aq1xsrdb$Ey?bf0$!v>J0}H=BfQDgqrHW4z0Q2w5Zc8 zhqN@PAe`$(n=gfYZm6j~Y4|3r^VS+pgqv;vQh8r0h7ep9z5|Mf_puibFqQ~KYnovW z)t6FAwup0sjxSmBW5yMkVJUuU<*a{cgP$MX=bvd|i14E-Jf~-L$NRn!nMo72)91|Ui_oQ#hIA92AIbXDOA}|K{K_U=(VB5 z&j9J8NkHx`1N39aetXwu!zc6kU=a=?N$cMc=J)=B{<{=po2r~b%`zkXGvU2VZ6xTh zOWo1%28T`4ESnZ?%D8&J%*gl2w=yQpwtqQ5JDbGxZf*l%>I>yQwF_{Q=u@YfLkj+w ze_3G1ks*NP^khW|6|j>FqCBdX5c*E&tcqwxT`uURUV7Vg~ zj>TEqUFEMoATB(gww&c^9Z!`&%%CUDxTx`Uat>21con#9;zSFBo%Fve|9(xBDh_OE zf>05_=QgZ<8RkrNiY{Tz|1%3ezq6Zh9FB!jMKTGe&AaI+8`C1uF2DYMeX4mo_Ym;d zmNqeGF~{kqA`N*_13S!1vj&|=f9UyT+gs9~D%IU0ByDpX#YZmtXyE56eN zra3H-ZL1Z>rT+SF_JqUnxK zvIBC0DFi=T7pajs7{7$9a=-+>6%!M8(5ho3Ws`>?M&rtZ@sUC4*fru^R~#_Gr}!8n zEz+NNgfY0JEJIFmVJ-QfMwJe4zz|l0KU~$VL$J9VAH|7z%7^j@H1Cw&W?_h4!-e^y zHt!~wIp4?J7t9fVd|tG!SU?Ar`tkQ6<9?lLE2!(Z?_()V!mXeT)GxN!=ATWGzy8|c zGH^S{G#X0b?MbjQcF zl>eOOzEL~keH2nuJR3^C7L94d`_;0MsbR`4P00R+;N>7yY5KWxnJ(^LeNAdr>=eoy zt|Mkx-=BDzRjN_Hp8^ySLI6fRn-LmUm*`LB{6Z}P#%Qrp55ROFGld6(c?cy1P(s+#Ob*+hL1dQ6KGWZHlFyDZf zO@bsC0I!Eq!IjoZWM8RQnXb!{?kIn~tlqVoCwnE3?zUun74_SUpq=OLgn`xlAOTk-?(LyS=ndL#F zn3649UqQ;}7<=y?hRr z0(=OTiryb3y-4+xO^!$yovET9zUfJ!uv|~*8h!Td4>To`)p{+OyqX-9LYs43J4T+J zcZhh^grz_H|_*r!Z8YwH5k=*eKE9$RAr4H5F3Eu^pg?4p97eblHkmC5U_*t4~KP*PJ#sASFFKRe0|u`1-qLYLA_nDK}} z_}w`2W>jUSiD@-9i1#kpRNO!i$rw^V3>>xG$Am2X{SR(Xi0jd=6w~`r26g_DO1aK7q6QE#D@6iz@FLE$XVBO`=$u!(=B?;)q~Vs<$gR zQQq|A>$ps+z*q-S=dqm7u*O|4F@RPOG`fDQhZuO=n;JBFNRP5Sj6S_{@Csq4MNB=E z+safqkuC7%)XZ%xRYN=@BAb=_!EhwIr78mgYaqMYIfJ~fWdKBgoZR2;?Uy7q@ozL; zz9AiqhuiciSx|m@=%>(%lc^p%J1}zYO?FM3$h@_;>rH<$Df>_RABVK3XF_^hyCFP` zrIw{;0@K(+gly)hTmAHA2srfMz)+Z4{$qd0gWoLS!B?1MB$rt!37i!Xm@y3GZ%00) zv5dRzZRtS%tym5#ISd(g8@(~U_w&7D`iS}0oPogbjoVxa>`=X5G5;n)h_Dtp@)_J4 z1>?sF7W`s3%QsyaK{>>Q-?1IEp$9H*Eds}G9=t{lxypAmpE3ufh>N5fZ%)UiXF^O- zNxyH@?|l?g?c6FX#{+}HHGn49fR3 zfkl+e{@JvHM`k0MQq`Ehz2RksT6*FZm~`Lm4%L%g<}j-y@1g!So88-Z@DoD@7dADT z&S}re%rg-t3g(lWMoefLI>J+W68F90Esyz;ofk4O9HDN*N&{W6;4ISXLiA*vuG0s0bLL!|(DgajgII0B%(H?6A%4TUFZhaJ57Y_aUas5+naPiCq>b;zm9;{< zU`aT*WFf)0uoiEZF*@CAXv3#)p4+QohW2S{^R#O1hR3#w#&cSPvh_!MuJ+!}KpQDq z(HkwdS6B%qC3@bOUv3Y5NXqL4p1A~-|2AM^Dws4MC9cXm(;^;a!b0 z7Op~;Z_+SD0f8z?7d3QrI2--eB(Kc!XGzD-HUx?4>9_?34zr;vM-S!PZJ_@yBVPN0 zvJnVWsJ!Yt33j)o&QK3%mGIxk^&kz?L! znmt6-d}P1?voaf;ZvTFq^JfG#is;o_ub?DSa!RRr=(yZOxwJ|cw#5!z<5F`PrD&)0qR-t**jVd-4Eg{Mh{0SaNV0tFud6mSXkIc;vD>?=>6|eM z%L*%2V7Gk)7Wob!_88<{S{nmu2}F!eI^>vrD(TpH&eoCpUr^drQ0~9iaT1c<9k~cC zzhOi`6(_3An)DnSvcg{x>k9_K61(L>ZTaqP_9sdF11MJl6j6Qt;Tiu zEJY*=DE+Bm=(DvYx)gKzyV#9t0aJsj4UV2?bFSZ)ZNRH$o4deRu8los&O*na6 zsNuhzJt+|SG3^e5pTWiGi|7zdop0}8GYgc|fh7&XsvvgYiZv1Fc5O8diT{1u>BWTx&5Yf9F0RB{&?5 z>W(4cRj?gX)EQB8um^jpy=OM8B=GY$Tj6G#O#Do32971*?GUkpvI?-Z*X}QZ!>+kf z!(qlwTE4#4ywudQve+8+p{%v`OWW4`s(UoHI(3vNAh4(4Dsnt8qclvGC{T(j!`$j z5b`%i2%H2UNTUM=1_)NXmqlzCZHiQ<2kkaOxGZ}AqOhM2^1omTf_d5Szejof+6#wL z&4)PHIh+%g-~#OHbz#sz9GQNN8EQwph02-3Hy$92TXR`M4to#`@(9zbGU@|F88#b@ zya5N}e+T;i+c`1*3D1I+21~fJ;er$PfGK>cdsMrsib^(GaEAQPH#;*j2z4B-9>mO} zA8`5`S2(e%(6#RB>gq&o>lXjv-^~#6Gu;NetP=R{!ytMvUVw|Nx-Z2#EG#bhc4uqY z>~T~U`g2OlX|wDz9{N3f1jZmH!0xXByHKbERY3RkiuJc{M_mZ)F#rupqIkHN2P|kt zTCZClNIuWtvEQxZL^a0i#toL$-VljNB%}gQNy9Mc+xRrtuh~Uynh;8~->rJtPY71b z!sKuE{QH6Od*~myYJtV&;w8K1K3hS{=X`0LI!-=!hqnKRqrZhiL4Ea?ZCQ=(P_ya zA7JeRm+}c^7w0{|-m`_bn_nMW|2F!+C((#uVQt-WCoo;%)egnv)>(_VAS}-r*_urc zVr3nwN_WhZ_csDb8BT?dZ8@sX1;zoYjP*Dr2kr20=vwBfEva+ip=xSsm!>~>?7z1k zcahWw+7MLG9ezC)E7xf1f1L7)Fn=C?nU^mQ?x0(Q7WQ5 zp!P#Rt;PG4j&v!u=(Ksvy3H=jjgJ2lb5sMKck4E}S;DS4HtRY|%9nFlO$_gsE4P#{|f1z-K;_uGM})NFfoXoRl&~lzpA1|ZqU}#F|Bk4 zl!ER+n1SF+c4wPBTUNKAbH`F7YAZCB)^kvcE%wZWUF8t4JBR^EQX)9ylsy$QfavL% zRI8se*mCDfx`^qUKGZ?@tn!qcD#fZaaTwW5V1199c2f!)I5jd*OgpJybsnYtuB8Fe>Ur+E5^%7VksV=y5DtB)ea?^XLbYF;WcGU4Nih_! zh7xdDO=pV+RT^m4S>gOe`T9N07s$^rbf^QO1;2+j6j^z8QFh*@b2K~5!=u`yUbJq{ zBW_h;;c3py%w*C>7E=Eh1jHO-I4w&TfpGF9c%U?K|EAlkAFtkX<+_X_HcX0)xy;@1 zy`@6OB62f|t(iwp)5;28{Y}!W<38eo2*hB1vXIh|yKDN)n<^obs%rQsmKqO)v@l*j zM+Bd7m<(drR|%FWj7*6XyM-HF_gHU*WYi z32e}ob>3f%a45nVd16mwe0A5|IgvaZHq2qY@o0Nkos^8dTx@n_(2gHUpe5sT0;|3M zA!%WNrZx%N?Rke(EDkFMQ>50Y!il=sYUfcZ`f`-e2mI z{0BtXTlIb1s!|87Z<;ySYDx2ykn1ofZKZo?12wqw%CzJgN!w4c#e8j^*8`z^B?|LP~@KMcs9I5h^c z+r{?e1SqQVZpp5`{u9ywa3)w^_;E6hu_>x`m^A%knnk3O4uMfs;;_0D1%@~H{GuMn zzNNa4u{e2txN`i15B;ePoV}(*fi=sBv+p8;Ca>rATFjr}-fhd7%bMRFC-g|iT(RsC z5+d@Q!hep(h0$}It=*|%y)Xl#zgSKdKFK_&{1hmoHst!=7m_yQwj2hX8LWR!ss>2K z&xd;oe?M<8OP!w z?Hq1T1h1PJjLO+^RcRZ7F9Ice6`92t7WxeA9b}+JiU5Q-L)P-jT(7VIe=itI7f>G{ zXj})V=L8iUf16z$OgY3ZC~th8F9s+OZGcUHTpySLZMt_nt@yp(`R(O@p^P*1eQA(oe>Zi zmg=0mmHGhCU!}q1-ZGZ=^3VM>u*_E)ncqUNVVCV`3Vjdo9{Co=i3J+i;#9);k}-0)W0B3fv=Q??$i3V%qHrqCC3qJLabCjL%_DD zZ)ZF(?O8~g3%rdsb@s--m%VnhFBcWRcdV8Zo>?B^?1s}55?bZed81!rfiKChN#KA+ z%msH<>cZ|GYS^5 zf>ipPY1}!9?CULs^ygmc76ctdF|gpNb6-E#mNA1wsxK@JmM<8koU&GZ^6|Zj2rS8)iFHgl*ehR22S<0E#S`p7cQl{V> zF!4i6WbN3kCe2X_xnG=XrwMa_yEiHxZl_vT0s$S4`!>2RZ==z8Wc^tERm`530!b!z znIw+ayOxhFk?)$D?fx=|o$C{4hU!@lWv7egSxL{S&MSIzz`qGneH1jukdrF3B=Sx-Ab$GcG-6+7bt^Vw5M1=TwIw#o0Gn6f%JvtjiTn zP>87WOfSYi-E5esfX7B>hQM0uvUgM7Rr=XSGNe&0=zb?c-0zdRJ3#w>l4T1h%UReP z)k{ICJlLKEJ&!7kd@V%7bz2jHKexlARnz2pqkt!t6)exyVRYF`m+lY64-F!#)Jj ze!`!RqD?BTl}z#w>zwh%Jvc&S|A866?I-Y7!QzbQfY zq&=^0xC|<)KdCxBJGeQT8@WqGlA$8EcGfNkeNyi*IKj(?Q_Dje`@oIH@NTeDl3*tE zB%5{Ed%-}c2?DMydiY{^z^M1#J3NTPqYCpc)vOOx>sCvYvLE<6*zbe=k=q66gm5a1 zo~1iUXx#6gD?U)uNOZ!jkt7DET~GLdOPvpjb>f7hLp^y?h{ucg~RR6G}Ux&YC?Jx6n*Hl=7R{i6&NI6;1 zchT}#8_g;OaBWF!J$9Xe#3E3dNVcErYxOsxmcZoOjw)y_m(Nh;d4&fZlm+OF6BgpE#m{z zeg8S>i3kB?qa1pv+?h?_P{>4 z0fdBAz8dq?o=r$5;Bs;DI0P$5tB1Yf1n2YuzB|$worV^5i>b!Ivc&g~td1l`5Bh~p z3qqIk&DK{v*((xS^RiPwdcUu&wDq}zJ~94^FOKzp zLA?u4s<#X9Evehs$xo)(?pzbFt?PDsP6541b5;ymIAYtRMxXLV_Jn z=VM3kg@Oo7?A)_fhfu2H8mMV0J%tE z(dU-DIa}+Ioam5^z75qMp~wCtx>`0gmPpC&0-Y$hWJFSnlu>Zj^jGQTD&P z+QaYWU2{t6HJ%$0#W?;17@A>2)&W}-1bff(ST^_O)>p%q&{2u949oxD3xJpnYwB@F zL7#ulW6X%N*={6d&$p?$b+KO#pV#oVm{0ic_8I^)8B+mkM^vh_qRIG)H{Wc~_)kg< z5$jmLj)fqZD2(-~h66T)3#7N|c1Ci|P9Pg8`#tkw+Lp?$rVs6m$NP40;oNp@<)u7E zMFavy3Sdw#_Xfhy)il3MWQzh_{)h{*lDd$?dMEHr%ZlB)0^u+kpf}(7T@L};?%{Fk zh9>;u^LMY)oUc%NR|fDU4p7r`h!@yi2ADJ2QG6z^wT1njp9Kb1acYQz5f#S0AAfNF zP4LvxRC}oN)UUa&9|c|gwZ3ZqJ9qY~+PLhpaLb^m2)XBV??nAO`I3OU^8WS|QsD}X zw>p)8$z5WlT)(irTjh#NxcD1!*D_tf^jco!zD4TvXHx!O=ZgUbV?3M!#Qw9@w(=Vr zTtc^vZaA6K$45Ou8nFcXu6i|>)PfSMcT$MdYQZt;T(+)nzyK zu3^PkTJV!pZ-=w=D2Hja=|FX1LEA$FfkHaIONs=H7g%SgItb=tA4FY=b?lRCJ+90e z(Elwifdc^l27*Jp0bL+MQ0(>zkoAezQejAqd#<)x|6vkE{PGrj2?IPkA(YL)3uH5t z9}$bJriOm5^%pdE{u0%af6Kg@SDOFYJe2_T24jfni#{=O?cVVz;QaMvp|P<{5KFcF ziF1LquA}h=uJ&ic_j#v+9i#=|=i^hmLc>dF2b5+{kMdg7QxhOv;YdTggWjh$R+eEqO{wwyx%$!-P0 zpanjlCSZDr!o&?A(=59WuWR3{<@q@8cjwqiO^v3+IS)K7`8s25u6kxh11`q_T~;3# zoV<~1zQ_49VYfV1y$p~xx%B>PFDNlYC?n@efUv17c7qTBn|Uq;R6BM|2l-UifVKs| z@B>;v$f9Z7AN{km)J6H05D?_|fal%@)k{<_TI1?tdR+6qG4*}Cwp8Al{;50=LnIwN z69ONc#$n!HOK@B8^_>I+j1cf`os&ABNfh}gK%10vnnibCd2)aYjx%dB;dgmfN5#m? zJNhQ-_VQ-572{_~+9k&^iHGanqU}X5w=-dz`zMnc>7u>F$s2?m6~+;ISIlv3H5w@ign6$)6;m;KH1%|wym=^pX(EFxF|lC z<3z8ynQxVSUtLy}woPE=ne*OBmc1x`_WO(7y^i~HMIAhsbOA(o$+*uVFlH0Z&ff>H zk~UN*X-uUgh}%9@35}6_yt=RVI*uI}+An8(8h7VtC4X9Z2fyZRlKmuONa4>?42Ujs zW1=OQ=YTE_XmyEEWyQtcm{sw5E&_-<;NOywp6-BA2gtllnkFN#2Ch$*?vl0bC{TN^ z#L?d;av5QE<9s2KR&!r*RS!&vWY%lV@#1x{XduWc`GrcoTmtX9d3y8{1A+{A92IeW zKDNFfmxP8p`&}LkD~48<{K$O3E{}`g47QViv%af%xSt>J`TgYWx;ketzsBJvq+`1+ z10BL#V_thXt;hP|RBERg@q_Aqa9Y~Iq1!UvrbzA$MoU!_tz1x9QXb#e`}PcX3ahK-iJMN*K?;U zh`Jbay63Vf$)!UC`A@3;&b-iriIQEP=G4?xg!yl(eu@`iI&b0Y7oq9z#-=o^{H(dl zP8kYEjb+;R-g?l6SExER!#bm>qz4K-xxXT|6JBn4eLZe9Mh>$HQ_+3DKraNz^L+gu z2-=BuJCP#}2sj)AWyZ(?ALZl8L!`s8;|Co*DyUuNj`go@%NyjB%AItry^iul z6QXY_${^Zf4(g=)(K*;Z=cSuYZRPfWbNg}zm`UnPb31LiL%-T@Ipg3@7d>g8>D|{$ zVn+|A%gEwj*`3*Nv|xbe8JPw68-Ng|GdH`?&}Ma4APz1^oMsVi zjUkVc;1K|qk}|KzS@W)yh-mJSwhL9-CcaJoO=kcsBwneX3z>o3Y)z3-uVchI!?J|?zd zfJ(H03X!MUNMcrPVTmSutOTDTUI8Kl%AB|VeE_tXQPZz_m6hi%5Pia;TSHMipk;+SPxxYH}bY~#V@Cvh&u*_axX}* zK_dLlVVHzpi0ehQ;r0LKHRZXl>Vre4U9?Xfb?r`fTCoxsH+dwS!;OR*{%h-K<6O3E;8}NaIQHj$xaUgscB0PGaxq%vy5}JtaDL(Ia5mvvu<`*KHv7LBtVf)MPOKR}x_{cB?R11ir>WLRfk zq4g)MgquJ+B$UbJoGvJm+H&seF@I*13&Oz~^h30GT} zKQ<8z=+e)!L-$9U*Hqf28J$w^$ibKBz)`|7<|iLzxH1w&5EHEB&7a&T=Ccbl+sz`> zEHls^(YR!h9PyhW)Qb;`V1CpECLp|4mL!e+*X{W?<_yVP7EzS}Bl z{{#+G zrRZnpu+|!>G43USDk99WRK-?EWlhxG=rh&9KKDqVEblu3b?0Ar!;M+;CqS3eUY-bfO#w zTW@IJ=6W)ZHl1Nx4vfmG@%k!msNw{>S*BSKu>TKRe*slx_q+kaaA-ujyIZ;`i_gn9Ot6akKAa0{|B_D)}Zi7J#tWo~r|MFSi`*tNGt>46~_G~fkP|I%mD*&!_hi6C0 zvnB$FAJ+w!Oa4b_J*?n&_m#?M=1iTDJE&@;Wtzl2=C@;+X?8y56vt+ASs?a4L6GGS zOr-wzb?n5B!uUz9Lq1i^Wr#Rn;;Yf}vlurf_@bGnC{Dv!iii*c&Tcuo{b$~tOS>G2 zz3;Lf>tD-zy64amqvU$BJQi*@pZaKl;k{XN7%GIhLhNg1m<7Dy5co_<89V7%aEl^y z<`{vH^~~ao9X&%6uwM?GdX}A^he5+~hKZ#12G0vaGq%Xrfl!(EQ*~dJH@dN1V%rEl zR531SUwQ0JNSEeNF)g!`qQC1J&ImQM|C77xO=6cW@WTi~1g(t&u6bR%d;yhoPS$;N zV!SFrDOFF$opFC{(NrG3@MV7O>ydh9c*#lCUTjoozb$RA z()m04_YblUXx)qWL-qYUji*K}vt#ZV_I77LI^VCV!KBkxOnn&eDA!7u4e<@7o6`Pu zU%2L0bppjOAULS183Iqy31@~;&GG`YB28MIXl#wr7pxDsn~ybk==14`g z?6(nvQbU*&<0hf)*)9}3Ory9LpGJyZoaP^{H>qChMK62N(x2AWBnN;0XGaoF5$Vn#OnY%}GL3d7P`wo6Wbb z@qjp*%JVJFWYXUT-S?z8B&T0@)<~9Za)nPqTvFehDQo)$(beu;Vm|(on(G0Dk3$z6 zvev`kMmoV-`PJpL9Y+;sTBO7}i1^6-kSuG-!o{7T!1Xd6rM~%l^08-V(#G<*v^rni z+FW&tWhyI`Q;J4GI%*nRzpjp;@lBeZvjMqhATl^Ii1=O)ICfBY^kXHYF*BF_`Dm@! zj3@*hGgbVy_0rg^3ApJ#t&#Sf0-VN+tLJhE**@i-R@e)0b}$BmlRj__QItOlut?~3 zoBcEL;Ujj#StLG$2IjJpL_opWpGzW2`|O_bx%C9Nan5Y!|D$S-ETK4eC({-Tb>T4S zYW>m9EYxntSETi&5q)M3%^rgV`d$_fK>$7o4PiusoNp{fOCMOry?1LgqRMe5ncaW* zJxiU9;zGoNV#3~&G(0HD`J{;wE3XM`Hy1|gZUaiU;nE2*wZM-fZ38~st~aXMm?*#7 z5S>2sTS%OHo@gSWzwDS{JZ~;-!*iv^A2Xz&`K20&)~{cUh&EX;AfAF|p_`HVO5G^E z=oypR_+d3#A21RJ@EV_v`~Cc?(}<6kR0i_Js9|iDqf%>MqWSMO$<}sepu_Sv*-d9A zkb%3HUI#`NNf)nM6Q1Hk>+jj0TQG#Tlc|mQiJpe=1_$=IhRMesOU9ZGs)?!}k7{Pn zf#RIy?*S#e48q0EyZJSr@qcMVF>|IMzha5r%@$s zI}@#?@wtGIvbIM}dVzS|-cS41(Ed-+76v|9YFB;7Us)F{sK()-(JIqABM7H z{y0`6{49IF{Oek9gYdF z7`d}cg_qAMG4tS{p!3o0Ng?zSVZdK-^ONS^O1!rH7=@cvdB$bo{v6?2gtw8TN zs@HS4c|pA|H7R>6F~HZ}O-GH@)h$yF`Tys64~H$_c4?^^X#M|s>_J>q6z2L0(?J6G zJ^;@B|2pM^rLaNKC-Olpt&IQs|Ns7|*At*DMgrY5r*etwBFB*+q)|q>*~(cAJ2or! z7Kb& z6*#)d*{8C|Kb*caHa0%qV9ne_ebaj3>xZx^iqp#-Y7N}RMGQsUl+3Ihs7arbK# zMSU@u7Q%oDA5A^*#^?B2PQ$cM zp3OPI2xW0#NS8ONk;7P-nufY$J3;yNQUBwOZZIsdQ^A4fLpCmYj%bh{q#ew&Xb(q8 zBgt-f=~o`YVtR!=yEaDo)!A+y7O-;6pNm*5>ud{>_}RoW`yyys=Hh^UU=#Ax55M>4d((=VZNT%XZ3g53krE zRG|GuY}=Dv^xMJTksQj=8UbPiC3KR*+=X4tCx?e3F1w0=-_O8+q^C$+Pe}>w?`@IZ zKaS=?an&{`+jZsruV0}_4Qv)7sJ;HFqAnXU+LX#A*x#olAjfY_5B2X}T>aFI!2!mb z@;zCQQtvx{PY91xx+pveYPxgx4|H|kFSt}Vfg^q2ErM8;QHc?$?a|ykJkmE~V=?>h zT>u=dsioxt&<`*3Y!P6=m)6uL)L&y&58x6QuX&9@NH<>?=|FWY0}xgQm#6ywkUqH5 z|H4C7kIs^=gO~5cHqI*!0+Hme9!;s6aCBLvwkv&1(oCf@e4Zb;c+y3m6C5W$(@d}6zYb#YuRl`Ue=QF|!$z%P|EaHZtT9m%w{I_GJnw0+x}lUsgy+T^ z7+1?Em-dx^7}Jbyu@Di&Kk4C^3gY-YdC+UniD2<3OBDDxTr&VSBZ)m)oL^b_!}<4| z6Bbd>?YFr5+R}~G@XYs_rkZpv3VieV{i#T2!Nu&ArqujPo%4ZlgQr7F(SIvCR=T0} zP35Hkq+xS9ttBFaM|ePX-*-MQJw?IEe48NeIzmLO8x9M<{lALavV-Kb`|bo}cN~5N ze)}&a**@d2lLi&|9sc>JqOpH{2;7S$N=lOdPe1U)1S&xEO>%j~u??R61xG^ipRS+P z9jfacfW5ZCM&spb-gvOoGLU4n)$p>e=g&c8lbJ5hcmo+oGnz zq>rrj*`i?;!BO1fxKN2Bv~Q-2wnyo$;zM z=~_vBe5tD|4pN`o>;bngaAGqA-VFlI>OBp;MLe2sY>;%(?*6iVKwmhO1tGO}*M*HB zo)e~@6F|Ae@HX09e*oz5d11?8DE2Qb_$i9NwK_^5%(tih z8W5ZS$SDAz5y9R+pxCbbre)1#0`QgE*D$K3^S%Ab?iz)yA2~}%f)oq~7I>c@8&U=o zaw4fCj9vhy7ljqTg?&1#Hw?K|`AztHI%PM4JiSVNoHLyk>D+jn*)^8kIlavV*o;wJ zg0t!0l>Lvi`BY;O0N?7zXH-%+_{V1)=RSY_3=lS@5zn8zik#&TLMP$@!wnJpbLcAq z-d^}I`}{wx4G_2gNLLrCxQ|JdT zFL4@vCH|WT4FCO^j_rTv0;r4t__9fyaw-~>-+(9DYnBWO9D*8fki3qzZsXN{sCL zAf<|%xO4&P17L*moh-GKrj~@M?Ko#0&iXjeiz{wDSoR1^)X-p)23ed;KJfVls^qYh z{m~50=Cz3(tG8QZ^$u1?7iYla2mQ^Xl;D>L1Iq@2^}5gg6PEuGq&H}h>byk&y;i0m^nXij2i8mdX$$$T2#6~Ju)pe%l6OGt z(#s{fOAxtp*gBhMGeHf;2Z9ohnVFvD3IxSP0@3b&w#iMbRN5(nb)%FKp=qmA5G(qW zr?a6UnS>%nbAdt}2M5Qn#}ZMgGkT0aksJ-;$^L$6dHH#fJ^?Kv1ZX*RDP-v>R0`OF zkn!;UD=Gqb;x%Qfs;i6hpMONQ%4=8E!6Aj*_|52QX}w+}T3TGRg2eVFqy1i8TT{L8 z+QR<(N3j@yS#O4jOl2F=yD*s$0L6Jh5YqFfcz}B!XamTbaau%x_E4w^`fq>8(#@{8 z;>7V`k_tszI-%<1;#_^;MEM&{%nb?83u?``)^!%Iedw?H0_j5`LU~n4D z4P#z@nKa^6{x;+z==)Z5lg}Y1==547$~o;g zm3aT8rDz?%9pzir)~}aKfcuS$7D&&qmUhNRn1;uef9~)P4g!)U-kAMsEn*g;1~;G{ z$jZhSVsmZm`FU2TY1H()O$)6a&L0rTAV%oW09WTc86Y`sGR4R!h(H4;Mi{r0DcmmKfA;Ak{tsPH?BCBVPi?paS`r`p8)-zhBS8 zK@|#4fD-{rRInt4S{MqtX4lG+0$ZcpUYtPWJ@hDAw_LPu(!~L3a;H6iyYr;A0!(Vn zo)hkU#?Nbjt1lmgfax)SGtk{-03W=CD-V-kPg6fZC6?P~*GUQ>fM*DdxevQjqJmBT_VJGLH{c>&J&87<;I37PlgTBWrC^$TUoe0k`pR>31q zIQxf(D**3M(gfokbZ~Guq3d65d1u;Ww>U!e-tTgME?fc4u?{#ouk1e{Mo>9``Vm#x z72Vw%#egsG8>pkuY!7wWnkpty_)z{AT<=@OMzeSXDkS|bH5me z$mTXQTok34n1`s9pKjg^3o5Iqs1X(u%i6wxt`AniIF)1ct3Zm*63`Ajs7r?Lx)#-D z)X%};I|y=4R(1%_*4kWwAzGXtMit-%0Q!!$5d(@5z)1<9uyBfNU>xyY;5QJ{U?hUj zdqv${9q&Yc2HeYx9OGi^t_ny`t~0T6!qB zAH=LxSxK~4G)X5*kGy~?6~oUu)z+^Y=&OAC=M!}*a|CHe6%gj;eHMro_f*NFhw^PJ@7Ok?4 z=#x$yBux|eJg<^n%f?&#=?ka8R@{Q*yb@-WUS^(u;K-CRGp!DhmB>^MWwiqbRz1WFAJ=gPPl z3gF(Lngw@_#zcoQ9$IV0`MIl+#vyVYJmXd}oBBJm;oYm}p6Pv89)yS*u_1@D8d%3N z{Eak)IHk3)mC++TQOHWrs$r}Jx=o+aov}NSQV+G?7BaZT9eFBxB0wz2HB6j=$rB!n zEvH!OAIHZPJ=1l)pX_?~_wgn|*mY0Ud$04yr0A)rYqo!~z;J%OITg=NPPjw@@OyLH zl#$&{tk8;C&VMsJWZ~YXG30#xf#>tCaI}`V3M*k zG?l%*+uk?NQ8GO09oU;AT6;dwnoK-7-3p@`)(&g{m0URFhr-8Rkg`DQyu9DJvx%mq ze{1;6UK(i0v;neceHH5Q-f6J-!{eGg%X1csUA}t5_LSsHmfO4%4riak@;Wzh6YpBm zc!>3{v2hczNb%cTTv*iwMqE=lLBa0Q!?QK{GK;|Du^{fhRw>3XcF3>UYnhIu)nTIF9?#>3M_EPTHv+!x4rT{-0YFW>rI=lEJZe#5$8R* zKdKk`o}?ZB`U@H3Mk+za&34p!0}Fc3+;0`Fy>S642-w*GTGr}KvE2}OkbOu0aG{Vu zv09?@!NS|Wi&GY58 z5hA_l`u6y!Z`dm(kM?6y24*LSkoWDRb3q1Fx8^aob}HW!+~20uJcdbZYV`=oYUzhB(djbytPO0l>z*An?RjRc5SQ4ooUp*GW?Z7Qhr0H zmwwswV0oHwztXUNptlXRP^-=_F)J&~#L86XgtU>EqQ?XZEQV0Vn>8O=@vzZYd^4=X zOLAoo^Bm#&C(?k>`F$4q7&1tifqED7f&3gmk zmt)pr)Mt+Va6?{)tMg>zraCxGBRWwvx=Te_(^9^(v^QpaTJZMo7Mlnc-TMMK*D)2& zuE7$0KlT?Aq1wP5_J&Y=vrr6ZyJd%os@gcnq!4TVQ>cB?PwNDI;)k_a^`mpK*79%o zX7uoAb=9}TWOc2yF~d*(05|#%YdKV5OTW12I!N+Ud*=lWV;kP5){Njr%+qn<+9=SD zY1`^ixPs#Y^!%iXTt#$5%Z$)pOA#_^_`R3jMzril`-;ss3X_?_iO@)3)Wpba=q{P~ zK6_VOq>^$TD)Rx=0^OQ&iC2Nl>@&S6Wh!$E$jUes`y?_c1V5$@F^=DZO;nJ?JVS_l z;F6EI+HiC54u^8q&-|k?^bo@D3x^f@5zyC&NW>FPgWp@D5sV$;xg&ml4wd1Tpg7vH zPTmzl29eZ0Yl zEbhU`5e0T8;og4Rf(?Isb7v?FHxo6K^(?<^EsNT(wQK|x?6hEDQnE_`c=RF z%i}9O^79w2YILRD%)K6yGaIk6=en5$F}YXs*;~)v997;$wQ>?=7f#wI5EHCFd_kJ= z6D?QptbN^CR^=-|k{)Sw^$de*GRO^nI68@Sm1F)04e+W&uO~cj+nRnl@^or4ERGW~ z7H@K!I_&5^m?d>iQjeR4!hYsVs5jW^Tvnid>K5w2j>QCP8oY1l~gThS+EN?2I%-)hr&0x6KjVkAHsp--n9{54tcz&os>gUA}+T z^PzQ5HGm{!(chN*=%Rxv)n~YQcjJCTvra}gSBq(h!GI&i03o}WA*ARc@fFU;G5wck zeU0elQ#r?85;uKVE4FUiTFP?^#9p#w?x}B$k5h4w4bOJE+gX_CX|lJZ{-kx0>ymJ5EETR%P?rOASRB(b;&En~7^xOHngJ zuwSzph(Joc1N5C0cpU;*)vWkPHy|aCUz>Asq_)m0+QNA^p8I=$$dgaBjEa!ccS}Kd z3O{=0!T+fkW z9~rZ}AxE4$A-)ruA>Yn2Rl(q`&VZ{j;E83|ukH0Rfu^=XDxJ-H##=eGwu#XcJy3MhYyt+x{X!}|{B=`M_<&*aAN5eFw0k}}z{Esl3S-Ew5w z+q#x5HU$I{NZv46B9B9#i39s^>BHB2GXgV4)aWpZpYgYCxIa+9)isySS+6Y?JzFAne`_);455U{iaJbqtM{N8Um}+)c6O4nmYi=Judj=K zm@D3!e3?l)TCZ))FFVkYCj^hK_XtZnL4r*50usJ&G|C(P%QGb4>|g&a4%Hd&rgMH%#loY2&D-aaZ&R zbK`C%o?Dr5bL)xSkEvD-7Y(cWaWbT_R1%K6Cn1lo#pL5n3gCG2hAbe)PmWhcd9~lv znhQ)ftmkBu%yfoksBR6S{)*&T;Z#?%Mqk+bco=dttKCUsUx@b0(q>&J*{2K-l{NLOVce+e>@jlH%)+!B9CyY8oT#iq)CLc@zcqK}?7NzK2; zyjAgFw?vYq&V*B@v&Q1>izG;bGesL(#av7pOcj}85V~FwjAg>>@JVjLmE@audNy4? z2wax&KV0N-{L)A9Y!`J3#8+-j4+FEu9P9P9V<7gNyT5-(>Ras|1YnOcA5Pw7G$msA z>i-$p>9>9Z0aqZ-&AH#Q=`xK5-SqZu?hRD@$~+~+I6)IHXGBKa-@?~VMe7^2q$qTj z_tHI4gclH>dTcx~rc}9Bz&p~fj!8yVTb;zEso2^?iSy7E7&Hvy%YF_oB5nATaEG$l z3e&*Fo_5M~_*44k_|4PaiQe_!<*R>YBlgsw4&%#siXLK>m2ZAkTP<;KbW9ek>MB`7 zKJ?&HVYTp4DSITqXXnF?zBKi@CU1?v7Gze2sokazCy|1jY@Vb*e$;(*fHK$VPfG=1QUz2oAvdLMbL|p$@Y&RDzUo2SO5*MY6MLdFunc8IMfE>O)kAQp~pzE~v?(X;80T8uiN@gX4``#p9!Uu)uFY zJc~(D(yk?e$h0S7Qb&1D%}A_6%Efi4pCWQihxHt3s=q2IcSmUlO3xFMKKZ%TWsS#^c^~HJpsIb4Hg>e#C7yS&A zyx-FK^_uTZJwGR>A@@(m;X%TeFAW)tQVc%))W(bE7@k)lmCwF%EqpWg2K}lcQ{N5x zd%TJMqFgTL2QGxzOHEDFc|8by*s{I|RU5vc@3XNtel=VJ8-d9h{czK2K;5Fz+eh>8 zVgAotR7vP9X&o#CS4^6@=v*Q;EqKzXeysRri~*Ge{L?dEHFq{O4BQEilcwUMLq8fS zYR1ksewyh%hBk!3X`oOtRAWw#W^*gQbY@;2uRV~=5WC}}qAJm0?oFVe+RLmr(LI@L zX&Vf9(VvJQahOd^QJe%j)jMHIhha>;{~`YgfQfau%)a7Gv=x#;y$X*^Mqn#n!(F%Z zqFY@`gZ}yABg?C1t^KMhm3X^udu6d%AO?-Aw($vWL|^N%I{MDgFiPg~{by}<#_f*5 zBUFc~m(g3!IC6JI;R=-AgD3K1x^E6r!-YcPmkbt-*}jbBGZA5r|ae2lw;q?}9--)P!EA$IM5b$U-;RTrI+mTvwbPyuUm z+Ir^t4a-hsOPp9XMHCUIM^4;i`&riRuQfjpYjlmu)J#Td=-GipD3V8L_>yo`>?C`A zTX2fcO{4hfoD&g$4y_xjCpeV+{;1dQASjf`26p96{n7mhF@3YaSe+bRx8%Q`YPrSD zp~N@rv_8Z5>nPM}2ko1l{b$3S2nj;H8SQ>huu{;&Z2qDF2?}S!5<4ShemZnL9&m`Z%2Q*^!_6}1PUl!@ zb)n=AXEk_|;DC>3NGoq|ktIJcv-GUE2yXR6U7`1PRPU|uO0yjHR@*cN)o?5Xh{@qm zyX^>bv>v_0BizMJrR7MED}(5xpIdW0n%U`4^8s$}k@Me&Yu++D0xgWZu{gMO+WZjj zku)t@G_Nuef|Yi>0=PuD85_b&9~Y^h_aFh9r0G@2LZRgn285CO4`qj3DBGBQ?ogP{ zu|Up^*d}o4cA69%v)Q`H%zb~WQTp@iN1y&+wAMN&!X>Lj_c<-7#gLW}3Vm zUalnYl{<$|6vn8s2jcvUV9E~ns`{-n)Xjh-M4N9EV;N(9gHj_NgElxdFtyg?k6%1~;@zyk z12b1pEIQO86v=+2uQrgy&SV}2?0L7=n`5z6#K!0WEW2d`IaQJ)-!NKwQ5<2m7l_oo zKX>|2s2T}X@YO3a>h7m#r3);!7j-FOUXGb|h8W`Ln>Ag2ki;cCCPZ^oHw%N$}? zHP0qsG}O^%M3Oj*(6}M1bFwYOw=iL3FxeDR|M2Bx znd7S!w{Bk?x>NGI`ZMg;2{U%4d!XieHv2(ZV037H?)EJ?q{mIbpngXtqlHDnmj0W= z__m)s4Uv%({2(@4M?KD&B`y(ki~n7Hef>uw!tK4kwAU=gCEK*XsCl9{IQYEFbJVCF z?^7+U92|t0K2@dZ4?RKy$oNk1jPzb_4G-V_F}%!$SC2zrCQy#LS4ko9;m( zSMaC)aa!#_*s|lLSbW`7u5HFOMHI!28(!j@WqYIJ6hS(ZDfU@+*NfmfO5VU0%z^kd(6))VsW>j*e^TgF{uF@&ES2G@F4j_TERPhG*P?o%<~G-je!YS6)Qrbt zFuj(y5*w_2zVb9jOwSt=fjmP?WTg^HO!7VXI(_|BnzZmxVjo7s+#-}`n0e#gNkILJ_qhgTT0%tQX zR|2!IiaYrP0na{%iJj>zi==goag}S6m~%0b?KJxeRC7J}E~p6(xhqL9a0nr#28nw1 z;K2Qj-5r)gQ6H{a+MB^e615rL35=0CE;GU+4&!TB_v;5j#~oODk!j8l@g$fD6${fm zmBo?_A)D_0=)~JmIotmvub+jsH(N=}`^RM3&U>tjaX#0&J>hIsD|*q#OMU8JkJmjp zNEi3~hA&eu-SFNvPVzjnj&;J_3A;&+a1zvb#tk+G2Rjrk-huj*VoW_yJiqch4o;bq zmHdz}2eo}aKMGr9K^~&o%)@HY#Ok5*rM5cHKt)B|m9~TJLVkyx1-|)j>Z4wv zGKP^FfrHkca278Fi854cB@2ujXroziWQZ3(!Vxe|q9jLCzQr{hkp~p2oK`Q9NF6q% zd0A{(sJv;Tol)-+Bfe}RqT+K?n$w0o<}%$(`1fM1?Qsf|UN*nOWu^;KNoDMFrVV=1 zaDhwC70pE#!0t4~>-)r2$8*!a#o%g88%<%KZ}S-xdQcAR7kbekGaW`GMX7r2z^rP< zC~%FK#nI$H4tt+4NsfTr_6dzNY(_EFa`+~Kaoa9pJvo+TNY}qZjmiwd2vN-{lQMD_ zaEFVkJ8^1L*x^TWs^q<0ATmR%-76bK|L!+ms8lv8oGdma+mrTbi*s$#6fzt+Y+7t4 za(wisC@Jt~Z^S@t++;~3Y1st1mSy@ZbC-RS&Go*Dam09=6sn>JdSSuhWG^^wAqwRf za=CD5T#ZkZ??NckX&h7U?-ly#3p4}H>=M%E18hk>Y*mf!x2qc3%HYgLWxoU{Av{f_ zzH2gF$Upkah9uPFAB@0M{iPVg!}ferK)gv;$?BvvrS>kC1zzDzgM%*X6P=A)4XOIX zzy6hk#V`j3ztQIkR8Z79Tq+xT^aWC?H|7rGr=O_??tz<-DzjJNoDtuBT>TDbrZbO< zWgwkG%f{4xWqC7AA`Fl%FtJ;z_;ipu?D(TWX(d}(DMauEVG8^cic93x9AuMPgzoh1 zcv3O6(z4l8|Ed;Q> zJmrP{ZdqNiafNtZuZRIX)Dh@#c4O2U@rkZcMSiOl%w1zRbv2oM#4~VgS&nRAputr7qeWcJ3#`l0b28vMPjg_JFUI248W~*AKs=X4yPsWbY|@(2E|lEu7cj zg9e!Xp4c@hBFa*Sp_=uObD2q9=?SS2nDqwD_yvmI(S$BxGx1$KaHwbbUqSs$S;f>$ zhFfE2f&Uqlf*hT&F4rofqlN}mSu(i`>Qnw;hkiQRJv*){23%2=lJFAPS$Hd)^bN1z zhvPC$udZ84LaXssKvLs{`wtt9isQVWs1NfhoMSXo30k>^z%=Oym#SYlUe8D5_&2yN zDty4)g&o9)Cv`VPaKUL=uLoc3QW{GfCR+1Kj*D__cf$s?OelT_YX|QkRe`^4tH~V5-X=G4(Oo?oxS4qT zdJGFPQuTN-4VFx#BtvAtOnc+`;os+|f15&z8pSBrH9v>){k`)>t4>pS`Qaq*%w&W} zNl5}ID^DimL~X@=&`%yjp11H(JWTo1k?c?QF?BwkNI@V)b0>)ZJ`({MgvSXC#Qz58 z_>lkzzG2I>rQ4hwQJs-+py~y~6Tbt_?i=$R=^pQkJqDHJU6qM1v2b9g`cW%;_`AP@r@!iHz*9os-S(tH3;xHKcD07`y4oqFND_1`j~7-^!O)wYCpO<$Il zei-8|=E{O*+KU*W2ZUO$8RL)KR!4r|Gy%U!ynx?19AWkbRghenM8 zyg1ge-k#mY9DHQGuw8p@JSpBAu<^iSW&V{=^@&Q4|Q#YY8l1O-mcC#H>%d-(SaF?&P8WRvt%0XPCgn$ zRUU(B!Q5F?MR@hj5ohV0i{?F_ZgJ)xHt&-t!x};O{HA0HDhBRf5f$Ki;k-3|I+&2v z`bV!GkBEce-wub+P$LPQ#ezNEc;U)Uev6Sno@W97`P!CV^vt8qlF7vUHr)gS_GXkl zwi${qfiTm?Un3U6F)a^O%TWGnM8eOEgItm0)Z%^Q=bUe zc;6vgpD2~1?139ttODyUbq^ZoZsGfN3L%@RbbgoJQCSz3(YMH0l=hy%0ifL6E~@bx z&mc>tEy@(&K;n?(NIIfli%ZA~o;#3+E#s&P+q{yT4znqJWjfI42IjMnW7QQEBS?+)T(*%C_Xx1lUh#RxPcBZ$y6uQ$I zMZ4eaIiQA3odA=tb|X#WX*1p4-mYir&mW}fro;h)di2n@z{w21cLExszf=Erpm4xi zxRnJh_;nqe^s@Z;akDI3RtE=aD$mHW8?^%W0A@z#Q)Lgx!K*QKYGPM5Jn-Xz0g8rL zz=H$RuVU0-_Pf=MYR!T&b9Q718JfuF^8?3L-;Aki8nZ#sA}y7Z!kCG?`tKj}#P zoBaha`y+Gy?>f(h0q5To(#owcrqic{dZJ4JVzT!LPNs9Cm-Z%Lf}+)mGKj+WPd--C zi^5-hvSgA<9ao0wW0w@LqxA0_H$gs;*hRDwkdq@EI{87LqIo%*UcsO$IzCZpK)N8R zEG_p#00e8kJ%HB8Z9S}+p=h0^z`~)^4gJ>_S6E+GtUOSrA0O`C{`jF^=AMEX)<`20 z8ed*jr6wxvslcW;qvG*m&NU%RM{}xKj6t+)A7@ORwUkxk-&)duwX`^7UqXw?%282K zv1nI~Xr~L5A)VeG#_<8wmTNlCl|A+L{bGTQURXLnFUkVR5Qi#zy!b+tkcA(l+x)K| zY@i>Wo7yhYKeM%PS)3F_fudAJK0R zYn2k-e!-BUM{L6A?Oz5_+Y{+A_4J?$p?@Qy1S6um90Cls*C5Ka={!OK{!Ju5TWz$( zi-?Sj1m(~#ToFx_6t?26J%fjF{w(U5GEDfMw1t_p0$^^!vflstGo7>7i{ zOtWDyAy*!KSI4Vr_-})O0zVrF!*UJ3XwaSMB-$6EhLmKwj00HLSTxvW-n%$A8PN`L zh&r+LN(At$+F+S!ViRgCnb1=+Ba)9S%PZ(b(`aj4{>}AAuuQ&N??4<_whnpOJz$uh z=xyP@zV`lUnF6~eN3y4|Fqr)SDKMk+c$OmFQuwg)4GqzkeE*4F>eLQZvM<)ul%&MxY=W50=KEx(AK{MsE$)T=g9 z3P|puAZ64GuT~cZ;ysQNMnDZklSxR#NyxhN%&=s7nAFXO>1Rtcpr1ElYY}jK&4I2u z3LSc_qB7C<5v2&ukmos){@-E!;I&!zf!nYVMBVChv}7qS4JD5c4CL9MkxV6yVMUMm zO`td~dG#Xx2`U5PF61FQR_f`KzbCPC0Z^nr(Y(`KQ97HhF_;OAvp|!%P799+Ns_HN zCc8yu&KOfeNx&=?f=x%#M`@pzkY(mQoC+G3*HmWwr+yTq2)|ul+TQSW3QrYF_#uq> zRa}V^?Nd`j06PdNK;{E(7rT_lc9I27KxyL`-4B%q=5QJ1rz6WqKh5vT9v^5)-bo~6 zc}1{v!G^qx0&ftwxe&L7fFnU5^lqaQ6p}VsIrUX6`ep_K)=S2o>jpMiR;qoDXs?iu z=0~S%Etx(7vucd`em5%lh<=7^i=18*tzaSK-+((1E#U(+HocK)0e4CFul^@Rf4al8 zSg}$aQvq*qFdCj%v6)6sVLke|=yOQ6M8XmMDEHh1y=X+j#Jzh$*5nm92paBn+hJWR z%mI4qnyUo?XCoY7wZwma(H^M!blN6`!a{9Uyld)v+(cyJ74#0G{oyzTKdG4gBaPXe zVm%(lY_Q9xBl_TH14KY_*5HkYR6OeQh;&e3`&AJOEb)ktprGKzv1u%~N!@lz&&V`O zk_m)a3nu9|XFEjf`kH!rOUuinC}d54J$yuIV&~~aoqLs?UyP|o!lPt%Su&aBDp@i? zl_1$1fHem@$cR>Ru($(cNBC_u!z?^vvD-6JuJ^X+uOI7CTsZjI^4bB6zXR}M5QUls z6fdG#6-0fY?|ryEtf=8IYW)R_^C5nb(aHwSnx>5OuU`bJDjZ%llwA~P3Sa)jSrM+? zg+3fJ9i0kg+sI!n`EBo?pYebHa?#IQZc%w$fBbC1qp`EDAVexU#Zt&XVI&oSjgg<- zSm2w#D61WhQJPnzq^%zMj=!t^N+`&!P9&lHEr9ovc5^@)9=3UV;xQYsR4uoSLj5XomJ`>94{ zKzW zK6037D7~|LC@24~f`i`oehp_|1<81ghUg>Tu~%%H;*wb2yg-ml$7F|&*UD54Tum>_<6bE@ly*Gbg8kXs>}71_F-Nsx3RUdsq1@mosCBYa)syo zEJZeL`*D^Sai4E`Wj8`u9x8~K?f-9VAUato(s*bCEsEKgCa{VGYH(JQI(c{^# z@XPA+s&9-}L8XhK^4p?G!z(NYsAQq@gj*EE@atguIa+}z5pss<&95%gxMvouCOfBd z^nWB75D&59mRCjm2Kt}^o=3e*PdO|cYGcq+vMBvqbTJG~zUtLm&=k?n*1vl*5#(qY z_JKiyC2!NXxNcP4(|f=3UsolA5?~f=US8k&P2@w)^7;5ZQXurPd{wPB+xYG%Kjz6? z-+7KfbF-7)&h(al*zbycB9rI8)rAuIMr4n&OgB92Yhw=$g-_%fHxsf^=8ak)?3kl) zoIg9dy4p1Ewj@HO9=gA3tBc_>?y+g`>+>*oja++M+n?&ODPI8|j~B9VqYvYUw zY}5laDs9C_A(LtY@VxeD~A(dDaHPD^$1*hM!C;*>M zcNJR}-MFdhjxNGgY-T8ulUM(mtoIjNi;W6h7JO~T?k$F{*Fjc2UCR$Aj5N5tX^VdkJDZmXR=r_MWv{fXVl=aIp^51kq=)s^mtv2@{mMK|kdq1_aBg;I+qE zU8iJ`!ndo3jl~NUu>kI;Zx3Z9i5SLd)UQ;Gmo>dnFHP}pp62fTcMAo71sim-6c@Ua z^=j_Ft^3by1k-H)eF5)B(R%S{1ft(vNnz_?)nWFq_%X$?qiS+k#dyWI@ zB@2OxpQqa5zhkj7qjSbTZUk-hZXg&ZJac%)?(b=F&SGS&W&FzTXl{LNrL#xaP+~f1 zWTq~>MN{BfZV)|8fiUUF@?xiY){^P|rOvxqAgvL`9DnlT*qqYX*#aD+wA02Ax#B+aOyYMAlf3pjALPj=G{W?TiJjS-$Ot^ z2eTdy=;H}EWSeHsWqDvyd`NNf^tbvj(qIWhT-o;~ zbE;We30)_Lvdj2baP?J6Qvxo|7h$~VHv7Zxcqr*1^a2*#H-e$jQbpAB+oheSpTeYVS!RMNj zm35>0{oSASpD*nVw-PsgdKR7pgT(0Ia5?cbB3)t9JJ4{vyTo_1{d zZ;S3FTrg$5KiqH2_NzbS!VcO_e6{)h(D(5e{g<=CbMC$5Cb0CCUZ)uQ;<;x|JL$$1 zbd2E)^{d_CX9BTB>>0)NHs7Ohcq2OqANiX2o^4#_i(4Voxf#HZWBSeCw?>^Ik3NIe zwWPsTh&Db7v0fTp9ieqQe_N%*NWfv4&+zN)c-}&ohC-ab-{Ot{PqLJOGqeqFCX3&$ z`E3}<3t`K{+OlZHlYpISV-!!QpZv?1_CLE7Y~!rLrS4AzyLbU_(gkIFW z8$r9JhgvsnZIcIY+XKFiLd`Z;8<*x^p6cUE1ey7c;PaJU=RS$+dE?eI*Ssc<;^~WT z_B%EN?;Z6r#}k1TuhoQ~2M9ee-lvmg!oMq>Z47pfH)MRp(QDh;w^kXQ+=4dY&;-?p zV3&(PRDFoS7R2FgVta4e&a_*Tso^XOzpHgzx&xL4+9U<9r^gga_0`*&{#2*iFB8YF zXXdk)rdp>L2Pat=Pir#Hv`iWWtb$M=M-(r=`@0Q~^ry+BZnSv}yAw9~jO72qLR=Qb z|HA*?Z6l^6V0bZLfGu|e%?8(3wbq!>-m$KQ7V8Ca09jml_3dO?&d^k!T@MaR*#xb5 z1Otygte=wWCFqFg1ovHP}uHx}( z)(~qYY1zd=D7iO^%yg~QhK>cRiQFe(-4(0GZ$FFT-RhID-xRR{Z!tB<1=caOH`Wb( zm+Cv3PeiwOp(QpsJ7I1<{x-*)3ce+6MF(aR&t=@#tD9!3d_A=FKs`sdZ}45nsZK$c zm@!3TH;WlBNA3jmp|2~hGSu(xJGn2!e4E3TSWA4FI)w*!OQn!}5oGBTfuA|Omtk>T zezgQLHi`*F_9dGk75=%?AURFyl7sHb`nR3k1L%S=H-m5cAG+Z0?~H9=MvWArIa53d z(f-14rhwJH&RS?0^UoR&%C&pyo! zBuKt8Q3Y8Hk$duY>!^ko?;YySrDHS7%+f|)8efiRX3<__x|Fwp@C3f#;=Ao>(~h?e zMxL)~>t*KB{|sYz2uUs6j=SIVpli&PN{sLC zeS#B=i1vxntoKWqe$QB9dYpCT2)ajXVkFr%JTQ;MM@{M-Jxd_HT%*I-#5Wrn%xxvK zp||7v{ZBcGk5dpAzb_UsC=}z)x?ht&+~bxvwPk+qe$!*$x*bq4R;4@mf(JJZrR$^B zWB2!K(*K9Ow|=W?-NJ??q*J;P5Co)48bm||1f;u5Iv3sDAzc#E-3`*+-QBS0hV@N! z?|t@p&-VwsKfGLA>srK|>zPlCagVykA)a4|?JQMS8=ozEa!HejxE0$6d*;d^V(#c? zEJ|wKsQ94H3G~ZdD?RES=o9o>kTH3A#mb?WjQ4J)n1Vo~gRW1CvxKX48Hgfkz|Irt z9xmV`Eo!6$oVw$f{6T|(0bSX}OUzWM64}sa`u+=EO`$in(?oU_5r>H>CT?eIC7Htg zNsPgw_FjkOt)ox-Zr`f&kJ9ZBCtt+XT7o|y7iD1CUOMYqH1Zp$)8M&aUC$t}(q_;o z6z62xm7fZJo-8qwrox6t&uE`Dll09ss!k8U@h<0W`ODN;T*3#lQKw$Lc%>=m`(FsY zBp|aiu+-8>_!gz*uyCh7Y7{kqrne=eWuf=~)hV7l1sbn46mOGBSd%<07p zZ@8K9%s?9_wX3vWg0s_cQHu)UBAnVN!h8D&Ux@mt;Ug)Z%LpG*+kQlF6V?)v&Z*E3 z=oTmnug@YI_v?&>#ktw!Q>CT0KQqKS?|bpSkn$zOXMF6WVdBlJ4TV(ioow1!9J@M@ z;T_%Yl*8)L+%O{Js?(W!e5}b|`-3mms3XjgkFTMy29oXoQ$c%tx;m`72GnRK0KqCd z0?zr-&BZXALYL(2Zk1G|U&$N_n3qpy%_#CnwWzLWHZ#CFCW%~ydi5k8J3fOHzytEO zY3f%8ULaBqTJ`Pv_9B0ocE1afEd#G=&Dr&yZ`xb}-m5NqB;Jt%_1D+pZ^mNg6^~5j zy0u@Pqmd%zCMNB1tyK=gckL1F8(;V2N5+151Et9B6-5$DL_R9;b6W@e-t-G8f*kM= z0fk>ldKLFy>xj3<=1lR;jxDv>tfX#&l5&I{Mm>AJ(3$HBWcgnbX=rm%3)a8Um_^L$ z_Y#c@B5FM}^8F)EUI0)3e8zOeGjyU>HqnWzhV(=lmCN6J>8{v}$W4>ll={@nrN(ot zIQ&(j-s@+S1-y38*2N7dTn)xuS;Vhq64t{Uj8u!_D~b*oKJ0f{hH?*O zgT`liP~cAUK&?IIT#8>>`V!p}{8rAw;Y!2>G@T8>=}wBn?3z4-FJ@0#dToGxUi}&! zmQECXIH2|`ULakY8vacKncKEqyACczdx|A=WQxRM*UhN@N#7T8b=)`g5{u9BqA53j z$8Q`($QO}Hm0^j@mKMqAqn8ct3i40CNLA0c!xP&?%Shi?uI;PF&Jm-L9xe3KEsN-KYJ`LVDHc#qdyC1Zks4S}1HP{(2lrE}{4F zz3rY(b)><#fiu4DkFuX9hlljD%2(AtYgUyFO9SSFXWTqQ=!+8R4eL>kf%9qkI}{@Bp|@VHpQvsczYeBP}V4k!3HZ% zG%$&lV^uX4V*$_$Q}sJMbW9{5eh2g@4oOi1o16|PR%`WWWNWMh?38sLI*X!|;+_XH z%5n*@MrD{h?V}$+eWxlEXOg1wD`v*N8O2bk$XQvsOB4Qz9^y3!U%h7N;IHV`j&(ee zL5@U+VK0sEs!6K)(=>52d$CM>vmc>No4;zP3@wpTTWi{E>r52hZi<0D&dX`zq{svo zOFKs+EjTK;6N2kUocyb=c5nGO#u@FetnofpmWr|Kd?&GHBh48tzX%`hTne1Y;KRuE zvG52GV#QH*-|5K?JUQ|*r%R4MpmHIiq2!f&3mgT4S*Uk9>aooA0qU{Gs1dX*GIdW# zPWmnESjjUm9HZu@9nQ!7G~A)4!Yuz#KJ#NM>c+bjMj`@nt&8HxqrJ7xB;2!CA-nBT ztkdNvri(IU21)bE>%&{?W%o(EPQ{_@P@IJZf_qZKbti4>j5Ncgz)+JNFH5mWW!L5OJP@Q-E`h#M9Ot zF0ZR z@l)y*CZQy^14R7c;1Z>M#`%nhoTQgYH0f7~Qtwj*E>W-zuHK|k7w?vn4ij<&yXQ`& zek+-+o6FLycSI~=(}q(`OLCTN(GaU-Dqbs0GUN0)`vD^71JF0#RwGN!$#7TUV%1jI zG>4VimB#>0EB6TipwKE(k21O_|s6$ADE} zP%EpfVa&GS+wG*~Yy=$=d#u5U`R1u>PQnE-&)Sl|Ao-1 zW5Tpbb7&3i&~yn1XOZHRDnQ)6c1m5$ zu(Kl?X!)4n&{y|#$nG(69e_jjnw9kCxX)kEbuV-w!o@z-eQ}BgV>a-L^UGKJ<1eg- zecp`0iuOhhfWT(rfiMPN!R=gD%RxWw91KrDYRo@R{497mjgD4DV}D0yeWlq zb$!IA1oA`{&?RPWhv-a@R4|)%b9GMXPKG+)@s=iYEg)ymi(D*Ex;hQczF|ob-Qb7c zWS1qeinrA*@h&KuxA@2|E;{IOkmsB>fL2=hjS(#2*WxXgPn{Q4S%|-M#MkQi6E83g zn&vNW^j75k>}~Njy-nUHQMKWU*oPKXU2B3`*fNxgXX+KKT`RFewKL#GDZ$YYx%QS8 z0f9ATSAXdFFRayvJ^*k{=Uz+L%~l!6`IC$53@)B$4Bm%hu1S0RV}fNR3e_{TR>F&3 zdE&VLvg9eW#@FOz{k%(G#Uy6BhgI;X&`=Xwlr%~Nva2}<^^N3t)BDG}q$)jd4I2oe zc(_&V#5n|`f_XD2Cx)5_1!6a#n!*c3T^OP=PcLESEgUa!71+>r*?hjcSZU8c7Lf{m zn@^^h=JTu%M&>GzlKEIoGGp*Ti_;TvCEKLg#fa<81=Yu4?cysmq~oi!TaAOl(^^pV zQNcQ6=VrY8w~)Oy!#K6iP<=xy0%Cd*?-mzFC9MKn4oI+d!RO$~WjFWvMYUElt?V}1 z7q3%aC`#P@FrL_IO44+_;&)wn~#EjIZh7gI?IX<;BLCeO}_pz z$u(6X@C^w~Dlmr~?aMu~;AY^Rcr)YP{(UuF~==)R}-p+Yq`NT0^(a; z+rshkFH^gYcxu@a?4RG&_OEX%z1?WNdsmKy#yt~n>b8LM{Q!2ps88}~me*cI&>&e( zcf-|}?RSM`SM@IvW}=cY20i3c$*w63wotM44)ss+&dxGrEBv}db1ZKlD#jA$Y?+#} zGL3ZkjClO%(3eQ#qcP@eG&jAlmzim4cmp4huCNT_dF87|Lu_5j@|QcTS~Vu=$vFX- zEUFc-536sri(QUlox8pLzx7$ zTp;8aA$_UkPxj?GOge>u^>6+T&QWL~>g7R8JY<*!>lYLY!6D{?SP@$0GZHxadIn@_ zaAduVV5y`54?ed>d9=YwFJ2C#cxt-!Z}IhxGQxA19EVV+O7XY7`Nv)P=arlY$oLrY z(~u_lV`cpN_3IZPAE6Jk;Ms2}_*;uU-S26MO`oz2h@6D&{`$qg{`ip_h=S3I6+-{} zR)5}?djjs$V<50np3agF1KV~-!S~?@t7z+CaA0P?8HdUxdX@I`F^N`R3*ODFm-^L& z7ye=ZVVp7bN?T2ePD5Z{Cdm~d=X3)T(eryWC&ZeVP=OBo?j4jYs*9Mg9 z4$}cx3_!sW}wFVaH*!cE2%pHnvihcZ|tl-6KJdkRWk#GR(+)mg!w*-y<gjp?uTbp{BUS@(qA8%Y!e8@Z)0}+5 zJe#QqrRFW${_*{{8?PJIOU9%7oD6VySdi-W#bj4i*Cg|=no5~X@{*D#@SL9y`HXHB zkCTgh+HEyHO0+z@0-|}6B**x+DR5;vpWwgPfS)bs^4(uoY5cGb4Uqa5)=9MMZt}>Q!fF=N6M^Mh3+h-fN#jO|lQ( zwC8bA@RpwCxA4|F*2+r-iLzEnvQ24d7psz%V0y7e|=U_ z)^=pixnjA{=l2)@BokOKIYix2f-q;{+&l5z&8>O|W^^VLvJK5pU0w5aGvz%2^4+mvUVIiebpu=SFqVR6OW>23B`g znq^Y>=}ZE7jh(TkGIoQ@)zzGB_nXsWV*nFe-*RGNV(Xh$Y&P@O+AnraVCpHJYLG2% zp(Y{F8L+-S9GG+`oTuNoB6Yh`98ySMivx#1Ot8LWjwn8kt7E+_me z000|NI`n?oi|uC?yq}0cQMq~Lnu`bv!pcj zJ5v$a_E^^>5ZQtmB@ROJv*ihGf5XdKofwOQgaVL`VT?pyt&>=Fp`J924a4V&`|$#0 z*}dpM6lDR+uTDs)dwzM~6BjflAS^sY#m`dK^Lu~g@bdlapx{jvw|)I|fE)nRCHbZ%pA6jVRBr4$coUQ z<%P~>zs<{YKxHZsP=*a;r}=&*H+~U8(id|L3*9m=@ggF{{Ok{o>v>8LvdOKGlmB0N z)`18Fg5iN8?~Z~z`ad3r{}Ny13_!j~&hv`j!M%S8yC-SG4#>NNBd8dlNz!j|@SoQR zT;LPOy_9|bM34Ucs+R~PAV^$!^!-Up{I?kWhz#7&_*db({~Ez>am^|TNaSp1q0N5> zrT*MD%NMv|rIy0W|9t%4qe-9!g!MT8kJJBsu(5Z*4ON&6o+v&4``c3jD*`%smsI-Z ze{X2~l*^W}ce=dQ_2!9`33d@kzOmVP_ z+h!(l$!z^vAlK3Sw37Ql^1FXl>gEYVG+&j@(EtIc&(I*qogidAPZpi?KvMr@AMZzK z*S-Qb$VB+k%=OrM{}J^vhw3srKukP;)-p3Hji;yGC;s>HlJ_HgH&Aw!BY7HS4*_&A zxZx~M@P66@Gqca>yeHu^o{OlzT(}X~=*}Ja@n>cx34T8(xx?EAWUH7sRX=#oGY7c$w z&`V}+Pd(dFt6#u4q4PU-B;iG)EediehZl6}g1LxXR>Up=h(E)%GzpG_-5@XNa36E72h(&)W3fpb*i zbZ$QFjgS(}OW6vCw`=l2*ndmn9)>4{RnQ`*3P?|O%pD`+S?B&m)SAWR=Boe=J!D3L znU?B|g!aDaskGa^)M|BS6>*WfI~=YHMsr3cI^G4AYoU@~WCY5tr|+FV*~48!j~qSB z!(SDr^4mIyjaP@pX)f}C?pF!~<}3zIW|dccOyp|}$MfsC&^WU!um0lwI!qZ;B zJ@^>%ooZ*<@Hl7a!LL%2OaZ8)`LmV)vni70kX-^D;NFg~hBmvqw0Jn+LN8GxDWhUv zii#aVhg>d}=M)v%qZj$`eG4k%sSKoW}Cj}-c z;7?iZSgxu1S7>pc$=+Idf4g=z>dmI)Fb2O?DX-gE20LCoC>A}|l$g2OSizaZyNp&f zOob7?eas-3PUgF;%-4X?89uJ;#S0;EQ9Ia;`4iCtlr=dGLJxE@)?(IYz*UkJ-s&lq zX09?;*D(!IR9WwYA8zEi&F1%JfM)i>109#r^80#idbP-59ZRGA!NCubzzj-`@bDvi2NGANnc} zAd9bGel(p(zu8)z*%vn7AaK65%{DbZT<9w>{6yf+n;7oeeNA6QzlX-<_-)MU*s^zd ziQ8q|-MsdGZxRAtz5>r2e2N(KCm}h+lYsQ#*a&E_S-uOoIGD|(==%OALG|g;B18bF zuq|zvmk<0zOEYq6uAV!1q>uC7%IuIKsRk}m=GgK4&Eco)K{JMXhCRoQmqz`oU+i43 zooiaG<=4Vc8^j^obt!o6PKS5`4JT#1-X#i1*2#Q2x5=NjHF)m)(1&j19K$spUwy~1 zlQ_isHrGgAxqTt+T!^a+;j>ftM;D1B^gQPg=Xp2zq>GTCN#MO}rtFE027;=6<-O3C zIID=i8SyB|OjX!Q>x&S;kley~4xW6A4b3z$ORSFEsEo?^JOM=Ls5wz=2JeCBT$ zvbQf2xX4a2e;Kp%*cTa2GnmG^RlYCo6x)DU{<6`0zVA-+@!>~vI~{%g9{R!DQt9&? zqZ%FEzTjqovJ4Axo7_6eiGp!v()?w-Z{N@^4zcpzARX2*Ri34fTtl!3na7(xYyTr` zlPLj>w_PvsqtWG9{#88)WIUQb^%$>eRNk+4)dlpPQ|{tN_|!H7P|f&@u{lIfyV3-CA%Zj4yUi^Zw24%QVV zq{DlMG0ix#`dw$D<$Ak|+io<_VFgaV@2-Tb!rc0GB#kCFxB_C;o&s^24R7YIs=t<& zMF4}a)z9b|(NK5E+A!d7} zZF)-Su@XADsXBUVt(NlgA-;-JF01QGC|5sIL21Zkm}F;@(|ObN`1#jb ze|aRR(0-blF8m;(uO1C;p#OG`MPcm%g87Zx;Z4H3zlKH*%`FPZnJ;#6bbtvNp#v@2 z*~#A(f>MfMO*#CG4TZS?NH(g-aq~bSQ>zPTnVcS=E#}#SAyz=?_h`D)viq5LILA@n zbz>ym>me{nmeASjzwB0aa_kwDs{Uc36s3Y<%~B*+UwylInZLn+|JK$FBeV(BojQ^} zWbvNl26YdQaDi!}#9~_8#Awh=OlrsdpAAP0R8zK>6KCe`0TOrv)04DY5}}mi*A}i; zr-TAH?QvCu|F&?Z8A@y?su{<-*~RURXl>W(D8KWA!^C&Mzu~<4qa;;xp<%nfie3>s zH}GNHP-=Igtm6FYs40Cd;|m*j#4|loBy7g{ie%^9vDo76SHt@ZWA&0CY0>E|C4bd- zCXd+P7Gi%ge69KiX}dg?M093G1Jma#_Tzqo7K)Q+W&ye|zBgZSSb!|yilD8gv0BUB zMi)FzeSUXsoch($`td`2bOlW)Ke6TbY?DFw=J;oKwNw1a>Ey>7QguzDYQ%DSOD$0Z z-&ps19W6)vftB6~-Su1<;RdVJ$mKUT;vU*nKY9ZOtux zgG}5_T{v_sbEV?At{=OGq@XG&E6XfvD(%g``;OM?Ng3UZUn2dDQ~?M1lU=K}OFJq1 z8>4;pM>*hr@DW0yB(p}QDwxo3`@F(50I_co@K!jQIq&auH#Rmo1Pvy?Ty8F`z3p*Y zv7|ASO@R8zkMINTtR~AHoW6Xgb>VMFYHsq-nQ=FDw8WbAZ6jxR(231hCLDu@L9hZ2 z-Q0S3uT6$I#l z*ACE4!@QIjaKf8dAN5^W^8odh6bDJZs28s;ydAZv;bJG*;RoI43c6(MwX+5HHQ4(H zS)VY8rM=C~FTc&7%{0GxU!tsg$$oTqmGfBXGGD*91k-HQIedGG^qBNjhAzs;R#|_0 zO!KmKZ?WnDC(*3kb0qfgWiyPa+fLK#R)afwzMrG9-11`=d!~yiMtSvL{u!n&Y>?7| zysE0If`WpIO5bu+v8I(33*;=$DWT;Ej)tJ15b1X*UnPdods*jX5WK9A(L^^k${4DzwDh$j#Yge<)KE zk&cm$L`_*h|9xjj;*#MPx+CST0wCnu4|Z8M#(jYf7TltbySFs5i%2rgLF#O{^0B|a zQQC5W3nVF}>*JNzahq{=zC5X-UPK6P0k&poIHV; zuB6s-WKdI*+W%a$0V3TFPQ8EJC&HCE@9?B#;&)pLY^TO1TU@PMw=}u?)t(-evDRnJ zOFuWaRBhY8C${WpyqL;UoXG!JNHh#l5DjPa{&q`*VG(F1nZ8|&AkVE}zLeM~z;e&Y z;a_bx>%e>1j*}88#!XD?Q(*()oe=C{7+))D9{b0WqXd9`hCdsh+#H(0ZhpDx94nB> zv%l{Hzk~Sf@k=YbuiW5qv9L;aIx zwh2N(7ek=9sUm8LzF+d~CA{lkgAiVRWEBi4oTXds+}+M0j#Zrs$Hero`e&?%05)v~KvA7=ll%%(&b9dPq zPJ10T{q@zI*5lhkwRHmLT^A{AB=h!VTQuypLb?9B7@U5Z18oPcqO}%aX-NLiGa~(y zysX|9gbRXb7&AGXzx-G~o;{Us#Nj?e;*fdT;xAr~fZ3)w1kTo0If&MpYP&ni)QG_H z$}yz!mj`)dD`d#t%K&8FE9yr);4WPsS^W?@eZHqW+cNwk@}c_+KIdk~Xb#%GVzhir zcnK^2o$u`+`sC8QBTlTwgo+zbu~n~Wf{j{$!MG)b{roRb z?3qmQWOq!e=NAH-(h0G$QMG2Xo~fFPc=4XrZ!EZgPHf?_zg(tW_hSK-vdGFNx<3yg z?_vGxrWt3h3QSwEZ39IEh@lk6EHSeC?z$ie&#Fx-8_$1iLTpcbhSpn@uh~2wH=p#{ zGpO?!ZOFrVl;?wPAwBIUrbDg|YRcbD*91Z4F20xj=p40`{b&P^!}lY0peg@EUO|w{ zhCheH$d3^plQ7c#|LDu)mw+{#yh|r92S{2T>X@aMeDf@UpDa3jm4XTWR0W-zjG!# zJohp{WhT-*>7-mo8}yHyc??=C{eW_0>Z6RqcaR|+p8v)oI{zQIgYiX(apX5c!_&5t zj)0wd&7UVobg|pHCcbNx7q-tKWcUr4q?KTG zY&~P2*Or*Y}Yx#Z74ehh`&gn0J-H%P~ z2^7Pd0@t!9hjU6MzqOjomj?>O;rxM0SfC_QJ47g!QT=v^7ku22%Bbs8(0)gO(h(B@x3{;R zFz}pQTqUYCVMr`lrAWs>+YkJbc$F}KW*YZ-ZF)M?B1e~B6XXH>RG;jili%?y_!_9C zzX+l))@t0$);<7Q0EH9t*aq|foDJ8nTTT5Qla}whnyTlICAk(mYJSyA#*QOC0(AHf zr*SQMecswTpyo4@>cHXMt=v;!3*}Ll>i6Sj54`sEhjVuvVuO7A;SPDc->d;>AU6Bs z$OK%jVaTq9I>cUUW+?&PihzRMPd979Om3^lngN@>I;>YWcc=YKNyA!=r+xG+bghke zk7t6AaT=*%T7efg=#XRd)Q=>9pa$j^U{e8P%CcBO-s;|DL7MBOa-YCS2UfdDsr{h} zR^anm#S)EG#5P@K2J`u920j6u0bqJ4x>SoAGBR`yX3DaJ7wsxK545ng;Ix4jD^v-+ z@CDHJx?ICuT_3KbwRv`09`{>D+=PW8O8`fk=H1-1kLZ(Ld;>MH2vmp;lfKaMo8Yh1 zPNk9p3SJXAOhR1a9SmqjJ=-O-Vjf<3LwDAWIVK_|QniTRtOvGMj|Fzq$aJ|QT+y&YZ9bp6Cj*GM0o-_uAwbc%g= z^TsORqTQZ{VXn$V(XEmmr-(WAl6reRjMuy)05k9PE27i)u)ujQTWPx4O3Gy);2B-Y zlg-&Gi{|T_0cwngDA9)Fva_j4tsD}MTLX8ZqflMH5AiT_AV%Yhcc6Al)&X7SQ|Y|n zP^~G;NSRbBd%7&cS7pu6DdCv6!w5P0lHXx=lF;g~aW=B*pKuFLPhAE{DAAk=T5kVIX;ff4Ni|?liY3_4k z!%Aojb2_Y=4kPAf)u}6_Xs3U>mUka~Fk7+xDo)L7H7;A5q3-^)|EwgmwbNOZo5D|*; z?Uy`j0eXjQtZayy7@OF2|mAm_| z=XixmWov~ML#O!PPHQL2xDI!7(jGgR+%N|dv7H6DBVV)=3LMYWx0?hnwY{*~f<85k z-B4_~og@J*U@mFs#6R};Q;V4vN^Jn7AhRyNsDvK0K7YUS5wp5}m7ABf;i^6LkdgZX z=1ezZP@`GvmzR0k@G7@%Epm7k`Q?HRH;Zg6K#mEePV+qVSYl!*g}0`@KG7s$YyWaK zOdp?V=h!P?RX#xVeTQt21dS(9?@GYs{8>*{!fhec0c)lEyj$=wa{e^!aW{=G=GW(Y zkoy&=@uUv5mLNW5H8M<nQ)wK#>TtR}+kXZqCwiM`X& zZ`YVRqp2@NJ}?*VzT%%9O61rdc7J5kB{8rW{H(R69AsLfUN+mMLuSwa^U(J>krq#~ z<}Xe`mG))VEA>Ly>v=Kp2=|i#7v;ma!Q4=8q?d4RWQ zWNPNK$E)nA-8h1JJ`?N4D~2yDjf5i(W$Eeg^3gd`9KZ|x{J>b z;#%(YGhbA6`#5IVyp*!@m^Y!E3PP0D-3i`FkU+!sH-XbXSyz2#Ka0Ey^odEPMd67P zZ2@ZhLuyH9WM3;f>e$Ac!5%N^)L)*ZLGA$xVUxWs)SHU5hmb?glQ-83+XCAJFG{hA z594kyL%O;GS1QN9v{!g1MLVwpMe#$`aQRRCCjjf!^S5@UHg63pCHV0b8GSF7PNL&M?J!R>Q^H}?lXOlM5383Jv3ZWs+$kWvRT`mJ=o zTr4gy=k5#7F4fjIE9CMZ9x&z`H(QD%N$g2D_#+`zc|-8YeJkdQ3F_(s-r$KA``#8zwQ<^Pg7~UlYq82iiS2- zrUkrUPhpCe=hh2mm9c3+-yS00v!_Gc>xH*{e{+VIw8-H|Id`FQKy|VaEfwW=5+oyw zIN#a6+Q}wDM!JjDUc++z&hL}fe%H&?di#!b8^>$fJ!8r4Rl{~&hoW0mnl`51bo}Of z?QIUzDgug+P}S@LH@#BU{&TM28nJW`Qn6;B~nur3$xO8tPO?{1M%w(LNh^BBYk`AjJd|i*d$=+XZ zm`g5HZnO$uZ#78jwGp8{bbo&V)||&Uh}b<(|2YIXA%h549U8z+5OZAXlHK7qSk!`c z4F&I%RX@%1Xv{)xOCWo*)2JZZW?@n4mWM+N)A2LL>s563cq}Gic6&B7tCu4f zd}t{2q$b(w3y2bhN%JvQ&+WCG#b^BJpP+5YleRJ$rk-SM=*CGK)2CS8>h!4}b#ts@ z?sS_stFa1-l$Mvrp@Y)y61gne&3rbGqKopQ(9hyxV2N&EO4pk;;doC6iBHAPK8c>4 ze;n5K&M1(wiwGyu*KDc-lO=)TEb5tv>Qtb%*-Awk!t)Ppjiv?w$H2p2a`~7Unky9} zUjcEbBf!>aIykzFq;@aGy@{PIAmi=$g76Lg%fY)gBe5B|lDaLzWZPpZ(8Fg3$sa@& zPQC?}Ioqa{!+6SbsUDLS>d)iZ8~t*n}Iw)J5Bvv}Z7WC~0qTR2ia5r3+I0 zYDvQ<#FIv5=KJyuzY1#5Wk)%$S1CAvgaW9Rh5JBuJL0cX7%{0R>4ZjL!Q%6dhzl(9 zfyiJsJO$Fl%Q#oB&$=4glN8>%2wEPTq3`NBrhe5* zKJ}IF{aQU0O`4 z*=_XnJ$-c$y_7{VC|@%iTSF6 z;@{?eQ~_&LdI>`MnBqU`#_Nk3%rr0z9>z(t;vHtyEX;a3BMVzy<~++xVmI+cf;wLUnwQ*=ZO{^sk(0vSP<{nH1ZP;iGg zQcl#Ei@{Vvi=tf#gWB|xM-oSjX(u8zKj{H}HlZQRx0ezFb@>Htd6cbp9#DOaj%N!L z_{61$s1}&X-9V1tBHpLg)B*d-U&mh4bdimIgG6!Zek3#mI1__d?SwFRNv)~xMmM`( zW>V0#dmsujqFFRuC`7}F_z(omJ{VgXt{K`K!%C}#p8Uch1E){ctT!!1)9%IAi-Bm#HTk7 zu{K|mmEw;PX@GypD0)dRJCIBGd^o#i*inZspPGr1S|lNVh2Fdo8{ycTwKJXI^mauh zzfT-VmKn^U!BeYxhFejO;Yf%~62sdwtk(nazKexAC?bKHl^y3Ym9i{d z6>76?R{;yxA64HXWiG1_cHIX{%fhy!U~6T*>{iN%0*vYn9>7GZTk_7!7lc93z`qta z8LH5qnbbVMFj<(#w`6W)KEXE*{rcI7_k;FZul0n%Us%^;am_xwxC+Wrjr@8M9nC4k zKxoFsA}*gbKpVnl3Od|6mW{)7W(uNYd z(Y)_6JZaV?GX$FNcHzSbM*C3j*RT76yQvHt%-`Y*KGnZ=@LmZz=(_e9l{utaK6d;t zZPwkaW@=tIzUA-d7i)5*^S%t0f)awO{Bs{i!-P;~qZD`gwterWML8k{%$VWkM?Ts^=Y@`>Jft=)2??kV zzFXV0uUbX!+Z;H?d5(zZI$7PY)fb=^CNAYj95O>?-3bYwzYHYmwhaGm9tG>~!&J2END zZt?UH$tX0Gx*TV=bYaeG9}XHr-y^kA9?n)iFO^A}&M-`uu0*ojFIU7BL|J+5KN@w@ zVbgvVr03M;8hx831)+6{R!1DwszV(yot-D3JPvoi*S%jg(;ra(c7!o~FAmLSuI<%M zt?-n)0e%Xj-!Xp4@*GG}m7mfWo}ewzf2&WDIx49AK_3r>E$y2>EmZ{`HCZdtAa2;| z&2gKq(0Z3H0jIw~M289Q9ESgmR)Z&Qi9IfD-Hng`XmZh&+_iFNfKc-bp|8!47bU9N z!bmf2oGpUSeawfWWO%+GepgcZ^;vwpP6BGBJjuP`8I{fc8euC_0w)_kpkZIUv8_84;Q3mPxI80*3Z2OjW|y+U|iykML-F z=$O%00H{`eO81KeNgL#M=SY^G>wq?Kf?6N0#oZ3e&4Azhr{NiE?agP^MU_0v+{l*y2q z(VO=7gu2iP@6S$Ic}wC8uKV|xh{Es@6n=&)`A~?G&jb0^ZWjfbRc&68K1`dqZCk_D zln!$%U^79gb*s?oUL?*|UW&PjNJEPQ6$_kzloi{WTi`U z&-uXy1UiQ>+ypbJ%N)9GxW^mp{ibc`Qv9o{%S`7w;6sIe&EEsAv{YpYz5Kpntto(> zf73CIaB|vr`B2^*x|wXX0uh&O*YkC2_WSI{go%}n1tt&w`p*6ep*0wL!>d+}${ z>W;b{taf{H4T!PMi1ZGJU2=`_3mmzHmaBGJEy61rn`B#}x{U!R#^twEaTsJvEsf?O zVS}oO42#h=4k0wcEB=*<@A7aIjC9-2Y1;-6bg}KKG}F#8rIf1#wK;G+tqocDelo+a z27WpnIgiURnRydxO7N1|nAP!g18B{i`1PPUoLV?abl)0qW~;qiaZTL*+$29Nhb1%_ zq6F(UMTyBOZIKf=ZGIhi%^ z*=RN*7(8ZV59IGGOGBj{WbU!)BQ4*pRBkV<)9%mGg4q#$1I<3jb@qmO044I|NcX2& z8%)QWtf0Ye9;w7V*Dba*ZNL7if>P)<9$y1(thJ&S`4dlO?bd}7hZE3DYK({U?M&Z6 zBPl`oiALk>7RV{KGuu}GjYi+=j{|~h@;>EG0RP>ITsQTwo18>N`Pmsm&4Skg3&FRV z*IeRud^0S2b9U@cw4rQSwpRlQY>Rq5#dG1@SPpQ67bR-8|ro?XV#^L)G7ke{juE4I`FYC%qEG!f|DycLZ0pnYbJUx5=ldq1nzX-@+iuXJMiZ~>X;1;3>SfSWtqdrUS1^?@{~ zP9H?M0f2f^8BbKy3Qu1$%_H|f=e>-~uh!3B)o|LSuwiuG4?qoYZztQ_lJXtR&h0hx zgZy|wx0j6%-6?y(SANdjQeMqa-ne$kXCU~A9qs{Zd2(c&d4`0G(pC%zLe zvEa02iX&8nw7|XJe1S*XX5NM~ z_6LiX6jRTctJV5SZ)C(veQ1&eL&L0l%tqY;(!`_`8Op%TcQ2w+Ak|M?v;fnmAy~=} zaj2>FPn|KT=|WvuS5m7;+4__=bgi5;JsnLSNr7ITzs(I}1Uas4@gp*`ik2T5 z!X3CpP`hamJ#8ozFk~;u1cRUlC7_tBX%ui>PZWj12|0qEZ){u(3XjbJzTrd|D8!X5 z%Q*T4v58%PK6Fs8ZnB|^(1iK5wu7h8MS=kYetmjN|AlFDk_5R&CP>hUz@Q;;f2SdY zJ7D`)Pj4(W>(r`d1ELnwxCgAr>u+6+ts_AKeu{6{mJDDR6a8f~Zrn^>tSz5q-URSf zDx3Adcv84B1qcgm7%IFy6VWprRN6>RcuvIUU|IcZ5Sc3twwW8BO7`v!aMZjodwmgM z==9MH*X#v2sG$|z`K$b?O-Bg^1Yw{dW_#k!JXX0;<~NGPoS4R zu?dc&fOA}+Zic36tIvRST_XC)uVauGFcjeZV$dFfJOGA<{Qz;VJ2Xtx|4VI{!}be@DmCG=6AUr@dNR&h#PrM7n?p^!lg2z(ukh-Zvl zu7%&2s%H&|aMxHHh9RBB>J6;ZLxc=*9qi9F^EL|kPFo+RSyMk=*so)!NrL7x?EqKX zS;L##HL^xT6%^?EG-trK@D@4ey{b<_R;yf%dO@<{y^K_TI6V~KbxmSnBR=;++{l!A z$QOaw?x;dlD zjdH$1d&|?^A5)7k8lFSPks%{S%4BQ!Ut6rvaou&W2J0(+&CBfJ8)(6|j@(4xHZYcj zNUbjPx0uFOu8^9R2$#c!5OJXe^Mr%S`3Gph)$`^KlDg8JXai@OdzY_$xIEpufv zTWW64f5U8Jook)owy@GILti#0rShQarX$GNj{xjdZG?}{@U|2bN+|D3HX)2Vo-A&w4qsV?1+dm`=E*=N+{`+!|m_Y!gTa(S$>;XzlqL?cxhJARosNXYJ?&{PbN->-wdietj5R%Fn$U7+8}#*au}~>K{=nlEgW#4Pa;fe|3EYR8?E| z{RKhkl5C=@yVuy1TnOm2OZ#LO_uJ<~{Yj@ArLw$90T*$JKkz z+54Qm*P3&#HPd7k%8gY!@7R7eRfzQ{%#
    !YU~ZS!^zIh7qbEW*clBrq4qCV+y9 zLPluz<@2zB)g7M!cEKwF;|<-A*OPdYUO^&xwH}4Dqm)qGIo~Mj18Z;iQrPM^o%!{O zmiwp2#sd(F+9Z3?23n8v#vf#L-`{*t=q%77%4xKlsu?R6aEK>4mjb4Sr07Hz@3ntU z%+u+X;x-s9VVReB5}!l|)p9CGThwVRf1GadbX^?#ppn==Ya=mzAGB=CwEyHz=|agW zMEXJL`~8t)v6q2R_?<1MsnNbC9aJx7vX>%iLfilkg(jEc8y8=PQiTOJg|>vPqWfhU zL7Y_C$b&?2hws+Rkn&O@?)FWX?s@ZVt~nCQ2cV1tNy|Gz=u{ZS;J1T+Gz{Q&fWI6r z8{P

    Dve7b9)yZP z{rg3l1yc^8Kh_kqjnn|re>r=g{4V8DQ2x6N=-+&Ij07}ByL&UeMcjYW-97LxAgY8# zbbsV5e^d(n6yT8j%UCs%zmNFm(C^k^D1GRol(7D0QUAF-02jpi zw}|(jF@h>K{e2?;4DdU3u*lh9*Qsax_u$AqHWPRNk}E2B{?lRhdqZ-!dgp^#0_$q| z>iP9HUFAn@c*zVNuji!CHEBXgC^cBpfmXH5}>e&>9PoH z>Y*5``^m$Lt0$!#hgl`!Tz}5#zPSaqovpVQOSY|7+u;DMzBwJcdhbqPRsPIk=^k^N zzEMc28fKF&5``UYJLx5$kxj>Su}d*dgeUPoj>%iQ>qKNOuIR4w3t1!nzk+T2H|3XZFfI^fWYUjAj|vuq>tq445rD7 zFmkXe97EIlY6~RCYrEsjZ8r|H>ZW2TcuiHKd-r3Xi`(ps+4F@AyPWZJq0J`%_hXT1Gs6~?cCH%&#$~l5d)%m4lJjFK)|5g==4O{ z$LnxDuehliPb|sahMjjhTQnHz!Nd5e1z0ZS{OYCbE7=w;D71NQY^#~qJ`AC=TDS{QpjVWYEpD_ZZ#QgObW zX8Wzh1c8Fr@o{aK?IYKeGNW*8#><5MZOqK%j~p#-V8r)0F)i+X{1A6#(@={b$e4IW z^h=`T+ldLb;u}UAy+<#5?Y@EAF|N21mIb9E&nr~AXwb;ocw24_bey(MG@W#t&ndfn zo|{usXapG9_Ygnq29c^OBVfT?bD&VElmh0XG6wn*IFh;nntMCb(o8=w`_< zT7cO5Qva|$Z`w9RnbV#1{RC*9EfI+y=-2^(PtOJ8q&%+B9THnVFr>@TlP9gOf0(># za$8HMOm5)$dAV_t47cP@Ovq-`4~@uxW}=a*>q8OEkHNVctfJ$-1!$%;+w`4HA^RRI z)d*Zx=)HrN#unShU8pO-E}p*+v!{MynTYDw!})Fj13a0vb*)iRxwpjuP23wM5&DT8 z0v1!&4KIjlryJ2k`ZuOUpMI@3v=F41RXiQ=0_K5c5JUUfvm8}X=UYJSr$91+XROuD z%cl#tQ=Zlwo-OCIr=MBSu@O%Z(}}sP<{zN)mxDAzSqt)@>tW(~(?-v!w3Y;s+79~)POclK9>l`D8O=nP1H^!P4)J3?N zoak!t{0=(!yS?}A=+xe(G5C44vjT(xSArOm5m9jixjjfqKS?0oGQci_-+YoXwXqvR zTCcj`=qA3R-hU5iThIkC!G$=m^%L^Z)xQ^%kvNd9oi-mie_wwsl| zGA>v9ew9LnYg9lyhSrYMci`mR8(AAq!)Ozk2g4#hq#)|j81dB1Lh&?9QiYobFX*_evY%ER2U~-1kK^G!f9d0TY#m+b3O%l>4mCW^EfOn zNt7{O>D#4R>+qH&HP_jPD#Ci0+gEL0fZ|6Ul^{qtedUoBjoNnP7;G?~%i(;PuF!C2 zfXtF*SxaZ&0qQ?J5EPOIT+% z3ih3})6dKcuGMjoC~;J{Jc(>q;(_!}&usbMXsAHM8Erpk_pUIvoMEW;#i&`JKhc1F zWDz0(v7PKlzJ+aKf*MB zS*$^%FIL4zq;jmbtR26Zda=!0gelDkc|tOHZ!kiVgZ@k zx0!6lVa|_8Imvvk#P#;J4uaC^R^ZrDnuQ;12w@P>^+sM~(vkhzu|`tBR6Dl*p!)H1 zfF>*8V7=3Uuu{7kS+JTKizP=R;6u4l5QJF3!%3|&okImbLaCp>wvgYCFj$-|Ny|a9 znL9T(VsD^Mrr5FY`6MM;iu^!?wh^TZ83O5J`OIu{+Ku9CViAuu`$2V{m38%0k85BG zBih$C1Es3((!V};%L;vMkdJfS#TLZiMakkkx2K~RNO-VMaj%mqO55t)p; zmeE6rpoRmalS7T?RS2zs|4CwW5>#t9nx0EU&DQRlXguP~6IJa;_v1z;8wd;AUlOlvuP~i z486ENpTyFMF$K%%+^>3iePOcHdSB37A3f=38oX;fDDA>0%uvv5s^u4)?45n$jjYCJ z`B^{zfuz@@Bu$20C;X<|J8Bra{gz`*?1wc>H<7Yp*)qSLAX+-GKjthir0gd#dptKw z0b0VYS&My~1M+9O_q*{o>kLX+|M%{;2=Ye7c0rI@X!a{fg5rl7W*-Rfwu5-$*JNgO z_s`@M66vN^y?lhOHshFlQdH{Kf(Qzc!*k&Eo^B>8KdubyxtGz6kk@sgJe7MuN5uZL zhBa_4l8hlG@9yaCCo1xAGsN_2B1O`USr%~0AEA=I-3K=&79SHFKNCPpH`S!Pn0WDQ zo5-rM8*y4Hy3=L7)4KIjY*|2qo?DW9MBTCnJt|GDO_gVoh5))s<`PH8g&9E`Wj1r@ zio}s~tc{b+Q6T`AR6pq?h}RAN#`7aX)T9 zw~iPz#kA8lk~WR?D~4+?92GS~9I8M)eZ7`4P)d^}(HDW|?b(-GGzLjcOnjPrGdy|UuLwx5ds?Zu->iugR7p#&5{uU#1 zp;V8eRaCfw1fFpieAjad_ouZuj(YmFJQo*3z@v&0pEB#**(9RJ;BX?%J`IVUGLo6K zSCe^Ff_1;LVkjoH*R7QXH*o2j;TTzm!JLtanol+F^(4;sl_-SxMKOiZh&a0Rlr@S&mfABVeX9>+KZT*UFM-5SaB>R1Vl)QLONIL(OWtYBO)B@fWCIfM}Xn zm#@*pV>#`M0Nrxrh2TG{V(|S2*f4i~@Wjyd@nd`cTjj!Z+gc*8On2_LSGc)uteEP8 z2dWD-L-n`asulSWvnLyU1;kWwbw6z?2dD>fBc5x^8t&yG$CKeQv&|qFMzO$Tp*#r|D|n-6v%b91t0tup5&^=e@5KrU}XOgYDk# zENp1ToSATQ>-}&D+}3S0*A?wk&?IWxd@FsJVB1IND08-KcqHFp<7EAr^p|#C7&JQt z_p|~fqB3e(HQB{mWYgTJ3TW&$ErTu%?*iG3Pf2bw%BtszXagz`GNKu@FN-XF;cklNZnYf+zN*ihfv$Tj$rQWe$r?-;&unv;V zHE=vDdF>7LGXxC|jpsUqB5Ku1mLG88PkI^k1nf3Al{tS{Av$os_3OsEIXJLK^)DJ8!Fs6(%2)3!~;hei6HVhd5-!5_}odrgh{qws~) zaZ4ay5M%9$OT3;g{k5obkbu;6WJhN%w}6MZDgJzlr4eH%Nc8J*M8jM8O6G^Xm>B2A zTTixWxWu8?_K(WXuhGM-S~X?;PHyQw78_8h4_M_}c?sT*NWygYLNsNSv7S8Oo)t1Y z>qx{ zw`09ji3tDxtA)kG?-eZEe0ASNkbf7Z(1HB26nb;935Hp)!y)hl+f_n^B~^$KUI&@Y zVe>^*F$R(8=?FKCOht;Fy|+pI;m^rhHKs$}(&i=GHUVCwcs#z~V1~#<3a`>U`!I1B z>LEM$?`jvh9Ansf@(=tzoifJWiet<0Az9=q#+;OH=87uM=XFr~j6g>aeCROB>5%&Y zt!IhE+OBpWJ+5KE%*qUbvB9kCHFCtl|76A7q3UCcz&(^{Iw$*y+OxyKslWjR;e6;l zync+ji#N#I60Xs*5gj|wW3>l9nFZ4Xv`dzjZ-Dw)q_;2eBH;jdE$2ZA-y5~>9H3KuG?D@=T%+UC=AzyC_M0(F=#oFr~;XpjQRH! z9QG*aRT&AHLZ|ad8w7N0EbyK`sd(2cQ;AVLJCK`fSF|wKx2XQuP>x_mxIjkV;hZHu z9!q-rd4Z}0&P=z)bB{1F{0E#EL&JM;deYCKH-J8oCTSUD6^+gZDkcQO{)>jLN$!bZ zRPcK2gscWIS1+ES;`il1evmuuO;Wi_2|9uvZ%Zkse1JglLiVbxDITb+Q+Dj<6f*bJ zKi9A<+46_4?zRF6G@sq!@^i??MNKey%AX-rR$e&v;)ke}i(=&WV z6<0rnE1#SDbq#Mz)oYHy7G|Lr`c>*PUUu|jBlV&MEDEUMFYl~BA!3_ow(B^tK|hSv za{U~EFNqAiY`Fzzl7EyFIKqqkz<3aGo-Qx>#RO9r2c19E9kaj6AuSZkBoNeb7>#z` zrK1@y6zpu$>JkD??a~Q8E}Ib?s*ned6~PKzR-D0w5wP1zXVHg-{j_E*RX zqL8V6)$b7B!TCOaPv>*~z&Sw7*qOZ~D!UCRpg2{y&3|L=hKH@~ve;wx%fWfBO z2y{%Vzg!QOZ{t8^we=9RqGNW!=W{CDdAuR+=^FnFl>?Y1SoV2WZXaF{Ifr?Dn;(jZ zz=L9UqIRj)l>fxBIVVg}ab?5iyV(RvAL6bO{j2l8DgD4W366R5xT}z$HJYP1EHD)S zNcByuhN)E*wrUU2%{JVg>rt{jNp#vr87tAlM^$*=`jV-hhR%BePOsFj;q$)lYt@k(WtLXh-Ka9C-o}|+2~-a!kf$H#s~_NZ76W^+uZY$u^TDXR zn`oXU$BQ*}mW$cDm}8>*1)a1ju1}qIg?G)ZhKi-nhJ&b31kU6F20@{ZRE7f%*;tGh18DCxFs&nc>T-5z z z=j_cB$lfAOf4S9xg+wXNhc|37CiB(9^b1M9XNa@Rf+sTZu=T0jd8BiT0E(i1nHuoW z1dnC3dN@r_%h{hLW7C*rV1k96VLNQte^;lH3|-cq{^NqsZK1$O>(VOB`Mhe*DE=UW zim5=PhL5-bhQxuvp0Txy8>3v+I~=HV^ilj(8H@5jn*98uAFr!K3s>wndh@X9h2kEy zQ!5UF{hR8<;{~7KX4?Y$lt}B3iT&%8q7s!+>mcq%6`RP#-_x{pPZt3o{00Z=6OsB? zp4Zn-(8jFL+qlmXV)WgR-|69@)mhGszOthlc2IHD>-AnQd-g3izOQ)y0JW@fpYEOp zndU#WM4SkyPE1ITMf}KJGYoT4$|lX_!a6)1vm+is^fVEx0C)LI*syK3+Py&Bg+gCz zh5AHxj+R($6IpWwp@g3sdIgGlt#c|ZdX3|-X7bm5^jFk8bKiK#VzP9=a*bW@q2^|y zrrFRDyoyx#P5|uuaw}$r^lgYKTal2^BAbPgjqANZCn!9c;@^f-vyh%QC3n+5r`p6# zQL|)j+$Dz}`m`*_Df2Jf{G9^KAgF(Yls>kbp@4oVx|$tKY~afw?cO!CF+VJA=FVIVWyZP_e0vNQeRdr*zA12|Q7z6!8jgflgo#TRM+$H9Mr*GkIdavabA zgWc#3#q82y1i1bJ^d+cCaR>gnvucKs3#QChr{4g$jf~GhTb$^0_w%+6;1INmP5&{J za?hZtBnq*bgFXPL*vtW1*XyJQl;Az)Gz=zkNd{Gcxx+lR)uCUdM@L4I7; zdTjyl<~HEw7^kc*3%pIAI`0P%&AX5Q^jd-H|MXP+|L_>8KW*yQ9yaUas2jy5sA!l+ z@opEI?C5&zrZ1z@Aj>qF zZtGxoR+$NTnt4%FHKQY9%uYbVu`2|z!4nUZZ^8+M2V3uIC%d$2V@;T4~ENNXDi zFlw8AULSaRzoc1RXmYjPoj{y6x%!@?)3U8U|8I=Hh0f|jrw=}-#dO=xbLLDm+b#uZ zMoxeQ?hd@)1RkNz;HaLge9LvVD@@vrZn7wtX0;%to|4%1Nz`^(7-DK-$c;6|9u*3k z)A9TQ9tR@@|ItjRNl8GMs8`N`k?RRTr-*6w4q2%KSL~If?Q)XN;(I0l*q5uCGnwt_ zQ82*{`H44=f200Dv~j-#SQJ_wiqHxdzvIC77q8rlxMILWv`gt_>K&g?eXE?e?R7sM zS4KIu@@+0o_eropfThUoB9p(WMKYkzZp_#^Nat%t!1HA8B26Fkd2EQHN;H>5{K$Du zdoFrb&R~>x=MyN~E9pocCy<0FzR41NZ2>1;ln>aQhB+-qWQo`_z#MFYP;w{^B1(L@ z8xIVs{SzjQ6p(VsF-7kyIg1K1!|;c{yfyKn8u#U@5F0YLm@cxbyXkZn??`$*oXk3G zruU;}nacBaBU%WBnBNW9(LYuLPF3ZbhU6w(!B>LYJeRObz-CyK?Y(~ggE3}Q9ZEnA zk_l2gciq4vb(}BPztMWW^$)8!zIhLg-(YW6Lovbaq?l$JVQND#F`n>7Re!rD2ETQu zjJfY-2N%BukHHcwO9^r=0=trN%K?}jEQ`70$>eb0sx3d%F&IgT6dh#FY?88@g^$&) zUl2b^?X%5#-ULWWgHKhRg}Hz;T~%3P~~_yA1!|#w3~8&n5;{{ z>pj7cO&h@^EHR|~RIIEE-4D`EmWh8Ae&!abJXwO9k|1hy;B4D$62`4m{!BHowAg#HBx4@EY#pW5L8V zG6toHJ=O?E|Cn93>RQV=CWi5Ob7J_zfC*)mclv1IY?cPDXS);Cm@B~)Q9PXl!#Evn zh?(a0J$5cVVzX=w5?lfSV?u&gMY6Tt9_N@|WnavXJ~gnbNW~w|R~Slj=IR8VM)(<{ z6zaCh6+_QE3^DF`S|r(E_g5B zrltY58|0xreXJ(zZG0DaserNl{4^Vj;2X@Ei}nYo^GpZ4=yB=m5|W=>maLInB(2Ij z9=eGH1xtyIzz+=@T_+nxRj3VSUsv+IdisTf(0-3+Il&vNSb0`=jLg^VDL{jBXFllr z4(3mMc@bVXD$U#yzl3hQ{RZdKgzS@fLB{@>!K!?=L~*2RQFP`e)W%geYSt6re5fa~ zkiB(@T?P)=~!>sB)yIiD_tMJ@yklck>z*&B@!dK+SM6R7_48H_iIxL}w5jQkF395`mjKzDl zDv`&v4Y(NjZ{|5k9u;>aY-z2Kd;e?2)8(@3d4f27(E=RQssr9n2u~XTu`W%_`zK#d@1)r444d@nqkU zwRMy>wsl>;U`f#Gkc(&vtnGk(1^X*GK@Pf|TeX7u8RM-VJGRww)5grC$ZQgMCwLYe z$0+U3CM53JF&kgX;1@#k`>ckQ3~}W$`fK7+{!SKD_AII05O5 z_*t%a4W0)jbwHXyT-DY9iBA44P0AFL1uDY)dwCbi15!8mv;cVY8h=DS6j<-ZkAm!sId761R-4+1I0`)I zLYq-`KEAZRs- zo9Y^$;^996Ck8`KKJ}Q#doe_RS-o>fy+nW4MQQ8y`^r&8nZlCrkK3M39X;{nG7R6= zW*Jya4y-9YW{RHm5r>zEXMyeQq{GU!pIw$BEVJv2+f01Ip#6>03q@ty^Bs=&j5T~? zUbs0qBErEAr&@SryL+O!_B9jqS63ar)XRSp61=H=BCPPr(G5q|S3VsCv|iEmJ^5C* z+izJ&Fq?kB09?_(;jJCKk0Jb+!aoyiejj}?-lgjh1?zJ#mO=N5&My-gAXAZqV+XA; z*3SFoCQ8aM=%x-qPb)yzm_gUut=Mh4#8WE0f97o?^8!NeDZ14OZ1>^T?92ksJ`_A+TX-6 z`JeEh0?G^mjgPMS#ENhbyUZ~CIwZU+N7r?{XLoHPjkQ+;MteAUJ1M+p{*Y2B5Yt(W z%sj$ymk)44c#4bV7_UU|yLjHVi>!rdImm?;hSypz1zqX*Q}N*J-G`N$X0+>K71W92 z#abM6DIc}T`wToyVVPrILelQoZS6vvGb)@Zh8d9NrRYn#@x_An+D-B7Nj8wZ@{iSJiOEKQW&4$50S)v);eWV~Jcr+(QUfKBt5PA|p?wZz; z1cGB~V#6)iJ>ex?-(qu#y%%tX^fNh6wi5#2mouf5K7epGI7J`-!4Y+ul?aOMficGw zP*K&_EJU>EuPK--g76YFb$G6WLBz)?h)kWa>1kO`6O1OMomY9)?+LArei??Vn(Sx( z;GE_*>%PKxnwc&fSLxwkYR3n$v}c7@A5%rGFOUjKmp2C_AV0h!hX{xr&QO`ZuiJAz zujTuU_&77|do>HK8j~*~gd3$U>HTTdXF0@di4b42>n=k~HQ&ya*Uo==)_}yqA{^w@ z)}PDhY7+Yg3ROJ04h>~(7`*54;)6!ndx1DQ{dpC(RdiSpFzdL-MfWJyD1Pc z$WQp>!9%3gDE6)CGTkzu5z4p@S?ae`=o-2NT*F3%MSuX9Ya`h{TmBWP&gVs*g zR3C_1bvsm;J?OcMb8UAzm>p53%~#41zcSGC$`;rShkbVLNWknnZx?Izk=t}zgNMmr zEVTBpUY*^sA;J+#nnfRh)ZQA)VDe^2{npRx@dJ8-Hk6(B#*hWaHtQHE$N+0|LNu$+)dY?3bVRUJRgnO!=w>woIyA&r)dA`ZuonESclF+HM@NK0qMXJVn zTIZ{P@2P>r3`0B8&r#S_5``?LX%XjTWdxav0)7KZ!rp-T+1b(AETlp@3~>@0rc1oY zz72V89BH`w>c;eW88RB)!H8jt;Y%lGAQCDB#;YfvSF3E<7!UIN>*ub-C& zo0L-so3e+~K)LVDVruiI?$2*H+mG$61uj?pL+N`JJBCrBq0DDBf=5IzSU$H~r?MF% zJR7|lTqS{e1D6G~82SS@@7=Seri&Cy1UM34q;Dr~68%pw<#$RL+w3ys?S9lJEI*MR z=gj(WoW)}zaA}_lsho|scEi@q17JrGuadMJ$NK*H{tRL?ZYG{41gQ(n&@Jp2bm>mv zgY2X0u>N7zl5*gU7YR9r;Y`=ccvFNZ){!S?nes_5w{qaG#qT8(cMbN&PDDj`Ps|I3 z;*P{nqRFH#by+hi%jwce(uMs#;S8nE=PevVg6IL~0v}@sDRS;3X~*s!{B#ejM50P@ z`^)3C8y9x80wmW*lC!1SPtRyN^W&A9L3^g|%j~L4=_-5jL?ecjtycm8U-Z9q?g;X) zX!qG89}lO#cxQO8iPeD&g}-rM!nWRaHyK<7RI(nU$8=zZ>oQ*8Ju9$n0G9!a5FiJ7 zUA-={Q)(S>Jvj_PO5+%CMC!(QDS$!##n_WIK{>(c6wo%afr zZ*TnE0(LeLHVe6ys1#!NFkMbmUNS0{$vhmC+_Cn?Uf5TDm4|S*@tv`Wi=Q#Juf8Z( zk5#^r;I!U;Bl$5yU?73hqIm(?yd9_MmumO&q(-!g`oh7eZ$C!8B$O}+$_TJ>MT?qn ziFG}{a0*TPyy5k>p`hAD2VRfskZ9UcU1xpTdMG?d88a_N(o!s5@$J3; z_Nq)(|HB9kLMx83U{=vb_DXwJn$FzLGLi|jW0EzE8%mR`oZP=xSXfd$QHEaluPA#N zgFKtYbP2R~8Zs8_?IuWxr@m5o{YKLiQ0Y(-!ISXETMgIyA3uLj7OUdkHaKJg!^QW}WxdXGas9?#!XzP#FxpU_v!4j_NF2J*nz zs0EXp-7~tw96dajDzQ1bJnt6V+69No^jvbyON4pdVZj$RXY z%2$j+p`M@4N+NN|xU-{vg;K<&C%3OyVQCu>Mi(mUpV!o7h@dHo>7Z#9QPIcMBd_5~ zGwG>Axf-O6?|`>WrR|%6%X&F14XP;?3e%_TEpnNW&4Owon?!fe78>*kQ};PT{DjQtpqxr}lHS{Q`4Ky|1n-m_}b`mwcQLKDbFpk)&-7 zy!`O7v~tTuDexLu_d?Tq>&17;IBIuyLcI|EjV5`&B%z85KEAr;5_x}9?< zqCVAjb^7Mh(66G((krAmNETR<`F>k*Ac#Yr1)3O&8jwltWwclX5LT}oN$k5860Q(p zN9k|sq2M?NXQd+QsnqM;>Bal48f}J(d>zjg<;BWhew>(AVS;=<)Q`(fn{|Iregr6WF-0~?ayAV^Y!(|AXCtY zBvrB8|Qy}D%slc|13er6Epp2RfKiQ7{GfK{Z!m!I&L6tE1iHXM>)p)I#> zG8C&Y6rU=`%@)zU8HCm{UKOiPzk=Vul6l9_TIU4chv6NXoU#7YheU5TayUBVJf$qF z1t-8Z1oUaojbFnl0sVtia1Tnaq=R7(V_vhN&Vkw$t^6zFdss3K(X~WvmlrehtGR4( z*#AqC*ixHVH^908eOkdGMkUlTTEx2bSB(fPjsha`XUv7c zHYOYt>d`xP*ae2)EBT80eB|CK1ycLGWG3AZSaar*6oIanET5YcOF3IFVX5Xpg6_r~ z!tx@iuY<=TjaaaA*+Lu{lw ze7pX5>j#a7vf(^mE3fvx2JG@~(d@zozMd$fk{Owt+Vm5oZ^wd{>}qg=fjhjO7Vgcf zU_{TvoU@7OO&?k`kLcy%8Z{Fa4^->nO|{UmPz%JHvzQjeXpao5gG)Gtm8Kj1_Oy3y zm?l_ZEa>^qTrlPyLiUq&BU1Mt)YQBzAP%jGTW{JHGp9dUhfCB76FG-Iu-)ajM0T zYW-mRSt|lo0|xlM^df}v;cN8c_-=1Qc=pLf<$9Q(Z;Q@XJ&=*1CD5OTVZzgkAs10~ z$rKNosBv*S?0;`(!W)6J44IsjwRChU)W;G}P_kCNhsK@B{)16M^KFEG=K%jwikFha zAo^0$Yw^Qe5K*PHL09((?g8S#IeC?oMcSuZ+(&!U(1|JNq||&yBB?XwCWlndkk4py z9Z*tUC#*i*l{s8{x61A=<-`cBThWMX*4?`P+^I>k^T0}+jyAW$K%6e3RL3+>{wn`- z0>nn+XJg|(Q*=L9?QhV9F;nxlBh-fBY1}0PWy?E*;vVj;W z!J0}aX-@X8SVVI}%7{Uwms~OSfQd~JHQVz( z43!{mGJU`8NSbtl9-VyF7rM`%kQMWU2|GL5?2~a&uBpQNgT{jeNUX*UQq#$1IPRek zacc}##0hB#5hb^(leE=-O(C`@_lU_uUt|fh#fJ4TdIyz^5n9-=e!=Q=bqEK4q>)0q z3PsX~ajlxYB6O|~hZ?feBOktucAh$IG#TxSQVmO&Jt`hZg)eOKD7sC14MJU7~D2M)aI|#RM6l{n{rrs_%c6K9) z8o)o%+>hCtFMrE7b%8=gkDY(py{my6$vQ)Rz#^49X0M%3S&`qu9~xm2zmIMzZ~6ou z4=ZS{Ly-Y)^l9M;$XyLd8Z=*w+S~v4vj6M-DvU^|&)IXHKpR;&Q zY6}dznVJ$ezNC}+2n=EBo0wr=O#K!a{hMC*y+O0vOQo}7JVk{Xj{4%+*`$A!=OUo{ z-vb0@8{gNeWPfoML&Yr!B8xRRyIWp}b=s+XkibOw-z9g@=)Q6}L6e*eCXGpi^f$qf z+DVlt4(68#j>UyLF>sbje*nAR+&eh`50dRhMIV*wrZu7@H7ujIZX_W0JaA%p(PWrR_-F(-e~ea{)XC1!UZPRc$zer6^}>zUlA}q!(>3>PV%e+zamIy@p>Q&p zWJ^rI)~PZd@P$O=`cYO&(Z}w%mvCIzwK>#U%jrHr2G=|dbCi` zE)IoV((=DsW~2EcV6dPM(l~42m*5*a!0qkh&rZ1{!s`s)=tK1;Hnl?|dR^}oQ1S%7 zT56t9^BK_R9<1B%uO9Mr|DUr%4kI#PP9ATe9q^WAFYV?zJTp-3QJe5?p+d=6WV&x( zfd)x$uJ>r->UBEPEU#Sfq=Rv44F;-ioc`y}>A=W1H(pdbF4Pc8AMZ`hS+`NntKA-K z>yF+=L&TeyD*MU7wW0)eK6Gy}fpOV8&`5$Ywwhxa+wHYJ{6tgzu)3FA7wPpPsfX&~48bK`+Adc90*%6|{HR<(skDwq1 zGxJ4iZYHMk!PWm+3oI}UAoD|zGB%=`Z6x%j|3@HaRn=p0DR)aVm0mCx2IA|7-aG*EmQM z^Ef%5`{ngw0lePO&Q`1b@Xlu7{_IDkv#liEZ$XsB9cFqOs{}K#wNe70d7Q()$K}`i zcO5V?D-MlRmuJXO5i@n*L3**mXKel%dOoe6-VapNE|GX&LLHfLHDzj(Hb)dZS>5?l ztnlw8dkgIaRsuE4)g~>0?5Gu}Y_$TNy@7aSuqcu39If4Ww$R|;|NPhc7*Xgu)B~?$ V0|%dLV+i;oDJJ{8R7B73{{i3fd!YaT literal 0 HcmV?d00001 diff --git a/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs_mem.png b/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dfs_mem.png new file mode 100644 index 0000000000000000000000000000000000000000..6b55cb990651757d954770ab5175c4b33caa1bfb GIT binary patch literal 105434 zcma&O1z1#F_Xj#Noigo1#ygmjlk*U+VOgLF57G|~(VE#2K9CEcxe@YVPG zKi_xn^IRC_oPl%p*?a93zqR&n6QZObg@Hzb1^@ssUQ0_T0{}?yQ_u)12!0TwkO9IE zKu2Y%mw>V%vK{ybe^ZUuX7cg?M))}@011c#KzO(W{tW<<{J#0&78xMvzvs%pXMe7N zw;5ysK>U3@0H6gwK71L#e;)4n=ZKI6{7;Knpublm0kRPOo`XiFGJ#0ynd;2snuiLA$M3o4X~0Eux_kfVIT?gw~{g@};!i8hw* z>6_$O(|+^Jj|UzHch^G~UWL_GvbC>!YiUDsk{kgvsJ;MPR2)jc2U<`JO$=X8@*&?X zC_qwic+V>1=nzFQw-NqzGcBZhUVd3xydX^*n=6kfA@<)d&U7*FlgBd)3p(oR(=#(O zW3}07vks1qy=@l>@sAvNuLt6l|N9rKA*1Teko) zWds)&7r%f1J|!jPWj2ls zMlmyuu|H?D8PfMWMtZy4-2A-I8wyI=75L7ZW?JpL7~g?dH^K1#(wMXytVougm&JEs z<7Ino&CaqhNqtQXzU9$Kw)k-r=D>3e03SE^-h5ruBc;|i)rZHW@;@Y9Bs3re&ejyJ z2(LnYQG@^MB}NFQbb(6~ap1@9I7}kPPe90OrebYvZAJ(gb!BdD?*8_Y#{NyK9?+^W z5jy`Z($1N9O|jG zYD5g4S#r+o=;1<#Q1=|pLJgS`g!2EZOTRzTzNV~fl=u-FH4dZuD1TXH{=7WEcppX& zHW=HVs|gdzc74-2bSLX8jmPFOUr!8V5(FM>Vane$cF)lUVDGfgiOqc7o zn)W9Mdt5|sk}h9Kp2U~bfN{R>3CDZZE`7lx_(zO>OH>B&DU)x@hBy~xfy`pfv;Mju zCej)KYM1ZdV#+?#BeFA*b8J-)A6c@F)VkC;CA%X$DYjhP$NxuRw-NnZmfU&@7}JDY zSjX?RF;jW1U$Rlbx+9+md0rhUn9MiG)>_YBw4qR>x&J6qDUf?1cEedzR#sq;%4b`k za#1^PbDAz6)6v){nDL>#ewLG;8~EIMj(cr)cUScOYfr# zjSGohI!>yoBhlS{|D;%3Yt5uy)_i-=V68*CIn_!y}b@YzWU>cLXW0$?F!ftM0uTKxor?e}PRSS${_G z&dN%s(kLLH@;<|HNFX7KO|!W}ioz?7%xIXCB|pq_zh-(sZr_MmEW*2=rS6J)KRTBG zHASlLfUdb)kPK;RywM+R&5{DUJ}nH`hfGXEzAa5IrmcU7d|9)&X87GT^i5C@UKQFShyOvY#k77!*MYQAb{cJ)I)` z#4mT#3C2m=XwwDb(ymVgVvsh2Ki^{ARf33O4m%kSgENp0)hZ30cJdST-cLR=*YwO@ zJc*`c$B1{g`bqE+P*zeB?n1Ec*{l*u_-Fd#s8`<0$!thGi?6~Ti(~NkR95!AGt0UZ z{=_Vz#dU+hj z*Hn|;_~+LVwLGpl97!D`q%$=zLLY_bOOB*DPif}0rQF;u|5|csc(R5%Xl97F66JkH zW6Np#bE$$4!%%%bhVBLDvRaFH_qO1)y#j~e_V#FQ0v-GC&-+_0oTN4=T-}48EXWc9 zY=)P#OLUv2Jcx+8y@ZeS{6cUgBpK3lel>~<;>GT$yz3X^aRqe?KK-YgJ%~E{wi^{G zLz6Cua=UXG+yr(f1Q_8f!c@K%O(RhX5jG4Z)U;N>{^^SY@9)y+9#ZY8uvd(K2C0iU zpcb@1WWK8dsT4eUAtkqFVa2!Be|MVs>-kYZJ;k!~w60&Fl8bq!7i5nC8UI!;tA$(P zD&4-D7~E{_rT$bzB7Mz_jBOBlxwLCldOY@()aQC5v58yt!x!~(i1!@p5{OX8uFY*L zJ!nfHC=2e)_}Gm+SdP$dypFpFz0_+wqzRo3paMQWJqP-mzxK<~P z2y}L(TfSMeZM}!Ak;YubsVIabBHLb>nfq>zsrC?jeNUq`Mr|PXY_@6`tx?X*6c02WSGwLx5rPI$l zjK0}fY5UkB?Lst6Jpc3^pWO-!=r2sCPnJEq{UR9JIq9NNd%TAHBpovg_A9x!EQzOY zv;M>ICqH-{nZ;1{YF%ho$=Q0V-UtjAFUSPwy|6w;;>EeqK`bvP92b zBsA>sOzW)Eo!U)zBNM61Ss1V5)BIKeN|1#VOl`0w7T1=A%+{Bk6#v;dgU5U{l*lg^ zl!P{>OuICnE-enbC@A)NI+Fc6CW|RziPd)X6CjvzN{-Gjq^z4JnLHO9ROY0>mmxB& zHE08+ruVO*V3KY3_w~B!H(%Oa9xS&0JTHfGERq(a?awL5%E}H3_Hi}#ICY5?;Fyt= z7G`?SMf7(8BH1s>xVOi+tmeC48ng2O_hfn9&)%gXhPe$Tfd?2J+>4SI(H?Qp`0r%f zDa~EYPa^T31wviqUp!-seHfexxgTP+8V5?XFxyb$XBt_z`}65r9Yp3#j=O7p15o3!VRmp7k;ZUiE5L1f#{8bp3xtsf8+;o6;=sgzTV#&`y6 z^o;a{uq3`NMd%M7*ME^U3iZ^jx1)L_ zqltN@mg`M?cn?E(d_-VpCaKG1jc;a%Gm~bHw;o7pV<$3J&+YQ=oJL)BO9Ym;c4g4y ziuA`TQzC(rn4KG^pJ=@i_>N}?YKfTpbOjGqp;HRlFx?dgsufNLzUu0^u-|ey;|6|D zkUt5ZyIy_l^Nmr&>>c!aGgUKl%55@|LUaiaFNYQ4Y7Tm2=Tw##IRFcf&P6{ciupib zRzs_krSE-h>cV&p&~kd2_7sA^b&T1GA&12lyHjpQvIVi)3`LjI#_!!xE_m17lO$aM zSgxYxC{x+W^l6EJw5&%8Ww@?|m{`q~(JBChs!P*zE>~yGGo5?+Wq%>P)lZPB5bkG~YAEH))3> zl_;1fPD%az}uW6z#0h>7=DnUn4Y0pPj zv#>-FcJgHdC3N!V{6|*J(r9J~@jF5qRVE=%9s7Ci*9cpKU8s$~qfjf=fx?OcY;TZc<@>GbY( z{yL`k)u!2URhZ#kjN!Nu7Qw@x0mu+=q^suh;0h-+!~1fGM}p%Q1aOo|wJ%J{f+m=p zH5@V?Qkhq_t@+FggjM#VH9hm5K48BFDM^`N=2)fPpU>!k^rOq1f z-n7#NYWnvkr!3BP`jOf;k2=4mRVb?dXI`nJ1rg=aQZk$wCwsM-3fV*$eM71S1E5iz zMTkL5DrkSyo(!0`$=7-*`%kfn8ZtOsXzEe&|B7O50c`+CKs z^E-vTbY9bH@^42`_+V{L{DW*)k^E2(uX`pIcCFCVsbJgWva0YV(AMX6p3ErGOrDPK z<^~ROq>gheh^gcPgCTc0nqctU1+@-lC9tgKvVY#BfLj*kuD z2||9IhT=0Zhx<*-_wVUA{<>fOC7hAd5$@TAum1deLWO^!o95cw=feEiv%LWgkvFCZ z=<{q2g!n<0PR)-H){liDjOipn4Oz^@`SQx-J)ylY(x<^rIl}B1>7Zk~u7Y11Y(B)K z797vhr$3XXqDR@dx<_<}^JTzMe+^f0e0D$otNA8mo?X&ohu15cXQ6{5W;CZ=K4+s+ zEku~m!GM|Sf5-Tha;3Oh%+TW{pecx|&nRTo=rCJ0_mAOcV!(<||0rNk1) zYbZ@XHj&lYqC&lTma3QCRx8P`CQ-*BM0RT2ue19H>r(naObeDHfx$#c|0lFW1++2t z%G$860Q}=r+m;>bo^WF0M{KwqLN5`jUmSjT#EQD-0_(PUYxI*>+TuDZf2WypATuJh zXMXbMfgakQq*hE2CrdKd$h5k>M`rFbQqtgik;Ea=b;ehO@0wl;{RSs~E%d$^ShgBw z-jH$qRpQ;L!xtn}?u%JE1J#QfMw^DAPuBXO%?L&f>DQ{bFom#1qNkhaqm=>nxbF67 zh07W*wxK3td9N4qqZu?lB!9&ue@?WEb-)tBhki8kBbF6`Yy#S=(KY=<A-fDxC^GwFKJg?u7}^`R>o^2oh2*wJ#+( z0ki+nH9BmA40>i5ku%tUfyaZ5P9(2frInxk^qb9B`j!#0xx3eho*iBxh9efge%^yC zB4fB%9JXtB%bFsuzw(<1uJw?640Pd&`3Qz_Co3`TeX`QqW$hjLYG#|K`gYURa?)t* z4D)sVOxgFP+eUVUY1V}135%Q*`lI6?NK?E?Zj#C#DNzx2ex z(rTw{6LK7a@RrRHpU#DDL{UoVamh;<`3UK&$ORA$22V4T=<6pUGAWUV%#oJ<%@x$a)(~O#*ae-6g+6mp; z{lV{~Kc&+~;+sjUq~1=e^YI6=zSzTS`^?|x3nqCQ4v?fW*6JLcZgO~hUB6m0#JRWN zyQs?yn=wnAabY}S@5OHSo4%pu&}niG_bhDOzzQ-a2&QHVKT1kT_WA+0-&_w6uJ99- zfxU>wTx*%?z1udsLzLu)$1_dqy;dF7ZVjiKgT@tn>6Ci&B2S115;vQoTY-FdWb4)P zA2#GW)M~lo_>})0wG>Fi(6W6-SXvdJ-PA)Oci?|G()Blyp+_9?@QWROyLNrl@wiYv zHSn~$x-Td4s(HP|GXGUyH`+?sbJUlSNER_A;T)1T)a}p)Lo*4d)tj)b>cM&c``NTg zhQQRd84oalF7w!g2ygd{M_+(FZTIp#Ca&Di?_-w=uz4XESGR+tUDa9~(|~0;G*CD} zJ#epiE5jq$T!)+Vu)h?wYdjgatZND^_~vl>>G+Zzj`;OI&`3pARK>B4d2Gj3@Qn?M zZ$_wU{3)=F#rY}@%w_j3ed-2itEPzLg|a-vJxy8Pa=vbH=AfZ_qSKNHPG=)<=KM_5 zU52p70C|7AwBK2D4R2yiT^Kr0}{HFoTGwydpdJr6BAL6$`C`O z*IQKwlAxXz692N)srl5XRKNo14#s0r-tg;=W)AA42KInBg5uIV;Yt~!w#$C zA>?kZx3f?ieNg$E^qm6ZFRrPGsRA9u=le}>gZ>6)2;%1!s58Ze%LhJ2-^CYm!bxb% zzkvtktp+q@m13iChRc#YM8s;L4;Y`SZOgm58;W}JJ!V;UhT2$aooIen}0WVM? zEiqkd1}BR>Qlm(E4cZ|}YRQe6$vJ*{A^$VHFCuZcW?HWwq~KS!c+6^e<#Z{6!~-p_ zigMX??VTVGmXbUEr$@wMu;K<|f)#XEa^VlZaX$<2E&|)&3O`rWwtf}atfPHnlF2tY z4^92MhDi8Zo+!$|6NlGUAT#wU@F#YD`d@PGYDh87RPT3*-v(@Gv^go?D42Z7!5W5zZSU!Gtj5Jj<^rrwHnR{QY z20@WtZB)4#D~0Ia&Q05Onu>8AZyO>L(OeM4hTbPrQ%zuxX%v1bF0P-l*&l2pEExE# zw2&e1p~CAxgS#2_BWJ9d^bNZQ;~48$236B%eO{>n^_PcK$s-H~_n@UK%mB8g zJx=ZD-qOsyrZkYYdcb9X771$g_1Lb`-y%3nma9R}0f;zN+pXRU$5usoss-X(^`qp9hCRntLOz~b8;%GCIu6yqPf@%!C`9td99^E`Ba|0izzTZ{iD2*QDB zUiO{C9@>e9ATD{|?k_PrvMrbKQOIB7{3$Il(^CS|Hhci?o(~O3BY>bK(jM0kizAS# z#GfX=;lW!FLG4y_84rHhpB!W|Nj{jb#%0t4K2=Wib9|^}k%;j^Y-&JEjIhqQ)j&`*sbzbhv z411n&RLGs1R{NliKe7$DPKV<>G}1ha=})dMhA5~gX6!38tf2lITxj8HR9W;k%`e5= z?0-p=o<`WbY-wydVEQAd+1K!?6RPZE=v3@QUYB5YkBLAJ0Jf2(B9D ztFqHH52=nA=B7S!E}FzLj=?t1c+|Nyv6IVR^ByOSs zxKSL}5DEgZy1x~FD;oF+t!;INe|O~#sHqr&Cc%hhuv4V^EU@qI6ly-Go~+(ia+Qbv zDNskJ?A~XnN({f)4JH_V6NTb&3fVcQb|-_{{Jle1P7G8Z1Z48~-6ua*+;_xFUqJPV zw$rB1`~IW1n(&_X#wrW`uQGk5i_Vdb(X^SB$hx+Pdsr+#M%>E$8XmZ3&`3#rw zuc?LT8FLM534{r5leSx&JEvbRCiiwvIX?sbHQ6WBl#lF!>=-KlDX7K4io_M+s{6UD z$Y$LSBb~0DvHsU>L~E4G_2<4fh3EpN@2)FOa;lVUmVTz(dlX-m%+=tS@3LW!a346ZV!4&IS-ba~U z$3?QDIYuU}Io<~qi(-~}=S4M&Yc}$I@+iElS59rQ@?lUeg-p&f)qcd&!|*p-8vHmR z&9y&k2&aS(PkN~L+yB4DmUW4W&O`wj^$)9zBiJMiq%oYZcx4}Ocu@jAgFZT>I!TVo z%;38|$=(pJEU#i73B2DWw9%Qtu{q9Hw3A+w5_L!X^*stwUTjf9wIS6cdeh(F)3vx? z9J(e`{y{!EK7Fpjr^A>+_^pmor_MORN5&Sp$le}AUPH`#=386O>bG2e=ZSs z3{{9<1plhM@39fBTI_oE*CHPx7On62L448g_%-qcG4v$NM(7rjuwT<;M3?S*GABw9 zwZOvqjP4GF>%eK_3v*Fta&o@?^2}9Wm>k{6!ra|df+?%%ueG1oZas|5R@Fg2FdJ1~ zBa4XE_og~1%^JA4e!kGK`Pd&=FD)Cj=fZ)AocK5fZ~=PrFK~~9&)gi86aV)w+~Z-B zg?h$62!~C`Vr9`}%S1L(R01*W0>K3Pb>-SqgjY0>8g>m5Ns)Cdc1IuLB!jfi+ux$%}qRk(7~ z{^B_FWOK`wb+P$tZ4tz0|L3dJoWNzJXWUgrCe3HyH4YK3+X?>{xTi1O60CZ+Wc7aT zFkiIv%dxbU@^*pPt6S|4j%IAwb~LuhcMXnp0}(>X_4ju_Qg%faFTvFuqF+y_JS!HF zaFG?9Rxy!kpDcwkkkbDkoPw^fb~0Oi5}oFL+w_fUV-#Guv~=>;;ze?pQ`PMM)qA}M zz2EgJ`ucl{zaU~xQ=~Q&Hy_R1h&^c(M%E^6@u>KC%DZkA?o)NCCnZd`?cGDjbV5ft zpmy9G_okuidh~U%wUCC#iYxm~E5^Y&hSet4uPu9o?#8$lL{`0jfFJxFUa`y4%o^X2 zFG~BPO-YT!^K1}WNBe9KZ+#w z@pa^E>63dFL2-WNhpQ1Vqg&JTKI%~uEdSHy5D8zHIIEk>d=qO1%%Ve~)pU+!p#S`? zajTYmDRd!S=$bRp_juse^1`ux6&Cn~X1XHdQ#v*7Tk;H%6DuLw*{vSo%Eg-hu z81KwopRx2IN}VwRCk_vCn^ZL$D9lGr|WC& z=d(+b?6wp_f5;_}%<0w$T{~3A9?kGJ?ZTtTI)$T+z?=8URtv3diQ0NLeL zC*h06%{GTUG)z*08#^kkw$^TziZ#oiaHVp~;_CB=g#bt*LJbP!$zn)gow&YD&@N_m z^CzaqQ|lb}JxR@n>I=hNxe>&V_b~sK-{DAJ1w?|7R=}pv^?;rH^r&GDEyF%{3vi)T zraSc8@i#+;=j2gxc1hg|=DZ9z-q7nt&+w?pFBveFhz5N>|?| zW;9Wh2w&~ZnG@J*eW%u?jK$S#^?^R&9FGs9SDMr~pMUJ#WOZ7V(*0@Et~>2^mN4`F z=3YNLTK5Jz=|Q1!tR1>y8>P&j?7Z1E6uvmT?C%-2Qw z$i*+T*8ghx;WfCrO$|G)ot|#3z5fO=dt)|IqORRtVE*oy#eC`gu~5{WPUX5+qM80+ z%dPr?;s?Wx4&-=@Vw;j#Z76*u3TN-$$zzE3_DLmR&8>GU?7D{r{g4l&xWiU_*_E33 zsLkha{#Q5uY7BlZwSb{nW~6ybhVV~nZSk4G`2D7G%>Tfv2zV?ghGNJ62HVPU00Pxa zY#zNA#L7+d?=l$m-J4?yF*40!N)o=~!w5o*xt4}1F^5|``Dwp!#J{08*Gb_c<3sA9 z%e_k_$*tXm@V$Mm)%vTtvzHE?Ow@iPuhYWm?Z3Du{{i<*>?#@l#yw1^D|ArTaihar z^3M@N9Sb`cO{`qJj@P&3Tn>efvM22Uqy!4YRg0+_He-4ue^wUYuJOTRwI$|IKV*$! zfw(Z9n^Knk#$zh&xrI`D=b!JUi){vY?cZKmoU+-Ir2hjxA96qjILc)>7gQyB&}V6N zkkpT;=Ek3sX=$iai%UMp$Ok{y(r?{=wKV4b28<7_yFWPP^2_|O-|52-99N&)lL?kI z8M^<3E&%~8+Vs2 znoosELtS!k{>h$AI3JQl?AK@gi+pUGiY`Gfa9q_=@t^pE0?agI9gfh~<^dEKI#>azj_djpaOt~_E&!UYd0D9L|V_+S}(WU!-<2HwzPcoF0QUu zerX7h(2j+<1tsfe=C&g9pWo!dX-0gIy_waHprDsK1|*SQFT4J2+KCS?!2`PDyMJa@ zP&O+&yDel10VR?_Ti*;$QOuQ!lAO)4WnyHMj(TZ!T3~Bo@j{^5zBHeY>a!26B8(>Y$j{_Ew&0qb%Cao#(gZQ|d)>=Mk)!?VB8 z@F@iq+GxAn0t3++kCuMZhvN8W^_l`8cP`0vTwPGJW zesHz9rHxI2W127q0KDSl*}z!+ph0!|f3z`J zxmLFcVo!8C<4TG_%8Rbs7r5QU#=t8tWgJN+U}ThZ%UQ;MbmDz~L-nzN!DFEqVf@*@ zc1sx#Px%W^=HJZz^;_=#>Rf%e;LuF2B`LqC!4aJ}`uQXd<0EBExp9^ll*i=U90T80 zZ*Fd2&u=deZDo>9;^O0x3f|=XF_Yl`f_rbVvTkNueS9W{=4&}R`Hm_dh#gL5uV245 zNXZDxi-r?J1LCnb$jYNXB4bp&%o*@#jTXMV>eKvV)0|Ooo8E+~2*>=EA}cWbOAqbg z#r|>lm-zVj0{Yp)Nc+iw35~$nqrph?f^NR1K^fr$fzKWja_NnS8h^>Iet2~70N>6g z^$OQdj}|sjAB6d&iWKLdOKA6PvZ60l4YPB7h^|3iX0 z@ESTJ&JoWYRB39Q#c4D49P@ENNWFNqW7lTEZ#Dm}MVi_448Ap(c8zyX zI7%Ba1ZjMq(Ljs`d+!o(>+@5LP(KRB zrpbwQnfT0Iij9!jPi?(O_vwBn2tlPv;}~wa*d4(>dmZoAVJ4w}IgRL)VnO^eNm^{q zC2g;@e4qMqtqdzhQG)2P**n&ooCW@~#weJclHd4A`9-^#sc?f|{;{QG85lSFB$+nj^ISwb#%84W7x3)c zcf!^dcev!%a7*IMXz$e-X9?fY>d{HpkVS%Cns@cWP)J<>ZikwhnqOx{1S#tUoOp~X zmn;2SHt>0o0$sfvzmmhoU@ z9K9;5*0d_w4|N({gkO@|s9PwguiddwQ3?#5zG4s|o4tDV>OU4KMuN?bk%rbXf&;TS zGtbQl6}!YTkReCDPe@F3QH1n8bp(H{tuLrs{~Kh)#PPV~mNAhN07G-4qQb5TIIug}aRFHhy{IQa03P6v{_eA?!Z2EP zDkgk^(Xc2R_^2W2zIBg3R8EE0V-BczVfks`{S$7Y`w$1P-G59Jd1-v3MXHzr9(1_m zX@`l`VH=#4ci%%u2hWCs<3rPaS&NH{uON)dTjnHsX=-XdG|2{R?$aa>+25-kHY?$w zDck3{ZXVgGHqdT@u(;4yLh0PHT@IU+Z2s}Ji?Y6f&q!IuCN7P~IHebFf5fMvo?!s5 zMK3WfRU4Tsj|E2I9+*^v+rO-Iu;bYiL_7F1E<}Lc23(Jcao-u{(_`^JCSFT+`{ruD zy=vGISh;C;n5g3Npx}NY^u8gs@TE`wQTESUjk{&1gNnz;Jrt+-U8Nz9H{m`omU$l@ zvKss`duYdRV@owE+o$B~w{yaXhK*lCcU zgl`ad>{PzR_?~fv_q7xUp5~)WI{7cBVC+U|x^d|7)CvPDE9BAGdyI^s>VsaUWVKY6 z&Tjdmux?7L!#INY$&F6jEVZ7C)gTH^hPqOT0P=#ag4YcmCmRx{W(tRjqW9d_O-CPw z1b3PBo1S0y!}kd&2LbNwql7OsntLCu#h+U0-=^HM9-qF5>z19hdEQz;%w=}vEf1$H z@NCif@9q?49t`@g8hFx&tez(20VJi_w7v$D>N-!Vt#?P9ZYJInO+yOQ>EIYu<-(US!Dp5ev2o(93Eo(-u(ImP7KQL%~MSg1O0f3*_k7a zEzJeuPph}JnGRD3xP-G+TbQmrw}h6D)zo!+<)_`jO1p5wFbMuqeHr&?{iDf;Pyzsg z%p~#Zw>)oO3d{&z%ev9{13`EovkTZFlzC2Sh9cMIDwUebU4T4i$C?aUeC3-_@L1$3 zOT2*tO<@^^iLJ6WK`S<`-5g5S3pzd3ta+~%8-nnOO2*#9zVP!XoqWnyn?*q*Pf$|t zlR=#RB3d5$L}s1(=qSedezw-)<1$-rd(QL-DLCLFC+-Rpdy6Z*XV9CtxIHrW_G1&_Po*_5)nO_0WpImO zh#p%SH^z7jM~ijtos%a&p^L3Y90d&l0vCx!Pf&=Vv*%f-Z0l+uDUNv<@E4q+^*T+` z{1v4C9{Pb@R@OYXRR6Y zQDrk+YW;Qq?L*mTlLso?oQsN(ME9~oKZ{@C(dtP9M4J;?v>HioEL-~f!H>x=7DElO zjsl%Tb#mVYw$dEikO%i7`<$6!x8xJPc z^#_BPMzVf1Jzt{YGHy^o6(FnyOq!MeHu3bO$6;^DO}LCmEmAaqf^COgCuZ~A8b)z> zhX<8GU@axo6$X`pPQ4GeigliS-jug_+hN^}}xr@rmYRvRmt%7r7^bvMfh2cw=b z1Fhu42-zT;bXsErq{m~jY=$~UA41`D;l8uc214JyF3Wxiw)na_p{Z1=oJ11ry*-}V z5m~za4pgl>oY@tz`ENg?uWF2f?My19S5n?Bg=qckGu)9_X+9m2^~7&oX2o=oOKTkp z6{En(8fWnvCS0cA`@v&RRmsg6hF^W}_DR>F_vt|VIh+r%G9{7wp^ZZ1K7?JeP1(3= zHlXFFzW)~t1E@nN3PE@pWO%^Z$l_T}x%4s5xEaw-u!SdH6VW{$kSpPgJagfZsle0wrXTLgP|_<}ftwH8ABToW<(BM&JHO7sSs&oi48r z(mN);#ZyS^&T!!P%>0&LilUoD^m-LbQKpj~VFv|+#2-+-1K2+iuqJH}r$ga5Lo%*) z7QlrSPR9fB<;^L)d?u(|iC=1=GpD1C`n$=-=?hBON}35MSiF-eau!W!jmI@$R0joVL5esJm$L_{kr zQ+Kh!UBfY9-dc#5K2FwVmZZHmClOULWz1Z^)-a*ICkKTKBMW9Fx(a=%W;Jpqu}F!R z<~Cjn-+|hY*w!TN5;fPsH*T);2adm@< zw3Ux5cOcH!YeCgOblLFw`{7@A!{{E*gdV|o<_DZH+}XC8!GG0Z!YL4!?QLV zzn?Mcy0W!gQG=ai|Iv^BYZz)St+7WVE~>%7&?eNl zE(rJ&PDq?hEp;ijUgK(_js{$`xIe7ptSLneji;*a- zB*~bEEimsuITo|u5(>y!q;JLZ8+PU*HNuJGh5{W?c9L}$e#&rQy3V^TYFFS5V*u^u zt?Tk#snnihZ|i~HJtw{K=c&mRFcx}+km`DJlleA zrt-84HuqL4kjmEeZ})+dVHf}u%3p<_0R7$2TIerUDiMlDul>fIW>s7-xv83ibivhS zD2J$T6@HCx1U2capUB-?KJ``k^`yw3dSo7vwfX4fZgJkP*pRDFP9L+Eaw4v#v~Az) zKJ3qe>Tb8RP$)*I4N8=!G7hLelYR+a4BO66IZL0+Mm?hgPB8mKSS`f0_~_uuqmOes zcAUObp^lw>*BC>-VU;+NVl|yMmO4Dn3ZTRb*>vM*CA#sxGl^pZva-2^AFJ$Al9@Z%wYw-j90=?2vO6>kg(`msdVv_Ix#H z6u##|p+WDvLs`+r5bkk;;nki7Gj^~pL8&S2Kn!9HHQbcywA)Xj+#$A(@O9xJCXNOS zeVZ;0!trCpUNP>YcJ&OvCGrFbr(G-SLuO5N8(p+?B)2sdC(*nXx#A)4+**gKc! zY6*Uc@cM>(GQcOF+kK{eP1`vKOv35WbEkF#b>hq)bvQ7$3*926E0hqkYpup z;x0U}13+J&l{KHpl+ux-kSz`33j?bCIG@%Z zPJwQk4Up{6cBhBBPnk6r3N4wwvtc`6=Y_S4Di~6?+Ix1K&&AGKX^7~)FRbru2_2fS z`asrWixXY=lt#QePBSU_YdYIkkFp?g_iwC~+)KhE6$!@5k}o~hp{s{*1m-8tX`A9| zD6*Gy12x|{L3rZr)%R*hG-!T24y9dqS@; z;%4>bG7`1=ESXNRE5wVb&6#qW_++h2{hr|^Ll7%?4Za%Ye!m;Q?$CHS;~|gu^@Rm( zWlZ21`cgGVM>3!3_C})qfew%uD>y5EcE82pijT8*Zzp@idBlDiaUmF ztXEYfs>~;pGmDG3R;K4G*uaHKbQsaNUOJE=9zRLP9rs5)^}GiA4Px?!zoMV4fggRQ zXKXfgi?mQ)I!va$V3GsRSpF5gV`w*^Ug!AARmkz_cyAp*xQ;Q4IXICOHRHpb!<9E>%M|)lH9r4NIgjc zq_@1PRuhhI2XUnho-5AwsJ2PKqyV`nP%W#@+_Nii5>YqbSbmxI|qE9 z1*@qbvQmYa)+!B5T0kqvjHS-JvW4X$owqfQ1~*Tz-uA!s4+Zg;_+YwH}_GTQs6vjA8TPcqp;d8&@yfkPU zc^i0Yw)Tk2XqG|lt!$Mo;c1eAHZOMAQvlw?Vap@q$1#Hom>%V``LBKNr)qMyHS%s@ zC7o$iW&SgA?p)naQ)H`Ch|)R41va+ZCZh_~Oj>GAM6|OVn}SMEvVceO=MIwe=vo{m zm9<1n@P@Ewx0Fj7n*Qr38Ys%gr z&2CT0P$$zf{6&yGf7q|SmKDxhfe2yN_(nqlZ)CG4q)m^pBI-g($N!_1;bB9 zYjf7nP>4%i50778DBaQ(vI)0Vp_6v zf!C|9Tkc&(;j{h`+_bfs=(nsx(_`w=>C-vWRVD%1h0GPIC5t#jDWP>sxm|~HxJxb>+X=8tBhnot3~NvwzjuNXKi2t>y2QYsXZpcB3Z z9A?!|)ugRNpUy>AB1cakN>`@_R>enbavGCGpOt;)PB9)d`pRIeE4KI)Em*7(%b*#d z^_{wwXZE$_WwFFS;8Nzt{O|~S0W2}iuJ@veFh0_RFnV8?5Zt;=m;Dy51IC3Y=S+?@h?RxoaW76Y-Q`Cxs3Wx8N?^e6+U8D*e0KwtZAK&oOQ%|RYCumQ4t%?!<+G_w03_aMsYl@ zC@NnVg_}e4a{e6qbaP~fz4ZlMPIdqFXtF=#pcB;qXXSdpS8ti+3JVG!ncVgUVqw0C zswhxa&YY}zwky=cjN8CaI(#evJ^rc_SMlgHitjM}>ALZ^_ln<~w_XUlotUJCqzt`~ z?SNs$YR>Fe+8V##9NCp$AHrE#4=Kk03LZd@pP+pLkIT$gkJoIuUp6QEz68xYdf4RP zS3={vLkID5r3kRG4?EeI-4A*mj0-KBI$P|*(6Mq^l3osYSZ}dIWVcy4>0p_cr^BLA8dZf^4o-97nlX#H+1uMhCo)pE5!F4~eVddIl!Vs!Ix!swW4`&PpNcX%7BWNi3 z7JXt|jlsR;kPt-_a*a7t8dKne=6RaB9KQ9|y7y>plhsU*?y25#_zqH0Dw}W1OWB=8 zHl6wr4fkciQDcRo{8fszS%<6>5C@73#ii*_>#>Pc!l=kmvB*D z`AFglb7OZ4&mPRB>Yx?v!-dpzHG26B#V$NQReG_0qQ!5JCAY0(ju!~pqIktK0u(qa z(HIDrYXZ)6s)aS7!}P%xrEAZ=RS#r}W1d}&pKR5TcR$jfNP37wjxSm8oGx)LQI$7B?MVR$#E*WwR z-*UmK*F2Nj?ZR)n^!$0W{h*k|{2{hFTY^nHI((zoq5C9Tt4=35)Ol&a)3BsxTklSj znVcu}N=p7XiMt=?6dP7M0Ki&?ZMM{ubG{3x>o7|2bcS*gkx(|CjmdHuY`$EUJ$oVp zxb@wJ^R{7%zK^MGLVwPDin?CtIh_#@FtjBb?ft`Q!d5qUsjH^?6@u&{VN*ZEs z3xK30teUMBJ`nJxt6Zp@*Gg5+7^oO^ycq4QK36oPE?}5-Gz}X}F7KJQZI$08{un|k z{w=&Jj^Yc?1({r190A7wfrkKxPEvg~nQJ`UOozco+Q$ALu4hlUn%6>^>~PHT$-DzK zC>Q+|L;6E7ct=WcMtBa!3h@J z-GjTkh2XBi-Q6t&cXxMp*XiVWzi(!$x{4n(-REr1zPGHku6<=~z#`XU?tGgLYkPmI zJ|$*yhL#CnWpqfuj?d$Za^(_fzm9qOJ0JAAF!eS`kZ4c9l*b|r7S+8@B=g0^yM1$q zWP&KSBMBP|>RC~X0(d%?NK4K1@hqDkq}Z?gIyU!{>j34zQe~K~Z*s>Fq49*uja(!g z<-%`4^OdVKWl5AnJU|M2ZaiPT3U-)ucPJs^YN$?#(|K!u{Ym|OA`XRoauIuB$#UH; z?L}a=2#O>M;FYk(KLw}Pr;Ecs$+0CrgAF{)n9Pdb{{E%lXdb>xZ#h6s#~CAjsrq4j zK@1{+Ifervt_o2u#j4(EY;w1cHX4-?F?i+&SS}W9=&E5@HYijx-(RKZ18{;e=<0Kj z?80D)=<;e5V&*U%#l9eG`$Tt30|D1w)V`HStCp6qM|H7%B-tYup1Q!-_}FKkgs_yy zp<$Vv@F-`L-Z!!IVkrg@4_1YE+TZ9G!;fq67y{a0i5GEA27;rm5X1aTUevZq)%O+G zspEkXBMo7ar3RpY1SJDSfZblhN-Bfe=P?s`CPL6*lw$92u}kzuD8u znr!tiLx|d+qkx$x796JlPLNog_8p2H4hR!#*RNG=|Bx5pSto|%V zi5SB2#dQ7pR|;XH@LM3w46;ffvaCk?v#7O-hGH6^nx2<KE)(;(Y0Wm(!1P^zmeQ3F4ds!N z$pYmHndG0kZY=|RImeyEb|d>Ista+TZ3d2d(Rwc)r0inrYxvbYRVhzoaswadiLsWZ zr{IR!!YDnK2DGHZ*dfeYay?UT3XufMQ*W^Hs8IYPh`Se|O`Xh)3{Y3aQxGNuC?gUt zTO%jy`ulS``PjmKnW!64uxwWeA%P|iF*HE)ABAFMvY4C7PvHvOqS@iVL+7%}N;xbt zrtoE>9d;`5(FOkrT~*b6x(@H=Zbkj2zw#PLTT9{UHXRW`LKDIpNW#2|DCiMjk-?t~ zYtb^gQI7a$1fU0p%Fz!wS_v_a!_ldK#OS zE}O+1gO@4%r;bX-ffec_^vth?ge|5?bXunJthec=6-GT=fo67~X&*VI-*bV*LMS}3 z#uX@%(M9kFRg1s#Sf|3vw-wtO@pUgAu5q}ID3}f zMQJ5aS*R_$9oq;!rvZ>6i-0}ga0&(Dz!{E9kfk`TKb0b2GZ8aI{)6umO}m4pQnpTY70n?fz>w6S_z3#+7M{j1)Tfe96x3ar?QhXnSQ8 z^Rveg4l7Kadnf`bDRf!xe7g}{sg z&Vuqz-4rRfO|J{~zfLw|Lhw%2^T4^BB9RjtkPpanrN3R(f8iPu-_z@4RR>U~^3N)g zPX~^xUQTbdA)Zn2ODYJkBQVzJe4^U(c?-mpy7C9HrVw^QmG148nf~W4(uC$(S#Qo% z1EQkn(}W*P+)n4$g%EW5hSkn+|9hJ`AOg-{+7sHehrV3+EQj{fIc%+3Qj ziFv=u9ElPCfn`FP2Vt$4Zl)=X0uH;2`Zy=TbPMYY1rGkNBLmcd9X~hQGR~Mo>h|^e z)5X{rx-C{4Jl{hpD*kI^^1r5+lxUFB0yy?Nc$osyib%-`mW!x z!)WMh9S0M!Y^AHlRdP-!X#Our0YDI|L1K?fDcA>u_kTlw`BrB(|JKd+KOYGb$gKD* zjqFcH+P&OvbBzJ;kFU}fw~MdZv@9A@t3A9Bi3WAXm}MYxH-~vQz4<- z?@y67?q7yQ|C%a1qQEMr##tgpoHiqoyT@5Z>E8S;=p5!sRV?SrQ#q`cTMjbQiBskN z88D$h_O@}qadC0o9yRJr{|Z#PaIe)k!yqri?~WhF7aCZmi`dp= zD?h`-b=v>u!aEZ7LwX0CnT3&QaUzznP$wHQ!y-;{t^XCHG9rS{7qBYbj^*je29G6^HM|?c6!(z(&!2(~xQ);CNBQIS z)rj;{1~|%8IN|;AE=a-L{A5kJokEB4F93M-6?E-78yUmDRDkB?bXtv8pv|GMtTY9! zMmqAfCy<})YlgbnUtb@{4H^TsRtV~RCfLjwDPS|^Tc7wrz_{L_|17-IKR_3U+d+rH zE)BbR|987`QWw_CC!)K#GBxxDgd9BTLi*pJ392BMDEf^s`w>hdOm&ivs77PU9_ZBC+g?}pyjB#=C@#$=6w6J5qI@Hr7|=EnM4&mEC&ZBBYZ+)j*=bV=Fau zGO^zS8DZlkE6M+cSIjdxsx;zPs`S}JGE_#}io4BcwA-Q0O1;uvCP z^C%plT3>8{WWFBxT&DGW^OBlO2kG3`f@@=X|0L@b2Tje<%k&21+s!l7?snvoGJ39? zau+2Rwl6D{Rmz=v>&up%2{zRMf@)X$7-4DLt@Y8m-cJWK5a{!E@?aT5 z2(Xy*jo2E7Zl#vZdN?fyvfu&y;|9T-i(fSjMd)om%i(JphGyRx8Gx-6`}2ASr&{7{ z2On>&7Si9VcG}iCL`6Ad0PtV6xR}I)xqL>KL;~D4u%1%!52sBP7~itbbQs@~9dAwh zfl;Ql$L6t}b0X@hd`FKJ#3+pK)C8=(x7^1ifZb1j z<3fbNx8@q=ldK*F(DAfDOTvgr?Mj$lnwj$VW*|bsZa%@QKYrc$}W7xnJmt*)r*W6#>YwW z6eCW6(m2HfL|5CtkHK-fMxrRN^%tlfG?A%<@Jkbq&A*&!V0&k@t&0(LfvthOdD%Na zyE2g#r`Og^+FKli{b{Tk4=nF5IuGdngyfSQS(dh6DEvCu{@=jF-=)mI*PuG6!sEPr zztjIaefs~raO86POw$!C^^l&=E5`H z{;K`cT9j$^e);EG@R`iC;GUw`_kOAHAvt;dEixceyRzhbh$>C3is5CQ|H-Jo8Nk+g zr@AxE8D7;|df9%H&S^7>C%`vbmBqUf23C>#jHNe<$5wqf-LgKS#o;?1cz}uJNrrI_ z32I6YaEDNv*!e7!%=%+0LDm}0wxUYU=cD*v>zz z?eVw@optMfB&DVEb4V0*Fe*!w(VD9n92(gsK9ko^|I}m<}&n)<|BLew?7)szG4@WKrgF^=4m4 zGA4S4v+DXL+Aq~~$?hyyvCkg3 zKe#I@s3>f#5q+=39=m^ojBLd(Z-F_#MTUJJLvz3(bkZb^jQ*Td?U6GP?CvFMyLvab z(lliwd*hW^iOt!n`}J0Y%ZNfC?_U-zu=b@B=g#ST*B1z)Oif|<`xOzDTP_y zPoyPdpUWBkij_+I?S@g%cCD5DOF%|X&-=#%ks(^W)yDeec6e-|xX*rUe@KS-edIe* znod%dyafcL-I?u0G`@8gry!f1exKe)R}Dy8#3n>`zpxpG=F?QVJZ-Fa_fI0=OxfZN?aIEchXS?_3j-%B3ux><9n zvVPxhaU|1*XPoHi~m5wbrJ@Xz!HTGRY+m)dM%LFcuuU^ zl~3FKq{JGdr7Hfr$GJg_wUSP~uF!q#IL-HM;ac!b$rQ5&6uSA2hs7YH{1^}rz&cU- zROCz7erQO@&1AA31e=3Gc1T(L^fWR2@hHKTpdo^h?yhf0Ds_t1z4fM7Z?%lh*ulBJ zqI=t4WTCEt%E$mwnvd`pq0eXabjk{>z9^dgDcexUV|#n}t*<6Jgwg+ksZ?{tXNQ}+ zrBew%IuDKna~QLy ztapv}HewV@pZf{>iTm@ge%EyyxAT_X2P@5Y-9Kej_$Nv*!I}aa=&7us>*^OOv6A`k zB~Fwo!nmje!>NAG_4Q|$c3MiL zOuoX8MNmu*eIJ9Pk?p*4xnXZJX)KguKst-dM|rPUDI|lX-b>wGLE|&0t?$@7|D31l znWofuC%nBCD2jMGm6!*24Tqq4;9;;=BRzOhdr;qrQR@-b4`;c%RY6_vg>A`wa& zBL);ES>0N_FL$4VM%*;EL8qt{I$?0M=DP6y!l=bzRs42l`|@@beHzvBMl<~J$_QiRSK)d2g%FPF zc(tdeP=z)RmpvglJsq#1pL|sq1lSgQCChlo#eMAb9QoMo+Kt9j^O_k0$7*2Xq*F)6 z({q`4a9noOwgNr*`+6V2{r0DOM|km3Dsg$<{gN~^oqXQCtn5iZ%S17GJlZ3uAt(-*LjyuZW4g`Cj|V?@sQQ%L!Tx?p!%-TgiO( zI{Y>}@p-V1AZNTQg;tSM7{KMc5&1PK3Mkx?gE!H9G>T&80WavY`M9*hsM&6DrKJTp zMzZ~?*G#Nz_mnU)f#)b-{mu2edY8ff4CrR z@p~L@=2opDXyzXZc2?x$drcL2(GiMNcTwUc0gH$eU=Y7$=KSyxQhA=s z?MTNAVVJJkF3+W?HA46x$VMe%tuIJVA`k=gR z_a|43KAj^jq!8)l`5?0ZBc0PpeP~SK%>r|?n25lj_l&bEI9z{fFL@?v=xl+pDI3g| z)AGwagV!%Mv1IK4BNCDKv?PWx#61nlh>Jbt?MN5LC8Q4+`06`&AwS=mWPLFt^lolh z*++$e!iRn^K@aTN(tAp!Md@tMGVd^6#bFh zs$j{h4wfKhRGY$#>_u0WpmXWM&+vyYZ%Xp9u={%A$<3>;{*pmFgL^ytT24dw{mN*8 zSVExH0^~tPI;VLI1q_kEC_(uCID0y)4}YXB!U7JW8Bl)&J>~z=p$JX^v@c5oW`WJyRgstw&V_ll>w7gmgdQJ=6u6;al zceKZ@3*35dPqkSEYiv6H@)aOqg#-{$kqb>Q!ILal!QB3i+`UNj{ryZ2C6a=3HDzWs zV%&`iHw-JYB8Z{F(Bs1)q8;M9R>jW@m#^nL=8mT+Vl^ofpPiB?$MKyz1dx&zE1$ z$x!nwe_IR;&$Pd@e#ORkRU11%k2htndM~yg+HMa+gP-6DPch~&<)r@jLv(%mD`h{Z zc8>q`8;dJ!V?ULniE81|It^@))5v(L)6r-Rt9SgD=R;Go#DvbQSn~&s6;ZI!U9&0{ z+|Qpk;e{sj*HdbOu`;5Ea~H}V(=73Hz5Lw~;U>6~TizVP<*)nh-f+l$v7I|UrMWb_ zuMKdSVr97MzBgZiw_u<@k~xa(D&5TZUJkg+WjXB!ZFBNtcYe`_$Ns;28moVq(WW z0E|8i?%0gd{!-&wLS>%f)|%Fm23DnRqDo%lree#|7xM=R$edRTvT-L>5W2C@MG~wF zgzNyN>WzddZ7cO^-n@sSTRE_7&RKE|$)A&QQ3Wh}or-*1@t5Am=UoqEGFWp-m+RDZ zm3@}0XXejz%phk#vwz|ku>PRhwyrHX=CXKrDj0MEd|k|Y+8Jo`GK83d zb(pWb@?gZ05#;62Xk%`{z@V^)!fvzFYP7M`z3N=5*D+3mny&vUeEYtcD_BlfX@%uE zuPSBnj8h|5zg<=}BzUL1s*e3j;klveV0V9h{R3~^d5E5KrFyz2l@)$mAblxhO4!{C zXI(E-Ng4lf;M>QmBSJO|SKnk^_HQyV);veDxvop$NB#7?fh*b?Aylii#<7>!uuQto z^(Ari2Fg-#A>(Z$2V%#^^ej^Xj1$^&c5kwm8c}2Z@00UyFR9t^tPWcgj@U!fi6s}Q zRz2FG%_usaw~QIZFh6Lbcy8zTuiVzBIo+{>a{PIt9Z?*Pla(TPU}v)+%`yzexnT6; zm}}EIR^STdFF5T<69FfXi=PcU{{$aoN<^xDJfcCenr=|5&FC z2qm0X@f)<-Be|SeEc{|VMF<(0wvf$k&TeL3Z9The+*{dLaUw2p*U(n2D)%*ci9N%L zYP$?}W?bwj#VxZpY6RxdO0*>m0X2K*L4fwb(HbnDdQTNuBgACl(hrAoJ?;uE1#Js} zTYoW55{c|V(L5gr_dRs8N`IN01m?nqj_fZ4U=z(HY+oNULcSLt2uWY?vHFvHDbz1W zZ#x)cZuti>;d{Cb@${0mOq1jN%c(5+uwpxj z{m4c)mrMP{c)xJ#TUt&%H;h~+5BVzD@~^MkBcltO)>#hz6id3?B*c^#!%*H$@MF;) z5rCJAIR8as!YXHY;BGI0FbJbm;TCf4K>QK3qI}5Jc2X)Hv<93N9OqO!(hL`ele`%V z9SQ~3+EYCi0GDN1KHFOan=kGiDRsNKi<1YR&kJPmt=F8uC77U{c-FprhVt7@mIT^p zI}=X=8S0n7B`2q=T_U|ixz`>hj>hvv6kHy*gGbJwC-}io+d|hI0?N1qx;vO{{Inw5{XAM$Uk@^$<8H(NW@N)9I#!M~30b_z%7V63&1nJJB2 z`yhFaGreEy`r|w+vTTKIo4kAsk>Q4Z?n|sCF5&9Bfv+$Ux$CNCKT{xXyc*>xq^$ri zha5kPwtXopdkR^B`4y*SV}S69XmNSt70QfsKEd!y(6nGPT8HMxVcu%4$<2Q4h0lU> z=KT7Ltu^G)Wo4X2_lpOUH1zk9Pl8zt$o`#_cE1->Z=%BGhZQx|c`7e^Qz+DUyjqbv z>INoUm~S2B@!9N+`v&#e9ra|Dl-()2ylXGe-dk=PGDnzqTOY|G$|p|)l73(fMaj{KxX0w(=YF%zfl-LrMnBe7YPp@U5>ih?q&Me$_EhV(d z_R*kGlZ{tOL~3b2PlJNP6yM%GRU3G;)12Go|DqP*p^M8R(zHB2!@-4`r?gl4+}aExznnRwZhC>TE$`HA_dm zE@ib=Lo&(E-2BKR>g=WRAoLEC6Bs>HlanB){qqZd?H%QuRi+SG=jEV;N+UT1$#ce{ zXeVrEHDqOC+YJt35YspXVV&((`TA^({$k)+1T#y}I&H?Hr6uQ3U3)_FMBxUMaYWZ~ z$LM^>Kd^dFpdPpxH&`Msw08-mW57Tpa(%6NoDYPq5q8w`vio|`TDF;>lp2|PJ|z)_ zR8Ugc@}PUJ=r-?^(C-_sY5WBL{U^3EV@m#>R_cEj1 z>!jYA(>|KgPT%vEVOSgZ><5NXovIr>PvmWQxAXZ=iAdo(*BPa$X(5|G2j+D9^%8Ac z)OoM(V=gRK-p{Fti7L-Z+Jb@SV}=c_j)h=e+!WtouYCd>UVVI3q&Cd>2I}MK07Xvx z5moX|x1PTB*Aci8xAo!zv2>7Rh!0=`@38qZrEH)4##I6yN)c3&NUwHU$wq*?;WdL8 z>3IGU-|(zn-?~Kt9AH1g?UO!Rkbpr(H~B3_8psO zxa$Lg7SUK_eL_@~f$-pL-JIl1v?8jV#PvN^(y-7Q|AUDc za)0CcsK~42YUno8WLnwM5n8TxPd4@Y)@X5Dv7yh$R^X{+`uAUs@kq`bYJW#XgYkfB zv+oePpLZzXCIs+|=qP(z)pRK-dB})$vCHz}QFlH@MXyBnlMb3R=?K2o3jUDT+F0ZU@U z=rnBynTKRejoaPY4~Kb+XdqWD`i32ZQuQPk4AIGfKnWxDF}IiY5i|Nv1xG~2F6ArE zkGkiJ_ax~Q-rglmt%vi1wqFPu^DJr3u2vw3Qj@jyyH$tONq^%|M(R-zQWU$=bqJza zTxHT3FPv$H(#I~^YW$ZGT_q%DDvyEFAugK}MqYu<`}~75p%Qqw+!;f64jGY#uGe5H zQs{KWr-yR`ee6hjxc;wk<2Br^hF>jdSrb}^TGf9*hC?HcZ9m;~q~?op8I_WUJds0X z&Z#JR_=r~z8f@>h4Ul_zeM*7E6&uiEXt|S?fgk1Tc2dU3U?=F(^J=#biW*1JcN`(o z6ZwRJ&KoQ6-13@vvD16;8;-L+<@eB|7;BwSrddq^6lDP1 zYEOkIR)(kU6~;jSFMIX3nXKID5zt2A`Op11OWbAhj_@OGFms9p9X+@8CxoB-H!Ibf zVy(QqRUhF$X+HY%Uga6kMn<;i938X=qI#^~xoTL^ujd*f+@>0k1h7FSMckhzXDRgo z8VzptqxtHfD3%oDkt;X)~7Q_#O$4771f60hLNn{ zIQ>MNfXMk&)`;7j5kR_05u>FW1%b_sXiH~og#nIMnIaU<*p+K?FrFtUuupsjq_>#N zLJY^6smSEC;d22)H@a#calF1UH>*}g{WFcL>)UaAzj7vi$Ji;f6IT$zV@6Re2gudy$7i9R=4H8$~6dlZ)6ClAW0`(jwDoSV+C268KQU%iHHY3)Rh zUo+6TISL@0pDt&A{ znevRZ?dmII1kIJS1zQQ}-U0^}5Toz~@O31QcFz{=-sV-cgXIO{4?njGd91w3Pl@iA z#U&rS9Z#2JZF*q#?PYO-x0XFq310O(8DaaTPF8Pmr_0r|@}_dD4B^@Kws9OUudUBt zE|-lFRXLYvK4cVQ{Xii4*$H?T&;ej&??4cp7CU-!MOYI-UmRMY%=*z?zUK+%OrTWd zF6({#RB{q9e0IU|fT#(Vo*J2?CqP_pb23r~G~*f2l+BXr2kMvMmhz~TA2&)B6%Ve- zj1UZ)8TS=4X5H;gUJW5M5-tzFsuK+VPRF)?-%X>U{YeAb9AvZqR?j>UPYmM}NpVTCW=d#VaH+R3DfdOJ@>EjxcGn_zakU@;8dfm`JG95$`?xYc@ z^ZoqR*fL5zy>U{z((Ix|Y7@cI_nGpVC-S*;mApOneU`B58F$VIadtv1B9l-r zf*axJfKAa}4YoiZ%%?f}FuUO2+${w;aevizlq4tCmqEpS!-^A<6wZagVH!h5fLgKt zL2%kDk2>Izx!_JXAdk;$yUx`oXK!^n&>w7rkWvXJ*+WB2LO$E3syTP&ZTj#wPIz>9 zD7VD>Ha_Yr@RG{!c-~U3=dkpk-Hfwi$MRsK zW5KkKK@d~5ly?UzG8D|_9PU))IB#4%k|1(L-ey5_=1^<^bMDBn;#v=R&=j~h>nWEN zWf;WR`R%F0SG;%lh-A}ac89w4sPO(Et7+CxO67__i#YDv%1$)ahO=uo*VtabJNbqp zOJ)|u!+Md8#47+CUpp~eKXYzwxJ2se?O}&FjYZOw4JSf0P0%^77F^talAOgG8VRU| zC;!P|Qy<5zyZ?0J12E=Dmt=4KlMJ&!{hW{|0lSYa)Z2OJ50YmHl(HwwRlKa0qVcPv zqxA6Mg5;JgCl;N$P__g9E0heGB|iLDV8XtxH8G- z9iC}(R4aRoxPVExCM{AsfgvSW1(y?gnOcMx`r&15ts?qETW60KwBc}$H6@Ck=v*LUc}_+giFFFE@W;9VcN&}zR!}%H!wRuwD4XwM=m^(H2&H>jY*m8a z#2l2oDHLL9&VaGzt|n0V^^X@<6!JO*EW%GSgP*-|k?1_cykbuk?HT&gr9bDh`1VUG zr{2qY=HLr(k!{>=xKGS=2JC_vcVE1KH@CCO5}%>!KwN=l9oXjO*4LemYq*agwBnT# zD>|f+&f9{0-xp3GdS)q2??p1!M84iq4aL$IKF+562%pFx@^!e%)M#ID$u_?mGwl-s zfm1?efCAHmkh~FD5Zbgp??m7jOLH~ce2R+C-DREMf{8T$qsiya8#z`qU+CXkz3wYQ z1z6svMy68J;8j!1-kXFjmD)@GO^~yF{Fy!Jw*3*RH)uwsEOuzh9+#KS*8FA@$4fD@`DLImw% z%b-@Sx1RVqskgYL6|dCF0^5o8^D`p-h%*Ag@$#_(_LZ?!wt`?VIkd}Zt80{RBmu*4 zBgBF1;N10W<+t6Q3i@TO92SSzDFq{Uv6-gstSEz4eJdr;BqOCyTcANC!8Og z(wpp#X4HJ4sz~n6?xJnqxqEcIF+%5%Mt!PH^~aKow3r@(e-iSVuVRQ)PD09QlSjB_ zt@E4R1}|VPkFR`hGK4KhtZ6Q_jSV6uhEYfUe9Ak0B(v@7v_*sCao3+-y)*Yj{%R)Y zD9-%w?gut+H)4TFIhNXA-*O#MT8lm@7{-;Gi|i>L&G2!&;N-&x3!EKPNPJB2AU5G1 zDE>`)C7y{a1NQLznsDmjOG}DAF6DLxcYDCtrFb>gb=E}X&qV0%9s$r`JB~sbCWE^? zTSBu-b&LfId6ZC`3f|L|=#~xiG*;P~rZ|%qz>!4452C6_BJ#=tJmxXDKhCT(8<8rX zFG_1vQ$QB}*(20uQTNx5^OZ`@X)g}og8a-=A_?SfrZCyC?A6lATNmTMyCdlG43nbT zueeq?ph9ItM{leOd$|;D&i~O_j`o70ZG92=y|r?*rEb~M+SS5bXl z>)m=;3C4~>qKvnaWB)nrG+t8%Uv+!+){cu&T@@t4MmBJn$vg=xFNL`wmGkXV8p@3T zKVoEXf4`jB_qQ{7!Li1(%2LwC{DSHa$;}Ew!WS9CMLPWApF*({AL8!5IoQSqCrG|O zAXH|b0-kO)$FR$cJ`YV6A=ml;uz;~$opY*GNBJgp;~{MVl&vJ2-qXH_P^jhHfxstu`i(pD`~+>UELNO2_=(@BOKDW2~(+ zFbHTZ5*|lB2rQ2!DR>5;%z*Eivwfre5?v@quU)G=G_;cCF3#SEQg=JWi0U6mY1bh9 z7PLK>#SVXFqe4v9S*I@zFmr2&)_v&H@3u{GgYez`*!*>oSC}xOr8wU%K1=s7n`QeH z{oxm4+_yC%Wb*8CL7D>$qoKMydvM)G)$?VDZvVmjVWJjz8w0;Twj;k$fr_;aw&7lc z1HGsy{y{EmYwP3xUV;jKFr-`MISE-A*@?TdLbi$j`%GM?<*>31uBt_K0n-;?o&58X zW|IEnz6wP-dyW@#yM4o^IOZEDFNWGb1Xmcn3E zXgVdl6)^aGzR0|bIAmtZDpAkJidv``=904F)2{gqwghgpHG$^lw$;ju)9yn7YJJwz zkS_CQ#iixbshG#EYdOv|?Pz;aR%L4(ma}xupo;UfLoh&8AtXMAVKf-WYL53FZfWDy4xrLUr=C{0`7` ztm>L?xOzwtL#e3QT(%rdiR99z@L;+H(IRtmeIR1FAu85^ja=8ZwQZBkXmsB35hr1+ADD^ zZ|tZxrtU9S{V;Y7{ZW1VyqT}aHNpsaC}Z@h=h*g~nO>w;Dh&-Nm4@WP4Zj~yN0UaQ z=E~6bOzHj9x+c7mnq(iwf58+$P@;ZOMLpbx%D__#p6vWBCdk*R?<+U28nF=*WfCor zmaK<7+ROH}FJYG+XLpSU*$qhmAbcv2Y_zqi=P4eMKQzv{;m3?ZJoVxgrosEgsjOez zo1 z_wpou@sZWDS#m(#jkh$~33mHIQ7_7w=>2ii;@^AmNk-P;4z&elKYrxMK_^2k26Yl{ zyQN)XMKt zw-}71@5g^P36}57dLG@hMeCC8&$J)Ar|FwUplkWMy%h+@p#p*aBPhfGWE(zb!VrVS zYxg3VDYMv0#1q-%oXhu$bM@OX@33>7wnR$2caEtY`evIPQuA<|7RVRf*8pVc_&?h6 zj`0trC9N1Jd!MOB1|)5u$DD4Aj-7+_2Wu&?+=m)_HD%s?OXQ@*l8Pt@Cx-XV04ub(bbbJ;F}aPi=HcADcP~ z--Hl~^eN1Af`G$}`|`iWvmBk3flQnsH;>sTGWxksSzV#KufiZMsXv%PjsOC9U+Q>d zm&8|mCd07HE&5hF68l6EO#h+u1*cT@DjLivuuq|-t?(G;@4_4jf|jPbLljNyBqQX! zN-&$oWOR2$Xy7nwST3(gWNU-hvDlNsoYU#D5{MWCd9GCHa<5i56O$J`pPMRM7*d2e zhWBL&n61^HApb;Tcy7!_)k@x&C7p8em!AsrvuUoVTT=U$?xA<|oPKv(viso}z&gjC zg>b|Wj;D66RpaqNW0eVwi6rW!v!DpwU{w;675EiON6;s1xX8{9d{MrZ@z(HS%&Lh0vPTQ!|-So*Oihuc<1%yqwX znl0MRwbEE@hxtK**M1liT#ZpHA@9ZpXQDG`_v+aZWn{%eYN7HTxvF^&P|mj0QJ6ei z_GU~5JS`kWJ6+}&r6NSt z4d?k2zLWcEHSvwXNC^7b{axW>rsw2vh@x1a_|pW^Ir+tbAN0VtL?&K-wX9G}i?G}x zn%_)CCu@l7SN_-;&w+6AeN9~zzT$~?i6Pil%Iv**>M)y+%n7&|QDCCkeWx74(P4O2 zR8E(d*ITb89_s>^s?xfr=WCYFO>F&)m6s|$)E8cEha}rN zA}dPgj>7~xV1gS*4@#3EY_PgjAB5nMfv{$FkL4rizI@XP2mn-mGaasX(dsSjW*!_m zpjSOAl8P4BR-nyDzRw-~+j@cR0<3oFv6Cugt-f@vAR~7zg~@MA(Fy9DM(4Dkl9TWr zCf|LPUyWSGlzPxI4r;2P6L2Y1a`n%(rRDepAxzP~w2!#a$Xs<`jyj(NFEDI^97`E~ z={TEn{pg~^zSHFIJ#*293Ri0n>XlGw@O=mk=#HnLLQBBQKFu6TT7BQ0Mp7U$HV$dY zp04&HtA?MqoeU<&itn|h=L9J zBcQ*PxKLxhRi+M3I%uW91qKef2`1R-U?(!W);%`gqx0Z;Mi>}?0-0YHx!2cVrxA2# z9fv^TO$4B^v_lWyU|Sz-9kDkuFfPcCSL*I-1L$6eLGS^o94jXwe7>B*zr9C zfRu+0Ja`7Hd<=r}ziTcR>DJ-SZfe1yKP<;fMfgk&b6NZ-KkH!$=zQ2CgAo-HTBk?p zj+G7&f$6DQG$NhHK;_$T# zRKIj%0;Ie_Uuk9eDL=*$8C#oYk^9%sRpy}f?jyuz%obU(y*r;U1g|kq#wsEqYVk(z zPv1TE!wekxM{Q7xaL3c~53vK|u39;m(E*F1BGWj2LY=&V5FV*`I`}HgJe*K8c}_fl zrx~+d_YNH`j&O$hnIan5x1HaI>ve5S73?*$jCry2)Uryrxl6`YQr9-H;Pu9^0FZ17 zYP`AyRH&hUAKG0@o6KKx2kRRx2=SaYr*qqSLmS>3E2BDLFODdO!T}yxQmWvzs-%*r zur^UW{_Jw5$bT=Fy1iRT8v+vtMCem66U$D%TpvzKu#+?@V{+GuYu7SNM`@8p{sHHY z-d;KO4g+qLOLBxDq<@-ik0FL}XMKk=xln$dTUF$F)LWpzFN&ibLekBhD7X4R9uep%y3t^9(&_%i-bTB|Nu)YyJ(9Tquv^D6I^WN&* zpMmgZCQ+b{^=n>cHICSuSFPL+g!Ze6oQBV?XMouy22`%EuY&;KAV08Yhi)goH&N5d zm~ZD7QSUbGN_&Jk z+DuV0_xn2N_6~t4v)`FU79OAx1{OrWb9{89u;J5{#kS$q+ScYRK(sM8Hzy2cws(3k zna$&M^TivyFAQ0@76sm*n6Hkq{h-QHb(w@quU!@Yw)@GaAEUDvhBH-AEKHtWDG zR)W$-iVpf+9a8w2XXlN)V^9tl1U5eIo~U5*8Srv{2EyuEdwaud=oAzbEP&vK^%gUt z6Qm+kYOQ}OLoCYRm#+YdCQm%n(62EYVaCn>!YfVPPqmw?n;$}9Z z4o@cAl6dd*DA-^EPyd!q{*3$t;uvF~T$VtG$K7`XL?>QQ04rb>ggm#^(fKuwFW~hE zis^BIhylOIVE+}uS_E(kpwTyTiGv?9iwdlZv$OReN;wF1Y3_i<+LX#_mWvD{<@Npw zg1m8*P)H|rXU%cP{lEm5eWu}kGPg~>{=0^T00@H(fTgXH!jPl-&(F_0pUit-{)TRy z?&;|P9Sv&(I-PfMu}&0(bA7sBv2CiaKLo^6%5IM`>kDlMsvJ|2|0_|T;}HCqx+Sp< zLC4opWTds2+uCk4+wXwRy9_1Cu(2ObpauakRe3)>rE}W)Jgm7g$0$KARyXrl&eQF| zi*Hbb`JjBuXY3PQY0)ds2MsOipk|k&%T!D!Mur4nFFB5Hl26Ue71)UUjxg#4LF7Ow z!LgrQ-(TM!x(*h9)6)zIZne;?$_mlm>)^5o5tTF9?%k=HcTjVB`8(LX22eo!78Gtmsop{`F7C0L)NS zQe8mM>aZbT0<4GqUA`xnTAG}!uv!p4zz-7*4Yd3(VGT1w*BlP4+&^4ydn}`MXxzmR zngJuYkwjX9jvQWlvGo{E44dk*vKhXF|I!v&Bs!WLFL~k^I@(tznh!MSkOV zEkM!cs>ShyR;Pve`4r~*NFl+@R*=e%Zj>rm+04VkKu`Y`48WQ_#)}44Bq0QJ1;e_n zmeB0JYBbwz0&1rPjoD8J1+i-$$IQmPt>uhW%^)4_Nfv%-jQ>)oF%M`|EY&t1URnUa zUJw~zvS%?RCnatAfaq-+p3%$YYfSFX*Q=lXAn-|N02A4HzLw05N7dNUxXzXryy0cy zU_Z3(m~@mw`9%r{SFf>T=HhKMMZD<_<*w=OTI39V9_v5Q_c{WUd6N~@ogO^_FzieC z+Dbbt|5s8(`V|Ky={A^4OQ)iQ>B{JV^>twNFqXbKBL9^Jn0=E`AM7`}A%{0YLSVfq zS(xKs_fLEZ8?qjghozq4fz?;_!Sfr^WBz#J^X zl<9nbLDG*z|WN^DY=Qa`ux`oA=@y9X0Wt6Bw1G|kcwAc zS3I-(^1L`RQ)@OwvhuaeD8~|!R6U5d4w^Wui;#EjPp9`YV3_Wt1_g%_M4N$__l%H{ zu@JbVshhnWo0;t`yTb%JS`AQd2cPN8!Nj0-ABugPI2C&mV-mf|e<&1N{=}8luLvKY z*}P%;1FQ=T$bN4(#0&N?(P-GMD}3hMC;Z;Z>qX__1bo5&YOAyWKhG(W)}uciZLU|w z7(#a|0w$vePp7m7(Z8(!N|m~wmPBZS`Tgd3%C!6P#6uUW^{VeFy$lty_?-c5(LF7%BrlUDnEY@C)-a8yDED3e>o1)Kx1-JL>zIueP zjwd2}^2hKqc9EUZ4x#_Oe2fTL*$=zwbJ!tjA3+?Cs%~16l4l;NlH#;3q|THLwAj&etEIXKXZP%vC!dTB%+-HOa}OA2jGi6`$CW#y+iEEE#zx* z2UUC_r&#!;6sbBQ(wXn^=>qaf%25H#8Q%v=*Y09|wFxV=1-3s< zI2I=hKh;z;m%gfZF?(#EUU!~(Rl9EdF*^}tNapaTg~PdyBhSD1E-K}N03>ri#!#DK zK)0R(_tTx)e;uj7Bk5E8BX!Le{!;C$(IO`}CE3SsrF z_dJgAUwg^+eUj>FPixJKX*|F4y4Gv;@RWPIV@0&lExdk~rHf~k$0YZD7L{W`1R0?F z35C+Jbd)HQ`}HX*-RX3bCrT74oix;C868LZC!#!dlN$kP%w9@G{yS}1^w}mQfi_cn zUg>Dw_~+1&=PIuo>UPO8<^S$A@)F&JL7p{gi06$1P1a^}DP1FYh$PMRCrZKu|0A0y+|x9Zh@gvNW%K{use#nS#g3)13X69Pw_6%!Uihf8^1KHnUvU*wqrJ z6DhvgB?vk^CJQQ`pEERXwvO8RF3k1~wvin2tLq6`fBEhE9=Q7@EC*Ls?+1*-M64E= ziwda!QT-EXl_;Y8VQ}=gYuA~f?M{|W7PInSGM66>chSBCK*25kYBkM&-dck);mq1D zg;wkWa)|`AOykKRAsl6<(I&+hd@rQ50d*ESP=FIj=jzEdUv>%~0^@3CdXk5|YSzB; zrl#E<)+S=%&uqrZyDIOy402WH;y3U-wrk@)g?cB=Ize6Vq#vR~$HpMwZ=rngH4YAt ze|+o{Fclu1nnG0*928t%3#^RwT3!0fx$!t8*!Q!__0QWaZDp5xV{?A{$-)pNx82Ni zlj|@VP=!EJFF`~81sCd|{{e(v5Z;QcrwjF&z}+M`m$ppZ!L9rK;h4)DJu1_tDxhm;M5wOd_I4o^<$T|8IB#ob!3vV>x_ znS9k1g3xVsUq!X{xDXq?t0^wGtU&v+(>5?+uu$2 zFKE+CZGfo1+X zA#)UMoi1=~vy`m%aDxdd+B~Kg2~u6nw)2rDBK{$4(8Nd(t;|yh3T`723BCrg{=_e* zZtUaRslNz(M*QijD7kPNpp0Dr)D|cU-4lDGRad2!V0nf zi;OJfXvRoLnga`Km?bPpEvXL9&d*mMf0rvkcof8OSP3ry=n^gWRH+bTLfAtA%lXpC zQ?prLqRkaRq`TCvFL6LIw5lNY=4MXGs6)_fGLyGy=_O4CtQ3sa?D<#v(XlwK`9Ud@ z!TkH(?k@J4Z(Ev@mPJ?O{1Pbg1q&J@5&}fkphA3>j=QFLK2sp=j%k0ePZ%lsart|x zEgJcRYcaS7Lm+svI{Lx0w5!+3-bbByS)9b@#I!bB&9nZ9{79RKw~j0qOU@~?$mMLM z?9q1*igB|_Jmq>~0!-pd|w8Z>?A0k->I*vz}HHN>PB;?auERl|TzP~N zY{qy@E;PN&Eb&ievD&qas(;)}p8I^XudOxi4y|#OSuux=DogBY?GlMvQ*izA%CL#HR0C_UL|2SW!0zX6Juy>H~hf!mu6t2(SjJ-4@ z1eLVY>(Svryrg9@3K>pA3(wHrM+{vntHKS2&C)BRFIilV|Cin9fH zbT;QFSIvl~xi zBq!FV#=pP3}K0d*94R4__4Y zqE}~{XC7*&y6&b+>`NqV^n=9L+jocKDczo^;HrzD+u4+|WkQeU?va3KB(X1%J=du7P>f4L-mug$_HQNp}jK@5$JYhioDD=QCGNU6k4EW~qJ zck@Wt^875(^8WTbBl5S-B$D6#BH8z>W#8@+X7}7q0QnbYD>1g=1xMx!krX!$JrfwX z&(SZtzDQeUblvcDC`|YmLwcM45`J%8+GTjT)5B)gK|`v}sNrve8d^kgiC7?fv0A&` zX=%MMi?TX+dC5QbSydxY5Bp>+5r_F?EH54wLWWlpHDMD-6LX?cERQejBgnIV!EefY*sul6O z2a{Q)9&W%vE3Ex=Ufgwq8dDo669V=dK^EGL&u|ZK1b7? zG9^`>t+G%d`m58E=F&!K7D{!XG0|-SmhgG(Z#jSX_-E|{)pXvOY zIOskiOCEeRzxDDiJ5_F4`C@|J6o=HJ zG)lM(L*>{A5@cW}(L3cp{Mt_f8>-D4n=dkjk zD#X`^<`Sk zLJX)zPblL~yh|R(-rERG_O>XdTsuG7y)iA+`f;DTAi>{O)>|dz)v3%OfB@hKtCsYYrluz7gc!gh3!Ca(0Pr+>-Bz4eY1= zcnN<%UDy**Zt%&lUdk3H>!*1owa$R#*&q#%le~ebY*W`p>_>bI-1RE&Q(^B`vwNSW zq2n}-#P<<_Fe~i3!ZiA}SSct=TU&y83i}dVlO`N`$h7x#Ha^0whxTjNe9+7{&L7;=<2`dUI2G%0KqS zK0gczXRlV%ko!%zr#P8IFV}?cm8?;=_i9vaYG~%2E9tR(t!^KXAIP}6x`IK(F%Gqw zNVOk$dIy4*TD2QjWNBy$Z?)==KJr-RH!4FL>RO&Ug^-m;jK2}{G(fyY5hWvkj(IAS<@@?S#0hv<+#kAXm^`+l;YQJqV)9Utltm|%d5I>RO{q-;IpwH?`aYOhp13ay>yVUt=*t`eZ&Xc#% zb%TSyC-*H)KJy-Zw5!6ex|a{^RijsDkM(VP6PAyWtj&Z36)vG4DUUpQsh7QLY=ZOW zN{ryC<>p7FY#ZOBqBHfSjkz{jR{ghVM#&{PZ5AFLY4HxMigle{KL!ianU>FFBXjVS#zlNrVk zhCQg=;`6N+3ldRuvhC28^~FAH<;jNWadj9pu!_l52!? zLMqQLXYMt7Yl#w)V2#Szvf;%Ew?GL@7)7X-_y+_{XI+v#U%Gn+LP@otYwA32zTJvf zu5oX^^ohskHP7<_;o!A4KCglfnB8#g6R-r>jaX1u`lphzTK;M}y7=UW#zs#>2I05g zihbA0_6GjTS|SRtUwG45zpnPJs|od7D>eP z_(WSGy`o^h19m|`mc9GC{`U1TC{$1iw&I(tm9HuhFR44#hTrQ5ZH{VFL7aJOSN@gW z%NA}qvm$H6M1Nr6(>Q2{y^l2MOx~`yeY6csv3O%*O^4x zy?*LYK;@Vu~>{?M9TC z4y^+#@-;Ie7j|fHLPcpKiqM$;iGy1Mk(6MP;|y)Yt7|yapLl}$BKDujjV~Veeys(Z z<hy>eT}uAIMJR(~Q$_Y`RR77RlkpRy|52T~Z8eve}4 z=|S3r=Cnef9Vg(8AgkE8fZx~=Z%i<~+w|535g)ddBGhTV@64Bn^CaIP^Q`#Er~2W_ z)kBnVSo*cv4F|bs7gZUOM$&Npa2)QP4rR!}#Ddc=b@j>L5^bZKqc6`;tBz6o_Qeu~ z3N)UMh;n#Mfz~}LrobH^WxMii^ElF|ebEt?rPS%>tEyvdSsQIZ#Ym|NRriPpx5}|< zCf^$GpApDNe|6NIsxsr=i;cd`fp~RzD$EsbRWr>;Eb-A%W($1bC>O7-3<6@OgeDA2 zRf|KVL2iU^q|eJkyn&zs4+yvW9^#-jNI1yIMyKd7T86>_#pbt*pHj=8dsnF+f!ie@ z^_tLj-n_bty0Y*vA6c~io_62$w#MH*>BoA2nAXuDUDZLrt>rUIU2p8aWb_slXG+*x z7PUvLlCYFA*I+?$vksra`(qt&^P6Tc&u0KZ^d7o)Jrf79d%394;$Y-A)>kwZc;3yA zluTxeU;k<-96s;j&3j;LlX>N+3XsKQny)7KKH}ZvDtu=gxvp-%A@ip8l_#$5`I|^t zAyv6fY8#h=B!L*ZT*6#%xq`0>>>{k8Z=>;YGx|dx^d?rsff}>8%s18e_;fKf*k|Lq zes4A*lU^awU$RItmBIdwMypQEK`n0d;?$WADXe-Vb?q_gL8Y3EKH@90?vk%!v)lGk z=z}-NVp{RXer2)04~43xwH=G`it&?(qwnLOreel#rWt)b^L@NWDe%jSl!M<9#> zt|7J?))a<@LKpSxl)4q+;h(5~PVbGa-7CGE^7gM(X!v*KtL^gZG)ItnG%*=bAEaUR0hS>K3Ui{IfA@edeX z=mD8?NVF{9Fb%|8L^H@$t&c>rsV3LDKZ5wA9NvO(t4&IpX^IJZplbkHHuIa>W9fAL z6PGZeiPR_eF(vaZ{k)1V1E_pn<_;*5Sh+@jHaVRWP&2f6+x56 zG;Pg6ZVCn7Cy6SmUlr|3ZqE5XE3iyUp>1%^71pvfNqd++^q;I0u?1}fn;=qtwI(6@ zxE!G$Ms}x!--q!YR}5z>ki|2GX1J5AZNu&eBp@(IQJn1eb4*2U?vB~I)#Hl#t0#&R zi|26H4Z+k6eMiY;gU~g{m2RotlG!=4&yIg~E9Zc%?ctRNUyB&(LN>o+YbWHYS@n|l zPNBg1LYv)QVH}a#kEI6*Bjrjv{_y(J>LL#o$N17 zJifge^?oNG5s(`U6e5O$h7}3qVjH`u3|T6+%qjY$bJ8%2=_`?|5PC6^xD<(p)in(9 z%F=zdxiZ3y7!Vh~d6=rj8V6B9BJ-goIyxCf26yX&ZtiJudPQwZ)@?jD4|n#SWPX8InMJ5#~lMrt|1#Y*x)%un58DsHqyc2yi?1U1c0*MtXE{43xT}|nN(a#HqgoyPSnYxh5R_ac2 zAVU?~kTR&wW>~`!-h#b^!9ek|xmijt5n>8Sm;HkU;Lza^ltU`?SS2zSKi{k6gOzAMk;V{V?4+KnGCmhLZGS35pc(R=Pd`m{(Rb?m-r z$7b?&IG=BlfObXX3<~!h?8U5YEUm3=-vJ}`Dm+fZ!G6ovBXN#!E2I^JQqIzH%%o3~ z{aemrdFWF-v~Mvidg9vm4;Jc5ne3=>Bk;0ENI!q#@wAqY?N?M0_!Fh_N*75!?iLLU zoOg=A5xl+FUavjZYy|39%1y95#>}^pcoJON4qb>+ht#Nkm?YsT=BBc_%8(&z0wS6{X8a4-IeSN+yw@sh!*u|0ZU&Iz0pW z4|+^yYN=|m`wX21uZewpi5|S@WfgloQ&bF3cRYY$%Lw&z%51i&%w4c49jw1R}M&A<&QJnsjCXevqPNTwTC}u$?K|&184>1xA z+m;2Qip$Fl;=*irj6hesj+VHXb!};0M1MkTI^QOj(0ePKcDWWzX#Shvw)>V(ZC~c! zm6j+B*vzag)^~#`v-8u}A<{kT$C}dmiiz|4Zvt&Qj5)4)s-9nK=2elmt6M6!zkM?6 zgqvGmBM*eltU#53JUKi1)ABxiCf?{s6Pv~60SzK@%2r?@#0kl4cbDcv+Z;do;_av{ zu3f^@m0Mo`6`|fW;w*+53dNsRX?^<9SIJ0ataGwp40n89T`hmhD)yJ)&Ax=Eji!U| zuCFhT@0G-rz5Bx5$ygPs@WSji;l{|+`|drn>&RIF#HMjkLjXC5y?^)T031714P}rr zgO+l3s+l%~s&Hd!%GCjsI3z=$#wP=l5Cmk4g!pnKnj3H&^Dg(;$d1Yd8=F-o8k=tc zN#q8eWPP{?>2JCHHBL>b}Cvmf}m;c>-^WLRz0~BZyhK4T!H!gu>f*Mn zD_l>w#M)@^myC-TFFX>xgqoxv;J!H8D$_iR_Ji2d=ob&A4e-U?#YQ9Pk7VrxGp4Mv zyIcBvCy#&}`BjLu`untjJ|^A1vpz!>-#~XKfh(q{QTpxos+GStH`fDxCB2amQd4MmCkL^+`OV!uTOC2 zFK1_dKTyZjtf$Gt#^mJW0F!QlshkP{`=6UDzxM{85@r(Vx6T+p_}suS+SEgx%}`#G zdN05C={Y@n5$iqVR1Sx|3K8K>dlfH>D0M!MkTypkOlZrD0lOp$F&4_jp$1*M{j>E|s;Ys*4^3WScPb}HXFR_rYEo+MM2HZ> zeLT!OW~StcK)6x}1|w1J%5~K;*I)ZnprhsC~M&)}Ug8Zzf%E1Om>cYAY4? zlR8>&knti}?yrXr@=`;EiohI2h6xn&#}$v@D4lN3#mOjd38X0z)1~z4@Q}o*O*r{T zpg5T!5Q|y4HAIaQvGTJrk6YxmZu>%%8^=*k=sZp>LM!QL^LRpld9;EL}`bSGY zMO{dz{ho6Rmc))3j?@dF(Zd!h-pVj}9XXI)oqaOli)7@fe=6wW{y~|X|GD=wpejE( z(r_F*@AFgaNYkEUZDwPaxniohXu-qasX4J2UGLGX8u`N}Jq%Gl#EnG!3e~VQqc-tZ zJ_37!o=-95IqBpfjk$!DnoGqPVQrTIis=Rz-6IZ3IYi&tWWGMqNoF|t*ck}$fj`vP z%uydg&_qXl+Bdcm18;oK`VAe0xTxBNa}g&(Pv$ox)`PL&G#P`3Mv@^(Obuz2%s@fW z{cFg_FM~a`$5LP?3q>f304tws;Sh_) zE;-soG$j_&eLlOR0dsg4+SpoMW8vUNXD6tCBcHDhVSIxQSqhd6HqTMV!u=s~>meRw zFn9v9l;N+>C*XqK`Y2Y=Ary*olJmjZa=6}eQz7!<-$1}-ZO!}VqCjV{-AsEz-K>uQ zQyVIp0c;VEUh0e;-IjWLE70KY9Z2dvUf`S(#I&(=YWP$XUb%dU^#NW#Ov&eHfuM@v ziXQ4V@+q7lPz0vqgHHAmx(jlucNvi}+p?9p1HqQd?>du*pJH%91z~$2Wms=pNq-}W z5R{fh7Ad3na;_PY6N>B-b!4g)ZAskRY&#RD_h$vW-14p-KKC4qApaT49 zf+#CS#IN>4V<^F{oZR0zOKz?o`GBK6b(T^i5hZ%2~B(+QsSp#AcN40k=abIW9cos$1X$#>}B zVc$krecSAOpM}!`IGm|;ewX`rm7&Q@_15VfS7%q-KkgT=LZLR=%_N z8%2j-Ri3PEK(6^RzDE>r$*kJSHy1Ync)7!t572Z(=;O>(StKH9BDPqq$)rn1&g4b^ zG~Q4x(s-AVV=muFCX7ddngL*O<^1wOqiT4w7S>ST9j+Hn!!WUGt?>h@h%TFp2Nfvv zaBFhdEOgHD4;L!bwQcs9O0avGQ#f8r{;P0gPlV+Nw5%z=agK6PtSl^FeKr?ugwKGk z{0$n2WCWG!h4LbWS%1@X$DPC&j3x}l;qsE;*3*F5#-sdvJusv4s32J2Ij9H{~^pgF+)PNxDi5;{7oT5trwG#-PM9~4d3M#$@BE_){E=CxkS9~5MF z@Zu8?zt_=v_6nEG%2AQv^ab7g_~h%YqfDyDK+x&mUksJ%LJAiAa z$k^D@K%9d?$u>(2q6F7hx@q!@@9XW7%8H~)erdX5vjqD9<>=Givh#+*g~Jkt=LhT$ z9dp6%LYl_?4V8sLLXCcVx~9ZVmFsc=wo^XYfflL9C4-J0`+jSQD~Y^%o7 zpE+RsT(USAaG}ZiOtXFVUjNjH+oAP+sXU$bw?B5X$cyD!{61X1fA5I?cs8J5q`E5F zThOy4vuQ72%TbhpZ*E$s3*gzhT;AF%Xp{`clpFAO;6nJ&OzDn~YAWsoDBDmq$jOcS z7INuP$;@%BkaeBm9b%$2aOUqN6xLQGI(3!!_ zcIdid4>tt+u(!;1%rX3VTIrW^S0-mXs`N&ZGGVfw#OMN-UCqzs?^#^I+_#Z?oL`N! zaJ7?O@ob zq{pqvQR4vJc!PJ5K$*Ya8BLjo!1F$a(*W%?w0wCDtp)8wS2M5X)1{z@PW=KoLda|g z-k#B6)YC|`*lUP{6bB6C5JlXUxAb;*MacF?UERj|9K;k4Pv%;s8)gP}t6AQ+m`uUz zw>8R|&7h~ojl}kxAgjCkV7dHo1E|r*ko@jzHfT2Enk!Cdn1CyWWEGk7Q_@{v!d4#w z%WAoE6g5Qi(ev*6^634VC+lB+OFfrKpJCKMBcA>j89Rt>YKAL=BdR;-R@1$3bVa!9 zs-V_V{(}6UrkT>9uV5b7>_mgUIEd>6f2N^(>=G=g{(5%Dy>ak-^${CCSroCFC6lM9 zIDMc59wKqB4nO=k61Al+azmA@p$+uZh7^5q>x76{d5eAIhbk1;hwy2}>si1{#p2qm z3XlMpY{YNk*8Pg>@$O~Qal1IolN-|uAO->?ScuLT;80H0G?GL&p_7)}ex&p^Qa{Q{ zLa+c5&g&D3eL>~da6h;H*@>=FXjQL%&pg4`AicCscc>NGcn(6XBlzjPjQH5?|ADW^ zGD`DjYvR1@k5L%^1~OowhlnVtFsqr?Xibad{!*D+Nulgkmo+js!;Z*efI^irkIqo& zO2sjv2)V_Y0veD(@Wkq2T)*J2;*aVXH#0$HI&33(w)616y|J)TiEHx;>>}@r0cbUY z2}`Lo9nVf0UR`g9X{6cvF4TNaTg`x8S-GfM%olr^5jOR_Pw0Qk4QX%*L3q8`3Z~SR zCdr!hMDT9Txs)!uN$MDk&>wJquu(6ZdWkDJM2J`+-Qma?+~mlIz#1V2q?Pn`Ml*g% z``7d;YtFh*j6(U7@zAQg?UW$Z5}DUCQPOv__=Eg-2!n(0GiNaS=S7>MTeT~9UQ(|M zQ4B8;*~3p6F!R)pXSn_%5R4C)f@t!n>+%3C;`8mq>ybOQ-E%MpGmOR&({=+_R-BZ( zMV;j3j25+^ZM#MyvHAz{tQ0v4vTj{E`wYqr0IUv-mf^v7bSy$KJU5V$^EeWslCE zk?c(*1Xo6a?492bN!Tr30DYDJEGaRf~FjvuQBBjN{&CkTxug zH=%XpuxFA4vDSFuqqETbl*C2ilR~^7%SHoqXCwL5+$`g*AH5Dlr@x|i;zs49crVWv zd~&}z4gq*8EJ_5IW9ZHdjK%slhAl|7z0P6;xSy;$K|MB?PZjc20$7ma&KoU;!9~wQ z_2u8(9gS!HKhAXkOeWB4?Qg*|JcG%HL5eaBr^NmAR2T_08WD<2_sM>}kWT}%5 zm~(kpfHVAVq#e(klW#r0;b&P2VujzYN#p(JO-I5o88o;oxI7R44~pwo zBGG_82;=G8!is|C+pT}w50MRYN;0CaehI(un9&71mafVL_ zP`Zvg+jP(|*@D++p`P&v4sj2G8Mw=4x!cHy%v9E%1JTXIq7S6A7qg$mlPCWhe-a@3 z`47`TGBHf2UGWB=(Qc{G!MQPSW*!E2GB>seM$%=FKXB``ZsVZtd)l|gDxUHdHoBoJ zGodz2P7C*QQ=r0=!;b+=+9N#_escf0W%7E zT$dt}Ux(yU)LFV_gS$6d>&KZ|kD*58PE@*XlI!8#Y~4ug=1RHpH^;W)p(`@OF=>mf zcxS7~;<&*g3C2OXtD?<>4Wg${#{ZK~0fX2nA$*bHWZYUgm|4N+(#uLGVwCEh52{cz3%NfnLr|G6 zB8EDL&5JgV94kJwesdRx{lT2Z$^C0Dd|h0SFoy2QrS~Woebv@WT}SsHuLCVg#ABfA zCX`2by{2iNr!}>Iv!xRn== zTvwT%T@keD@7{Ree_dmN0g|$ztb_8*A4ow$4b!}Ms$|VHj=)(ZX>vb%NGn>M#V!N{ zfChRiL)F14GYj6_K$A3&`gze@dJ{Q6JaK^`J0wOhOb5L&PT!}Gy{&kwgAY4|CeK(+J6n$Ewe91n{o$q?)&4uI_pdl=v_?VW8* z3oqn0`tq5w1`3k%;6d=x5TA+WZU4aF0a#Bj(=1(b%R!ktcl(ms0yIwG7DaPYVR052 zt3VXZ8^o(b$vf)=4V~-57jUxDqGD00oUlZPGOIrrX%_T^MJXBLQ4F=Vt?Y6kd{;>x9 z^`VswBu~My9_fK0dB}ez4+Kk0c#zKWc8sAxyeATrT89PyoH-H;_)0X?QspjQlSu>8qrbLlR%o(hYYa72h4~(@ei3Q0CRfZmnVF(eEr=e(QA(9 z7V_dTqK_M?TZeMC*rl19dl7b^{1yVtdND|rydT7>7$hojwEyo2lb=iJ>Y{}B7yz4(9D-U#K zTK2VugFn2XH)^Q>b!yFw@>yGY4yp6G)%lp8pZ~_ZL?OcksQ4pC(_keqF3BLw&_A=K zM{8B0N5eu30Q4D3bfpQ0YGC(`SwNY#Xw$XNXjUSzh6mGmpOTplc{<}+Sm8y_Q`NU+yMUx-{3vvqu2yv1C@v({smzM37@BM)2KN7|N*l=-b-Xe%#+9~ZO z8?`AMBStzp5n?b;Mn1|9HO|xdo{J6iZ|O4Fqs=b?|E42bk`V6a zfZ?@uxqr>GK0;5ZM0?}dYk(gz^s-+Z)HRW1J*(o!3%$Lm9M%_H!UtJ&Q8LMYWH^dk z$2@$mXpwudSB55A%VD=bjdr(M-k{-NxbDwJJLE5~=e}JgV>l$A7T6WZi9{cUKLec1 ze?)>UX$VHW@7}H_%Xt+P9LN;w0IhP;B3s=Fl{$U^-0dStN$^)+NGdaW-RpW{i&6`U z0@&z|f4`~wQeWbndS;%L#%=+a3(ma6r*i$~9Kz?t_~vw_u?6w7PZSo9@MY^2mk(XY$%_0UZuO@23V5#2!|*kS#}nRNSp8WDEL0=Aj3$u0MkJU&w3 zh-L{(Po9#?eKgN9ILjxYEANSHPn|d|CrG4!1Pk9U;9j##pg()WoBrXBs7=lOu1>l3NTgoCNjJ~<`58N67XK~ zS_HZ#I`<_(!h?Q?Ku9MGdERx8P@JTZ5H>)&Ak(D)^)Uc!lh^==m=9qe0aC5WQ;+mB4Y5`==M+1 z^?+AL&Uu1I!TRJb&?GfrrpSK1ifMG0XFy?5V{Ib-l!3pfU;E=fd(bm=%E} z>50a_LBcyQHz5;xOvwR%Wt>>xkI9taHPUw(6p8|=Xn=0ZhQn0xwX!#de3p~D3gE6H zRoS$ieyB1%lPTKgi-Z`mGIZLkK_<{(-*z9a#T+!EWvK(0!r}B0#njcqfHA8mcE-?U zCX50s3z!NOa`Df(&wU#}@vMimZxE>s`z0feQU7cQ`m7*+iLrEF1$d%C3?wX(mT98N zV1Q)2FGV0*09XKh7W`ywV`Dl#E@aw~`b>emSjVwU)ZFr~51oq;v>Pw3B(;c~RDxT^+9PIg zbm?~8X%rT8_2-v0p>!Nj6_N;fV0{&;mNeG~RmebW{`7=47C`|TNne{)Rf#Edc?7=} zc2;-XrTS<;}ZE~%IKi+QhnxJOK?aQyC)C-H0 zEDVMq4I2#TCuu(@l3@9xlpC0iX-x^}Q ztt55b&z%Tq-(H~diVBK4Z@ga~E)N6|Kh0jw(Gf?XK=OV}#ZA}cM+Z6b|88}zK+uD1 z+kbmL#|>9;s*$r0?Ir;=C%bJ1C|Dg@G7e0S7Zj zDN}|MzM=&n7gs3QcDec$pw;pfx{FEPTMZ~BzEb#25^vFX#cWxt$`;+Q zh39a2+#CRQwf~m@Ma0mQsZiJVLSafjmWo8?ga>V$9LVto=lT~bk+fE$LD7kAlgbbp z9)1EsNF6HVN?do~5F4JlMo&SXNH>xQ!8Z|yf1QGlAR-Q3uS?0(4Y_sjK;-;AK<*Ev zn*MjKuEU1RWPJm0!Ed+Kz6ymNIFc7^dOnXPiLzM#_b?egI1IswL)lq5%kE>g_njt} z9mNfN3YG7jnJ-NU9XeUUCr!WZDWALg;dYW0hmS88ea&LS0X?H=(Ec|RegnuZJm^m% zZmGgAeZoJyX+aVW^oT&jL|~bwM1@9W0@`BL&40q95`Y`!cM6D2aoGXeErCqrnk7@` zFHGX--LIy`M=ADsIh3|*NLhNcu5J4lG1-&?p0OuXn?XLUh{d}tV-z>gR}Z)6-rLD% zQ0Sw)*9o2{TJf_=aCVq?9Qesh`;yubM;24BecOtlg*lQ#4dY(W>)JH0TXdyJ+`7D?7?n;d&oPb z{^u|EfUir;dtnxW%2%mH#nIf!o{%`X&3AlGB07W41!>P9DeSGhP!fz#JH7|~;W*3~ zKLz+2wC-0*--nkxY)^Hkd#4;Lp(S67{T~OTjLX`~$A=S)2d5n`tf)O*V|>IN(}_Jr zfzoPGRZQjI`0dNK`DuykqW#V*OI~e-tlQl95r6HwlZPziP=sgbz*JTRw6(gRo?9!J zO>J3ePhMVbru=H-G(QdME9I-+NbB|E5VNp#m9@oPpKdL3_g?qUJezH=V*6p=%%GCa zYDHUQ_tzYWmPRAtTHyl2xvJ4V5zC$qL`R6>fPG&fqxaxq&RV369|L4n_|$Ikebmmn z91vM^oZlL^b3i zE3ui|oHl25rl-x<9-MajEjR7Me_DbhwN&uVHygT&il{3);sz6gA;^2|W$1Mw5`t|} zmaGyXa0oYQriMp57@M-)x(eDTzr%;>4Ucnz=UJOO9S^MaV$b&&ZC!W=XN55RH!xpn zN9&vmq%N#+5}kU{0p;p1t1ERE8d+8N&a~>N>CQ>)yH-IKgRw#s;cdKAAsRV973ce zrBk}3q>=8HZltBV8$`NMQcAkJkw%biX^@a^d>8kA?!DjNc;7$XF%Dxe;OxEEntQFe zW<2wm;X;$P#91RN<0n<(wa)0>dtAd6!7z1^22De$RuYmcs-p-Nr!zd(UYpU0=-pK= z%1Ayzbs`AX9~JeOOvxcE+$lT7#l^DwEoVGGFzl;X<#{W`A*rKV`3EihCPzD;Q7}^j zV8|J&OUSP{$0g<7vJyLK!zy%*{%$XArVNjPYEDN(eMEB-CS`k_zRUH`k1OU;`S+Gv z2o%6yPre!yV_zz=(|C&w%oMa^RMLlDOwJzo0cl2vrvAD=L#>Up6_Sv_J)V-MJWLd; zYPMMC&6_U8-5N?Zh(7}O;}3p1r3zb}M|qtiwdm!yDZOLWh4woYpj;E#X+T%ep9vXD znKlQFxVd!KJ7~&}Af~b( zP04)s!WU(TR`L;}31XJ$=&=Tt_AZhy*_O~!!*qUypm&g&wbbpph3O1PBSOSJOG{Kw zm`y@c=Ii>&a@+K6CD9?&S6g|Mg&sxJ$B*|UVmp2+`QZYcrSla{@A8r|y&)9+nAXr) zN<3OZ=*z;fN5tLw`NvYgw3R zm6aexnI8RJ^;2K-H_et9QK{g`RnoT+B5-mdN%;9?9JQQAzx(*guq#L^o|>v9dqZt@ zYN{3hy&a*z4t%VwI(eIr=7{b;lK$#5k7FOv)yd}2>rX>LxLZ@hazg=wi;eN0E+fGv zxfj89_BS-`PLkD}SR9vylK7Sj?%h_eg{UwVFu>WXqskz}>cE*C_}Tf_lQz=I9B4&! z078}K=H^gTd}U>2%x?6?`s(T*Ev)M;)i?JKvLpfF?RoM3Fbe&ku3AP$Mk1;Ko0X|J zM3YW2L}@85JJ?j%bf-tE{s0iiR1hcRybS#9a#BIJSncu0QjcVO$vQ!=)o$vh``3pk z^nsJE6Q10yhTF4zKv%i^x{6GUj8H3=UP6w5_nqe4@2W24`XR?Jo5KTJ@!D1^M`yY& zFLJ08*v-{P2y}VqL5S?rmy}hn0-t~k0_NyJCBt8~&q~H35=3^_%C)Fghqs-GhfL=bG$8s9$=+BMQ(7Q}JB;k$$ zD5d4W4l5cXvB$531xtkKos84cZ3eiCuf$KIDcJ~Mv7@qUTo2~+H@iWX_#mPW?uvza zzQFi=Tj2D2wG$8+)lGsD?fQ2=zI^&56U{fOK9S!)p?6!jt;uq{$H-K?8o0-7(=y{YbOSy+nzXw7xTASzmlHlAGmUHcQ?W^w@ zp&YHHcEZT_;J5ZaqwS_i=}b$$QXAM1Bptq0gg-ksNHS$bZGCICqmZJn#3-aE?4lUl zz7PC8md3_LI#pyROTfDVv@WtGO9m!cA70rmTF?a&?A@IIN@mbf*3!cFirj!g-idW= z1!n$!F|PxJjTzmdniBr#B3EdY&=&Xq8nN1I$-=?S)}t&GU(UI?xPh9I12Se0rnN6n z_?)JipT9%W{Efe%5iBe(wO#K}-v|O?*8DEH*Ay>{v+UCO(xt}mcvMu{zY#>+a^D@E zua}c(vBt@m2AiscD9mZqvloBKuOQ7ruV{Cn6>KnJd1jIP{Lr;IpknVIPh6K(7^ zQ$maHUW5`p<0DMo5pb(8rxX%WG4^so=j?)SFOM`PnSDachEx3aFIz8{1N($XqduKx z@HpO-3L`@ybfO&(ftj;RbNY|7i)@)p%6^=kU${26%g;%SABu!u%IeDM*h`Ze5h>#- zPM()NZE(Inh~Bo#aKnnzPGCVoskt2flz6Ltgp~))c^ea98mo<@S(M#m`A(8t4IH zF?fZiTRV_nl)^82w*qbIKE>3GY|^37lUc}6Y54&)dMv+ow`XPBvPP{dBPA>)*GZ)! zlsNV?P=#+7(cD)$88k)|yG;4sP;yZ^k?MlkBX+&ogUDWDcF0$4x7miH-*PD9au}f) zLYw;wtu+?~m6l-jT*r+i%Xv&6wLQ-`-slkJJThT6YKkL|!-bG#sfN-48J5oqQAm;9 z>{{p51@BX|oSEamZyIW<_x9bswNCI2TwH=k`h4XF9_I?U{1My)$Ih$%vK{Py#&gzS(EmVgh@Xo}PSEWC#HV)`52Q8$y;2xuAs& zIVO#af1wXxs?vSmi&H2h`ay;LXQYV7ba2be6dECJ#SN?4m+T-iyR}N8=|mawmlz#t z3)*(A`8?DNaIKcZ$JF-uvtAV=(^zqJjr^ve1CP!Uq-iU#Oa?D@jT1EcDFeXD6Q3wrGQV4iaFi-^s4Uj zm0v}Zo-4Vz{dL}E!)ehA&2^VXwO?UNiXW)uLTeLqBP3fa7Rjqa^WpQOW{kd=;>OgF z8jz!R&l`HTw%wl3_pR?NGZWQAi}9908#sNAl7m;7v$frCuckcPH=1ip)LM}P_(-j2 z@URBLv0%}T_lU-dp2IS){Q$uc5&p94| zwvrO7?}OJ1-3CFi)d^*dLZfW^_Cx_v_ATh*LB;F@hiC={_k%z&^HI>j@7t zL4tBvO@ECU52OtV;6RIS9CTs#xpNCMWJ`QwQFBlBfIewg;~M-@VT?khebHi~o;KO& z4CS@xE2qpYe?4<)5p)<>yw~)a&&eWJKiNwQYpw2;l%pZa|1B3Cl7Q1#`Re6vZ`qd~ ze2-@gOIKWcI+Rk`CnXR{FF4#U`7EQ^Byx7Q*YLW{u zx;v@-RbsC@>lt$l#HLLwi&JXK3$(+1F?8x(>pwI}2Bnw|dw}8(I12QG`F9rOuXcyM z%`dyk-}lcWy~s>8E<@4cqh^bXehTY?JKD1tc~YVEqWv3C78(qO|O*{N#5?An?P+1$ZL$a~gcxz5;okX}>f`xUS+0qyI+AV~k| z>+C;yDfiZi8xo;_y})K@ky@*4JsPaXT$N=_PTgCDs!9Hsd2RLuR#<; z%8cLg^It1Un{8%_KGH@xF8$r~21P*p+f#x+<3%6SnjsLxbEP~PV~O>P_=VW6BH6PX z<49S@O7-i{+R&a>7p9IWncuADz9t9Uo<3~luJeX$Y#DeSGBV)Wxg662&+^vh8T0RrWP#AM1 zX?FO?X{h7)0gdHLMdd(NOga6M(sDX}Arv(>3*%I9Tffaq)bQKFy3y3A!+_CSp3%{D|HXK&Q}8;#anb z-|D9ETUSYESUtsPBi}gS#fl?bi~r24hw-leofmgJ=^KojsV0_iDy$@?LM`fxeAe0pQ1WvRo+i-8b$GphmLZ!=g=-S!x8$h{^47OYqseD#ABUct zr>{r|G7!|yiojr{HFy|s25(2F4(V4YFSfCJs;M23_$pv^z`gcx>U@rvgsPg{5LwbB z4qom(IZ!P4N&;&abR_!KzMN?nuk;(ESmrt2@7!hk5LRJ_SsvIQMHQ#+@kq=W&iB7d zkJhtq%`FaQQI8y*7<`IlxB79Iq~+B6jLzgh^LOm$_vC&U)&P9-A{*q!Fm`_1&WV>#|J# z?6ww+!1c~q&FpBo74BT^om|rD$uQEH`^y`zCP7uhR`M!0Cne;ES7PZUoap8{`SJ?%Jf z`iLkvdzh!urYNU{*p_dS2Mz=4==0fB`Ajsz%Ua`sTsj6rPWe zlWjX&0V^YNw<03spcvv4;#7lOeX0^)WM;_tDVQE$E!RfnE;oqo9!(T>>t9R8M$i^O`#w*94ea#RLX zPcGq^9x%|!yJ$%P^0ZNu)+SV*R@avF%Co8=FbjK*?km0PDtv3Z=-@TqV!t6XF`D2Ztxg%6nX*u1#(7->mINLbXg zmf!hhOVksdl1cc&6+Ctl$yQ`+kx#!5-(xZvonMdMt=Fo@Syb~LkVW6#y!lmLpCU}I zpx5H|URc;)OL-!|58fXH9lC#Yp0lRGGr%U~xLea=oy^HhN+W4C<{NK~`DjFFaf{4v zog~2Hz=}2|@Ik}ixR)D4zIz*fvvWU+Vju~JWi(;rwB>DLHEDf$(c)h7llDbH7U9^1 z`3;|QPmSSbx7zEM7j5r&Z?zijsC>!Np{YC#X91~eNs1AJ1#ji=K}&=RGa@~%j5xTI zA}ay>(ao_z@0X;mNSI#%*GsHkS%rYewpgu;JXhu_o%kkR z?qbtclfhwJxH)8Iw4w^dD&~z9q1A}GWc5aS^B!N)u)Uy7S*9>mU>Qlfm|B+zibndz zHaCPQ?ON!D{6TyoUuS7F#gOv@ER_zdFe61E)ftAvYQOyZ*yrz1w2{+s>#QAJmz9`;TWrO$bM7|rl$Vj^F_mY@A=FFLgN>Vp+RoY}=HCT*; zWaLN&57u6?Il}x${}sT8MixpkB`uuyHMoS6lM^)6`_rkuwBwo|v|mcaUJP8H!GKS9 z!@?j8AwQfa3OCQODQ=d(WGZg!>2_`Zpqg6$tYs?hdGO@S*VLJ{ltI>w$@;0IR$62( zltPZQ30h0b;O$iiZ)lt%q_fA5Nyun1t8yJtf#orF79`g3F#prV$aVp#IkT`lWR-vhSjg^L8dB zkt5u{6)7|?J`iTxXOmV^qudL+`+1!D`^Xk?0nC zcM3PrN9E3{8naq+Pa|T^BdCU<5W3o*FDoGgHJ}+a(3i{(6300?FVdzkr?6KVd=vSW z)0fUlo&XaL8xpR|HyQ5wN|k1rudR2*b3js{$O22*b1B<3Zazqr37EBh>Una(8>^n8 zTUn$xf2v)nS6$RL@7pf7^w`FbQn$k;MN-IMMf_NDq3U9cnutX6mbFu8b@x^a5^qaO zOY5X<;Bjy;V8zMcL{cDzl9EvWn4l;VN)!2h`}p)fo~Oai8W#fQD~_YDyO(e8M&q~5 z4r*tvK!iwd<)*C`7q_4=mVKEqn5Nkarye<-C&fV+u(8iT8oUc%?wcBSc25()q9{rbHV6(pRD#dV~5;nHTI!AeD3nt%P= zc)LxN($$n6VnfzJZa6K3P@`re#ZunDAUWDV%~LLhtqC_MB=kBTH6}3_+7saix5~yT zQD;**n9y2<;opO4FkeW{E)fp@B_Ohc6%y8#80RyEKpsO1*`q_UWWZ5_ix3g|Mv4QU zi-`e?f&Tw~!uVj3I^NrnB`Uy)3H`sHELu29nCJh0UsW6=M5usMj454I^d*gtQ!e#u zge1)9cfD4=%hA=B<b(Gb-YMbLLw3+TwQ4SX3} zdNQ0YbWjnmn{hg$vDHtm;b-ZBy0pZ0yZTC+lReyh>E3Lw%}sq+CVg$K_R@^x?v2#@ z#C0vs=)7q2PSs*%bVbMHfZ_@3L?Bw!FeYNpcOV~zZ`)te0=M^zpviSdZ<#+)+aGR^ z9~O@)uRN52MO=L7GtBImr%Ls+>%bBV)b^t+K!_iM!>HkGigxiL%{xpY-Sw#K$cO)J z4nsb;J$yAA5@rkx5oOo;Z#{}8=bfaVAuWS4tBNY?HRC@9(+eD4#=zhPwvCfY8w_;7 z=P+;(bUbOgyg?CEwJ?g%5P1v|p(Pq5GwFJfH2#qG z3rRxYF8x4w>QQ@pGt=JYxIOA}Fb^pB=k$pf0IOBPa#DfD0{MBH?hTWt=Uny|QC7Ql zr{_lj7%HKc1fLFhzZALDF5Pmq-Ze}W-uee()mxVol@{?RSC@1ny(>0D{S~m|c|KUX zUAh40qS4~=+^7F@oc)7CPmJG+>)xg9QI(}YaNZ@+-XeQQzMM)dy0@pG@3SWTie`_Z zBW~q^^NtHL#=BUdxN=Uha-?xoivixyrs2Qrhkp(WU*sE-|?g1W2WB2*FVYkBaQW7?HJuMtwr%bRKcATk&{LqagsrS^( zlwj)~oqCQ@WICq~za3Fw={w)NzK6c*AT8g0bo9$NkO-CY&)v9jgI!b zPND}LQkIP9T*wIf%_}W|j#74ywGT8dhbxUfGfkzQ^>r_J5y&iEXE7C~_(+~!B2!QamV zDhggmMl=C*L}VN_S3oo@sjBK@F2NQFm1sr?fFX|FgDQVCeiZ=_WD!jAOY zX`EIWXPy^x=GqycsDw$oj-lWRnG!N3E#zJBR$K2JG)n$kDHbvT?fv7&GM*ra%y#dw zgX9KC706|9k28_DZ(v-+NGhnQVPVkH!=7j6tf?A34({&{x<0saeSzyb$r|*Hs5`3% zP^DyjAD(gMrX6xVxd%XtoXTmX-iAcq0#!uZwCa z%~7Ho@A;(gF5{?V^gx+e!le2Xy8jx6;tJghnS9EBo=OUNPeTgJeaM&~bM5tHl-0N| z!)Zc_X}VA;SDt4RyIBm0LDWLx4H|}@-%a$3C;zGbLi)#veqU^ZrP*5%O7s%lks$Hr zu-8Z?R~&(xbS-W?S9AU6+y3;F;2Fms+CWbW^Ek5q`3L~+*r&}m`eQ){iYje9k!l+C zK7$i%|2?v-HkH>lrD+j<(6Ik~C`e&vvnBJo@$FsqXWs*V2g7+)m9m}!JdZ+XS3MD3 zaiu6OM4eOo-(#x-NMp7jmgN;fMtRFg1P;OOE9<-c8AQgXCb+JSoxv4ZvJ@Uq@HW5Y zcNa<~*%`%0{1wV# zu$|%zAT&*&2i>uJrQl%`^Hs&k|2=F&UI=1;_?4byVx=|7=JU4?gc|m#fLiyEP~>Q- zxqdYdTM2GW0X*=*CJ*z!*DZt(nVFngm3bo!8gW-o*cH3*xpG?t*B096A)>NKp7 z-_yD6Sn}MRbTQwk%q&ey|`C$sCnPtA%z`?WdQk5^}LPfj@_UG2A$%`sm$8C zvVH2zSFQE4R)F+n%sG>ZB_JSR0vhNgVX;B7QfI!J7)IDLuurdQf~Cp-(IPPr2-HY7 z`T>0;CODQD(>#^X6&A1c2*BQ><(Vke2A2B}GncGB10bo4$#4oQ==Wi59Po2zR#hQ| zrSookI5oZ@lv{`5It&Jg;|%ZLRTn0-3nhgbXGB(n^s6`NM0|4i3U5HkYLPZwzr(2@ z9Uz(K0@$p^0MDUMc!W3LuT8UQWM)sGWu|%w*?iOoFGdH&|CxN2B@*cqimzl~0sgAMH8@io|<4eJx~#0{3%-`kKqRL3%6p!(fsEYKz``c zG5#2famrU$$Q`njr1xw^G1v*>%rt|uG@snM;sC!o!vg2?*C>LWAOpf9x&vtHsD-P! zeHb@zVZnCXVg1i|_OHSaLfA^i7TwWA>6l`;KWP!TKl?!)Y;@HDJ0qtU!-}6o6m?ICjI8s=&H0X*M1D(`?76Jm;@V4vIG6*d`AEB zXpj*SPMXygA7-XZ_OFstegi{gPi`$~Ob=7A|E|BD_d#COl*Q*FGL-BLWE)4e=gdFX z-q+v3I7A9%faTuP=&Z{1&vN(2#xl6x16j2-{mvBR4q#~*Ba=N`Nvn395Zd41@N8!| zg4Z7Mky+*nb_3Uc7W4m{!v8zy5FxBMe&e4sN?~*`p2+{vI(wQuC#@0kU4W=@{ZWmK z0$S<+sS%+CBqAOM6Tmi|;dGV0a`&aOk~L-(6lH^=;>zpO?e^=fbboD(K@a1zobRq+Nea=CT8Yaz()zbQ9hfhVWl)Z3G0_TiB=^H#RZMT^^I-_-J zYir@fhF?aMI}yLK)hTlI8C9?Ie}ZTPy&rs8jt(QbLnaym%uXKdRq`8V?Yd!K*P1kk zKM(hG+9jRXSs&ogz;;f;A#QoPP;a-=$5unoI(Q4P5S~d4S`+kC&$G(P4ii-54eN%k z`~P`?un;*pIe?6Kr9bQEd$uX}YH6}emk*_T8!4NE*Q-ajO}!w~Qmp-q4u3e)fI1Cfd9JSb-{0MW%Q3vTq<$`<^>ked<`ZW(bx0_84Owz|qikR;!BR@8pIjf(A zdQ{UAk=OC}=~+qc*W~;yn%IU8^D|9NmW>o_c8z|!4-@>P5&-*noKI;SBSAYrHI75X zB9WTZPm6;CPAp| zZmugu@Bouo$Tb&9H4+9j_~xc?@Fb6b&dl#hT;=7sZWL7tZB8gF7ru+6)jc9P*}*zxYg&SyR#xnAF|7 zb_nk4J(^|aX7#5{ zF-R2=7AAzE@tNm5kZ=-k;*JHS&r`4&jNe7B;ZtGZQU#_^2FQ92G1i;BM6J91!rz&* z3Pz?cod<0C>}r;gZ61+BmS>ZQ z6C;%f+>DS85WE@xsKc=0kLSDOJaa8DIs)Z4n766_evhqjFM9mQ-YZ+~txh|uO)S%p zLp`a)2(ho*GV}%|U}KR!q&G{iBI`#&Y*8qVPPArLytkkz!!L-*${_9og&rKb6E(eY#R|3n}J7a8`Z|b1mK(L z$iqygnt#Iojike-wA8n$=?~{j_t)8^_!N%iBqC+~4VPnyUhccBbn(Oans@Oe`vE9S z2IUM~dZvxcf%XVEu-90fcu2UBrari=6?@2?oaNpocZ+;3`(g033(^Y#=5JDfm-RvK z#+<-d5M%pIfh8~FcKmoJ9$&66KEIL%ON7%YyJBaS^bkXq zksWl_ln+BZD9InP6fF~LU?uqtI_8S^HTF*6CN7_5+eqw$;gI?_1nZuJ^q%qZGU;kS z>^X8Umo~e*rJbqOZML8k_&;DyN3zxplsj(|5_x2+j0Ce z@TNfJ;{%>Qix>@FuF7N`lPOWkvnDoDadwQ4oNP@@DBU}}9#;{l!~H98208p-EfMu% zX2(V-W;BJvys7TL#&rO0%zoYGb8ZnBGonPLP;0`?n{e!{Hv+HbgT)nuzZPTp?IN9_ zo2PxO!uasAb^L4AnBX6;7wWuZ$x%nU#$WwRpJ)%C$;`~k2F{6LOo*61o_G=~=mbBa zl?g@0GWn9-i>!lLeJ`UU(Y`c`Q+6!Um&@>?@bShmTws`Iyho_U0d(zSjT_qMu^+vH zn19GHxAr`bei1~d#&X)=&ofVsE9FfCgZ1;m_s^y8lTS_9f1B9;by%y_dYa@rNi2)v zg{N`}Po&7clY8;H<1n)FMUlef;*T~ECBM&Ne#%CDTspy0*!LphwGN|jAK9#i#DcTY z{1zcQJKOdhio>&nA4~&F5DGHiePa?)qk7{QruI7r+V`@oowKkBY9n>=XeA6%f-$N= zFtUetke`umNXqg%6?68B1)qI*sVl{01s|l;B4K1#*58YriA$N5ikJ(j1vC)uJd^!X zkrpcfYeTb?7+!dL)c)DPB2b1kng-<;O+xV+pP;#?-fqrweJ|D|GC0~IGLGM|xm7q6 z`(txfh6>l%nJ_Dq%d06*=8{74#uCukC4Rf63$03Nh`cbHLMBV)-CP=3cGyQ6a9MkT zfJ*C8H0kP<*OGa%$H$@GQ!t~TW&4+rtJM&1weNdlp51KDbk%>Rr|+9;UMcG~n|T%0S|crYw@ zDctYLQ5^AP5J9V@`0=ksfxB?3$pQ<pe_3NpP@e82UA$vtw$QBJ_z|23TYr4P`gzo*UCcD-M7XQ-MbqaMF*T2 zQD&onydAkiLVF_%-LtdE^elXlQxOw^kM90fWsP4ncTYcwjOf{0w#dy^2&~`_22_{S zh0XS{$o1#wyX38+B;+zB&+wukhjrnO;9YFbt~;2Qw5@B7fPfN1d4_@>+BA2izNwxP zi0MDZ+ey#iW;bVLO;F0{jA%aN$@L$twcHQdH z$ZT*TQLhr&oQ3sm2#kAt3Y6?2ELd=6hTnKie0-FY+Lh$iLKq82;GUt`JP)}iD(TT> zgs?nAWW!WQv@1lnSdYRT<#)%kM-ai?WWzwn@R9rZ1era^jLUyB3)GDIBoHkLZ8@(l zLwWj60>xWk?+qNDzww(R`9ev-<+Al8X1pP6ikal$cPPpb2YJ#>^s~^zbp*IJ-)LLk zr)tqk*s}lg&m1T_pm!58%;J%eZ#XK_k~;&D{G(W|ldh4fda2{9HQA~+29YA*>}4v^ z*~WY9?}AeC9i15cZK#N?%Aw&mgH*G7>H^mKYm~A*cBta4t#vzfgz7>}wc2&ocsTVc zoD25Y;ux@!KLTR2v=ki%q*={T4)%KW=y7|yZhiej>WC5@41QP(4udA5aY{kQ-TlO* zia-!kT!H4F#zA8jc-r4^?8@6k;@UV3;vbE4eLAWSk~eyXlk!DhMOK55*qk33M+gaq z*@jP##VcHjGd(Q7)bDGY{$GQgZ0=L(I5b%apU)nSDX<&qbw8taHH5noOFfHJikCBN zazbz2;j6*chi<<^Ti*Y=FhGlkD6-{T)P~jJJV2FP@nV5rt0G8fsG_5_wxZqWM-_-? z#hD?C)eLp)JNL1i{H>)6xggLFg|Dpxa2RRP5V)c(|c>PW6BID_Z0WBO+XiLnK~q%lZ) z!?z3bQX#Jy?$=d*VRC#c_r13o9e?UJPc^|ngOvW&_Ew34(lHVPqe165EsPu?1vu5=7fXGD#w8^Ijc=-v)AkHkUrHHVJy6^4a; zmk$tc47E*_Ng_%wxHnBHyDp6ogz-YX(=C6?`0Rs){-C<()(hZbb__Z{ZL8hq?Q>0b zcdDl!peDU4ImBB1^0&tEheEtK(dm#n%;D#7tn}2&*a%0tD&4AB22sOe7&EheDN``r zzG#+ltPob^3TH*1V9NB`S)syc=J-&#P!~~|<9!u_Ko=*d&U3vHe~;Z|OMRIc6#iz= z#37`4CrK_$+z|oUb>YJ#4FwBTD*r?XU$1O(Avk zK`N`KI9ow=PLmc~C&adqcp^Muhj9pD|e%w&;sO`XNN zoKmS8FsOugjneeGSWcY2#gdCt&^4&=jPHONt#YX?l}LUCY$DpII!M|-TmWvRlwl1I zLhz<+198DP9b63#FF$rcSelg$d*uvdo0q~%ac3TK!umLHULi^cJ8$_&HUiJB2qtQH zZZ8O(xD6ZVAHmV@1ncGnh2gdZnoHtWy9wsJN3DOUm}4Ah^Y9dqI1RYF5HTp(Dn@sINE{Gd7_f zq_xIlVQ=FsWqX(NVZHp1XjFZ+UEP$_l{SoT(0&1_YcyQ3D$HoCJY4|U{k+R>{>$T) zYaT+_z7bhDR^g74(CX^B)7RSlEq6;VKe~Pcl^OS!6xFsF@kH{BM&@7a4X!EEK)6%v zl$F7rUCe7#%gLm=A2;};hui$8Ba2xvMsk!T%)g8@u>JSpZGLH#uFZ;^avtZ4CA2<4LAi?KJ;*fwgWkQjIE8Yw*hUpK`&uN#vh{+#yh z66CEqH=;ZtHS#w{Uh$WrDvQ9~3EuTb8XGg+AX=p2i}n{=i9AT;+e2Dn`>cyu8Pnf{ z|7zM{H(}nNWzA)PtFvb_40EC8hve0}C8M{H3adj0Zm(u&)aU9E$;4b*o1wtu=h_r0 zWE}UQrLJ$M3hUFz9If+2S<=hampapXZ8%1n*}~^f)U8TkcpgJg2*IgIjD|V4ETpTs zqh;x~QC)80OEd&ohkyI#aL}wH@%mlQoQsB|w%Zg{(8o05d)-HvdxomU>%KYKSQxKK z*&vs#9%u(#$4M%Dt3NjSDrg=yP&kE@fg7jGZm(&u@!BO46=lFWt5>+#HeJE`(^2F0 z3xV>vi^Q5qbSItzr1wi&iXUIRo_ERVbDHbQYA4goD%`18;#|j%kT=OIG79!CFR#zxp)G{+*KlXD%qJLV;gh-uM287keu7U#3gk>>=sD_%_SZN%Xq75_s!W}5V`I4x zhSl*H@~)_jT&2XsBoI=W=pZUSPIdTGjf>)pvIcF{Nenr3@eqmO!L715H@ zAXdKzG6dBWPMsm&Ta?X*Qw*6Y>V`xS%v_%8e0^t996$oU%4A-3jA~s#uZ%)(q0=Y4O(~-!?>192A%HaSdlXvJ?3)SWg zn!2V_{mwIYvBM^+zuev!0SGN+ij4CK<&$b)x%%5hc-!9HtPPA^zPF_|VMp$;R>fE@ z>FOQaF`(U1ZH~&p@YeF(4n@_e(tU~Bzhf3tbKv<+ge=-t#657dFWKjs&1h;EqAOVI zM|i;c#Lb*#IEDQtSL%3{V+ivnMWk)2QoQ!2`2&t1%+v=&Na8uOBhq@PLQJzXyupi0 z^=Gfvjz`S=6@{Lhx)^9MMy6N*tNhRjRS&uSdc@AHv8ETD01IOQ|MLLP!cz3BAyDf@ zBcIVPf=2xuf6&-^U9ZOnfB1kQzq@`z#R={EcrvdtY3^-wP=MiL|b=e|q0KRl`&=B$g$Y z4FP6{bSXq(94>;Ji^4TeYXJ}cXVOvH!rN3f`$&wUr`54jb8LNZb`NY6!`Kxcwu|5` z^?H=NWQp>59~`o+r^t?h5gl@hUC3uRyG5@pd1_9Gh8&JhaSF}**yF!8^4w30;zPJP zHwN-E_dB`@cDBpFY}HrU*rru<_4m#0jqg^*ybm4S#idkMfvb$)aC)YV7-lQ^LdRsa zF6F%GOe<$~YuJsrBD}0Q(J2&PWWnVW9;~WwFuIwS*e|8LNLz&6-miAck}crPDX9H7-=gWzLi#WxL!)%kbwr z>S>EPHb%2(7R+^t%gS(jK{-a&-|zYv|5{8iut9oa+&zrZ&?eB$+4`RTfjh<8RJG4i z?W{1&P$Jaq1AcI2hhpbbB)sZM0~*3@!09m{rj?+^go;(W^A7(lDS1^hn?A>39H}q< zOCx?sN$Y~~ksRsc_K+w>BtJE54`Y=|%_(Rg57|0ClP(hm8f1Z-BI0=|Q=Qm%K?>b! z@+k73frR5v5aIhqU}dPky)`hO?PvONp=73(C3TCI7M%Bm=p=}66a9wfbaDj(on(7q zL>_vPMRhgg*zs)fA{qN>t5rJWgq9Tf2PK|7HahDkC6WMJ@Nm3|Q4#oL_;xnbkwh0tJ1^hl-{ ze7OV6N%=iOmOo`w>DqP((G+<}s3Bj6u}RC>v|~Gr=81C8od1iPfP>k;3M;G#W|%)TRVsm4GM3$B3f2XMa&eK#mTMBQApWzogL-rjVCtnCB&Z+n;Z$ zFwx8vUebNiX0G%0^dCNZd#+S3jdHcL8!2xA7bT2Iecyr;`EULT3L!$zgzhHO_q>Jai_Xx8+i_bydCE;An-Vjf`MJ6iga`qP z0}sBV*%PWiEa1GX4+9Rvo!ful5YkU@F+TA*5AX(F@CL}CJ7JjI@7XHsI#Lh8_2)vU z*;>6Lx-LOUtma(OC(p1HSCm$e1aVIO1cfImfV_Z zH#rrt`$Z416j%BYgIX3aq!c zZI_zV3XZG)xMjZY!yiTW0DzTwjQLOs*GH7dGcoUP1!VRA-q{OE8g?vtD9*87LD*9R zx9juNn9*neq^Bw;OUui*nGpaL`QF|>v%mdmeO(&jkJJl1L7M369>=&VD7XT-p<;QoVrLGp*jY%x57za+xnosO@!4$AXheIN z_I-SE5{~NOg@D)W!b6jx8wK-7~S>4csNApv327Y}TBrj_?dnbF3X3AGK2I5QK?mL72a~G3pN;vdE;3g%Lldf;Y^I-nFN6d2o zm2&?%h(3#Nf3Y*Z-i5+Ej4qKqsFj* z+qg++Z@Im#Za~~R!V2&+qwuPG-GxUR3q5;$*O)49|Gl_=NdPi=!%*r8Y-zm9!K+4R z6Hd;$l2Uw;a*%mkZgFqyQBFyC#7hBqp)0cA7ux@1gia0cOv8E3B)*sPpnHHLs0=Ut zrWBS7C(nN@R!;%CYYQ@p?pS&P86y&!mfvhy5?vfQ0XQIzuJAoo!rJ@&)D*2t&d1l>r6`lXa{|FpvAqf27>Xv&qesW8Mz|BIo^LI&Wi(cE!B8?^aIFb{Wdmx0(ID?EpC1F9P9 zh_g+HQ^OUD?wkuj4lmQ`v%5X z9=utA8U)Pm!P7=NK3(9bODOD|bd`ovANXI2*AJSP4n{t`pn;H4%5EtNjp7;OJK~q| z1G9xLAtY5s7PqNPP%Ev_(dkh!{c|+ONP`u@&hc`)b~ydjSmwyOOL^k4BZ6k{Z(abW z0MNQGU#OLr8Xaj$D}W(mw2{RzC;`Q(e|QL8J^ z%Ai_e20&r~Z(M`f2mxejdfMpg%k^#4mihh7Bg;s#Gtk3EG7e;!o|W2PF1gn8xKk+# zfc3U%?9D-Qp2_w^!{y&qFZBo%oDBeo4^Kcn`Fp9|S5UhD5%eNbaeCU}|0FQmS(g&x zeRIb1=kAn|l|3j#FqxwhX!6K}@Hjd~un;FsM&dyB5SQiTU%Vu`KPu1~h}qW5EpsiJ z`mX?sc_895hw$TStg!$LCvZda&)zNoZXp##oc&0deWl%1CLWRSF^&!gc1C%RFJB~s zO7}dI!C>QW^ZBpyQ5eWGP>joF-bC*!?8S%AtRv!t_^747$T18Kc7Pf!0}v?yY>uTY zg|V0U|FHK~QCV(Z_$b}dA&sPTNh6?AN=kQ^(hbrg9fGuUNFycP-QC?K-3@24_m;i? z=QqyXxjAE;F}(POAH1x0uC-=9^O-XNiXz?bx*`L?G0N!_&CILE3J~LZM9T;-azpX< zZ*UW7a9j7-HCzD90o)st)>2;`Af`sORZwW-&Hw40*s>dAZhM9iz;WYh}e03cmJ5Sd=W>3xDJJ*{z(c;%O%A;3 z@pL0Nw9DI9DAEKELC{ux^cpFH{(6YA4!Q?`tY|r}MPa&q)mP|7&LjjOR>Dxa7`#H0 zmjkUKMx=um43v3By*ydR>-3y`?03?Kk21~;0wP|So_Vs2FD0GCe>rPA@WhnjYfwrM zCOV7#IDn%7$LK>p3^n`&6IG|wTU3OE>H5VOOR@Tf$K zJK8_rYZWiYG%%j8(lJAR$p%mXm(lF^faVYYaa#EHYg=@)rafwEL&C%?e)Bmu(pk%% zCSpP>9X;MAkaC8s{Bb+~X`4}AEA_f<$mY#YBn+@5a3i^m`-!lFF+J%=Q42unXM{KX z$QRflX3I6|N0spusw{!G6dYo34zI30-RT|npdAj}$)RDv^1wW2wWs@)0zi;klf~)S zV6F(gF8_;T08kQprqRqFX-5dQ?H8ohhxFY2$ml{jyg$yi{}V_)>9U-h0L;UfACYHx zV*2_k0IGPwXSX@{&BRmLQvGzZM2s+=pki;1fRK!qL43cz4&I({w(iaOhxDv9tvZ;B z{_a^*)ypuSvi|S8dC6***gv%E%sLJbCz{9?L)bPXQ6#>o=t=_%UF1U)|fT&5%LOqR?@u5)3}vseiJcLwY#%im zhzz+He0BI!%OBn8u5M8I^{Dh-V~(!EICjQ6d^ikWOCA-mIFr{hLL=Y#{;yNDF4*wU zZ>8r?uPxz!r;Q^*?6R zns6m!v%n#(d07$2-HNB6n zzrRp1!YCenYIo*Hss4b2{q7B$nTxUdNI3NZ&HCORaW&U3J`0NC#A}Dlmftc3SR2O7 zum#Zb0*TibVT5R0C9V_aZzF2NI!I3K84LHI!Eh;oB0g#y$wLMx2YkDJelX@W0O z#+uqKdOw)Mb+$V^!Q675e7Rq+Z+7aYx97g^GS8QGF5VF4*nhqCo548Zs%d#&X}h&5 zL~fAxw)*7yWOd*oY{EkOIKo-*&`oZIWZHJJ&v;ftU48rG$?r_%eU~*YM^#m^zySo-oO4bennNf40ShcJYb{c!w=rn zyD!{$%(CU24%tjJc4t2pgqCp{g3V=exbblwaqj?#td|NY>g+Ncu=0%UI%Ep8*f2fq z+gbbivh*s>e>3X|ues9z257{}Ehw0&bvf=XpvSdTBf=0^9#7fxZa*b<)QfS~oOb+~ zXI^1A5FJv(#iVkNx$f_!^3+(G!0Gqmo;SgCV$uEJ-6`LA-LJbJMQ+p`cvFa?G{5+k z3YqSLNYFyopF+YXX7={w5Z}bsc-i@T+o4g0jS6fvr)bPUrV}|c}u{)sVt|qzCwiR0^ zi1F8A)yx+3hw|-||F}#kpAcGK1K9Xp^>p=lqU&xeB$lSuZOvKo3oA^Uo!%epiN;K! zzSX`G?HV@gVWqAwB^!R;QlrFLU8Rh8oF2_t?z_$WV6VImBb2Er)_)_m!JanZ|JGaj zwoO9N!*oEszqFfJ#Ee)Rb&D0IR4bXsTxukl=YpYVf3l%vNNJlJ8zXeNJk)Ts!ghha zLz^RigC%CayiOe&sz|M)scYi>EH%Bw$IEe%rN`YDPN{^^(}mKUr8AT?|J|8Wf3n@2 z@bV;~)Hh*`$<|tuMBp+%)%be2FlemCcK=+)siDS9d*LoU%VRXAbw_F9OjGdF6)9W3 z-Lt&e_pRf)KC7fj?((rbUp^#$O}KmSr&h2cE6f)-$!lO|cV}|NxZyq>WP{Uk`_ujMH4}*&;!Jyr^_ohGp?Y+`n!vW~ zNUG4qW>2wO2f3A6EbKNOf5TZ-#C76MT2a51$=6g<$9#K%Ayk|DwdR_B{7-w$Pms@u zCf_Xj5aQmg{c5l&v8X_u+f~3+>Qj{ypF3x&#v2F{)84`C(E7fsK4<7k6ktJows^U) z;qEkck>|KK)07T1##3fn&%aA=f*rylx?ld%Nlo4FqnulkRrGgxQzBvl-Xa6LzB3N~ zOUv3@Q7suY>uc1uMOU^@j(LXSCff%2%4vTK!1ezo07%HEc`HF38q0K3TVJGGq)k+j zOz!+^f~T!LE`EKj44dO~=s1f_wC=Ru z=FbZ5S3UhDkt*g#a=IHDE<8SWws$DuKaT6WRxjRQcCwZ{wi?q-rD!dNqz2!2Qy1jE zD{ez{a@T$<4KY7axnS|5%s!jgsNIefy8so6U?RLu40VU|0Di#TW1wkWHJ-@hPF^jt zo?XA|O>*4Ll(!#R$*xkNbRD&e&q+{Z>-b^M{_NX*SrHEiB7B}lS5P>G9HyHQg8io< zpkWH+Jx_}9@G+G7@MoYDPMG9US7#w6xTU2`dB5LKbLt-xRD~a*v#smq` zHN=RLaaNP`Pha(Zw?fX=8e{FzeIUM?Xekt|b>HBPwr1P%R_L7_fiAooA2JzRatuj3 zA--xJSYPN`sDnSB&Lh+Cr7V}O#J$K!d4qEc*@^Hj>Cu5Cm*s?7(}L#DJ>R^lHp;nf zqCzX#+FqjYi>WPZ4ZZJD*`QJ8zZ~t(dP=WP{qj1*sEs*mI;FGwDR+F9pf8`*g^&(E{at1rJS)4X=HR((?7?Iw`64qQIg=?%}~-F>sHW$zj^hZi>17nOUZbU*=4AGhgd7Uy zUlKGpw$9l(Lm3v>6WJ>oPJ+S5`6(1q1<>sdWxE!!C(p9(3<;o0)myeBWK6erCM)J* z*kc;%f~z|+_JsvkAJ)n74y<#ihn<4y4}crQTA;@$QXQgPq$ty;r{dE!+HS>eVYOqY z48;6OPQ0YhT(M=x=Tqh?sGUAz#f%^y&%-vxEY zq1a43kZd|9wef@;Ynhr{^orG%h%gKb?9N7{t=Yz-P5`&^aBMp{Tv z!iCrCZv~zj`P|&YqrCfkmum3=EJZtYtAxY~hTB9gt)Zi<#XGgaZZuaPWb}Y`d5^@a zAVT69%(C_$^(yjA)&g>pv(V z1R)+eub7()Bnq~NC}Ow|eXx=eXi!nn_hc8E%q?rSg^u+`Rc?oec**L{R?lS!xVt9x zlm}G%%%s=Tf0mD@?rDaS$n^aRN}O98@wbAn77catgPALz!3Z#}iE171Qi;K~%l-`1 zztDfR9}bP|o@UCfKbiEJi05pjY5G`$<;J>C6TZijU1_U3nnb5>RTX<~GvMm0oSl`> zhiYt<$0pSnF&Uo$?EadPN$+iyoTwWBoVC%u^G(I#Dlv4 zXlmXJ+X?=ji17wfk7FLIP0;w0Rd4@$hJzb{8+ctE3kCa;wfk1R`%=cUy`oB1cU?h( z?Xu0xaPNlZ6FrR}=2TPpm8IzS)PxP-cL4O0ahU*Dj zQB&-nyj!{23>UDuB*MZKjHMCpiF5K6Y)`$EQS+zD3A~)Avx5%AJg16Gy^+I$psCuA zDq?KszriN4nIUZya_{afO9j0hk@4$`2y-V4?MB=9T1iF*Vf*56FnxN3-7ld_5IfjQ zk9lFZb_Ts6X1vc4ip7t_&EaU{lf&;175;6SwjCRmz%De`pCHfBwhXK!d*8Ir^z5ry z;l<=w5*xTXG`in}OsP!EkhN%LAN=*D!fa# zr#zek>B-s$?GI5r`uIFQ zMoP!ZrUd!;cV-)O4Vv8@!V73IJCz6^s$osNHKA#ifqS$ z+mf%37fY3f#dIB_sigY1jpmih6}){bK4?A<2Sl}M!yHYU;M`-bj#~R;3y(Xogn=(t z({r!WgX|9DmG{J!*7>b(_R8Sbeu41`)zbv#;}Zm2*@^R z%YvBuKvK=n8H!c1(s8KpR&2=H-vPXoI8p>s%;NW5(}WqKYr=Cet}x`}%5iI|b7Ek% zN2wOz8w=}a|8q9Epw`Ag;SXkRfuWwXZOii~w2F}F>n&u{6S zEX=p|yKZR`Cd-n49D_^+RQ#>kRLxpL@+I@x0>PyDVEA|HGxN|v%JV%}v$*A2N#E}H zbF=!DHa?ZU$wCRi*EWs4nRA5D4x>MLOQQt?D}xXn2SQ(Bh;!~vefhPs#|v*%?eH;s zmuUN@XGi?gHW|bM2rX>GtRN_uDc)=2`dZ4~(coWLQ!5wL<&sZs zUhr>j4)awC6jYJ*D}-KaT4rAQ3QeUKBkr2lhXc+0Wy@VJIb52T-g7h04R`O!BW%2> zMtin*=2*_`qKrc%jU@2_h2s*h9_Qd0p9dp6%|I{u5J?VZhK)L*T{qW6_iBk5m;V8|mz;-Yezp-D$!XNxf*& zLfBGN3sJZ7RP8Ye*>W-SW09aVi`TSTjta>h&tbUVoDG`cUFG=I!E&dkbqE6C^irnr zIS{W4+_kT`f2^t1?{dAzu2jXr%-)%CsrdCntIcih$HmKW+|EGyKPfVFQYcys&#xr3 zqk&pnj+=g=-__H9EiIX)%n8janU>XRP^{~QNqK_w?}hBu>Y&mva3SDS19AUs;+ zh>EH)eA-s&rRFfhas)U1t3dUs<@!Cj8K_o9Afh zTB{12dmA@stIU4%|1lomOmBN~c;+_4xi_s4Zqx5{n;N(pKPx$w`fbL6uIe%CwT#fJ z0O6|-C|NN5xT9bnvIMGeoVlI7?-Sx$Yt4SkvRabiJQA#salp`7p~!ul%pw2xp8VZj zNd|@ZPy%=r@+0jrA^W&o?1#?kr5)`5=)8aYW3pEu4^NW~xMTmMH~u6A(Y=~MJ%lMC zQSX1BNBLJ4WElECW{VnqW7=m%?9n#oErig6Zq=`5FytT;D+np~+b zwfUdL2-zTFi$Pdqn~F1ij*N1Zt!VNh$gzhKqI1HxNJPRFRfR`D*=1!|Os~g+kZS_T zLZoKp@qb=6jQSzn#QZnO%7^7|gVm!G9CrJIFJ+y|q#?wAFtL~SkA}dO`2*($^l+*XTG$; zc5AA2X<2AwqqRw54u$t{VbnV}8q|V_g8kN1Gdl0bwUj*M|44>RS3f|@$SAFiD5x2c zQlrIVD&4CQkMTyCZ*PvjLQIgl$~7y_qj9|O8*BK@XYB-$b}dcxE}QLfdntp2+%SUp z+;&c`s;{z?U$%W>WEc;OE%_Em*7!gvG9>?OO%z6cTey*Su7F6rBcm`&NkFyw+w-+x zm?A+j=m8?(_L_Bo8HdcWtFYPO`$6w1_iu)f%4-^g^7)!I<*BKu_4P>&yP^pt{E9t9}}XF9pd7w;2T;C4MnN{Sp+};z%1h8+B~L!AJ0HWEcgbS z>u2s`t>AGTgRjBuM@FvtHzfY^Qh>ez$|{ghS@GXu{GXRHc?Jp@h3;=VAZ_^7U-vxT1~Scu zpU!)!NC@C*N0=9L9`4%ibX)gpotu=Ay~3QT)B1s#TcfuZ*(|{du5&!ePX#Z2 zF&zF|;SGZWw`pWBFc)Z0ArXX|yPCe?eGZSa823H;iu&%g&Uu>aB%6-7##`nW^k2f= z3k5f@BC#r>l$xy))TIpnIpki=;J`5j$LoPhq`(ImoE#%?jxxOO@)fC3FyR^vUG>tT z^kmz8q{H3nsuB(KQP91HFf5Gt5N^|AXysUC=#|Jqgvvnr=Zpg@kp0qTwG$wl2nh&& zZ*O1D>}?kQ{26+GDKWQWX>M+www=@bP~Q}?Ml1qQqYDqE`j@6#FH586u);60vRtK* zmJkFCE8CW}7!uOiuYR8{uq(1O?cFpdY&KT^>(_8WA(Uh1qR4W-$DsVaZN1@!H-RH(B`#v}4$1L$GfC{0?dH`< zqQDv2a^Z!@@0!zkSnay??G+g!r4d6(-|rcnJ<%6N^03#oko_Ok$_HsbdMN%O;rI^p z?YQk<3Mmc%&h{#I``z{f>p_a%fhIIYW~*btc43F*$d}apYHpHHjL0P&RmDDg!&vMj zA;HtFNb)z~LRGQRJ|X?U^-5B5T@HYvxmUd2x?ITput8E>7R9{nOW)Nb&IF+@>IeNF z#9kFRtuz(83O3-hVq#8Su77=r|hFdWf= zbb#H88fwPlU<7~+U!#~$tQA%#Im{#8QXmN zAI>sfW#N!@11ye^+&?YF+x1I6&pCPUxUH|U#M047h)zO3AQ zgNY`$Tf`Tv0(4QIZBy&_pRRHHm*{#PQ8;9C4;Qt@O_2}0Iue!+j&;XU*BB$6pue=`FK z9U{ioD(Bm!OXUPbNm!3u?yW}+?9Y)%W3dqmUK%Qi(G0z~sxB4uQnam<`xpH97jINv zZk3`cSdS9IWU5#JO!u(^?Qm}?f-?-)(0M|CN8-V1P-~xQyk*UZ5(13~Cg&TI$ZGAn zJC~}WrwjEtsxRxWqcouCga4RPq0b1NFuZ>}ULjUQGFi{GXI+6a(v>7cWWZb>AvYZD$OXvwfe??5UI;RJ4{Gb{<%-2(6 zz~d@Qo4Cn%)d$6*GdIRaVsrMAYRT*hDpUtDI1mpv=XiygO4(Qe}7AWF=ZmNE&c9cqI?d*#2SWKtJ z26vX2;j3=l@=m1Ab|WeC*Bo%|2H_IlqYz7vI{GftCPZ3^1&V&4AT)}%-F=0CS@FC7 zMmlcBBMrX?ANi+|PMEA8T)~dNNp!rB&^HgK;_1Rn*GdfN8Q2<550h0f;M}i$tyG&u z`o|29+1I?|PT(-yCS=i7V{d=zz|Dcizk9}NQ zlkVMdIKDEgl@bE8)YC`jPn-ap`M0L5QJ_ki(h4()P_2UO7!)NPV5)IEw%uUtJ7~dL z#0>2hlF+iIJI9=>9|Q_7uQ(9fr!S^{7a3azS(yq*kCy@9Dzm_>$TKs&gLZ8-^xH>J zw>Ua@{@=W!!YPt4frhZe$FPk8QlKYzhE9H-ygSr&b-;~v5^ZNT|BhoQrmZDAG~6Ze zBuIAmbKAsw!OW?osSJ&G0j{RxKehE7@v89bSp~1@pA^Zw{kw-Apb8NFpQDxLhiHX5 zLzM0>$RDvrk2-E%VEI+4IGACk74jU2 z9R?9|cxc8Na92xaPq`~Vv^P)f-bfB5exKlB@YGa;!0yzfp-%r!_v6o5Y}YsRvM{kU%26)X4dG|z|b;`9``mwfshttNu*s`|SQ?CpAKA9df&)j5we zii7Utb)dEop=&$z-hVRFQB-74tPC&d0hypzD^{bY^Zkto?55Y$4NpM(fES#t3pWL# zEwXNG12q~SN1=J*8WS`v<+@E{2#5N5DOqJzVzvIutL1k|re3W2M$CDrg+zP{{wDGB0=N@z ztNNEPi4Sx)9w}ag$!KBhP~_y)-mQV2jcf?Wei)L2bp8xgy9Q^ULZPTgLs2OGh9QPq z-cw7zv%XYUH)h8^NwjgkI0sE_Vr26`AAH!b528?aLfz7T9IS@>0_jsBI9(0wd- z*Xwn)_iZtKlG8Wxnb!pt`oOjjl9#!g}fJm5(^Fg;}(kh_;k>j7Z72Qh?fR#1=zx!Li z_2n^fgkj_)bee{=Kjvix0ue|0Hqr#j?k57t{aQ|2Kxnv+~K=)lW>fbJ=KN}MUk#GFCe~%wh=zzKZ8EB+L z^>0!5&zF=C+xU-3k;f0&NKm|RIdtEv7^7wZA06O5l12i&DzR*affB#H;7K-zfAj>< zByj`k&j9U|eTaytma|;lGULEx?+fJ34P5>qaHZd~hx|8AU z!iIiA7eEvSV6OeFO2S28bP}^hrPa#N0_d`o5x89SZ@goi5g0uWaM<MY$a2Qg~_ymxTL4NI^liR(cV=<`_O@wmG|fQ;=M^f>S)y|7ld_t>*d zUNI|D@}}Z#1mvAZ+V`j0lFlqBw;digV&wrccJ>OONy1AGOQpg6rM%HrkFD^W71U-dvhU5Go1(#r%0a*9v_k7A+X!2bx~AUJG3BlzJS|czo-DZ^Qh0aeZ?< zFvI$kx%S|7C=|X|N02zXRnQyqhAH&?+do_YH*3uB{QnS0p?f^2f(6ki+9f~` zn&h;McqFhEDc?-xv=YMAi}l3nVOR%bvp_a3U|+^Q8`uP`6G_2$(!cj6i&p?Z8Ns`9 znw&B8Ts$>n^J292h{qHWnM{^_>eQyt&>S;9;kh}7R>)G%ry8q6%lifWitA-Fd z`PV0P(X8Z#l`zr}AclA1j4w7z_p|y~b>U$DoErtOL$kICwceUGG+myf2g+ZD!bIT#JzA{Qz3Nesofyei()w|EU=S zw+{xGyvooW54e(!WqH;ndW4{ljLWcy8?7cu zL9DkJgCq^HEyb_|1gd^cB%1A%oua#IGs~N&fK(BrX~~gVZeZ);b0U0w;jYS163JDz zq7nW{Ra(U7u4nGfx$<+G1-lpNuCT8}{#1!M?1>$D!Zq7C$-2UL0)c<)abB}vc)}&q zF4McVBC2Ir7w25ADSF!}Ccx%&uo!$<5{Qzw7+f*DF$KEaK#MS38|cPVKe{__5OlLQ za|wBBb_;;#R}bT?nTtjR&WgW2jITGB;qj$Cz`WRQxHHYrR_2U1Tc$dw07o@n-s&#L zx;Kt}A3N8Efso-PgHfDyE(b7?g+qVTzms&V2TOhGs1h!7-g>evRPM|Y!H0#U5Y!VAo9 zTN3#{Y#if;uFnB&mZw*24L#`16Hz7Hi4);^Z zw^M5|+VxI{%gD^NEGgMV5Sk!U(r{_138z%XVa zGc2XNzg@U@o6vOh*$zS*@yN-yY3~+wLM7r3>%(J!ScaER8jL+T-yHI;lsqoG zvJhhr#C8eT61shJO(!wLnsy`t$8?#2VB*=4;qPEvQUlHuy* zN-#UyPk2;=cln-QnN~z0wiz=`?=QaIUwo5bDlRz`a^K)v(5;bTKo+>38i4bKVSp98 zGIWReQH+m~0Nz{0;lS$1Y!;|l>%5e13XsF%1)5l*gR_a2!ko>uO9e8E!4TU9P=JyCCDsDu0oM)16!R=sIC0)tRVL|v%zB?bof zP%iAdenP@6JY*=`E4`$Zy(If_Ridm7rbOuQNqlMcOFgKft^r)Qg2b0z;vAdyQ|Uc7 zAL;=yWmJ_~@8^V+en*%P(vZ$||4bxm+@;Hs*yo|%#*J(9xzH_6d9WgBH&vZu1(|Wm z5LaX>D&a5=JH10L#xK?ke+P2hyddJ0J1a04k2Snm3Xa``L?yNOl~eSZ&168|-$T82aE2h)#VQFH4hegDEepv;c2ei4GRwZ8P6 zZ(tA`L)(FCiI`+%ryXpEPs1A8dn?HtBfd-odWj%!6Q7mE_f}sl2PHyC+Hbm9JddlK zHe_HKc3<@T)wu55|CvkwWYk3uSqF-RjWUn&;+f|&hN{5QALy163mtRKUIuO2X&1xH z0<}w*m(Tj1m{U{q?hf0@P~3GINj6L?`CAO*rk(?RWpr3>Uf4#n0|q3Cop6*=lc>q* zkRtI_1PEWGJ-b(7y1cW;d4;AbO{ms=vuo)QuUFHD@SI2QK#FUwb&v2){wi+3wH;@8 zEL7&~{@UZ7xu}qEeKX;e2>Jc7;C;&xe5(;&l4VoFO-}>mGKGJu#R@%U__6-ls&!8d<&$}>Jw(k&+;j-+bN3ZgwQL##Ic8FrG7)Cm-a@d+13 zxR1GXTR3Yhzx-RzRL1SrK~Ae3nnWlnUAjE{pOGUCQ;ZWeyKHG4J`wM-2Q1x>36*VD zE9my`KtCL(e6oL2E#ed8cjDJtv36giPg_XuSVVbMzFj*|xl)IPsMft^=-sM{-@PCg z<6E>kDqx1d@*O}owDRimDeC}zLDumS?9PF*tw#2pjh*vfD08m&YQA1-tP0^3hlH{r z`|WfbHbSGx8p+?ty+TRhG@JB{Vung;Aa3`Bgl5_B!DeaE_K3i|xztIyFzkBb z$Y_7oYQy)rXy2cZ93FUkqNEp%@HPx#5z37GF~Jp2Uq1_(af}@DWxTdNUQp>URIiXE zcg46oTDkz_cx{vP9eQ89-(!vw``lvfzA2r)fi+Lu&weq4Gmv+{K)V{F9C-qJ_5^hc zG!4mp{~Z*1P`T$*h1}ZQR>Ho_Fq3!@_h9pXn_vgB@c< zL~?@MO5#Xu?LubCk-Da&n$g|ej8QE z@?4Lbwf6RZ(&)J;(QuL8b6eAJ&3BT(bgq4~jMq8vRHT$vfiuGLtXk}fs{3K4#%Qza zNkDVHwJ4Tpj14C|kI{@)==1y{E43*z+F!qAPPcN!8xo9Ubc$f8_j_%cBo`1pD#hf;xHTTZquE?|N!} zE&^3VQ%Y#gQoC3hNRzC4z05XhwL*#Y>h{I^d{!%K$XP}hO?M48x-=G)*rJu?Be1>z zpH~>VZ&)h?pR7yXXdpqtf6k*Ng}S@c;&lee7i-i3*9M&ZE}G;>2%?SXT(hacq|4v3 zs06g7<9P{VVs-2Lv{qoAhx^S6so1e=O%h}O%iBCUFuN|ezh|z@`8i5@*o-D=Mv0_hHA}zlcETJLDt484DRb7Z`fY=Wktyfp%%{%dvIepyB+AoJfi7qY`=yqF%Ke z1FkISr${CL=1jq)c!0m_-Z37+R{hGlr94h$lS(W^FhU=5m;?rMG4jfYPwvv?>L&(*_@ho-|mJur-9yj*5$X5nDY zIS#Sr)Mp5@ecub${V09Q3 zX<&_*sNd*cUKve3HJrlK%h-;$tmgr|5wcF2fkGVUi<9vW&b-A{fTGWiKbF?ZSOV4d^_Sk#-*~w5b&yJ-q@E-Qi8wVGVR;@8@1$CuEADw|+$h#tz>WAU zu|Ld4KB+e1`XYEV055p)YGjh))8Ts|gEmt9fUyPyiAnUsCbkvbm{8ln<-JMgjknAQ zF4Z-Lwkhf51A02CvgJ?*n>$Q^uVoP2C;ZGMZN9d(6gmMoFv6#)n!-MiMxHqg>X{a18cl14@_#cksB`;{EO~c$wx5f!o5gU7FPA2o`?t6j%?Jg_ z@C8NQ*fm>F8XLdFsi!B!5T}Ry)U8~9gfpSK32P=8znAVAwyVna&buTL6PaG+!-tR_ zYzXR_tr^%9#;;77kT=ucUJGj|M2cIy@Vo9l&^za7++;8A-M|~{C=nqasbpGf%!_vr zmi%$&7&329|jaR(0m5RIDf>x`h{&7Rd0VpQ?w4&(PALCW0I(}I5S%lf-jZtqpL1iY$t=O zSOXoPEnGV%j8d#ubg-MpW3ay!mXv-k_0Pm_ABjYy{MOgqjSQ_1*reu-wfQ(+T%u2CPb<~i5IN#T{QQutKbu-)*;yatXSY}DO~PIu|GfiLs)A^ zmyk%912yY#VW}RG_cf|S8T&MPTqoY#^UmJR##r_*$;Pdexce|^>~-5lPS6$I&Rg%| zdorbnOkj}e*B36@xy)F@aiTPv63T1GnvY*{3{Chgd}I)W*boyo``jk^={<~2PdKe5 zHHQ-?%fdmAPESm%07ACMTxzh|!s^Py3>g^!Li!zF8HU70(NFS>A;wTTa*R#oR+X+d zon19(um>+(Q}>4PjagmDNQZC1=+o|4He9iGw!?6T>1m;-aH72(SP--XHOw}~ym7?L zbCh)wgZVEo=FAsyigP;{acr$Ah!F0bHz zTPzeMMfkdrY4sJdG-r4WnZv0!s=-1`VB)0fw9|)O91{Xg2?|wF1j+BgU|{Xh zYFBj7%=r-1FIH0#(6B9z^!eIu^{z&rO3wDiqEgu~8 z{^u`z6+0`${pB89kGKi_u}zS<5!!WHN};zs{zTc??s7>Tzjwe8#JJV1k{^i?7VFWP zi=cbDSF^uf5GtK?n;mOpxUs>vSLyBEVURtOHXHpLrYL>Xe%r*N_E6MF{KFo5KX(h7 zsP@^KZjH}{vGM+UUn_XW1v0_JZ`HJLY%HX&WNX1x0<;;+>#BHbOQESAmtFzaGyCI~ zcs?DOevQ&gzojoLMklXiEftjpp0Z9yh-Ub)*)PMVmpBl`Sl&xQWI{OUn9(!L+)U{e z&mp2xD?zDyTz&upRLB9nA21*jnN)=s-*p+D@dPvpL_9Yt_Muj;FY6EEuH_Fv`zked zo&L5VA4PUn!(9iK&3eiKnUx}b2=5aF!XKF>$0TL{HZW!1v#3b&+RX( zdWyW?ec=-5?# zS=!k9?oGvick^M9sUm@CdT#l2bJoz-^(39tP{N_U+~YFF;}ZDNWBoF|i0yTay7x57 znCqF&MXVd5gqBbwn0!XW)DXrqqk*kcYoUrI8E-VMpis{~hmI+G4neWo30+Kf2(*{a z8~Mo?HXJAJPvQv{A-`aUC0gy*QXOq0hI9kD%~DSy(sPW~0#NP-rWI6ko)l=T@{%Xc z+YY>xj8?DCO0Yrl#`MKQ$hOFwa?{b6C!%f|(DWrnX!I&2)qHAHGc_@!$+oT|h$bps z{vx~>HQ52nQ!^Hd2qje<{}k#C{qN0<*cJD9SOWsT=U$iY^q9>J$7sD`t*O0qZ-@{w zahr6eAH-& z`dV`csdlnOj;fGi{(+`-)Y12V|GnlP^&+uY=X|4Kac*70Y?M!SJ<|UXZ7b!FM7k(YF<$ zxn(<9I^Hr%E%>zQw|f(yHx_;SlYzFr!}YjGeZ|Ff#=IhqCLK3R6k)Y9i|HJao>0G< z^-D5p+}-_IKVN?!xoJx>Bw%af(YZr;PA!?Dr=+0JySJQ{RvA>1c?-e1qbL0SebXiSFA8yt~x? zEDbClk^*mvOIW;fFND%?47<*@8N%}Awt3#ZlpUdX-J^>m483BBDScIx;ysazfut_+ z?Au06a>w)4XS}!{nk-A4QPrd51cy@J4^f0reZ^cmxQERlGQ7%kZoD!|MC?BWm3j%B z;{{RIR4HZ=D)Yo+P*{o``jP!^g)2ZZ^k~Ba7b7=S${T&1t}W=9Mt$qM2}o$T;HRvg zw=`RVf;*sk?fdP}Fy`c-V~f!t@(P)6*N9H{7X&Tw1m)2@?ynaP^Xr!b=~*tnkHzGL z<2#SK1ID9>Fl$LZ5U+sgFu|%bv+9*)s?MQmO!jWngzrPaO<)C;GQ1UjC41}{iyB5` z>$tg%d%g3$xj0kfkZ1|L>d%D=HcBMUI`^i|N#dst>45{8HM>J_?JKCqH4InFq_v1@ zC{Q=aE(cBUIX?1fjLnjAI=M5Uiv9F=CNyImQ4l0I>QbCq!9@L_*_x?6<+~pC#T9P* zTm_K;|Ep-yrtAbb(=-%Y^GewxLZ3ObcuGh3Lz~%26|6TVXVORdgl|=U$hq=ZxqPAK4VQIPyk>E)-c*RfM`kv(V`g6s1D@BRs?($Q^4#Y~%<)u= zmGCpBVfoO2=%_8Ne=Ttb&9=K{zn)KP_nA1<7*XB z(pcUNMhk|srM>mnP~u0bRO!_YAaug^^@Op~l{P$uYra>z+s8!qi!(ZYK>>w(?{z^M z@VV`hPD6d|Ueijg+>rC8;%tKKQtQ*e$-{tA^QT|VSv)?_H*)1AHDi49_Qrbl)=7FT zWwb2j;&9>n#TftKq{pYsI_F<{mcxW}W_`m7O9f9&jB+Dd2DANF*FTV{s_=kFS%k_{ z1}2E+65HNWikIod7#L-vPiJE!3|9H6rSU^eyi*b@>~QrqN)})LVrg z#%{3O zWn*AiUWs>>wt}16u1Ro*%FtTt`5NykgC$fw7I<8RW773+yuZMw=ShSkscmr0+vnWiO;T9no* z=YT5RNwQ)YLwLcQSM~-F3}c+u zl)RN}Ja|0#w}w7QX~Gk!Ao?$M9!a9n*xitXd534b5y<6EN(gt+BVzEej{_3JBpe1I zUBrN)S$M;MQ7(udkpQ3~grdNOQ%3xS3ew_#BNoV@E-5`>RlF9CoIj4zgK}vgRv5Bf zTX^oP|E+}c0~A94it791{{_6$fo`12W%03;e~=gdR@{Y1K;dWOFh5)7@h?7pzyUm5 z8rB&fW&eG#e;!Tt6$TrY_p7mp8y%%rj(irbjRrv^SbrO0e%;wLj-~vQ#||DLAzU3z zxPP7*#tikHxo6^1*7ENMAqfM)KqX2(p;~}3816d1e82zS#RtlOua}ecL;iOK5Kx$n z)hPe{;lDqt0eZegT_}CbT_nFTt!q zFb39jqz-tvEM=ArcUS*cWp5c*<<@l%1Jd2yjndtUlyrBe(nupA-6_(DG>CL}H=Axy zI;0x`LEpu>kDmMd@8`?w7q>s$?CaXsT64`g#~5?iSo?m0=vb!aPnW^$thh`m;Lyt> zA~XPSJ|tPJVR-W7E%@zo`pzQ@FWGL+yHbgPC#1SNaOo8~s1SeuN#S4LND-y~JEbB# z;gGaqNf^WdA}-4so4F(4MDs8MlBf$9!tM_3bYnby@K>|df4DdS7(xL-;Pb|*Vp~1J z8_80qt$qyxrq`-Jg74jFKZWnLzk^c6U}&p~C$Ea>lah)qFwv3p@D|(&m=g+NNUTkqv38#V!J6hiERmdy_EUE&S*^eh4Z!`$B--FoN#`-;+b z;kGxpqd{-=sq?X?vHl#5pViSVmw|_D4WJn0Bge&(HNSUF7R~*}xmOow09@^exnC&@ z1o{AwUb+4KCbCJAN!@~_TxoCApjF|A?dGdRoAQB)MRy*rb-)DKZUT0>p2^^;M2)iR zT>!H)UvSz%5K7x__biriiO_h5S-*&+%!AyyzM(v|>%}N(aO?WfDzmd(o%KwFQMj$B zNw0oBQLjyTYhO{S@{g4jQB;Q9&|dxtqDRxeWi5=tU{!a0T!%*=H%Cp}0mE#~aajFc zhp}wQ>C0-+>pkHLg$L7_t*V6+`nvIbdijX)N+oF?Mm@q|)(qe^jqrU$mDU$xrISm% z0qhNACcnRz*@Km()It(cJH&LLUF7?h^LT0UPIP_Q`=kZ+Y65ZMXiyM>abx@h%%B$m z)`X{Za>LJ4`i8X-5^uIg$1Ck`uWBL*Mr8t0Bh6w(@MFU(o_^~DWF6Lu4j7Yy{2Ms_ z^)MRcz&4vi4NU{Nn7u?9mL>F;Iw zsuGsgPw(JUx!LNgr%%?7MjD9BQnuE}Yj^AP^Uyy|*I_;%*qAl|%p)~x(I<*(=Y!s0 z74AZft(WXdmWS-!0v}&qXkoA)7S8$ZH@mr7>!s?!&hlA7#WAU5Z|~S$>kfT-y&9 zw#wwSBgXf|XIeF{1QWD#t{Rz6<&T} zIgu(Z^Dkm*&1t{na{|_WqQ#MmyxDB8zuBcqCYoz3)p{vw(RQ>F=j zt(>8(`O|AB@^y|6zwFe+onx>ZQ*+H;FxQ6RmIgfH>WW%jGI^>~d<)ntl};opu8Srdk0ZHGSNi4M)+?1PCN5(R^>+q)buu3~Bb!c2 zAjJabM|BK`wi`>gHFv+2l0;81zS7Pgs;E#twF-c)5xJNKlmR12Ko+uGAFCk@0kOTD`Yl|o?gwuM) zonR6rR?T|NJMyKK^(APAn1Nw}C@eK`q0^SZdteT; za6ENJP)OtcL=Vd@?#M(E6Z7H#(U@MpOP;yAZrwORfw$7T1@(k31ghGsghl$77v!YE zbb0biZ}x5NQ6qGQN~jl#bI!RN{`7C{Q;{gx2YX!Q{xoN}HmyC#HPfnc(PTnN3NYQK za_H9LyW9eGz*KSwTrrq{7y%?W+Qq~aQt;Y~PMQOF1{v4O+Nd}c)=`))4 z-N4#s7z(a5-Cr<6Ic?2<e(gkTW$dVq5O^=s38-e&ht zGdLz2jS4unv2CYv*thSlJug%Ub|rSM^oz{?P7B5*z+sV&XluvU49seAeuTp1H!OyT zy&K%QA@4p;(z72lB%zfP5mCj(A5euM)G}WmdOrwP?d&GA>`~JRPohnf^K6OBz1i&$ zq$p&=8G^3G!yXP1JGuw?wv#f*9=-xd+zhgQ`1&5nYMxw$$qG}Pxv8zyx?F8p@TJcOBC|?D zuM5jB^xg0>8*)*nvnbbwQBB_SoAGI*9&HRB$n{(Onc@MxWrL6kC@bd+3+AJCGCaoG z@I2e71jkR7x!&h?yor%mBxJ#p9gh;0VUSaJDNK5cY(m@xw+kFN{gu$eAYy!iLA*~S zp`JS146&jn1O$bgT?trZG}Ak!e9z^U4A^DSZ?H-|(Sle9hgLMtgg|Bn$oknjKw`9G zK^nO;ExQf^_f~<~tAd@@CDEMq53eY4v41~aCr}vGXE+zb!q0YoaFPg`A&BJiS zx?8{EyR?p)?Qjn_%* zRzZ<~ggnL??O1KULEMEUAD?k^b+RNqrvEV3Vy7)qk?Lu$#0n(x``H@Eb-D3CML4Q0=jno9e{ zb(>L1>BlidH7CF=R_fym{WA;`5tIKG5ReN6@x}nBEaEk?HpUkh1C3$PrGPKdd6XP| ztK$Cj1{}+{g-|v-4@AASi{UrEtdHB{@3ncYLm=EEz>}`az#;7;5eEv{v385>Pn`M6 z*Ea1q=R`#A^Ufpg!C{tvjNjcfc0%2Fp=Qi81$E zUF;TObCCX6>d*;};_KOprI>Q9t=~a5s(#2VnKn9ZHk=(*bfa&cWI=MBL1p(!ple5$ zxbT&PaTRK)W30uB+;`h}au&(~krnI(2lLLnZgD$xIHF@&(M6451wmVwdFGCWY+`wO zV$@nTsJcDbX(04E?Er3gB6UcB zahffGK1j@R;Rci$I(vPUHw5xMJq$YvjpRPj`{VG0isPH!11_=8UI)UXnf5{-dzf8+ zB4eEoS0u~j{H+z(0zGtlft@CDn%?fD7h23!y%+voES2RYoJxH?Z>28IT3%#Lv%kk} zYj}TJBsT~q>*&`yHqfdz2W(z>G=8$c%T9ilb3JOd8y1Bd$qjE!o0K|*)sYki+xPU>jS_n6epxfH7088%rXbYz zAF);^MbIG`bF~0!*`PG?n{j@8xcTVuQoVZfMDOjCTgiH^9am6r{~*3LchE_2aQJMQ zY~wq5)wNeUXSnnTns$Gi1!0M8k6_HscSQR(eXZOu1?NM(TGr3sdFv(Uq-cHgv?Ij| zw6m5zb?y^Sxz=vjFh#55>@jz0j@zP4RQt?FpJ1&1Q#WhcjeVy8zaxMc$ygoG=MQtG zFz2MqD+iU6T4BTo4J-gbO>(dRURdfO%RH&ju z;;A{f1G&7&S`dycYwRm;l>sRG2*;QbLIy$MLfOA{pwA-dK;~|$`RUe5h^`9962Oibk$AY#KYt&a>Cfyv$=;%ihnV~&BZyq5Fl zD4`RCGB!&Pu%xQ>24)pVgvYEc#iPY z!%rcrn~ZKXY{U%DWN``xQcXW!;DDY6(gbrEMT%p-tGhsS!LXtYX1t@RLX6}mN8`LA zu=orc!OrOqZ6f7%7Yq)EH=+lR+1!YEZBuq|Hza;IzW=SANdglx8t6XW!^e=hMtGmi z4+|5W!vb8oGWTo5dI8l_c;fI_2YQ?rZ6NwRPIVA?VMDA`%ZBw9@kW{>s%h=Vi|zvL zqDHXFn@P9|1iY@$hACkV1<0`o!{sCBJ~VYV;mL1^MNHQZ!hM(MWg~`$sIb>Hvy7q- zQOlFBR1)GOCX-GR48h9r3nL*!8~3$x?UBoUny{4+$f4y-!GJUFlCe#JvUH;qC^s3I zbh2j7FOq9qG=JAyZhoAyxIE=?tXi93r!T5UEJq{;1YuWdXj z+y@l+7^R04gK;1(f4^+kFpU%=L{NT12goGpL zespj;c;Y7+RLHCCOaJ&o?SdhBW7sc&(@+-qw4FK4E@RG@op+LUmo1>YctMf^!(; z$SgZ=B~XMX(Au7>tNh(#EtcsZl{}2kbRg$HOozo_(Qd!)<5Iz2wa$DWKxzYW;tzm) zPn=@W!9T_65b$~itqb*CawqM_kW>>6Ept|h@9)sUm3vD|EPq|Jy7Bz548*B1tC#q| z@?NK8)Ge(B>O&80%xv%E@Z!OVc9x|W8w@{922V%o^eT>~yu!+_rGB?;U*LO_u3K^^ z&kB_;&(tpaQ*Qk-&c%D+Z9U19_|g3=AvZU0GWev;00@jdJ;Y|viKVoTu`Gg z4_VSi?Jv0{kysV#pw-uI-6B-%R(bMYRt?n33NTLoW%P$>!uX<( zEGs1K8-wfa;*IlbJQTJ44iAM?3>Kw@6J&neOeCo=I5&GvDe$2YaRwc6NO!2}b&2A1 zqI_nf>L2LQ^#UzXS?%qsi%-AHkH4DriPXvkC^fK-`F@|tuwIeF0M`|Elsugw5_0l> zOT*{G>Hc!9NKDKnzvYkrIX9*SK}DkVUab@>tmD@m28y2#T7}0y?fkz4`FYqWJUm*df9KA>frN*D0yqP^ z15PNL_%DFvUvGQ%a2+nJL;wAG{|QNqgy1IuAC%GljC%Mjmy@6bltD*9f~`N@{y)DG zHwyfuQ?V%Izo3l2FDMRx5&Rz_o&HyThb#p`pD}}a_n(;ve+NatLIGGr#;9cD|K1cF z4M?#naEjOdeD8sW}Qcxar|EdVG3(|2=sMk5>5%sWzZrSq)s%h~g^faJPe= zRIdF2IGjKg=TKxA7)L+p z4Cx)f3GVtx94H5qKu3JRGJOQ&9>O)&a?*uT#3Kq0EbE5h;h=DK{lXh(Vj3=cv*o8c<)AA!N zpcKdb0XiPPzSk>#)+|+()1}{PTm@ZN&xo}j%IJ`&*_uB?i?U?EBI09OCblQV8VF>T zJ^PKTAa(V@od5&dfvIe?ScacYuDGoz6VWkE{BFSc(%TdAK#BxO7-~022Ef_cye=Bh+NNS1rUFTE6Xgus@o@pL8N?b}!Wx;4Mng!R)8)h67$J9R7%&2w*FkY-HSLEbX%)HZfIrl(9CWfgvIDAvwN^Ssn*lwIPf(jPbDM1 zXMa;(zx<}HJ_H+d=5~OH)QqezXjuX#IPCqdc}+v{&Y#&Fmj^wrk2hr$7V5sMinAg^E9;L#3A}8cW0}T zRTEMqYh2G=_{yL;!SK1!F-dj9zftDDHaG~63rb>&LaPCeKt?9xHUY;@75{E5Ueg!s zHyKS^?@@r**+xSElEdn^Ye%S&BCRpod<51>wi_BobmKLE7OF{0z94ovI;32gCWZg} zqykNfF1u?B^NL)pz4@??q)+5Yl;ACp4CXvRzBk*OlXDWfIQp8-Zp@#tF%WksnEo}o zU|ueaQ_ymXf?t+h<~;|CMtPT~RA+1-n2&clhQRcBDmD9s#qe>6g74+-R3Sj*HA50c zc{=VE2jXc2h9}_skjO5FV`-l(KR{N+6(z=?3cq`fEfVuFYx)t>Z^%O;i|An!(cI}% zES7#cT8@J1D2ec5pvQgtO6u{63>i{c#UK2q%?kJfCxss$frgryRYuatAU==RFiU-d zz}%IpEhr5Df5pz)Cgqj>T6G7II6Ek?Xx__9nY3*&)94%f!{wkGhu(e5jjfPfGb1|c zdfv+=67pIR6Pqk{7TEU%!KDd~(19Ubg2?dPsHQ5G3brxiB5A(sKOC>uLcKy8vF;2{ z7RjFG}u3tQSro$PajGTmKlC{NZVE#7Fuea+vy-5|Jy8+pU%BaXkSH0KaZ(o2G(*ZZxEzkjpKBOzSJSH#I2=)103h(ru>!W8Yt0ZpT-zqywkjOYI0PNTniW23 zu!PU@aZh*QOH%dbKgW5o45dhp6^~bz*;X7>F{eHP$yFReGr*B52zMid-eT9J_BOfU zsy|szDpU$kNgsTF2S=cfxx}eyMBv*XYDyYI?&-NZ3XTiYLu09ekJ=6crx+&wwrs?G zTr$Q`E<&+*Mb~^VOAs}PkRyKA)h-n5KsaPk_yf)AgfWmf9Rr!RLphUa@c9ENt|+*o zcBp`(mutfKqpmyFTLAO>cZVHF7;kL^$e^i4t6OS@b@392cU@qlZ*TbVG*o3gnI!gk zB7MJD`PXse@rJ@_fa=p{fg!QI9S-DU+na0sYi^anBEoCy8 zDk!Z`=V&sxQapGuG_Nt%rJkWmu<+A`=t4x_lQ=*l$yp*AJ9D6+%%w`j=_#{53E%8S zOV8cFwiR&k!+T?yO=tK-DW}BPoS>}n72$iTvlInG^^I*K-a|ZTMnmf7dO~a8LAgwB zPfe1;u>`u5D_< z@}qtVtol8I&>zDUj~{M}1jW|GF&?=431Hz91ZA)gf5aea^a2Hv>%lx$b$cV!`!p$3 z5*6Lw+?A{$*fki$7APpbDzzN$=Z9ew)VnP~=gxgdAvBJP@p_gF9bkAg}haFCt z^M!p>&Wm+0nxW0wP373G6&nEF=(>@Q0YTuU_C$`i>+WRmHzO0#K(raQ<|g-Jd$>E{ zkgdc~dSe*EsiW*ra^zg)%A6qBmRD2@!ev_(6NsCSXH3Q~3 zbaer-(W__nZLh(V2KEa;@)!wxOp4P0Q{ zxCM;29;}~5LN4J}69j{Da#llobD*%(M#2C>2 zq%Q}Bl(A&b9$}7$HOQLmDA#$AZA?q+lj7eu1+q$E#4bm1%u;_d?c;9OXYR9P$Y#nY z0+^5)srZ~H^vW<-zKG_!%!`*EF|-xR4b;h`>ePEVpjBqlR&R{7NA2^FSE8UvwlL22 zV5cPpO3^vj%4}-e9+#}bRUsLlzuiV-jj!jbvkyuQhSme!Hl5r7Gr9k3Gs-WS zH`&hV%X~}E7j%@FX+v-nRs z9(m?FzYC8}tBxH;Z99duEnFbro25rMF%LEk{V5Y)v4XmGL{jv)Uy{12>kw2X&q6$n z>|{tp$FwCOt@yo>rQK)ox#YeJ1j8noJg3{yK}gG)vg*aJH~NB8O^6>oAVjk04*OeH zXTZ?Z#ip9btGg?IvH;|lQY3akEwq9z*51VOf`i{@jg`=;-V4fFoX3BMt)f1`<0xZ? zn?hJrx_$aE*QLlv9iO8J^}TDLMkH^b@`7P)(Ch)RqCx9mVl33v4RJD`{^mCd!=v6U z>o8l_CIUubJ_;)n4u`>xrs&r>-gRw{Z;y=lhIo`C+kG|A@l5AaPW0xS2|Qb#*2c@>SHD%8$CFg-xI$i zNu023MNEN9@g&kz3mtans9VYlZ|^1MvoehAT@d@_h?p#+A6KHwfWYP3if+&xj1m&? z>=b|k>rq@=+#sn_-RU)S9=Y{4?CZ-xrJpOw`_(eN?o}a&rg`)CPE{s^iGVRkQPUap zU-dXpon7kvmEhh1xDDMJj0NgBQa7f?-q|AGBH_d?n-r3t**3hWn@Nurz>kTXcg|6A z`%=yvxy%#e%_cZB>X{=Lq{fCk^t`BPtZO?kvD9Y-_!rVH*pdY{yfMpPV^sWO@ zz-jYEg5JEh2Ng?Y0R(C|m)s=jtHwNaZb)Fy8>?gG-lyZtmTQKS%!XTV#Yy36HElITGqM-6Ap-9SA5(lf zw_FjvN27>c4`qMFF3nu2{xdH@(Uia@M5n>fR&(uJgnxh)y5UdAq2H~AzZ7?E&xbCh zpxCrva^~S3dg@T3C&Lsx*6-nL8R$PIt3UsJ&Qd-F{ep96R;Otk@`ErJNyy`*tL*zQ zABztg6Cba~!VJBhZR)HT!9}rgTj<9$ne<2p&rH}d+527?B(1JZMziD(NodTY=6_<7 zkjDqSwn9Dk<*Ub*7MCN%bRJ0OpvE}FB4tXnR#*hRh)O={)~hehO8U-g;IFYtnE{(= zHZ~>tkfh)V+T*3qumYl+brhJaqr@;x9cgF ztNqb11G;K;CM@>(z(YP*fLl(Pdwma-N(ixpWkh;VsNI-m+LBg((ct`BusYy7*aTi{ z>|fFjzV&`9LEBdx+HT@P)cjcV%9{N1`Ui(HcsjrINblv?bbWhkk-jk^GlEE}cx~-K z7962Gw$SQ~aoO2CaSO4k+E{m z4Tx)ZZs(MWM5E~MO2C-kZ%RQ5OrNg-3L1XeZG4)%EiTDCsLeV2@eSgS?#2_;pstatpbHyzc}YyiwhYSBPMO6biB@*w~O>OhFV}#B4AW z=!fcB5qJda3s^+FyYB#WaH~_^)hOAiK^8qM^)gk|LfjJ9;%mf>`^6jHvr~dqK_X!f zb}oASs1I^*FQa|$Z*N>OwY2`T@F+jRUb}Zx$0_*Deh$5x+A)9<7NO1*hnA-yYTk1V zb3)cS*WS8d_+44|hsCY|2ho(MnHp71XFqzB%0u z+$rMNAl`tAO`S|0^%R5cdTE+&fnlmJBg2RK8=XzHa<|7C*A(qOReIf1*q%}8g2?zd z`OO@$fQ=;6sjAH{_ArSg08PtzRxusH>(e8w$jAa)w)SI)y98b3X_E@aVq#PH+Uez- z(RTy1>*NjBbjHrGiv0jw1P=+U@|2+zV&hUa3i-DgaNB;}<2nk{t$o~Hz@RqCsMRpo ziuWU91KUuGsG8PG^9@tQAh}?pr#V!sPrE9wWgulat^WzIaFoWe+QJMCLU|aE&61Np zsps68aeNwi#5-mz@K*T~ZfcYptsmFwD=k52`%Ey(sLv->hdXkEf;fDBf)&?CGK0@Q znVDimo*uJ9%fMBGvt1p1%g2Y=qQW!jacX?*#dFxScZGzrB&0gIXY+1=cPm9NoyR`O zu_TWRngaC5N&WVs6P2=`jMzf#>PXL-M<=~Y@b?I;4q3RF{YOf*$_>#z>y79_=&ysG zf9T&zQ!>1AoqcTXj~^D`#U2%-IWSc$_Au}7*$g7P(3ms5Qc5-r2?mC+HRs^~*GnJw zwIc1EM;M3avoff>@}686N6=~>niA`#?Iykn2$FX#)MgGol_$b7fx_PFPt4Impy^fh zs;xA=_v6ZigU=kh(&>j@^K`w$ImZ^yrKTV*6jZc}%7^}1(e~O6QKXyeDXAlIXrc&V zA@6`#_$#xQXmBxXcrTNhgq7Ci(*)-qo6ACJK|4*LcXp|0wSVy~iF7u4Me_J+P4KM+ zF;Z|nCg?EE)P0F5P;Orv_8oR0-4%VkF2NT22$jRun@jfOj6t*{X0+7ZW_qp9;fz|% zcdp9~F%I9JmpNK@b?7vaYc_0~h6G-%_V21T%NQ0@{S~k13uXzX+-j}pxWKC)c1rhd zi5uzNT^v!QkhPqn6E$aO2r|I|g>XJ4zvr0*l{=UdmI1G(ggiW=T^(|})rlt-c1*h~ zNU}K|m3DLr8(cs)it>(=M^N5F8I}v9tjH#VZz6*JYev*lq55utfXQ1g>_`X*a`IdFe&fYMn?dHj+PGa9KK6g1|7tZ z!4){)VO!!@p&%hDffvQs)P{V6^pQH~?&uU?GKf@5kt$vFdBQz}brMkWG{fQ3U80j7$B(p-(>W6JUBN4 z<8iX(SWw%Gj9@oGV@1j!)0d-&`mO@goI9}+1!+W$)zFKXq<~Oq0<~}yoOhTfMe+ObDyg7WmE`MZB(hEYghne6AK48Xe~%f-|{H zT$R)++hQW+T|8as{`5A$i^p;u+;|alt=?D;S4CE_CYN6|Vx~AP<-HDE0?`7i!ZM5a zV@3;Hqe92vQH-lkuhW>Y5sf*mFk6S3g{j}Qyq(bBR8EJ$u{)|WN*Q=l<-QwKh^!3Q zt}&n`gga!1MA_3WoswFskToaQ$_T*+!Q>NjL)*TACOSrPsMf&-xpqKn68cacfkSY1 zzX>K~nTP!g3V0i&>ubvt?Ux^1Atv(Tll9=$){TN2hNKmj>;vX{*fY^ZS{+pqCk(FYLou{R^ zQTBa>-pTcxpFx2w2|KB?ZqMuFSS@!3i7}?9os5hJk-jV@)=Ba71Q$BcF)v;a?{Q*M zR_v-hA`0!RlkiZ~Z{{FYORk+Fj6Pi8&m|#3GsS!9Q0)211kIB4vy8jUL))I3dlW56 zJ!K0cY6-#+s%kUg>*imP^vh7&lBt^wVGBnyYH~{+R!XDhztI@bbP*=a`K0`2{waqq zrJqXpmBeC&0os0@2!|;%(nkR`Jmb_unpLLo`#x;TyxLqx#5+2*rOr!YV%Qk!FkC#O zCz2?i=(^=ghAg#t`J5KNcd!zDMbrzMoK@O)uK8?&CKau2rv}qtUF1snFC_=2e-l!c z6Ow|g04;@S?6r*dQt@2KVU@UPch3mnKFan%`zl5d0cZ#g35A`V=^Xd z{QCKZN))}csxBg<&G>8sHc@0^<$(HTQzT^C;G3cMJ0!Z`0LgOfvyiL(D7HWVQWK~ zV2id<_+R@6yx&7$9i2ZylNtR^PyYj5Qc@cCNpZeNxuvtS>;vCbou?Nj44%acYW_ut zdCi+qUaaV##5IK%DOsKyNUpi_!!GcAFd#RKVtf=Uk5!&;89_U=Bnq3@qkGb4-hlk$Zmu3 zJIl9M8SETWaoW#^k`^rld2RDXYg)wzB7&{T3CU1$ISO>m$)v0<1j>fr)6Ui;GsI6s ztJ37)+a=}1PZ)cAd7Ot;EfZ|he{TSl{w48SY1iO5l`bVT>`9SIuI_3Fk2#RS!Q(u; zf5 z%%Z^KKYkQ{5b`)a7=9=m4wPe~9u$Gt$to8rU}*TK zXY+ef?L&|=gprxpgPFyd?OuVa*b2d4RTUT!VkZMC=|6D$?-v=*7+)()T5k5>8?&XA z>>?=tslI;y5AdEdDaxviDl38Iu~eA(YGEwP5KEKtg5N5GKNvZ9u^H+oH>Yl5liH4A z%xk-ZBZ@lAR=a!iTafVw0RK-SW^SV~SC>@L#RcbBtX!uUDp{IBP_|g)^8W;-ziurI zF$kOIeVGZzRO_bp8n~~-I;T8OE7cS;Ns&;A`J8_`cK>;Re=Q&d7tyH;i&uia^1fty-i7(!LGZ87V~0UhB`8dC443Vi^lr9D=B`R^ z;$rE@hRW2rTA^@@uX&`s$d_y8LG-V~^zSb?D;zO}o=0qmO1|Z`QomBy_xhd!`b(4R zzN56%BKqozulm!>e}Kw=E*NmnpX)JCCgz$W&N~uRC3k^Ov|Fs{Gg?I4XWOH#yUn|`-IHIN=~gW z{oEyGBbD~zPtWTQH3AbbgDR}cCF%NrJ zN5Ymu0OIQ1JXURSyX|0im;9oR)aQ?h!#_*n85~jy=T_QlF(#51uzQgI}vCb`vV2#dsx@&x>$E&6m|hyshe5SR5uT{^8agEdDhB&lR;Ch?!I>d%Jr ze}lz7v%5I82X;u(>Qa9BlCR8($z^%^+fP3zprhOjfH<>2>7!zHh4^wAk-x!h(=44B z(Nayf+Sl(P3QYgaEBYNiT(Q~7_L?SX3oTUVu)UWMF6Ch8Y&cVTz)hNW;m4DZ+Z4E= z)wct52t)MRuN%ULij1N3pFtXNIze3v#W zn+;IBmvTRIB3T7@GMqyYBH-A6v;p==?9-nq8`)k8x1+e7?MTKy&@CM2ODhcW|3&Ed zPl(8>f`v>I%NX&FT-k6E(#b0V>BDxW*U`6cOfV_fmI(A%$g9ft@9IM2q2%DDwf`U9 z$zS_a42IRGRH3G9*jTHb-j6A+=3ckd1sWn+(g&oVYk>Xa2QRgf<8dgyDEgi%RD7|3 zu0&~5d;j*oQ@g*Pj6Xc4qwB8e6n2rmE{z?9R=cg~54t_PeP5li^fx{4VCHr2zzl3a zjS?ePAVD%#YmA?@NXRW9m-#`3Iuo*cpwmnK=IK-Ef5F?IK>>m z#RLvgv`QGj0mDs8K?YRbN3{lgcwuFrXsxOWVgt@`K{yyUL0Ff!0DmA1ik}Z(KB9y{ z`Omo)2FvezfHuQyLD)a<2Z0^~$IF))@PGNtuOn6##(!GO!u)eL4k!!j&pGA*a5pIM zp|l@xxaR!G&Bsjj=g z9ycUpNJqm5TXfwNXLnb6M_T7oQh3O>oNR+F=n~j&J=Sp7>oHx^FiN9fK#;xBc69(H)bwf4?Q&UqDa$yrofiV8-jmMZy@*J3U5xw&~Towz2cP%elucKy{c@hj8TjxuZM?ZZ(7!o+uKx3QT)H3-=6}7pvl3IArFaj^ zw>;JWl_pJ^n3$Avf9Njl>2O-mC2%V*E!Ba7O-wQ(17*Hl{DnHuVBI%$vZcld6VY-SZu!|20X(QmtMXlvh#*^tYLAM!92nHIsri z^0wMv5)7XjWMFz@nW4^{W*bKzwyg6D7`HH8uXBN3^L1pd=pCfKo@zd6eE%=T(GeF& zZLVs)FW;+tP;u?K>P(5(Y`=YC&Pgy4Zcu{LaK5TY++7p7mbw#t{HelM`rFHSakg!w z#`q$c#{BQ0#*x*&_hkPj5!ZWOiNrM}QYj0k+Tj){k-fAubPK!g!Opt}7Bfd!w9Yma z)^IHo);?%^iyrOXR!6(&Cp3>kgTRW8tzDCLhANeOHZq2R;Ze(+2woEi2K%X zw4Z|v)WwF<`ZJc^C5j7w13fWQudQ~2ZpWGt|?Y= z33oNBi@m(~igow3bV|(h-XnuBuv9|e7!~H`R_|2-n(x9lt0t`wy-$7hqdX_mGQ9ql ziBmG};DWB$UN?SFdeStN9W!=4Qm<9i`J2?%EG1}sY>bV8WH-eNR*Z8p zgz;D)c?t&-waVdn`~Q+A-P+N(Y(U~YwWE#CG{(QJWO2*EyX1}m2J~8xTP-XvPYx?* z%ZVDAq}LmQNa$$t@{u@_ukI23@1=A($Y$IRIhs~?7#@yt!X3EnpFhEA!h?7?+yj+E zq4lCOMYRs$lyI|p`y`)ajwXhGFGad<(AMS;yn8L8Hn4S*SNEoFUzyg>e+nghb3NF} z&W?xGyd`y*CGu7e%r~(&Wb*U>>AZ`CTe^@yS^PoKY&lN9@>8f1*&4=Jd#u<=jh&Li z?HaqGcUvNnljeFp3V#V6-6ST{Lw<~+zNILw-eT6ENo0WjM2Rs1H1Q_!(9*L1soKJ| zs@`kf|K~XzOgv3TXS-V}1I5~@S>yNL?Tk50(t=Wh#{7d+ZNi=0_-q}%lIpzC{(JR& zyHU|rWXLUDGW1NBYr5LL{vCEKjhWZbs!H#!K(gwlVi()ryk#ZfZY%N;bSXiI+teGW zR6PQ1s@WyZc1~dR$oChEGIswhaJllWBx&-{6b@fOt^_+U$l+^2SzVTAGN-BiN6de> zF2#-Guxj_*FkzT8HS`h4!F0;9&Msn0p$=mBZ)Hy`MOT5zl;NT9QQ#xL6@n538L~C? zGQu>{#e4ZL#Yz|Bh>0W?Y@{8p{ydH4P%&fp&gv-90V}WhFEWG4UJ*Wa1liwK`F?b@ zb?PZ3yjMbfsC6tcg#Iu2pu{LSlF- zbhJUq7$5#U*Dr&vxjGiY1VK@Rv5w()d@Aw(GETu#*I3>~MmeiqW5B4eG0h5Bd;0&P z7+%03kRXzti1r(QaR~PV2*bIR^40ub^#5Rx0&PY!H2ftB*iV8$Lf;4;?*7%s=?yy2 zVQ1`$%Wd{C{7TYQ@0~!a_)q68)Qlt+W`JW`R^)IRCy(_YG(T z156hl_HRk~77fCvkc?Uk`S)dFXAI15L4w4e|I+U}3PGR=S_Rj?4BY?N{y(<=Pul;d zWd5gQ{-@jjXHx%vZBiFc;t2lbCP_WUPVo}01&g9K(wDc?jQn$Pee%@4a~HUW3_F6dvdvYimeE3K{jHi=hnV>O-6IxW{z7*q#zC zf@sH?g{GsB;jx=*Bh>y?yvw9EJM?P16^}S5>qTJ*R{`+_hX8?_G*G{hWWQ`}F*?^Q z!g1$kCb*U*fKOXjCx(*$$D77Ymui={89#kgl)7;!-;$%|PIyr_4DP|u9V|YN6zwgV zUc3)$o7|JRw>xqq8hUm&>?n0T{*5@DI6JOoB&Z*wFC{-e`@dN}@NMO*0t5em%;#=3 zqY}4?`t*f>H6Zcgp>wAcTPgpO$iu7ac`7|2m!Xbj7(kF`4R7s#+TMAMSb9$2DwTV{6AvvO#xZ&k4KAzfQg8a01kCHf|C(&6EBJ*F?f&z@H1){TyV32V z?14erVjbk%yn7lmI}&)$j@~{pKFG@1DoKAXv7jQvUPZB@{JKP6bo;EGoQH8Q=AIQJ z0l=Y5fS|)#Vg7$O#AA>Z0M|Z#H{7%k#8#&xjILkdC~z+1J#{Z||A+yRtsJVFww<}d zx`xCna4&WHA_wQJC7f*JcKs2f7+V_uRN4p1`yGTWbLsH!UCI6NA6Dr}QKwF4zA9ZJ z6-7bPsJDe6BF{T8EYjZ(rw?Jowt6oZ_f-t1?OAL}OPdr`|0K8 z_H&9o)!)Bwb3D~d3R~K13k7HR9WqhFHs|o!>QrbYn%$RoWv^c6+*Ed=dM95ig^8ism4F8?aa{ox)s!UeE7Gl+_y+ln{&eD9bE7M(`X5(PP51~FQ zZ?kcC1rA9kh)5G(NOw_?zj-^_S&7*rR+n1Uc0Ry;p%i48B|G|r^wxi|Agiov60Kpm zpIDz(OwiY6>Z}GIPd+3sa0rjnCp|8}DYMJ*WDS11mM80OMrX<|nsV#Rz8dz-%8KA| zSvi{rcyroS*M^BnHubq$LEMz?E#faaz&KSL4c9~eL_wC3QmF3O$&3w85m7ybj3eF+ zN;MOG!P;LfvsMxg1_%4WtGx@zj~$efQl~RaiOC0NM<#UPzJ@q)uX%Lp4mHx%m}^V! z)5R148XnU z%OWFar=IM3_EvUsNN(cVbZsa!*EpxR9of6Axb4m?=ux|ErYuEsn$%PToc2`t(2^TV zR0$#5<5Hdb%dxP&EoL}^k+#CNN_HKJ#1mXnpr*MH#y@c`9Ttv=NZM3_TCZz?ZOO3S z$|A}wTwcT~O!$n*)AfKaU8A|KwCHRcig0v{x_+HTBKz@ns=)O%N7cJwR51B4LBX1` z0IMUkV2LDp?rLPpP)q&skBsJ`_o{Mx;?p#|BZ?mD*>9U@wBLe)cksBeMVcHHy8al1 z;0)}?o}!dU=en5{J_lEKsJ*iEqH*_B_SO{TM|~rs&V~9?{y}KCq_Y`qfr0=pTWWiP zy0Z4kLVdaJvMuvuHLqj!gM!@=Tblxq;aw>r=R`if(ccO%c$4HAV$I3xz4Xh@ z+6=QkB|XekJlt@y0|Y4TU?p*sr9?^-ZeFA!1ncZOG z3HbbLA223#xFZYH^0-&NS_2(aH=v6Cr1mV!TgyY$h6!!9@cTykMTfa&0%2gMX5Kw- zn@=u%qV>Y|+b&F*t1L?caWdGkBTpL@d&SfiIemjf4-sW$l#L{TTm6=%bprjIYK}*h zK~nO#16XM{dLJ?VlT@jff>c*SVzuFe}Tcj47zqbi;x4$__ zq$s)5aB7vOre?9F-q%~!|2w+2eZJ?OJP%-0;@?-H?nOw{GGmnBzA$cu42Hoa;%Wcr zidXtl^=toU7~gd8{gzZ>=gHTyeP*8$wj>VwksB5Dbt12;OL#3SC$o62aryHKUXobT zahh^L0s(8d|D&lRCXaVcTD;M5<<>S+B8$V`>q0J%UP^@k>gF1ztI%)#@Iu3^U$n2T zGc7zReSWa-18@4!4Ye}?oA$u8j@}fy$u17hD=I`!%hH&^Mg^<-jN6h~F6yd^3JU$R z4S)nGL37+oU4Q&ox_Y$%$)W0~ic<8GFXtI&#S8Zp&r39Yw%!h&$g$DQTglTd0lmM2 z8$(0aM-c$%)17*H{@= zA-YoJq*|#{%gthXd9-%9=kw>@$VeQQL>blu#63O2uLTuCj^9nAt)ghMn|y}pnC*B9 zX!a+KOsOeRL#$V#*kjA_o`-0I>vvBeAeUTNFn*O#vN9}6m3#5@HM8$#RV2cIf&31c6&iKHb%i7h%+sxccqUe5L+KQ62drFA*r$Ptc z{_>DKIOu`%M5xp}uq0knzQy{*CHh<3^yN-R{QfP-lxGnkPvPuheL;6Vfa|{;Zm|ej zd`+Bn>8>j?R&km!Rup?Zd+hG@h@!qA%bQh5?s{vDsjDrQ)O9LP`j`(O*(?&ldslyH zWSRSBCy*T3Cvu;yq2^{J%wZoge7|hQ=C=k-gCbZE0cXyqU0_h>-FI9KrT!&v>d| zvh@&$bd2qTDqj*e)!bPAhwRVs2hjQ!sOxm=oxCximuv3vQIWAibg;)MQxx_`i25|M z*!XByW$w6p)F%3~LxUN)O-D8*+N%#w#%BW_3Ufrt}Al{Z#%sS*-{_={xFK*w0rJp$@%g! zgUQz`%mtedY0`q;2;n|@ASP6L_ge&zX6AZh`76yV^#R9s^J*PKp6uiPhY|zKt`1*6 z*!e{co%QxAyompt$0{0om6^^|96R?84VB)F9XYx&0`BItAeG;y1AsvUkde(86Y;a za>@0p+M@7D)_1uK4qHQlM-PUyywp;50pTK_`=mqsYi^uy9EhCjSr4~fNiH=rgrEZN z_Sel;gVC#Ix@)aNKSb?gTNdM5{qccFZSu7kBL%Oc)bSV6(LpymSTSbd z9nS3K@gk3VZrFz=TR^>o?-es? zY3YD~X2iaTX+uK;+H_mV65)1W(e}re=25;>c;OBUYqPpl6_VF&_t&R@a4YK2a9>@iOK5*Gl{jq$%J0}(aEaCt^InF=9@-J#vfhFo}IwExCcW-~z znmZuh@9lZ;{$j(xZJ?RvLYM3R#34V~B(;jYC#!G%>+YB(Oep;DaU#BkKfnmOcbL^S z{e;$SEM%B^0F0zn@e^Yt#>9|oNqXX{h>NFm0~m4eeG-iW$U>x`wP9Y7(~nE5I&K|* z2cYVUHHLp{Rfrqb~*pS zApD7PGgWdquyYV)c=j)3GR7LiI#|sO%h`7hXKH$yl7eD=V`F1|y||>Lq`3G%!tck9 z&;GK9PkB^S6lc9x{R3v^0+_LbAm&~F&B{*sH+NhHq^~9D*64ZcZ2*a~4vYdVddP)y`K*;bZ0E>QA@co&OBne|&uWb^V=1l*_ica4HVz6?QZ~ zM#b8#K5K&YsGk!E!W;x9e2~fETBDSO#R6d9jUGQvNlisL-L5H4I+WKpTl$(1p7WA` za(QxeY|OKrK+s#D!=Sgf7bu+QHos?0Kk?iB{86xT-M=@5KUKhS=wnYZ@0R>6;+P_;TW+GD*M=nrx9W{!Mh_P9vO=Q$2M{{(vDw)5h*l)q+~oDRr3Vnn|4(A9WX ztoGRp;iJGmUe(cN=U0j#%VGOE)Y10*Dtj>I!=tL^P zfMepbL>B4B$;CA;4iDF#^{-nuWoubi;p}}FjU8TAp^!+3|4!Rg$+IV6xfo*`zK-Z2>VETVA_C{JsgN$7~A6m%LP=+`b#tF_XA7De)o`lp~l z2H)%9-TImePoKX3nqTMrE%>Sm{V4ruFp;0s^F)u8x+?+~3$zy>zJ=glA)v4KeZOrTX&cIEI|V-Eq7X2fZEaKB(LYDfvR*)UD9q z%S+ih%ylxIhq=0P+UQc-`cxc(oLbGqkB@L-rpj@|i#y|ZI*J?lM!So#!`nUZ?N2Kc zzm`hBV_E?U;25HW?xOFrwPbwH-CDnqF5%y(m?dE{yH%=i?l4`~I}`3O^bSt3Rcg$Z z(7@okzuYTel&24FFw`X~&VN}2dt0-!S>0b;GGkq?wbpE!ApOO9Ay$FL7E@-B8nRg# zaf@TfbCwu#H0#xJakf2fGHi-oj8mTyE%!KrP8Q-(_@uT7A0Dq{dVw{Wb_K)5HQMVO z={@Ah`jL;}zP0e7P0r8z>!Btv&$5qTxn;I8`2}{pah&`wN-j|Gw9u?;EJ(mKn6MdW zS(x=cKhL2(`{HI18n8na01M;A?hV`fC8#mKox5Z`X%Cw9yd7P2ne}p~bcQU3{x2>q zq@xo?T4b^kdtVrk@LjHH?#=87&UL3R?(RG|GmQSS*PKfY2RdaWRpaERtAIQ6=?@Q)A0GE>ss z0!EZ)#lncA=A$C;*kKi0q;H+aNqMuh#;0e5nl1IsI2F>c-p*?0F+&@sdDmJ^PmJzge zt}3rz(XnG1aJYYqvk}Tzg!8k4>)cw`Vkv19(8u5AWgD1l?_I za9c^aYe!ddO`or_;v+}3CRVhggN1!|%qHdBkC%F*92N>p)9}jM?#jgygKSqRy*t&w zN5zFqps`z@TPY}u2u-T>8Qs;ftrGcvjLEm))u~GyTdHoB4wXNyRZ>`MKR7+!mqZUn z+Zhg~tj*|Ucs7-zyL5HpT9A@__=>*z-hmTO!+d;$baMp}-mw@FL~bJenFjefNyX9Q#Qs2@=%KA` zQQfL+hOfCe%pA&=&D+_bb_=e5bt*^+qXK5HSs5n*nUsy!>P8A_L}L!|demxGnx zx@U@nOA1l?I=iR77!Kn&Yi_akD58AQ3RIdket5jWuvg!V;yAj>>R-DXCVATC=~6<5 z8>Bt#`8reXX+<)_V|ZJ>kx-jhFTrs*B`Qxh#4vBk`OLgw?<>1U5=g3mFbh5o(*x3*isBG9-tqDLmg8RVro#=iQT_vdv`NRgtFqNo zEoaVNzTLo!qrkf6Mn-9dXg(=fd{!X{h5CD=7-Y*u!0aoR4sx5^$~$Qk!L4{exmcYo zi@_62^hue^OQs^UrfNa9lMnxc+`0{}%w%$QPfw9gh~pye(Su$#&MImlv_{EVl)24K z2oyq9B>`?I;5x)^&A&Qf>_9@r7I@T2iXyVVSsHZE&oVapP%r#Ct?w%RLYDfSaBGXy zcW~>bN<%$+yMX(SKJ%}c8^!b>;&XQFn|y9cUda-DkrX&vgh9QUhxqtqQ@j1vN!g}2 zaFK`lp(bq4Q#!7+)@?}iRma_jW8wvYn|9@KTAOFc9FPv<-WW&f_P*?QE+$c1(I4s0 z%m>AA8X-Q%drK6CZo8jvU+hK)*0k6z^mCB1dQ5+Y2B6se$tl8zn6$pMoNa+Gt%HYQ zv6TmH7==DB*J{4|nYdPN{G+}$vBNaWt>>m+rC$?+7z4p+hU_p{HvOSA`j(j2tde$=}( ziwi{Jr!XiHiHCCrL8jJM;e$q`0;v);tHx7zuktq6bi@4-g+KJF4N}`jk&QH@pV+GS zjb;`yYvtfaDc>G5R^9+-_IC%j-Kb#5Cc{W^`re$NN(=SKI;JQA3+L_@Ub3L^oZng2 zi;LZL;0_6jc)h-mDd1#Xza^z3gmGa6O@48H)B;{M*LagVS1vALDM9}EkzocJKUZ3qxsNy+0bbw0-2W%rT#R#tXJJGI;bq%Du>}!JzghMxD6DxB9?LBl0O(GNw8PLu;flKVa|T_=!cE5 z>-wX?&IlnlCc_Wdr08PyiBGNmlsC#W&l+Riy1!>4(`Iw1*yC60Jkysoa{BQ7ivL-$ zWR45rgrecby@0#5&}@G%2{fKm^!%_sP2*E$Z7ef?j*_lJosHH4r*G0acmx zSGwQd7~s<%41~K6r(9DWd0sqiE}o$IVRdreWRw{s{-qNx*Nvx&+H=_4FBZw;pMCWj zvUTP`!lc4oQ1L!#OGr!1^Zaj0)h_(jf_26UCRqMbZyv+^nLwi!n_$Q-5QyG3Fpf z;OtGR(SfXv3jt!Rw#-zT*QQUWQukbRyGdLBLk3B{w@Ffcx<1##gmB}LlJZ8Sxd8Ou z?2}Cr980JuS~r&OvC>?RU5^a7a}7IJ#GmCQd49Z)epTLpIDT=CjQ=dBrC=2hD$@i% z=@%)%#Wt0?kpnn>rY}~kl*$qM(uXej)U2R-vx6&-5f>^ zYRgJy2gV+H>vY`Hh^{{l&0lNg&DWaMeJI@=kcf&fo_m?!_&a2#0yqipR(v`6LUZa-7AmfJ>tDl$y0fKMAPz)2)+59;r{Ua@u6ATBv6S6ylH zH%str=#;b{rJk`n&2?fREwyg+^tzZ*0~-lIv>WT5fts9L_1?W_S9N=bV@bhlG~!^O zk(8eT>r9SxygpL`BswO_F(}YvxvxDBL?8Aa4KZsPGdFfUdA`ze1mB)4LWRgXI-PMJ zg%qFYlXTg+Kc``}ar)|ffy8^<#i~AseU2V)VY@#96@pZ&&+T3Np@~*5Hf%;Qez+pE zYI9uG&8ggN+D^(+wCx2XG>X%k8OPTSaKmwT+T^%`g~?hdt9jkbT>TFCpIP*|1LNlprM? z%a%>liTrMZw5iB)wMI zso4*yKL;fiTdJ?t^{G4Z)P1hMwdxKjh1nQTqS8at(^OVQH`|I+PL9=;a-cK^x)Ctp znQkSslPmnnHf))CR_qDA_`ZFS%Y2EgfId?0B^hQ2{rJs9oLP~9zZ`}XQ6GfQZ>XzWHw3{{SXqGtRfk)x zK?Ve=nbEi)9@}Dnp2$YtzqJ&BCI9=} zvrTx9M;KQTc~-;Z^OJX0ifyh}tPPIRvQe(G(E_`qGG@N@J;{8J8Va+8^F`;0B+VSj z*xxVD=0`a-5IyKd<2?}<9yG;xSa9GH!L)lB%yntVZuVT=KrBOP9BMzDWym}YJkS=| zqSK8W8ZqpdJ4a+ZS_@q}ACoc?yEK|fW7FUk$Cl~}eB`igPVdINue%}f70u8-4 z%D(H+D};0QVJ+!RW1vlMO5Vy6i;;hQ;gQ6J0iy}G>yEKhAUkU7WdPL9A@-aQoe`MD zGeVSZf|gN$=lBxGGc}LZ>(KkFcwILLg{2G_UE4g+5Af%wTFBj$q00r5TJ(+|?sL?= zypa+$FoDvqcXaPB6JNA9JxM`_0XArEiX#Jua_HF$yCkfPmHteqZ1X`LSHp8%btdC- z-=WkS-ROZe!l2pqccX%Z-bT*@U`_Tq6e(4s)S!)ATJ%8JL4k3!kMnIQ)M$D;>v@#v zu)~y@mVaG}|A=w!kEDdxk=Y*c9kJ`9v)@yy73O%JHBKRKenONx>-R?v)ZY_G1{7x9 zhHWA3@fU>*zdgmH7nOE~{;WCbaYPb7vJfb^qI(S#%zi)}neohucRXa?a>)Xd=*TlM zj@h{l=l1FeL#P{k3#W=9d%;KTZAW}Dt%2#>sztrTyLIdE)f4Ks`kl>74$beiZAZ^a zUdM96547T<8yG+FB67h6-wNq)SBbuQwVW99SITz;?5ECMmx_FWTRFSc!2%f$)wc<+Tiw>=n)7 zj3C95Mb6BG@TcVao=l!lG)P?CYJk(9*O6ob$AdiC1~>|h&{-A z&XhA&fG19_EL+9D^kJ=MJQy#sX7)NcYB6UO9hZ5UJ0-`p|3FURhK_fmPnl-Nb zTwfly0Ry4R_0pw@WR{*WcF&Y=oBZ^R9T9-itbfp-E~;-eynWZ-Gz+JE^jnS2T}Dl8 z@1+F&b!DwHx+ngqQRDL}rF$1awD$QY>#m70K}k)W^U1F%QAMr$4`j~}zH5`#{^xcd zI<@0K!2^KL8&=*`)!4fJXr!0p+D#TF9X{``tOf3qWfXzS;ePY#Gt=?w`#RwF!b<7& z`KJ0M)@Ee{LA80^0mh8HtPqF^?XbObkpy}})Ry4!SP3l6BKqrC7_UcHu{sRao>R}v zBQGvekE|=yHum@R^|@Kt`XFCV%YwCo`{~OSDLvVHX0Q+lzp&m8dtsJ-i{QcMWRvw~ z0i3Z}60~j+crtqh|9U_q%XK`1(4G@oJ=N;wjT*6(p!4=PPDgjz`6dO#<)<@mr| zFtWfTB~s>j!pPWVpl4HI0On;D$0TP0(zH(rQofIBf=Ve3{fjKU1h1XE^_frpu(J51 z&LMf(RQD)C0tS>Kx_M8Wl99_x?oJ?8&qBf-ua2x4UFqG)?eC@Fr80gC4LsYhwn#D_ zPT}lc%z+PTo5-zn+`{%H9ni4srzSurhc|b%box194xia0CATZtCuqnPas112h`y#M zDDW)PXHyhApKR5a^iZ_UEqOUgmlIm;At3g+A?+P<%grq2jg^E^!UfK ztdF(4>iyBr1F)XCKgXk8>AiSgn9q^xG&SB`ZYM3c&kh>X*Y|bGhKK??(4;^;{6SL0 zIDAfqE;;@}6-XtIxp#I-rF|W|eL@TxnM%6%)T5n}2BBZ)tY++uWTd%VprdAOw}S!v z-~tKT_O$UBQe-|mD3J63uWh+5d*B4FYiIfb9w+~@p+ys_!SJ~%pId4((}0B>B!VRT zfJpLm=Q3RxZWl0DyfJ+cVFIM{$q(J1^ahr%>L-jE-?20G!eVu5qS z9-Fb8mp{^E%GLZM`{e;Unvw3hN4J(Y_Dx{7l-f)T74f6+ASTxlD*co~0TC}8c*=kb zNsT;FTcldgEo=Z72(TpGrNC3YZ-cZ)$;m%;hGxVqb}Pokzs_eLCzX~w{Z7Qc^Kkch zbJ!5aoy!D^7|9RDwaub_JrZeKv5J!>YXqI+pgV~C7RVm)P+R+`Cd67d3`fN@9Z+0b4xug%N)Q=o)^J&Vib!@II8z)n=U;78k>f3UF~xz-oW;lZPno60YL z1N9(G3BUt5d8MfLD@7(09t;>DE?jZu*WB;$F@I(ZfgO85shviz^skM2JP)OwR@j+i znpLOKljTtSdhUf`5Y0F4dK#;rn>cYRK*nttPE_GPcP+-`0T$43YDb6ivN<|_05rHv z5Qx7D{XS5Hv2y=c3!jMTKvBW%oWB57KjHzmH>6nX3`o2~>A3s9Fu6xk0BnaLM)<2( zmCgx6^zLJ~hb6K?$c0A$=YF=+e2x2BA~@di#pP#WgW$_lIR35s+@Ar^ZRoO^rC(&@ ze)STNDb<*Bu)~9L8(6&Wtq!S8ONny@r0-`T3^(Qh$c^zbL9CeddFjt@0pMCT19(}^ zyvaUKt)s2&Wc~Z%oE(M(>^k;$HwKF=SDySD$0NYg=fwI63-;Xpp}E4Yf#REO5o2c6 zh?M!wKofvhZ@}hkKWQe`z-SD?n&Kw?iFnhw0q6s7y<^eOf*+|n*rZoBKD)jBIG`7Q z<0tnH0A9Wa%dq`IRy{FrT7jlFBOe|AL;z&y0USEqIy>ekEy-M2+}`J>$HfIs*O!GA zljjg4KT#vP%M$iAjg_=t6uUEIaT`Dm((ds}>IoK2{DFb}Z?ZOf-rtdq`_!^Rnu{&yz^6 z?AYvf>0uThF1C?W%Y&-Lq?Zy3>sHNod)~5LJsHYz$E6#19u#1OiV2y;#g?Kg$jQmc z&d$!tqLY*~-S$a`&dfYHgL)+lG_5QyVrSKmV*C=x&6@)T&;0XqrzK&J)2HrUa^t{6 zD!7c$4h$CDc64-nJO(oXe)XWNtjw2IGC=GIbd-_E@N0ViDTtL}0l4*1KIh9XKKw!i zfTrMpqC&#S75=&K>-!fPfcU+Ll#l%lw)`oX_64e7XCS;$|Frw78{GGRH-bwUdG~MB z-ULAHkF2SJe@pD|&WUA!fQDFrLhf(W9u73K-7EBKE&d$+PeFKb=~g=Mh5wnZ-^~m_ zfMyV>F4(`8!=I7;zf_w4BtPp2^B6Aqf`RypUGP+Xe%+y`ptO@yUr@IBQKCDCU=5#O zeVtDq0$Ru-DY)OVZs%xk`^@%W1fH8EBnfS;et;?gsyJ#@YcuwjmzEQ5CHe|f{RGv2 zif7Da)phRPTc0a;*}>phzk?sIS7=0FbX@&$ ztw*EmGy!I7FFi6G0xw>x(@OZ3+E^@?H8*~*-N^hAT42IV<31#)cK*^PM{jQCbh&z4 zEG{WE5M4UVOps2R-a5xoQdfpWR0I55;APgnGFSB_$n?0knuzPS!q>7) zz5o{P=;RoVFq(T3IJ+<|;OTiAI&;_u?K|;MQ&V&2p#>iHhAsK`q?P_h-sS<3ikb_L z$EBjv(9$r2_~Y+4{?PO-a4iv+I%{plZtWHXJ3M=hJ#F>*Gq3!cs^e6ggo6A^An)jh zXe{Wc7JO2Iuh)NZGSmY3>i2Pkq}k_dAVfeO8pT4XnMO+9XR*oSaUPh+ov7)?z=_e3mYmpH(uR( zw!P{d&|6%_KN?OM7Jqm$lV*`K>|Mule!A7L0ugA`?|=8e{(IviT5A)vzOa1SdBkYg zQur*aJP-f}1UDnAqVDh_^P^3_b*;c?g=QUmoKRocY0r{uz=x#)XSos9QR{Ur#u)2A z+7{S7b#WI9uPHFzIjgI$KU>-A+&1*}l2u+Txe5>P9vcmNBazj>Oq9A8q5cGXXyQ3n z-&@qnw%vBy8vt<2oA$l}0CP&VT87y1isZ%d>R`@u)4k?|81zuHKg>16hjiS{+O)r| zX`MxP2T;M!=j7uD0x?2i>ZN{=#UP&0Sf-=Iy1MPHh&wB4?+%0uRzM#*&C^{64j^IC ze*VKReMp!H*1V*W%%VFRNuK$kGLtyb8|+-QG(yLTqN>Xr{mI@vI~eV{(8lFPqvg4I zFiv0qvKFe_q!({0WZ(1t^ zHv1hbbHUwz=gg@j#9aNYCnA6mB)coRiQ62a(iF$pJQOPdg%RE+Y&?ZsH;$atHLb61 zcACtO>@N+l(e@a<7MDK-5>os;^v+ULr4d{(Ahga->qED>kPIt<*h5OL5DC@`B#{yA z9DIu6pe+d$an_&ZoWuHdI&&&avf{2^EL1-ya*^0M7a-R6=^lPj>)ge;ms|^g|H2NZ z?DzNf9vc|}HnwzYx?Xap70dK&U0?Qb(Vfrv%{AaR-3;!q4;j2CzJ#+liowr;J^NH? zby@dZu8I9v-XMb4K2f}y6n{R?khw#(KdV4fSUQ^%Nh`cq5#1*ugRjgscD}Tfq~|nO zSI~9xhH7ib?KvsY-06_tQCuWqrs`PvXvWs{d_HYK?VZ@DTjE5?_SVe#Fi=FCE9|@Z z0!kbB*2Y4t_rl#2yD`Z4f)KXYp83rQJ`|H0s8UfSBi zn|+m47?tNQp7gL#zu_l?P@fgQ4+FcC(hixjKyZ?Q12y(K$H52Ih@ymqX#t(Bq4j}) zv(faEuP;%xy#PfMrR}Klr|p6q-sQ5&SDWRN2+{u>RpPu=tAg{Z&h zkUNH-liNDj?phZ5@y$#v&y|F_JiiinSm=|Yif$h!YFK=sEwMjtGz@U7He}as)N~fXjKFtV+joqCpJeSU?Mgg9I?NTdUN-zTH?5&!8!ubnXoOMK0Cc26JUalg zA!lXQ4`Blp*(%)|ItzWX(n?w&?ER@NJh)fU!^pwkEq9?YCr6fOmY=dSA)1o+*_*^@ zioo3DJi`)R`9AJ$re4`@no`H`g9^q4UA-syH`h%y12pH4iCAcMp7TP{E&Bl;J-H39 zSL6vXNGEF9_m-v#HFI3u7y{X815bB*Qw1*zzAQQ(a$0f;y1;~ne8#eK}6rstJPP;TrHdVFNccTZLDokcW-;P(p&-wZ=! z`$^4H2AZawyVSsJ26prcdM34_XGOQ-`vz^JQiZH^CZ($3+>#eDvGng;_0)&c1=l7N z=KLOh)+5@PHqxUK|GtV69pAEjb6shCW1;kt1oxR*5fzHvlco~Z-1jDMECme_+{Ax53N0a=*h;ToPcFThBrj31uB)eG zehV?%tSa8IUdCTi6gZyVAt-ia>9s=b;+aek1XCty-D*UQW4E$RPp8glflfP?;ZMuz zKFD?!wHF}E=g#atm!vEx8894<`)TEPh`MiM)U(*Fabaqn<9Cf27n_Q1 zOjI-P&8ph2e2HMr@RpfnO)?E=sC!5Fv|5m#QL)#s5Lg%~>>T_y-+wL)e*GIZUB9-v zO};5hYLrz-B$(0CE~BusD_O1+Fd7xCcM|BPxF`*6mzR6Vnlhflj?m~V4Oz5G6@16x z8xE$(ne$)zT5~}}5il@tE~|R*OkQF-z&J8cpjxD)kAnKF;5K*+-nP!t$A`HdCyj}9 zRbKunpWw#0O6sq~DKovTvaIZ4a(}*!(DUMr|4(KkuDzk=WdP#2ZiNjvCYf^Y`V)3SToJg}~{Itc{m`(Ca1H#%Z#} ztys;0xM+SiY3N*oNwm=8N~=)T3hIl}QAzktLFhI|4cQ=dLt|zgOie%az7;;Rs;I5H z#rP3@G>s6O)#q;yZExQZ6LM9DEw^bMZRe||UM45CbAj?bj_c$%ztRrfDIQ5WjCApN z;dJ|_hwvUo$LgE~z@S*8RHRIt!7fWnOr^$7nBRh*ucxMxfgrv?{7Iv=HtCSQt(GH| z)F3U3%HTiQRdAeOBWD1)3?k-ky`aXZ`OYV&C zWNbTtHV=X?vd^pEEbAuW$;W5_>f z+~zAVG(BCnId#2`S)a-RF>@ABc{s3(lQ_sg!M zh#A}<6*SLsIjlaPfYk{EpC~Cwdi;=XM-xq(z>ICGqj5Z=0}+p@eZmubs2?~TexfI- zZL*s#hNn_>cJx>)Uu46N!1lWEuL9)wxc7{tq_=@z=qeU59XIw$7oL5~sLaRW`{~sU z026)WiJ_Gyi6Rr)N}Vs^^eI^QDdax-gu*t@0Ut-8#x;LbQ@j<)_W!W=)=^n)ZPz%Z zfJm1}3eu%?gM@&TN`r*bjdUXdBHi5~NK1EjcQ?}AbrZjJd-R;=Iqx@q|9{_j&$!1q zo{jfrueDcP*SzMOE0D<_p>dBf4sJSc+9+ybZ<^bEx8?@LUvo|+6_M8HqKL$Ir5;g5 z$G#&vx^Ny7dFF)STkyQyNNbbG{G6ua#YR!XMJM%a?0IFa`9!}UqtlBnwRh*#4Mln* zk->Z^fljvv;ylT3W_1KOni83rZ)b7=?KH0aWywMNS|I7|NMV^vQ>gW6!qa#zB$N(W zZdo~2tLRq#TSvpY_jVJ*wo>M9hjLC|_rRZa+YRO&)i#f<&imyuY10v-(dHFNWA~g} z_E7JRgDtWbRAA3!))yjbm;?Da&YFq+Z%yy370zK2JhISq*R*jqdcSh{7C+JEqvU($ z`kg`4Uak6LDXQ?O@~FBg?ta@~t=#7I^|b|?CYejedv?4isn|3O41Bw#Be$0W?l~pr zlR6aEG243!poF41O0r@UjR++SO=bQdy$%%nBMv5*6fz4&RpnBgT+X*wG1Z z5cIO>7Hq*sei+G`b~GYgENK!J1$2>ND;<-bFh5)UbjJx#?*i_hI&los=6w5K(+eH9 zo|ojf{hW*6Dy!9=d)!vwQ@u7)yji2JQeo0zZE|*Q5U0z#W!%^PBoY1f=H4Vz1!r05 zWaR|aAa+vkOSw6-&umPFIHlAuY>eg4C zr2{z>%~()+I2n}@tqOCphbElb!`3l)WhRv8&+lF(Xd*OrXZ48Vah8heOAP1crzbA_ z_(ER0pL{i%mE?Zw80c8P5+*ge3c1=YEYxan+nFp{Pq3DCpvO&L`9I8kO9U%Vd7M|D zg%aT6?o~IQUZf4dBL>pV5FXMY2I`i--;G{RvYcNMkHp0nU(X{bacekmJU?SKAiDZi z)lp+TKCen4zP@UAqw}`PU?5(dV>9`Sbj`7EUEXq=EJmF&wwAcXj+Ot~eP`E^`!^*O zBKENiQ-mO|VM#Of;`evHF5$?jIUSa7*p*EQG$3(W8|c5h3~>up^$vRMUYccczSGh# zaF`Nq`bCYVmw8sTVx+FEuD05$tX*MSOYz41{Ot1{M8`UT=ya+Y`@E9q?#ULto5A*j zKEzCSrwjRD?r<_LI6#QLn%j5dbiF#!>xtsOSo8$sT)yjr7HF|5ij)hDlIx!AkEft_ zWDc+%j4r(!h;`*h-1?CIYzeD#b7SNC_wO_`G@Szj6V96JQs~^HcWZBa*7%RxoxF1s zY0O9w*run5zM(CpU4I@wD<40zYpWA>)D+_jOt6HCUN)VC!*Xl&ydq?!<&9vO7Cw;2a^%1z7^veP0D@nQul-CN!53pY<^%d zs8&36dd!IW?KT-*GwzIze(z?fq3{StXM9674yLyxbgc2g==k`S#cHcBcAt)iysE6N zz(5i|q#1=-dR1~nWkEq{B+36u%`f0xiYsfeYBmm_+Fw1*4%|&lK0YnTdK72RSPK3r zFW+*#zBsJgKKv?xs*Zt_*N9HyJlOlmcB_;?kn2?1y2C==#6;4~WA8|@h+W3UW(hGW zf=F|C+p?(cc%6}_0>{GldeCR(TfYiFo66nn>KMKmwvVip#YbjzX2MkWJw|?}_ytdX zDk(cSnXidE@*(cU_pZ+O1;H+=n@q`dDZj8-D#r3k7(yk&&gH1D4!JL$6y^mLMKCam zaXT|ww#IxMDA3GqNqSed?0r5@{BByVgr|no_r$j)c*{03$a=?%Hav~Tc`SC;C(lK2 zWuplDOT=C>tLi0v#wh=AeKO4_0^^I5L;0X{oL+2g=d(;xMUC+xy?6mDr|t)hzW)Ft z0YGC{g~g(ztuwZ_&wo{DZ?9mdDl&Hnsgobru)-$BNI1PcKU7joU4^PycvBZw_ABe~NB^&d26C@@uW)rANOp;lIJX&x{~x-8oAI z$mK7P>(9RgK$Q)GAj$Es%T0J7ytIS7=K0rsl0gXJ`uQp0e_!^{hFZkcxY7{+H+=U= z4xoJkFTzm&`?AmvC{!^g5P|le?E`M4)PRhXOxE>prOza|OOrY2tAAU>g#+!LT!`+! zbws8f*V_2O!+PG0W)*W$xThImNEc5nm%Q<>BCw~tgceWD7<(4JH*8z*3cptJ18BA? z^TEzNk&=0sPbtuL)Oi=Dx&TE`{}_8%Pe*UNoN*2v#T zE3}65uYTNw7LVKsU&H<9eZbhj0fMY7nAm??KxSxx!F*#Tg#6D)`+a+mQN!uV{tJ8! zf$Ew6m?a2P|G3`lXt2FlIkl)IsScC>BLV~RB)COWbJt%&niDZ?=eFt@rW-ZQ7fmUl z=~q^K|H!sC_-XpnM9e8}UbD7*QD`+|S6)1iJ6-_vl3-e8;_DRO+^?qW zNhjxCZwg4gaqIesE$Gr+8jO9vMpflEfb2mAx&|>7mjGPq%xVfbvq-L~;of1b=~J zXknOFwVpaHo@DN3pFc8$N&7+5{BTE=vV)}TIKM*b6-4XD(pMP^cC&0Xbyk|JA}w?4 zkFpKTu<{*5wyhoYZwCvtFmgp>|J)w7;|a_f(Bp&Q&n1*%v>Io2=c>3J1;4jrJ3URP zrCH!^YrB2LOU%wW|Ge?z$-rrjb%J>LY?)1~h1)wevtkoo|3BkQ=rw}F!?&;1K3hTO z^$QHcIkl=)`e^5LE#*(2z*GDLg@9aIP+ z+Uuz9#)b3P=?tFM{-?|@;fM>Qfocl>&(sZJg8d?tC6#onpCV*pV&dxR>gMKVWMp(- ziP==qc*{jBy88PuZe1cHmSOIc{P~H$uRIO5UZk?!BF1Lxv5X1Y)#mts14wC)j#ymW z3e#WW5pm>co2t%VZH?8@ma^081iJP0@{&Gne+3D+Ly4Kb&?>CN4u(9oK zS1qcFkiP@5hWbi*~ms;>ZA)0VAtpJn{ErIxCZN6ovp9xDBa?GKd%Bi zzRMkCLF945jyy*q+JpOF>o8zvlAarW&wVWLK!KbwqK+>EeyP-5faH8Edh38bPbU&la1m%Ma<^l@z8ZAIaz zo2y+z?JQ9Vrvk2D#g8~dDeD)(8__eP<3&#PGsG67g|m@esRH0-|Gb$3RZ#-4lq960 zN|j*x<@^~Il$B;P8a3tL&}PCdYpJvDRFAuTCx(|M8&=jl)JGub4Ld}u!} z^-?^LVRmfrcCaQ;i>tKM{e93>{3u+;^JCBZ`!`%{d2IVJZ|*7cDIZ(C9Pkta$rjsf zs3lJMUG+C7AXnaNj`A-A~7qZK>Cn+x+*d){_QY_pBwk^)6WZ zNZI-9U#mJR%6wLQL)A^!B+bRw1}Bz?-fFh91G#Q~AG){D(4es}ym!JJfY2^>>Gbk# zW?huIQGu~>p~;?uqlvXhuJvq;@Tqfd4J0i6%JnuhM>;qECxB}>kHKH^5+Vij5Avz~ z30VE%JgO3z)nCi#jRMW|`V`;Epa@WDQaYs?w!c92*YusA&t>e+P~MazLh#ti>}*x` zcIxsM}C%kbR^%0;*ke}qW&u>7#IYW zHz`$Bms`1o<1Wg?#Lf_F;P7uzwTd@(`jsN6v-L4VHLLizU-oGY0=nGFC{g^^#t`nVx z#PIbb_*;|$S9_uxfW8|@BuAmD2GtV%)s+*w$w$ys5^K$Dzfap=L2#MRp5NJCQ67mum_;zJ-6l7127U5(t1s+dAQ?9lH)*d*mwa3Qzjgx`unVkj%-q!tNv9eY^QL3wAAG z(*fC;`AA?P*llZsgHwgh%2j-rF*yYx;sB{AhL7vT&gbSO-%_7gArcpsA`yYgO~ry6 z1X{fT>MfnQYFoONloy@?2Oj9}?56bK??04~3Kg~)PWXJ?6#Y)cv5no8rUY$t$(8F_ zjL(KT;{bYP0ge3V@qnPgjbvsb88=9TYy`pTWIg}? zz@ku`sLj25I+7#n3Z@kb!pTND*z>oe<4HUt?v0(6_^C5h7r$>B)?8Jwx=wh*c}i)z zH>O(5cnIl-sO`n5R#^)~cKb!+uLlr>D8HFg8mT>}`mB(|-?si%@F3&GW%Yog*;qa8 zkGJnWk0u3OWsdWX8Av%p1e}`&6((YjpZt((oV^&IC%MFuhcYzQG*BLwWDj^bO&~V+ zUM9I|wOvAeohF%63eQ*yZV>JEAT4?v~5VW`6QX&VPN+^AuWitZdwC51I=k z;6hP|-eKGgN`jePEeM#EfF|8$--`2H&DvCj$lPG5n*5+OcMyZ)A+Mm)lAdOn9DhOg z&FX)JC&`z=m(-f^chWaZ2^_zwSGbw~3V54+f40xwKr|YvzwKq?CytMQmi!4xys8g# z6wjVT_1DUFK5C09!FM6a&tKKUaRmvO-v}VH}EPzX!J0}d+@i7 z-CsHAVg*Rx7SdM!mDgcNfP6)W1I1hTzfvrW19f1|VDaWi;ui}x1(%fJ9c z!GFcZ7a-%^si3+1*JXsaAc73(6Hfg*82_vB*C69PQ;)d**JU`Ue`u$E>gzuN_wU=g zflLv?_{!~Hmpz~kHM%9QX@2+UU;P3D+i>sbra$D*%^NvAC|=_93Y4BSqkOrFq9FL| z8AI9tgUjIKPkCGL?G$Q{wQ;JOZCYH&AL5ROB}k2Y(i3SYK@&qBdqj<|wXZo@7mSxj zne5%^%S(wE%l4`MX@G|=ctmRTm}@8&@eEmEV!!&ZC>MA-C~Wv>U2ezaTa0mzd1u2# z{B0|`0j2mj6P`ie&J^`)wNdxWt2TSS^PchY@eZv9Wti>qLbiv_k8?Dg&SQ7k)r;l4 zAOB_v34H;NW*{N*78N{tBR=4FwRwLVsJ&dYAi2>r(Jw^*z-C2x8Q%fc`3V7E=Zmbt88hY1`Q$aJesf5f{BmYwgv46))($}!+1XK{T)28x4_!C_vU z8{>(DZ0cYVz9&2YZ2LWq{>zI$#Kz;g_8OhUu75z#bs}R4(cZqeJ?j44ebcl4$AaPm z58OVsr9XbWOTsWWS7iG4F%@v+jTjrB*bv&F_Z((;=FoURKTvl#?S3N{{M<}Kr|<4| zm9@s0UUAvun@EAcey~VS^&|GgwJS>}(6_(l9@#6w)pv zglpuM%+p6OcREX?Y^+fs-LJX}uU7Vb-AZ3 zTf-hnrAe|qe@q!Q`C-5P^F0(gG0_(hL7z_Cs&?;oTo0f1N7dBWpU+QC?Ymqs7z`wE zThFie_9yN;E?Db#@cW!k@tz*h(ZZ3F{r>dH3c*!Q)d-4ikLakOwaB;Cf@m zUi$VVGIBe5VpJ*1;lp)1ZIr@u=+FQ9AR)mbFMhJ|iVIyp00$$j4J0>Mt#)hJO?}g- zvAZ;rPO7M{u66>k$+vIc@`1AZ!z;hGuu+UmjpOKn0vZc;-41>O#-wtazGfU56OA`C z(N};)r9Y4~mn!0OdjrA8$8X(+$7fVy1K=wr#mkco%-hpC&It?k3UjleG%+Beq^F~M zdyTwe^n5Bli#Ym22}-`Zd~4bV@=U(z?SxPh4^PPHw*8b| zG%69-v)yVNp>$+G+FGsNF3@?lyy%HuzwAekH5UD$NYuxN**%)r?kB0;*!bA|RkNan z%k?`tj)Iv|x)a&O2`%@V%y@G(U^xQ&USV}oC-`P4*bSK4MLbAzVPMs15BT`A-Sw!8 zdi-Lk#YgaLQfGGjXedfyy4L0L2!yo`Qg#IkwcPJt^=O`2#1t4zk;BDehQRa(#0Oz{ z(0G8#!3|KHCiB{xpYP3q_|D(NaJkjjr4E>NKs8`{K!4Lbs*B*@1(p7v0fqZhK^@Sy z%2?xdBWrb+$rBcuwZ`UH@ani@f#!;=<-jFCHs3$`nyNU1=ItoYu+@5}?7HZ*mkh-m z6nB`lLQ%6{=?n$2SB!w`71Sx95gz^7j_u~RXsXnRh8q38xVFV1KU{h=zJtaVG~S0d zcfg?nSjVjOaV|hz8IC6kL>aZpj3cubj*gDT^VLU6fT2$75r!}zi9tlgS1syy6z|sd z$8w|mK|~<|V}O~NL&pV&(DTF<8S{GA!rcV6LJ%@F&<{Aqd?6sn#bK#j{K^*I6iy{m z4Fu%A@_L5uogq*7$~w#>{MwdD2=}Ck8txo-3kPIf$+BZ@7dGFpVWXh4u}oh$sf?-N!d7jJ?1o5Z5T)qDG)rq6-;nDFO;jk zoE2f>w`r}^B!=*WtK*H`*otn8*zFA#%biP1$i6x4DYp|jSqNJVt*(|}d+K1c;%QCM zqiH3+>I|o~ksd6tHOjme(((ge_``Im(WSYDDdf<7-u>=oaE1hrsWN;wz>}BDR#2&w#-rV2~gfXbis%#qp;sd>Y|AQ*JsSXfU0kNr9Gk*Zf%} zT)^#i44t&bWxe|>zbDOcBTeaz5e{yhU1){5Vxb#FXru}Gs$S?Nr|6b31wUF~RM%bK z`L=%OQ>KzOe|&I|)LO$wHxw$lB%Rc>pN^Wn z>fE~+m)pIkOicrmO+07;>D-=^QuZ~4Pb50Q1Vm!ei6w(meL{wAsq`!d3v*Jo5ij_C z+2pS3tJNeS&#j5LkPk((6`X?ojq0B}cAfiCA$wz~GLE{3N1errq20_2=m9+y;?WX? z1)*nVY3-)@et8Qxs#P4tZaTV6fuVg;$1Tll>Rs?B01NDYk(#j7U}ID|p|amSn_B5E z2-zuFxL)Jzb*EZ*D2J_*;vWye*NHup5{bR)p?bsUXK_{o{O601RWk$LogX4+#;C7ks6Lv7bj~&) z5?R8$S>TwNTKV{TIxf&o<#k22tMy%)*Yk0!X8a>YFxaOq=Dwq0BEfrXe9z(lyN)TQ zcAtuiD6pyVv7gs2r&H!9AAE?gHit>eXd_c;})#}gBV;in=Ral;eSWK7&$ilotpHWuATBhH`Tb&6qh|DvZ zFpqyqB(Eo+k1_iZp&V@qFGXAZDyf)o?dv! zaf)ANv8VN+;qd-Xsxr!lW`+A9wYPNObvnAj<{pxVh{0hss(NMTJcw=tY=^!Ua`$(R zuASyJyGoU0x{en}qhq4c_L@gxx_wu>?Wt}x_Cck&TDN}6u*wsncd1@e7i?wHh9`0O z9#`TvfB%^(_rv?~f$vXYe(jTPZN*{;EEFl+jjc<4zwFVv#mZ`e z8WMP-@fbqTzKN0{Ae0|fr#rFIZD}EU4z`WVL`4>CP1JutzZ4a>m?Tez5uytEg;N|twb(R7Qsas^gGF_Z2fXgI)HtYxDpdr9GR_?fA2A*Sy=B0n%~ zGi(N18X5^7SDEoqW<=dhXWX{ty;?U{n-m`!!*jdy+M=w~FWs6*+3tkZ(xqBpC-}tI zcyFkWg~c90`sS!qbfsj}Vvj3#O`qbm=Hsa%=M4Fp@0of(8O19<<1_kKyO`AwuXLw7328&AcbVr2dZ zewOM*#gr9dR6Yk?#L0me%#amnygl6d1{YPY?>P^mUFfR5#M|fv@j1 z;d;x>MQm_4#u+Q*&|f~H2z%3jM^;xrZgl`ZgFcWu3aX`cp#O(;3(}5`m?zG2INwkH zcrU|25F?g#D46VeHYEdNF<d z0PN@K7#k`!-IyK<$OqQT6{1E-bTM4(ym`Psu%5`Z4-<ji^QUjj|E}3%oE?Ti2yG<~cA#(4rY*=I@xKG{HOa)0vnE zuz5Da9kHc+CM#fSA+LxYhetfbHir}8?6WV?jjQ&EHu_|M=%Ecvf_Tf5(iU?*KbKS1?Y+$ ztrYf{)NwU#{U-I}7kbp)V@Y~vKAE}Zp%PLMa*E%b?mRYn!yJ>HrttBlqa*EMyTW*D zcTc9lsG-Z8DFJa@G-B6$?M`_Fp@!j2Jj)|o91k;uQlu!Rd%uzmj|>F6Dz3L0I2v<) zG3#Z7mI&ff;rcC4X198f8_O_$K7ipDY%rhBL=FtYfbHtz%eB5u_+SVRpgG zcC+e?z5A2$c)Z32yUFMr{ZHt<$>Fls_jz(=%cV!lE66b) z2O`Du0$X+hmD> z5(?=PJrNk}>)r<{dZ)*R4u{Ny%s^kA=7{^OX@H%P+>_aN%>?_OCusyvf+nr}{R#BO z7?yB#asek-Zo2Zex+75=3|bY>Pw|y49G z<1f%;bOqVNPd3!GASf>hGxlzvb6hMGm5`iqJNiojeQ+re27!%v#%?2Dk!T;zpEl5N zfqu9*vC5(X?T2Eal%GE&ItWW%2j~2qs$zIhJx#&bpz_L>_x(HP{qqMlQjo{4=@hd5 zlMes=qbJ};|NE`qC;{95UA-X`f|RG$;Bs|cVX|%PIZ9I|_IphLqr^%gh0Vt{*Z8m1 z1qR#UV|#kA+^zk`9L8 z`6hqFfQ*H!()nKVusuFg&2D6z>s1tq_3MQeSth!1GNZ{Wjk7-xjR$rJ4ATLM&>9%0 zI5Zx?crr~#e#1u|8g@^M4)f(uMdzY&mVK>_srzFmetP61?hdbJ$n&7j<_i&Ynf`eg zk3r~c)rjsUgIhGycSRgy)KIotSSdS~)8wr+=zH&cdf6V! zs8(?vIi&pOF0h!L%wW=8j%gVKrjUZ7BG6b^oNkT5BccHksE&}j>Z87#f`a){BYD8j z$W_Y!z^GXP5>6^^K)d#jCtdEDn5>r2y;`RnNme-uUe7(B z$zJxNa3-}hy$)OI)2Omq1wKaOV2g6Qx%z12Ju$c&#RzI&YrU})F9U!it>bJqB$MM` z*pRUC21ki*;T7jBf^H%bn?wrPZ3w5ti+iGEI&grparYZM866!}p{#Q{tv(qBIw8nH4NmvlYfoV20c6i#^VchX23hm3&Ov#L2aDT%$gWQEtM^S!HAu>; z?RNE(>&k3x+r-4?d#h0+DVck7oZgSm3Sj4;kxcKdjPRb2Y%pSUpFo4wEWsP2t}se{ z4M)ezCq6`0Agc&P`d0Ud{U7K^fAL_cg%nUXBHUGWHGqIEj@_ho-1h$mH-gO=1-0%P zhrud4IlyXhFkc@ zxj7X_s+w3g^sm)bckdl-T}4hXCmv5Wy4J=rX;f&-Y}*EoAK@FEE*5 z_7{0(fm{sKPMK(Fxj}Us0+^D{XQn65(lxGEKr{6bpGy0JO67Bh(hqHZAY426(Simw z@P$TlfCZv`5ybm)-OtBV9SR+3y<}6XFqb)y(9@$tb2nexo$-0Ls0H#Rku&TlZt@TU zuom}LZqi_W#aisTpazC)h@>u-XHy31+AEGAY6QO6fZyT?{DQwDScYp*{)Uc_E%bjw zN7UTxb!Q++BA4`ln^JtU51D(b;a%6p*i)+{SYtP3UvqZ47ILdSQcW1`YNy^XSFt)V ziYKG_D7MyeX(1`+ah-Mi$Bxr!lf3;9mpebWR$`}SWNP5-%%tA~GJ7}R#wn95K>VD~ za4^M%^I7@Le<4X1-=0;a1L)FAgyMn#4OL8M=^U&s=*VCsV|`-T`;F=k5yeT*oup zY>cXE;3R3_>v@>$Udazqljgiq#kWO z7Jy%CI?M(wp4#o!;YW>+j`hD2uoxh|Uq--cIF~M8Th5$6eFPa0 zBxr5Ro|DJ99H0L?-0mR?U}I69rqJrf^N+yIXNYZNGaJsjHF*LN)o!@@H0?!Kc10`4VLxMz_lcG;z`ZI zP0&jX# z;4G@)Ztm>b(p|Dh9qEA9=3G4fCh)EIRwm~Ml4U4@U#SK-5CG`u_m9dNlfZkE z#m6f4=2P!2F}pF&zB_q`zU=gTDcwVd_v#z|+?Z>(tGW5trcVaTITxui#`vHO-v0dN zJtop~&|nPv0%;RtI8m`ZFijosFX?%KJ^3}Y%p6=IvkV-ARf|bDS7#KH)bU0wbNyWi zg?eN~c6Pq2=^Ejqk+n}R7#r5yA)ynUN6X77&JSs_lb=Z5YUcAgw4_#yj@!`E3qo?^ zswy?a)~(xLJ9LN1;pM5yepvbBdC}XEuUx44vpw)MF)O#Zpl%Htg96k}E3lBUG*1A z1#uTsDM@r4H}$8?tvTN6q(N~~*NEfrO@u%y?PJ20ibWYZZM8B(>(QjFw@0f3nY|jg z8q43ey?vr>3?sXXA}*EIiG%kmZ_Ffd+bAc_=F7~wl%t}6Q&!6>p7QCpb*D+R%$5=j z<~2K0f(;AuWyk$PZ7sr=+iR3^F&x%w((h&iR*8Wi-i7kO85jgUHD&f4G+o0KJ6vA2 z+taFT($`kuhi*QEamhcx@*Dsy(*(#O`U$zf@o7#PKR-($Y6!7{7lDz(HzN>manM#9 z%|PXL8W^cA+xPC-9=A{Wo>xG(J6sM~?Z8)b5o=ZRTvIl$)?M;3VlNo6?t)~4M>f{GQuaz_n@&3b+XIyY{P)y9beoJi-mem zM09iMRan09c{g2DYk;maxbbs~zNj#c+>&?=BjZ2wbP5IU)MC~|B$oWOA;Qy7DES?F z797YJT=u}B3AzjbpsHXV8Jh|yr$2k^GL(<6(huz*sMLJrZd~gxZpY~4F&Sqdu@et? zG_=!a5_LZ!x}57gL@G417Jak4OD@fe{>$0c$5r39{2L3Qy0v3&jaN6Z<7?;~`H+Eb zvHCN6K5GS^@N^WJ1`hix|KxhPChxEikzxVEg>xdR6oTzHJ= zmwpIQPg0%vs!0G$RyweD053{Bypg>-0qr+(*O#}ZO`@ozw5DJV_9COF{!959HBks^ z&s|8gp}Wx<^P7Auk4$O|gP4X*!ci@@Aupj(>)z|peNb17#;3}d=|lmvWMF{5B;Gm`{HQse>mzy7JjgVz zfjfJMxuWBnq{upe_w`ny+gnX&o6jw&ANJ&h*IB;4tCrG0#-ve!6PrDEe4zU-WDN5{ zw=+Dj#?!UAjbyon^F;lJ22)Fk)Th<=1r4`On%Ld)IP4Uz$T%zSi?l_IOZ++8rU9_t z-Q52XWdsvyOS((WlUPB@4+LEZj$ti*fz%M_p#>nS^>;R1(mnE7Jc}-fy?C#X9kj>0 zGaK3w6%{dRmal1jx5ADtU66a-T@$qYp$VDqdz!8Pa@aV?(Jp77f167y9Co}Mp~aqe zj}j|)SQuvRxbdd#_!x|8MI}|U06K!n_>*@^djNswQ}b@zFbA_!KY?}+U3R^)1@{Djr=dDV0go3t7A)-(;O;&bQtW|KQ z1XR~T*pf|W*HsU^<>ukZVRyVUOObRXJ}nRicjn=ZfRj@)U0C{mRNZrbWWfeb*pLL| z`FAk7O%%SPpe%m7%wE{<@&M{xv`lEi62vK38REMp;zjiSwp zVb9sb0AN?M5vsc{j4T@z&UC^zELX`*OOEzmclv6Wq;i|Q#~X~A-Yi*y#2C%~uA2v% zo^}d&h%ZzfFIAWq`j19NN04ypjtyXWh~x9*HRE#*nC&xWWAic}FS zm)8t~f1@G@5kh`&D%PLir`?UCpx4;=Jqq`45;dFxFCfRt1=ByS8(1}6m%`7{2rDN% zZtS-wi_O|XG_R$GpU&H!oPCaE?ss(G+Zv!%QhF<3zF-}6GhGozl&0|y(l^)&*0lIX zm9oE(UJt6qIF$)d68^bAf4UO^nNT#p-Vg9aABd@xnh>b5TQJg&CU!304R~aiD{MMl z?R`#|>L92l;Jy{cZKz#hw(p@MN9gJnZ8x)kMPwL1#`Pzw_xSvT4EEhK4Mq_$D89)K zhr60z&%ISkVRyHNGD70C+Cn{Uu6BBek8)ukp6*8fsz=Jtk6!gn+8X_~P9#qBUfFRB zAe`8)lZeRkM!xzFi}h8#+xc`;Ytm7h44h!DBgXS;Kk1O;D}n)e!vz zvwF$L)u{Y<<1OQEDYa6Qu~exU5{CfzAtm--&<#Luz~?bJ8SE^^lU@8@IO2EePmlJd z3#ji}MgBpFjv|CUCY7uA;r)Rc{{v7gMUsHY@|N<3cGZ7>jv_%}ul_7yn(-fy^Y^<9 z!2?lEsb%($-(%-Dv;*Kqp?c6o{S!&@HUIO5!{A91@R)G_=VNp#Ls6J%YvYRlc|$cY zW=5L7HUBe={t$&OU!|~V5f<3_d~F2htQ2CXQpk&vtt0dcKz0(6DJH3DzQ?ym(8_NF zerDmGR8Cd{qb?-i)2%RNdb(>985t~5fE&$@(neFk4~W)ymP|?`DKmzbx7|piZh0v) z4)L#FBO|39%ztJ$jlE?`9)j$VR$X0v4+Sp*utK7uj#S?^;Ns&S3nnWVC5wrP#VQo) z{HqV4|4Y_^fKi^1T3$XbU}9`6aonW@c5w}jK7(WdS2jehj+>ON|JhfcUkJf*J-SIz z`loCEfVnF%;JpOF<~scu?|;8l4{az2{!;Ke?*Hs8ZYt3EI*mD+|9;J)WRE}(dk&!+ zd;ibQ4+6o{`lBB?e+{mGh7ExK?t>wqed4M19}xJT(V+w0Cinfk|7{j{dH?&i{!CXe zI{tSM{xhA0{-2#&5o?uhip+mDjSy}RuwsHU>VSokuBLwwwBF@0i@h=cutGiIBBuKNO>Ro}DTQ(G{j|2ZsKu_@|>Hn~r^9rF@ z46KssPZj^fFuH+_gr|gKhX`HP8I;9JHFQ|eUj{TpH{`BW&SNM`_bxnEAN0fOD^96^ zH7bzu`_JD?FIf#j!qwPM9lNF+DE}t`OS>^BkT80#^n4i#n1RS_TST^Ecl~m|dd9qkb+2z_GAFU`8Q{;;m z(kaPnrMkH{*JHP>)J{;(TC2FV9@>>%@<6z&*C{q?-9o%)WfCPA@<{%(nZUEup`v?$ z*w!ZuxU{N2wln@L%^^ZqrzS^aq_%}6QSIuGEXG0}YZZE3&lz0KC*eOoAUo(+*9CB2 z&JVG0?^(pMUt=_Kc~E|Wzl~@7z9tESuVCM!@@w1i5W;R$7l!PJ#TB8}9ec`P3w-5q zYbhE-`BG+pW)azn%MgBCS17-!XZlbcJeW&^;n?jU_8`&XpncK9ze=8+gmC3*((ja! z<;LE>Z77g>_F!aI*+Q7g*=WU|F0Jy zEYV1H^JVad#M4kCU!Y`o^6NjQEScV8TTksMry?g@iIr-Os^v#7pzpd!8g$2Bz0gE1 z^vRH%)9AV2D`K(F} zO6HS**6hHG^6k&?i9z^4;pI8iAE1tFqHQYM9*KC|r~b1qdq<*AH+cV}C%tdd1OM!}PS86_oiJ7KT+pFNX~#pjNfbCP=xUOyyE(m?4m7 za}D-%7=Ic7bg8*qOIVRv7OmCu(Sa(3jYcxxH~?mg z%@kmDZ`2eQZ)~dwg4zWTdD-twhDxP@!~7)aGRmO^4&d_E;)8C{+ZMipGfnwtbs&SA zq71n_Hn*Eeg_0fOyc+?AeGN3nR$G99n;yht3{cyyJc92V16!QB2jH|79px@iLYjA6 zCx%qt-9o6}*aK^lyzK0ch?y}d%_{t{%S8k@kn7`tWMJ*$MY{D76luW0G}3;NP~qZY z>Oq>Ae|s?L{?BKCEHa_#99VbK+!;zC9!x@9F%9^6f&k?J9P1bG_M5}20j&MnF0hhB z6+e1JTUi>Llph}e*eqYG;TGh|&}t9y*GKJ90A28v-!GfcJi(-!(-e~`T7i3n%irLp z6Yvv25vC)Vs0L7!l-rxPhPi0QyoR!HL|NccnddWHTiXM@SN)RCe+Pi&AP1c97pBFe zVS}r52apeH$zxhY1qH80DLB}`YCu{pHJut(z8_4Z*2ywcZsz~l>Yk-@^$9ps)74Gb zOLe_JF~1R8jxV&y8SXGH2`~)zaJXes_@EUyY>gDwy69zsqEUorUQp#PClAp17-)LXV`dk+D_<4t^pim4N504J<;M=aa7kvR!O1DCs- zb8bIT$l0z@Z?RraUYPk@b=&w_UlDv4tarKxmp!`!ih7*9?i(0ZFlJWO&vkjszI>t*tdC|!S7HLL|h@Dc$pK{>E{dBC4Qy);0l^$WLE8=Qs z7y(Mw*R4o;ve%GesacRWNev3;By?&fP4np{lp&S4yARRya)|ifW0a2!Z4a zUqZY`8uI|VF?@5jp%Y5SJny1TKEe0944OZnMJGB6MNA{6md?_N`l?en>B$| zKM!iUz!2`k(PpJ+Ogb+2AhOgWEWZHeQDf_qi&w2&NtRfk9xDM#)_IK}2Li6`V#j>p zC7?J&Vl()nJQoy@x&~yQhi@bh8)V;${rSM|ealHEvUb9Ei7PzD@=e$y9pa-6g9!*Y zR80K&u}?n#`~W!V^k;GfQ9)^U_p5!)Z}1W+T9B}9bfbl$56%*op*QqFgs*LfdbXRX z?=3B)%N){*l;3oJsQLNrQr)01C#Y{=n<#c?Ti~=9{cF9SkO#g9)u*x)>)U@J|trS0!K@E9xNe7aWmfKjQ8d6Sv08s@cKVmt^#j z%S`17CNGO!i0RxSyVgKA2eEdHDIm4%oT{P ztl+n#a+-=4d93`y(bM%ZC^BB77hw)x)1e8sMN&$F^%brU;{!*-o3%44NwQC|*(Ucu zJwD1#bG0j+I{fFu)N6`hzUABlP=P5x)_%&L+FuUNYo@;db8YA$nz*n0g_fY5l~5n`4bVkl+@^sE;FK$MMwG(@FwDFh zkrSj0hta@mwQ`nxmOlX(uUglYs}}3kg`iFETT|$KzDHBBwY7^Ll+fv0eRDQ5Fiuc@ z^p$d7d4Cfw?B$7Y7cIY<_l4kQM!03T0DuJX!~&+pQ;PkL09fh!YQU*SRfpi5_CX7^bHs@H2Un+azE;qI73eft(?2lv(fKB5)aiCyrQ;Q;53_{R` z^82mxL`rq{=N~o9G&0N_hkjCvWo@AEzU9c1+6=&*ai3^nd-79h?yxDdD_NWx4yT_g z93{0704~o zR&6DXIzD2LCjH&0yqu$slLqQjK=tL=SEd>SXPR`)6G;K9CQulf!_jlepU-aV!r%7G zXPqOuPUd2KlhnU2gNi^HOvu^)*{k6&(BcPKNlWwfJlF^mZBtbxxI=RUm%maZh~wO5 zCJ#YRp{!&NPDOyrca@=h7Yk484oc`k0b=EhDUX*D$B!l|{Q(>O9i72_7B|}k)eWUl zcBLA}#fNgw5BaLdsrQZOQ;DQ7$T_`$Q|?}_O{`<69l zyQtr_%3C02$)wJs{kW`3rZhqfEULd!eli!3=C@2_6-yw5VG4V{>!LyDxx@gmjXcyD( z$P6-jxl_!mls+^vs0m50Tn4mhO^Ys3quP0fRi&KQN{We-pIW>Cy zi)p^paUDonQdY9(UUoNY7huiLuV`@gpxHPRCX-;GJMxIV;Xbw%M=pv67UG?g<)1#_i}Ml3-xZJG z3gm1cj}5>~NkQXUgo*n9*!$|gs3D5yNP3sy01y|LY%#Y zT2pS`aTUju##{V@X2L{iWhSduywB)99b};H>g>T7_B7dBq=HhqhI<&A_!mFihNdb+j3;l1qHk1Rr$*U9N+TMvNL#@zQg@!VO z%gx@veIMmi#ADfWg|MVhE8*~jI#_l=b40ERL-nRn$Ui&Q6H6>z6L!v_3HRPHRDzUd zn>54VNQ?>qE52BawjabqCfBhW)fA3|c0(+ToXhh9_SMZ^iDmu+*%cVyO4{A6>%UD? zOGB}}ah%^yq$=|nvx#Ty^g4c<7ebyNXrJ0PQj~IUe^XaGkwC{^HisVJ$w(KOqI~ku z?6kd2k}8GP2EbK>>Zf`+x3jzM)yyWzzpck=DCTIoM>oALXsz6xHhR8VIVfuF!Hjs~p4~67?-uP=;_^lg7Zh75VtH*hLTS$ayofoD^MWMj0$^t$b!R&*oX=66h$p5pLCy5-)= z2aTE7DpM($V3I@=HivMiP4n3pZ6+W}?Gzo=uI**m^4wrM{zQyD1z502SI2ue%1 zfObS^S-aDlS{7efFb3gEU7ouI83H&&pyR^nc7R74e zQk`-~U%YQTzZsVAYZ9(RVk#nA&5K!&w|}rM>#++& zeRt$vFjx)AnyepKVoXV)^seZr#inAWP4zbBauoP5G9n7&BApQ3wjrZfIM&UNae25; zB{+P&ko3%=*;^z?8lWB%jcY&Ys39v>_pmT#)c8~UK-~=`PsNsrbJ)4?-4N4%_&nd(M#q=pD>j) zIPCW*=ZaiRPdC-tn>lD1q^E6RpDg4x<$RneHPpY0zlfPt+V9V1WmRh?7&mo&c{)2iu?!Pd7ko+?ZAH`4L?LtPtyFNw)EIx7NK~R zx%(X_-gY7UeI+ z#p+Wv*&VC#-u@UU;%|Nxxvi{b&4g~+3SM@)c|#JxY}8FZql27RuJ)k}N_3&oEF8y3 zMFS2I{O}=7-4TXe zO_buV+zF@Qcnib*r-5tPd5pB@&GAj#t>`e=C-_HO(XWN5y$xAk7im9R>5IHlDF$p^ zP;}wOl%yqt$s}D(F50U?;U_tP=48rJY~}tHoO&4-euv4AuZP+Fwu@C(p1^MiJC}UG zr#^34$hxAT%%8o{u70Q*-MMDzezW@7ki!S$PpQ12#O$BAbejn@^)+9jynK_w(@)PB zL4~n}sG&mHh)8g99aO?13|%*-H@(tvu{R*6b9Dy?Lm#81z0GqwJ((ICq5ejHl_{oB~ZjsK5I;w8NW9*=>@E;`hA2o~`OieL|oy zWa})%^aaUb-!0lU>pbKfw?o%ne>_J***mFXNSks#X|8;exIFa*)e(f*q-?v3OID>} zDj)5%IIw%n&x7)n1KCBhVrt1ZT&3Oj-w8(CGRI3G!vxbsFTBdWa~s$0hObO%)YN-P zT$x}``6L$|Abpt9r2nNi!TCb=CKmTthBBzpYq$aCaO>@^@?7F0awq)zo&XvG1GExK zW|M5A9@W8a-bV&H@l2ny_v(;uyrHw|36V0~g5CR)kEq5JGBjMHwXxk&yAhsVLT z&p;Zc(*wn5eOkjxrusZK`FQYE52)41PCmSRI1RdH_@0j2R7F)IOP=n=#U$TF4+)@T zf8DQgf~UV-U5l5KCG_9=^)|prJ2;x4hr>FzVmZ5>1%4UWZ@|i3NNK+Es!r-}Gylc=`A311 z|A>|%R2kgYRs?5?N-gB{;`n}1QvdY*EmX&^(&#?+k8MFADMOJAD1`Dz7OTaxuR7r^ zVEZV`y#}eheus`sJqyYVXMn$0%cY8q8DuJH`x(RMBkc*h`UDSw__sPbEfmI;EeNxZ z1~2FJ+>9(=s4ViGqv0<)o0*Z`L6@Z61KD=l6-ICO{mH^**dd$|-L!Hp_f*79bAqr3 zVOln4pBe*N?rd^R@>1Csl^U>LykSoGv{fj({Z*i{J#F+Z;EdLbLRdeR2ZD_mMyMRi ziMn2wgBw#799;zrJzP(Rl>`!-^s0qvOpM?{fRodD$3W+uh+lfr^ZVID3pV{;l+e<^ z?#n}ClW>k%lJ?(=%^GvTmbq6R)h-7K7bzweg+et;#(7>hlAEjeF|JPyD=QS3@;!$O zauEDuKy8q^@x5q>03`%f#2Eo-cJZXseQL1Ad;{QP1<$a2E#J9U&ZAN^S5crb6jMgP z^*kqN^hKMHbjghT#*G<~8e;J(&lB$w6g48XMaI@=ev;Y{d1@YOe=s*Zt<63d5((!@ zoKYijD)gy9x1yk@+xqpTn|&a*aMDsd;Ag)}g>~rnzFD^k87`YC7kD!1w)};+i%aHv zL~%@8=@0hFP?lQC)oxdn-T0=@R|>mXo@B+~?3@AuFQs>oBTe)6sw4e1UyW5*qmg-9 zZOxasaygvY@ud6Zaf?`=i(Gho^L&uhU1&U830PWRGAtxHgN)dmB^NAncFz$0bJ(dJ5tw_sDy}1NG zSh>X4Z>P#F!$=D0&YypU=U;X|UyaZ07HQ=#Xx3Ly)NmSJIN(z5ZjI5ZXp$-}E7oXg z$pahtmLsX(RzFm*u0+Z2ZPy!e@oQz^MtCRCIaDd#`U?`8**|0XynRfKKfY#k6Cr*E zt(PXV)!+f@sRD_aj%lCBSc5AU%c&s(hJD-K9=^!* z#`Ls%{XCEK_B)S!33H^*Eun*?8_u3}WxR5#{kVjGHrvpesZlUwDiqc5zC+^U$lEgc z=O228T(*CTm*SpGSsfQq?3&ep49xR%wY^2FzxDEDCH+dpjj*+Npl9A3L;!!nzHfN7 zaBmyeHbB0%2|zwO*S)WEblI#ty*7GEzV$vgpDWy|aVZa*p1QZ2?@__F^rBM!xJs|7 zW4ByoHiGwkKdObH;TteNV58IVs>`}fd(Op1fQl!TO2Rt#t{`EFBGVt?1-;L;95QT} zP+*wA2XtMJRj>n}{qZ~|u(qC51_icZ9WWaP2Q?baE>VY?NY#mPDhEwHtD z3A*d%$B5{Ec7?f*kt}Dc&V*uC1?PxZu5}8>lAj06%NNk+LEkM?7e$$@UX%KxBsAyF zLFb#L;K#q&l2H+^+VB@c-@w?Wo2ao!@qOw|lrI|@z@zsw$v7?3FZkj0ZiZjqrd|eT zUsY}$XPud-K6;-P)|1X4>EJiR*=LjS?v^~T=B{jF9|-d@_idE9@^IJQICg{&-!EiZ z*nFC%NG~bWZl*Ag)UmocOUu{Dw_Mk`!xH&5pHRA@&g;bR%*radJl7Cx2Fz!w4*{a_ zu_EFx10Nz#hd>;E34l`msIGC?qdl~cdxv5Z|C7kQgsyhinQU9~sP_o!FotMS!F23# zVXL{aoa^!5Nov;K8l*9g$Z=BUtsCV)E%T@(Q=qC-ZYHWo7a^qgJns(~8`@=2=K%f! zDv)$ZK@I&Jou1aF z7beM5@X5tId6=8Ad)LuVO6%=>ZWM0)xo}ML<=DKK($;|%^o=hA@&TGgXSeDPn9qE^ z#W^T;?w-qx`?2yMri>lZg0x&_r9Vm&gSvyKQ<@=-@+c{0hl-Pk{U~-~A=^5NYJTfW z^gg!FJM`7s6HQ+P{4lhLxE7T8NZygAN@x2*_@9HrIqke~%SHordJtK=Vii!2UN)lf zuGps9dDd^iw!tZ|;O$1ju8I%6>G+~}an+Ylq{T0(fXJyw7#2ZqR6v{ZYYFZ}8SrGr z9vS|!Y00v0WUjEE|FH4$xA6ZV=!Qe)tb3{zPHeRO0ohxqR+ZE6} z`)SP!XpWwq6h2&>Os{)cWgyHZcL1*~fD9AMl^^ zWT>YeTeZ-p&3~am1V#|3A+L--zOVT6e+)v}j+fYU>=~qF`Ybpq;Ci+XE;iaime+ZI(-K zOHcKW;`=M4I})NR^0!k$4UgFhbb6y*Yig&8_*v8QW(#y)7-(o{7#VBi&;gw2GGs1i$yk-9kMYl` zA;9w;^bGuZ~ z#cB6MF*{UCUuq|T4$e!RGpIqt!|p75S;WbyDS7_%<&w+65|6t}kwaqc%;MtW>gvyO z?B)HscbmGOnbRr#?BsB5F6Z5{J%w)5T=xqr+`#2hqw+4#mCY&jn4X-B+Kw%G z)TL?oBF89>xLO>m67W~yjR$(N>_$4qY;{O3gY0o>5?=w?0b3%V*xFjZQ^F>Um_zN& z0k4`6_UEdE1VhP%itFHQ61{vf$<6TxNm7A$)lz41ILA{FDvwk~GNW2zG%xnw_&avt zHac&Y$W1z_zFm?^67zY0Sr{9Fm=Etr#8RsHx?yXu)k@@J`{01~F^l-;zkp7|GT!;3 z!ey_;vV4*od0w?Xet(XjH*}+Xwy4Q0BZ>j-y#M1ON3D_ltIbJ+u{sw$Jjqms&hkle zK&&X^@fdavXiK#eh4I?L`U7_p0GpG=NDTc8EP#Rj4(bC&$8_Pp+Ql7m0%5uVt)C1t z|D5fg4QGG}g96{>fcu}H|AL9VfPnpr@2(N<-=wwv3LDVHW`8{FfA*sP^*MtWJQ>)+ zT!Z3&uD8Fzy%2S^p8R$2UyFel238QD3y2%kME^yi|N9ukp$_o=cKn}bq(%Uj@=-%V z;jiHHpJBkL&ipm>|C~rb@Uig`Y#AT@zu%7D)&Td93H)&`AX5Rc7PPMl{};&M?_-F9 zn&$W$80pV5zC(T7xMn3A=KRm4^@fy}{u^BP?+-q{06VCOx7%74`pb#lM#mcs2jO`c(Lg`<5#_1`lThU~+k30r(+6T0PC2;@@zH^B4gM z2?;edwJsC{ z+SNM0hQ5^qTh|;yx+-DfK&VI^7Z3dc>c6nj9YWsxKL8rHZ*^SZ^=44MhM|Nb;>Ty! zjxGQNhCmHHc>*-eaEYFMyql!|_cxfMu3EkJIF{2UpK4I=d+ z46#h`(0O5nU2y)#xs`I;8xo|+3 zm>k3qq48)|AxI4B7_&7x_rI|+ndEIcUa)V2Dvb@5$UxS^g`V*Y+b08K49a@UP5kIT zS{Ok1kQR(dMnrw*KT;i!K)6Toi1J4nh%FRw5u5gCIh249QQHA3D{pyw_8TAy1f=1r z+VyBrQDK0cXEK|yY)lVmol*fjh$Ym<)^w@XT|4|sdd)g!iH)UFz=&?XUJt`Lutg$& zG`Q5e2IThmismqErZgTqBO@ndZ!f?G3Yo@B1oTJ?jp~o`v=(kh-)0_(L;#X>i^QZ_ zzB;9%qXmF8OgcJ#5*cc@EO86i31fiN!xkS9Q5#lXT)XUQwlxsfe#rez5`yI8t&OIP z&`gj2^ou~2*!bo13tV%jnw@XrchwpG6>E=XDmQlcUl3 zxc3JX=udRUK*$gzfi0c5G}-5m;0E5~y6mI31GW^00BCg2*&Gl}@knZ&bC`Osa@2Bn zysU+D4bVmeKp&g`;p|iMO@#?#B9LJ0-liJxnH}LhN%?5iaMQ3RKu`B;mtr(M2S}0w zQ(U@6!2AZ3(#<82FU>&YawE(2?x^SkU?u!8&xAw=u0j*o<$} z2Y8+_fa?L`1b|ZRq1il{D}@c|2O>zLihIuiwV%ACe*23V>!qZY%T^rFUK5KMz+e0c zG#sUYmY-fk+1GYE*m|f+Xa{z{&TrTV#J4=90sW>}iY4I(r zlMg=K>_!#mM%;*JdA@eOHE3hh!vNev_}Lv7wf2nGE#F>~c;l++3cYeXoMl;(gi#>+ zbijMkB%}`n23qZtA$DDe3h^Ia1QLz#?4$ruE@}x1O~mk;42nnX%3BEY2LyY|M#rPT z=P!>-e7;1ZY_dCByn_qy!#XEQ1gmBFU^@?>(Us`H(Lo8 z1);jD6;F;fzpcEbaG4fXbGayMx!qT7wJFnXeuuEESC|7(WXX~hYRMkdM_>dD`Uehsirj_&70fZ>^Im0z@XI%AT0LzeoiU^ zAXu{H7EL@8BO_;ou#0B9`#P=p)YrHZYvF4E!?ISh8zunl)RyMUxI~ztkj6pQ^pcfz zEM4iK-r>5&@!?ID)QAUC@nj1Scw9)lqmh7mXXdEt+g=m?q(Xn2fBulE70 z&u)Cut6P(=|Fus_urst?mB%#4wi(fqbxM*w{BVz4Fckq;_Mnw?R`jpFDd4h-?TX*F zjJcK3>wK%Q?eL3!Yc?>*J0BO(Mj^F|ciicN1SWv0jy;NecqcofXfv@JB zJEOc=PMP-=D3u@YKNU31(NY?T@Kr@b*K{*?eD#BGJb`!F2cAg$Su9$4DS$_EB6p?F zgZ+k~+3d(jfgBpO^<#?c3Ek(87A)tTKMJ~AKYX7u3WPqYi;@)}JE3?5%8LnSbn{c9 zxBGQ%l;MY5Vex{{t)H&xLZ7Gpu+OEv>pHq6E>F2{>L-ruIffD!0oli`pLYX zQ0H27528#Xn3r?>FCY_eB)fyoSCcg!T^KSXhCngj-G(xQIou<&N&AEH{1r|b9ptKT zh9IGa1=im%ORe%AN&1@tVymT722*=t0u$9G9a;`>h<3WkBRRW0Z}J3yP8&qDg?*7Nby%}rfqrTXc5GdS2 z=ft{Fo46!3I;7zg3Ff1~=!}=_GYiG+!4eYAJmz{qi1nJDpp4OCPV?a3LzRq zKLen)=gvMW%7$nVm$4zJ-ZIT8hl9Nj*z&fd|G!d2{v2?z`ld~$;Vef%OU^@ zTKAE_e|L{q*6%HdCx|Mu9(x_l5_SN(|aE+UJvPGV{M}wGVKqh*?W`- zbGkVsSXs5WZ@C6X1Z~cAStC+*xrPq$92N^fRds#7lZ2Uh36to(_Gp2=e6KW4O&yvk zv^cLKAnHuue%ijdk+wCl*u9JE-tpt9M<9Q-a zFZ>&HMtG-LJ{qF(RM@dJ1;vUMRo$>YU85_t()>}Qgp*B#&_u7ETH-0p*z-{TA+lj2 zw|_yp$&aaR)2a22FkuB)Q@GD~ox3hB_bfOZl+>b|`F$3-Wku*C(0yx*)4o<3K)1YI zyFt3$+owyt@yXFNkY2)|1}5ID>9N+R4Pufu-XoLY8gW*UzJ8nH)Ya%-Wa76a40l}g zwf(_%h)+8hC|@v!+0ihf_?I9aEUs z@2la{E=t^#x7z4yA?mtkv^uFMo)pf@P4Aak@XR+`+RGOBRFw3PQpDWD>~*Dk2}U$` zmiM0DL&k89bGWO}+6370am8**QM!;}xW6Odk<0}VIwlq|So*3S7bC3lHLT&lue-i$ zC46PSiVaQe>119LaVj*@$Y-l+3poFUfN^R&n;N{MS%Je3gU_)Bv+60odCo+ zQC~TRE%D5RkCW0Hv3JX)|0py$SZlT?R6c)Hr&r_UY&UhCjKI(7KrWi&JEbPWn4FN> z&AmAdVz!g`+Q7wJwCCDC&8RBtjdOgk8o7SlgG+dAWy(i~2pv10(#Ad4&GfCuo z27AUHcmZuCv6zjQz6W7$7*x?vS435d8)lTUq5#e(yrW^l3!cD2>S|kKo~~B!igI+# zCM9e@#-{gmoVaJUr-g_ituX01u~~~+i*2FZ3yehkLZ4VJj-So*21Slye zU!}l1F`~~lz>GPXted1J8u-xJ(@2#yHj}H)4z<{RChYsQ>9%s0Rx53e_ud;>fQZ=J z!|h0`Jp{u1%-}-yF7n(nqWQx4DlcuM4F8!U3#ISNnC{c-@Ly=Tmd^VWY`bFfJ*<%9 z7xY<189S+^mEx{E&1 zGRv6)r)>HLkg8RT9;(p>n@Kr$h>*Bd(-|C{e)PcYPD{<+<4qz3k4Q)#HPA^- zfb#MKf-JvZ-7&DwC;UO%?pLa^3g8USOe_qD~VV3{_;;fMqiV*g)k7>O$gt#}WTM7Dh- zj&$1!`FtNJ!==&g#;%LlE$`xJ9`(Fo>X`4el*0sAVVq$ZCaraode7VcR|eSka0 zdz;N@v4>qTep|QUX7ByBh&_*PYP=qxYLZ0IZ3iVch3srx>{+-ub~`-p%|B63pjE^w zwfgS-8Ok*+iQb%XSebyj4&t|5J_7nc&Jun7rx+{;gmJnqRl=5Qv-(xZ`hLXMTunCN zn#g=HQfU3(5U;ok(4UWBc~a#5?sGm1hmiy(Vw4M&Gu z!J%r0_0)egFjfR$=ms)gZcJ|s3ln5*_LG=~pka-3(zSDa>?>&#d6~f%tBDVZs}m$R z35m&3tPM-%k?JCuuXjR2*Ym=~RU@+){PdkOK53Jxrz~dH)6=Qz)9RC}hW6NM;YN@a zDNq4k-z&Pf7d~%E1fr_Cc`$@_A?1KI1{Eq6hQN|$7&4;w<4fx<*d5|LM*(yn+(x|w zt#AAn&YQuj=WsS?H$(E_rrG$-250W*MB6`6T(3B-@P=az5Gk%Sm#;LYeW8*Ndh6Y3!&E<(Vyr9}#WDCR1_ElQ=)Edw-s3NO{C0D&^5LDnWK`C%Di2`~=51{Zk+jJy9PE z46b1FgjJHlPy8{6ueS)ZiWR@!!MIDdk|nkrpA$oQ$d8$CPy605S2}jdx^i4yMS|=u z+7(ap)BN@eVnd4;h*q>7z09#E6xx$Q{%9vk*|MJ3_i#6Z>nNnw7B!GoWu3??vyG$5 zc?1U=v%B8_LHA9%G29|Cd;zxBLw%4D4*rNsWTB~5#;Wr(px=up2vnxKHaGxWbFe@# zVjY60`69sI(Gxu7$0@VzZXI}Ogf7eA#*;`H@}5wcd%D@g>(GzJJE^RD`x~Q}N!mH$ z$!0XQqhR*XNbulH*krx}YZ2R0_81$$JCTWH8o%o(J{fq+ zEmkPTaDwhOUl-~0dO3>K^Zc~Ij;>fag6`AZCzRZJ7hL5)S(`VGI{FIt^YpD@VZ)IL8oUh4HsxS@ld*HJb(`d_hGY(ns1jBVRA2){ixE@m$i@@)*EE4aIE3 zQpD3(c5)7GC0%xopoL>KrVptiWq>JoQ>HEynB**86eEfyYPhDYZ_0?~fvsr`{oLKZ zX8m_>cTjTy_D+!WfiRyKx>H-0;^cD{+8y+D(@mCyrwsb6)ZTlI7M&%*@8;L?-Rt2q&VI&gLgBTCW^aMvfFn9bh63#bga)@Ao7x ze#{rv%a_m1DrW6ES&`KvCf;_-55?(6-++BJsAA;fTVe04%XXYZn&Fpmz$A9>Bt9A1 zw?VG>@L56=E}1!22a6vA>qn(i2nAlQWb>~_!ZXd|Wy>AUq@lTR|p=`Hz zgr3`Cb4qROhrCYXzoe`V-nuy6x=BQo-7G&0;!3aP`QQs4g_M-dfF$XZ$RZ4PILhCl zI+IzZXOXP(oGxpVSLaPz%H-?RDJDaLGQUoXKMN0GK%|0sO;fIh&*zCs=_s36nqv=q zL$@+Yn!c3P;hq~q0+=u_dwqZX3skxM@<@`k& zE^1lJ`I|Z8cgb=S4R~Y7=Er7>{)Lk3wJtK%8V14bFc)uJo>#mi8@wd8Z@#g|y6|b{ zX<_F=vcwjyVS_pYZYSq#`3G9K%i}{z+E*MuAuO|xGDVpqix!^1I#l%{1uBn7EK#rS zt}fz~lDW}rj&a;ZMNFz0MMkX>(UM!nB$8Lf@7KrM`0q$32fmAI_bjYv$?on^$xt>A zLsG_ey{rO(kq$Z@i1F(er3c#!3#&6qhkA>>TDdJqH=UtcO+pK#Hz7~-1cER#IpAmE zB>@ZW4CJ@?qwhF^1+GCTdJmUVfrX7EpUQovdK?E-&*~)rzWU7#yy=W<2cj{8UHvLmJi*LAozj z%ciHdY>H(VV$^9upUImqK^b<%USvN=F@{<=Oh7b&^~0xLnJygcCynEw484tmLw5H) zBVm1m4l=*g-JDGgsC~>r6j3w171k}Zv9;`@ohZsDHID6WzPQo;?9`Gdi;{gfJbo6E z+=fKnY=wQ&xNeiaqJJ1?2kIJgOf%0a+VofMf-+vvd6&2slb<%&c??_Ygsg4mDvQ5^ zFbBFNpbm21dh;UIFk8e72A0;o+ISY9Sz!`_Uu9RKx7`gqTMkQ6cxnTVWn4wN#hA!8WSOWaUFsWeWv+5u`s?Fy?Ic3sWLGoBH zjW}rNq(1>o8SeTM)wJ*=Cf9z71XX_+=qSsUiZTOQoinfVvBQRClY0AJw8f$lZj+d) z$r$#KxaO!j_gm4=++EWrnS~XN-qIUOOfR_k@7JF5bWtkI@13%;5Po+uWfadrP|3F9 zVd-G?pxE#fp~9zwld+EMW{9<<^kYJ9JBy_7mm%dMzTy13PH0n(!GgYt1-VT7A{YUM zw3@PP-L%c+@ydmNz6-bXq7Fr$f&E+&*WyQ^P9*7h-_$sJOT0d8!eIGHWI8 z#K(OpLGP$)mUURyVms|{`AGzm9SxJ{gGp(pK;;_F5#Q9qx_}|)mW8Qyc6>e=oU-x0 zq052)Vu-B*CUM?5mw7N`K_f3Ibz~1+gveDcm3mn96Gof#?1lv$pT+`mr*M$!n-;<} zog2i{$zz93BT8>vj<$kL*{A-> z_5$+-Jxu!jDY@HsoSGT97kVd{iy!EPQzW)aQ5+GtD(av-kof!Ol5JIvA7hJ1mL0{O zm<9={oHM zzO_kj;2@zBiy$*X-cr;LO8;;#Sbh zdPix=EH9a=ovdQm<82-FsNb!Y;&`gB6htk=ogm^<3iz&n>@Ep)?jgVvFY`!Xl66^MB?P^mc04D$kcdG)zc zqR-wK!D>)mnbxWxf>fplU^Fa|UlY>7tdX~o@#0M_AN6-qe9COni>}ioBD{V&tWTjA z!av(YJb7p=NrJC`NsV!@{<+%S9_B-H4%+khPDQ4$l~uRRJ^s>{f$++1XaT%k&!n#= zy5a~82anc)E-snc8+f7d5;mtTwUXX=VZAY*VJA{@;l(&z_iomQP@D~1y0}i<6Ww9j zJRR23?>JXXk?%BLTIQN*ek)8_(&#Djw4l*7cuvMKq-Qvv!&Cd64?(w7I*rL?jbR3p z&IdX+Isw2JUCmCXQ>cdt^oa&yr7DGcV@}?;x_6U|exi`LxnByHi0gTR)KP40CSM$k zRjW?qhS8mY?)}9TnYL3jyo336TMH64fJnE>**mif0e3k9@AtL-FLSd|B^|GfzyO_)_bz9>;BcMi{llNtmfd zHH)A$el`kPh5P26KH1SaefaQM(ry_(%04Jq;c9Juy$Ei;y=#GVfhhH1FFtV4_mtSwm$%WILTEa^ z5(!r;gxB7o&J1*7eV^ouX%IspQgBhO2?^sqrfw6mBior6$5}BmdY8zUAH6(S?BsPQ z%yB}akYcs>np_?)4gsDsnQxDw$9L!l8sP`oJK?ondsNt9_!0VtD8d zX^Yzur->u{XDF&{TN2PR<~jB~<4mp-WR$X9tfHwv)Xah`HY6^5&lcP8F(~9*9?-`> znLlHI-YQ|U{9=lzY49ekG4xQF_}A~FRJBiG$CF{KLk8iN=p^u|%eXP+*$c4p)eWT?eYx+I2IQr&(JJ97Qvm+DQ4_FKN zU(NRSC*B|LXcClEjV_y={7Tv1cO9L7vYREY_>sI)T@EH+smtrjG%G?#Q`#!ack>75 zo5nne>iW(=mPtOWp^e-pWl7GXnEG#`HHLaE5ByckdED9Ac90CZ;%`XxK0>WvR!5vr zoR@!5-`*4oK#j21c07d*=pkcUF*;l9?|-JyG(v{PZ`gW0zh>s6Hs-EA{$Z!b+GA~u z>EU*LP5Bll_hf`B&18SA}B9c5s3fQBbLc*lAn zH14pq*6`l6FID}1z)BG>&n=24eop!oHFtm~eGzF7zZR7lbF}aIWmQq1VTS6kTba`q5Z{aQ;+w-?b*#cy;m=T%Z zHcCV$XVv>IY@vTat(|{+cBZQR(!f(RjG2^-X<}9J`&;lfadAAW%SKpRu|2cK!dhPl zVM5I_@6z-jLgl=OayjZ~M}Z1~eaC`a7O2pXImj(!NPDpTg2tofXb7 z_p5=BGZ#W$w8e(1?4aXF!rUk7OU1DY_e_#u*%%_Q*oh_8cYP9HR_$%I@Q43B5?x1d zzqq;TWLms?*M%W>`jo?HkwnTm*kAY57#7Jk0lFWOiZd&!*5S1E^jy-2sga^>@aei% zrix3pU_t0kw;!>+^2nV?P9|qG z=m>@T*{$8LO0`KoV8gn_W^n*r(r4vi;F5%Kcn_3fAF@o?*cdc=^RC;NllZT!jo!HI zMfCRSuz&k(_04&zL&xCz^$ugQB7Fg9K~8dek=`r~ zy*Liv1^+t)E8EkcDAe8=)^B^$5(qofQMz+0uDj4ycZ@+;>PUp7!BlhxYj{VLG!HMt9fxzj-k^_LrCX{sN z;j%lott7mkKg@*$2DBju{yIJJ=fAt&*zqC~yU&V9N3gN{_1j9-)r^;d83nNmwaaAI zrg2dRrYVhB*4JOO;m6EH;ZB50AMV;_J`Y_>nyW{(7wVPKI4rNrd|9u<{ugMgev`c}PKuL=ML?C2 zoVgj{bn12UD|0J~F=!v6ARmX-IePRc@Ec-iL~#Ka@I9Z=bi;Q(Pg!)M<0!$1dc*(h zpJhb+5d!r@S-;XU{eDjr2K}Ru_tdAGMtxUP0t;I=4RF-s34HJnslCnoE#Cj}BQpv@ zxy*}R_T2)l&)yKYQ~|@dMnS96qgVk32my@NQzN?Qwn;Qy4_+09FR$*W>HqchiFhy% zSLt}i)2~h;dBiY7SrZj5>0BLYJxmpH=FY$RntuhZr9hYT1AY(mF z8<`Bcv$gYj(%kMf#B>SXdH4&*VQV#Ta&^5mV!$BrMqUOU;$PnvQLcAYJ`wS5Z2@qh zdWvy@*DvtN{x#}dCTP9Z0v$?$pED)PBKPg`V)gvoRPpCD4kT z5^u}-%4FyJe|JPjqv;?zbqw(RdF?)sw|Tid%oKSBoz2Z{&m&AydGC$y?tZi&tjY?( z(9Lljnog^udyV`I+6cm+TJG%?)kH#+qCAKGue;_Aaiwhg{bSZ?*roL~TlMM>l6Nb+ z2&8SmR&(9!Zrv29jsJ70Og_E~P@K#jSJq$a=Z}@}<#8pzSvvgV&%tk)IY8J_ z>|9a@{D9<*Dg7X9z^HY$$%yIMxG)TLAnr%r6Ddql-~~ekP88TmvwyjJLX~u%`eH zr{lVp3*12R4o4q7#)$XC|HVn)x zH@SaAb)n#MelE!+Ls~@-29f{o7tROCC-X%cz}Ou>0=FdPXcHb?U4ViWV8gA9CUY@a zp|N^^p6&TLAU%n@QHPU>MWX)j1U?~wrCUNiGok?D__te>&*=l)H>9t$s*Gf|3Wrj8 zoz^-6{6+QY?H9Ew^h63h=|lpL&%s{f8u&WY19g~8vwD<&E){ZS1W2G96k9khie_aa zfYX&W?0*#(e9{W6-O>Rd5C}e5HvgO!H^}+nwBivojI&osoSkuXIEO)FoMO{+e{(U| z_~9i`!vLr}{XCX5S1|q*#(Iuq=6W01lRe=_r(s z?t1C*Lw;Lg14sc$;6vYEZ+zM@#F54U42y*C{c%6i?>DI)Ek=M0;=C*1)d~zPAA&y6>$`GF^AwTs>-)6zO8KqoXMno zczpnGtjL>@1L#BWtHeQ~6jkBB-%t?Et0~4(B?%zX$W*25PulzUm4W0KO{T6RKj|y= z{;_@v!6jDw^~*KR($@(q=W zYkHox!;6nex9;U7V=-wHB-9PtBgZGx>bFW$wE!NT#BbmMnz|8%h_->cW7uAf`_2OZ z3ny^Sfe`*{wsL;OQ5YPmq1Uo1cz;{Fe|#Ln^wIXs)q2m2n>4qYAUGoPI$Iq$RsS?j za|wJM#rlNys&Ye&VrWoxU3^FLl|eKER?EQpBGOMONjw|u{?s{1KbO8Plexr%y!gF1 z#Gx$)i~o)~iApd+G~Q}xp|H7@O=qH_uw1#Fob|waa|OUjCx2Kjbb-g3HNl?p0*J-o zSP{SrWeiG6nD1NI+p~{?olg zy?g6n>?(!YA3)LZHz1HD(xl>V8FhT_;1`=%?+Qg*D8XI?dX|&@1`hyxNO$1O@zXr( zpD_X&@kc^iInrtdYVs47rwN8hU;+OEZFu~EB=50`@L(fS%RKv!N|(5nT1dG|>QE|J zTcm--P?iysjsJCdDgD~sJ@c`Iv~MnJ0F10k>+SokX<&6tN0X`<|Ma7X=jTgjF|(gi zXPVrvG~X*P?B~4Rj{fgALkI@W00MJN%&z5zr#L8wqAlFSFI>b!uZgN|ER`H0x&s8= zBNaNLn^)onSVZGqr0~eMTEN;F{qHquH{(je*RJd*kB6F={gL|oZL9*R zkU)Bv@I-fD15|6V07rH?sa0#aG@dJk2wOGIiWdhxy0t${KQlx+L@^8~S@8+d_>`JM zq@&=aVX%@7WDBDr!ukXRIYkEo%0OB361dxL#Yi!JU|RzS%kovM_t`!tO&7A41dN1v z;9+}|iyK$JgdPzNx2?)$Z5B#4LRE+y!Yup^n7oxlIJJqk_Xj$^8-#+4V*fTw@R8XA z0XYfE$T&L!pKE)A#9Zn{E%N3zLpZ)=oli)kAyTGRmxZbLCWfHShI=LuQ0nqNaPUh-C*YE47T>=eZSf3gxyAFD0_Iq7%0 z+9r_6{o@4$jetCd`mv)NWWG8J0jb6#DK5a3b;*80CeC>1d=O$p7vj=?*;?+i2*^WB!c6ckDnbdv9}NQxDLOh#YJhi99{r#rSgC)ihQUYIr)NXof5eCU*ON*9j^AeuA4ZFx%*PM z9Q3W@KPQ{`76$$qfhsT(Df-P!R(WAm{=xK^zRG;A(l&PitM=8^$_F6rv0pCXmp+Ve zy2?+cLH&=>N+UsY;x5%%y`)2OI7q!oq^GL3+ZpMxkknY*uZgd@IVdi&&<`Z_LtA~| z%Tskrx&ZHXF;UTL;>+GL?SI^GA80b`ZcLT2EC!95N6tW8-nY8<{XA!bBL@Isjbr-@ zh-`|-h0O3y61`~q<>F3{6|rlfDb~Qh{FC%=L-N@Z)CCY=07?cJK2J`t77tk#Dw_Z% z-n_2^i+O2TZ)4oHY~qI+-nAF7UYczSDA8tYI_&^;T&81mT+wxa+e}EmJtVXPult1T zRH^5L^YG7!x=8K@%gao;%^O2#^h;(+G$CP6a50WsA^Ycsa&Yh*aek{C;xB%#SZvZzUIP+z)de+kTUYS1U2G4;s(@-&k{;e4O5D|-BRGDk{EJU(yTryp zK{SLIL=z_UQQu!JPXfi`l*rdy#pRCqfCCQp$ zJLP#t`ZrK|f)xm@7JQatUmVGAs5)YTVFeLCs zOqLCMc=iN^yegMku@S>@%g+S#V6hxbCsJEMf>hS1BBI&OwoT`?}HI`zwexpK7r_@utkz zN{wS|*rzSD$CriV?HjqKuP#-*vVuJ88x`3c`3yVOwup2tO?iTc)g@_d)kzE~Pc`VG z&KQ50(Cu&*7-*A<>(siI31_L`gZkLA631dS(X%SB7KsVi-7kRWVia5e_}@)gHI(5g zXNZ_dG-_oPm>Fh!=pr*dTwZ=Ob)PIAK1ZMssIv(~GwZzj{B_>a{}m#LX9-77&t_JPAAM`(dVff>v+7-QTCI^3bM)>3M|QkI30sV)8B3 zMS%^T&ZIe*eY{W}mfjO+G?(T2fZ5AJ6Y%lL^HK7--1XgPS*C|%R=X3_mt zYXyX{K}LG3;b5LS!g?C2@K7c2TX7W9kU~h?CwGBfJ0EU+SF7UF%_W;{PNH7&zxwrG zDRWprf4W9P`Ezn;oZd0(iM_QaL+%LK@jt+4sgBzct`Xq%#%BXRss4xNh^H(MSfG#k zMkf<6-s~Z&GR8?c89i-rH*366+NmT>8O_-jYI5gq4WVi&5go(F1&ta!Bp5#EoXDLb zhu7L&kBQ(MW7EIchFpgfocQq+DEn>5+IdD03wR37Xb+C6&091yQu!{CLEO2-SD9(7LG4DnrCeuz5X`plD+&=#?Y6Ozv0r@+s73Gg z3*;Vo{qxZt_P~@*JTA62St!BvprTXEyyyVJDkU18{xa3Rv`zBbGBDzHmzKcxd%KuF?;P$@MqH;G&_x9ra8NcSE^7g>~6(@gsIXFY8$4KtAV}g48{+p35Qzf zWGXu1uGja#cRjW{?rhu@#6^zzE1w}4_nbMYMJ*a$P3u0ubHfQ|=5}PkH2eO-j$>aq zU^}?Zv3Br^?3wkHpN(%=0fjMo;ZJ$;LQqxMb$8&U8Y(xWa7Xziannk0U6L$l^sJEB ze8MwS5!I|EQo}nOA^gm7r@F=|V~By9m-wy=b9JbTB}$J`cC^#d zr@Q?^PPbPmXKE8Z@%iX4Aa`rT(-MLu5_iHmt?<=?AYr|G6o-4{?TJ;*sH6xQXQ z_SWs|(1p+7{~5P=d)v}*Na9>x9PVRmNhoNmM9 zi=ASQ+Gox_+=or6q}RHXC`EQed8M3ZFJaAcJ4;EQweSe1?(Tu#G)ZfnC8?dzR9L9I zkL$+Bjh3xNL3nSE{K<0sm>}NIC>v{k(3vjZMO?Zs(UOY&F7iXH`!Bchjj5wQ5;8%l zOj_4z?S26DIY#g8ns+-&dt*<_G?-!-kHzWhdhoW%?QoiK;_KIYf1;MaBhV-uT*dt+ zdz`i#Ux4|sE-GE{bsY(-Q_2UrJ{yKqsu6ixOGQ@u+^IIP$p#&VPr+AC*d&d>nzY!p zAO3+}(ZU&Aid05)8n19u!poWA&quQqPOq3$e zhkhu(Wotw}Qn6-naM;;wblHZZD5QRKIkY}IAQ8z`5eNw%N?Iz0fjC9*QIK`||Lgnp zCREFbiVI%Ee`8Mb}b)6GI`CM)Z;MMlMG|W)M|_%mSk?q|EFp zU?cs{sQCM`EJ6QlWzH~$%G8x2=kA>LBxbAExX%m+yBlwK%70%1)W_yp#z0;7?xxH% zW+)b7@`f4PQ)=pNhVK!i4ix_jBmzHBXvrpSZd*d2VQjDixDdp&$+qsNCMj*VuPAf5 zSRLr}m-5IDt5y1UP|5l*{p327A`pGhcyJK=-y$lS`&A!j?*bsj|Z?V z84byI@%tzcfMO-}`|q7_Jj|JkIJhn!94@!G?I6$P}eT^F89CSngyEbQy zS2pfoe^=p9+p9TAWmwPff_+A%C6uIqs_w>#@i(IO9npU@BqQGSTq4q~Nj8v=!8|#P z#_Y;+!{hP5ZX!tZ5)pC_~AGDmzb1OiqDkpu_G?g&9{F`X@b4H=2sYiNUqk?%% z&CTFJ@nZ=bRAaIhD5({Cv)?k8k%4X)Fn9E1Mn2WMsiy^o>N+%g4K*h$(iKc}kp?L!ubxIsT z6Q~=1kzOh~pV^!-*6%mW4;9O4LP>Z?*kHJO0}8`ct!l(Csjd^wVD)jMHC3uhCr<9y zCe8u-&TBc@Xpb6Om1~**$wkKpSw*GQ9KigEE}6g#ZUJ8P*(y2d$13B0HORt`{^j@^u<~EmLutDmYoP`3H(vA{HWlN#6eMmJaM7|Dk!&gRCam zOSa!*AX0{IiyCI?ds;@3UJM|g0?=Dt9%{4kaz}*Nj-W$DosHbcVIwor+OHjv4a@di zg#L{#!CcqEz~vY*lN+a$;=cp`koV!)eIxXafbIUIbqIOA*k;j%w(XT`xOV8?vF`PjJyXL4FI#FwR4cKDLXs82BZu# zo^1FZzb=W`!s}(+5s$L(?Bx!e$d>Z&j6L@o>m-Kp+sF+!HBv5C|Q(1dn2Yfe&#S zc@*FS#aUhUF{osiavS*1Yja&W3ne8G3vi7ELPsG2q5gOT_yVDj|9tt!EAl85|6Hr1 zF#Uc8pc!HfLi_nV2&4~u{`hAMeE)do*C%Qk%D)uTz<-`b2c@C@xdx8{PlNoWBz%Am zY^NuB&L9x}=O5oF(X99s0N)@vDG5yvl-)(lq^DZQ>+43}N}j}u?dA`!(Xi6*2hcJx zDWOG|8p~c@kV+w0zDufJB2?8>)moXDXvrSXDt)-uoGoF6O@H@()|p*JWjFA7kl-a!XswxqGw6o4%(KMxt+KuXoTXRe}4qJ-0Fq_=+Bqq)`7=9o^Cv7!OiU~+ z{R0C#xdz&)CIv*3Q&Uq=XmwSUnWg29;^{hjJn!jI)P2eS81@I5v}-W00}rlOZE1LT z_~qs0r%#_MDk?-f$3?-$K`q{Ads^jDA2XJjP|BDaKIKy)OEH*l{wM1IjV(1=t~Ty( zIbnX)X&_yW%I1uW$0kk@dhub+-G$^EP0pLjCy0%;wVADHD$tHcF^qezFjnuOCDbNL zqvHRFVhhaMP}QEJr&?EtT5V@!_|4J%#=_E4$Pk*d4waDG7D&1^ z-C-G7TB6E#oP)<8M7)!5My=8&^WV}%LHkMqP@^1dBvUMP)V)y@78+RX{`t+#prD|S zmb-UBC8edy)wZ#ZpGvpxfo488d)0E7=dSlE&b|bJv1kMS{SmMU=Doq2Ub>`opArV% z^I`@~Da?lu(3G5)ci<3Gg9`WEFHYeM2(8lcZ3K@=S(vxdsI5?-efXa749URWGnBuU zFwhT2DuSI9#zI4Zn)vX?$kWIs3xNun?|>k;YwTt-9x;E_I<{@}uTBdK^*C`b5{xHP z9IH+{n$;pk%t8IMBh3DbL4eiUfn{~$uahh?iLBZmB%*h9>Z|_b;fAoq;O#SH8|LDj zBE8b6SRq6;r<2+74*qj^g<;z4f9WH@<{OPRv<8Ops&6@Oy`wAjM%^YD?TF(d6Y`T~ zBdF~MoM=H)#1Y(^&bmFNcfj^baAy0QJ#>u+i5$3pGxV9+cvY3+g^dKo375g{%L}Qw z?4_s8ti4)!wy9y0R3O`sl(U_^ebLyuc;oAI-X2#BSj2krg=`xN%$YlT zE&Z*v2ik3DDGVwUuUTk$3>3`@+`G#5j%?mN)_=sUkK*X*S%2z=+2-o%+TSbHw6R!X z)866|5Tlgi)}BK5;@x7`|K*IZth_{E3N3~Aa`NR{*P+JoICP3777Xn+Tkwh<+`--+ z+Gf>g(6f`RRhj0V&ABg@)B=b0Z7f{lA6n!CEG^cLTXIJX_Lm>+##ukapflt(ihpNy)Ut z#6+dgt^Eyd`Aqbqk`(hwW4CXA4=(-{O_9*1bG$5W(qa1XjDkt2JShxis`F#VsWmie zhTRI9#X0XvUfET2_X`$e(ta6@> zL5~UK#>g|UXS$1GVLDSFL_hGiK<~@aKHv{~%42ZsV8o*34zladEv`v(i)V+}cKoFS z0b=NOTQ;xsq9@oBgH%CwkU6s&n|E;88gIS7ET(u{Etnz2Rknk-gU5UiKcI8m8scc+ zW1xi-{ND_JumS6A*(rDsl(oOGG>KU>;KGYS-1`r!1ibQHo0iL}8l}kaVUQRh zu%qD2ycZpx5H&R`R=1 z8tF{0r$dSU8jqGxx(%(KcM^^tKqqOIcu(GF_wnCUbigaCi{jC)~?Bbx^Ba94>zg}9xA_RG_K4Jgsw&*Qd(5WY?n($wA9MP}> znxEgl_1EoGTtd)AQy>%df2&jg9~iA|iT@MsU$;llu>yRT#7PMLTbOMWC}3kp%0BeJ zZf^oqL>*~EL;ia)SHeIjWqjdle|eP^XMjp7A7R&DuK7SF05n4(`$FQs1$$5rP#K|k z^Y!k3tAGa`Ah0mH`v3G-KQ!YhKxGq8>f2uizUd|)uV&s~~PAn0!JXZb~DYn3zuo5-+;*eEc zok=+q2x74UkhAPjIR6GUfh}kW?m~^JZZMsYT-*80t=umnmS>NYzqwd`XI8!{JK!#& z`|uG6dhX2bYLfr+9_=VNIeB=Aet6H4DmSx{dX`?@b6gR1;Xq3zd%d8B9;)KB`swMkR)Pt2FZrNCgM6q0+Uo>~zcH(7wPVN?@lWkS^ z$RRMYRL)=UF;&;leDcRlem#LNeY&!=kwDP*D8D+N^`9_Kd=!21tP$>)>!6iK4No3! z*i7n%rR-#V_@0cJEb%H?j!{5BFFl~Xvh&UfCnC^$p^+CpK`Q$_-#Yl3Kb%&MnQe(4`2OaKK--;(NllXflsNfi)6qv&ysENq=DE#7 z0=ovo&-q@|Uc|B|O7bDni7=IK`(&*`IlAlg_TTZzL0W zJ^57e34;TTO9f!)XfBDI?oVuM{b01LefB(WqT(WrrAAu_F?$vhDK?ldyvUs3G`b{S zRJG@tz#Agp;GU_$Jtc7bDNdnSvW5enO4LQ2iPbz#=cY3QWq^1apwy^K0VFj4h^h@I z&(wK(HNCp%ajr>fC4h01vRXZ$+PRVVg(Nx{ygKC0->Krv5?+pZ?wMXw;JN;_Aen-q zJ3`;}`>v{7s-90rO3dtVs4FV7+YYJZ;j-}gE}Nj+dFw6v1>VdTK4V3zA<_7KG*28T zUx|Z$j6P#Ua_67XY1NHK)Gf~<(}$o!NkbgLDW0|vs4W%J%v{Jc!WAO|3%zen$gE{? ze$6D& zDG!HOgKvAiX}GZK=K!uXS~M_&IEbjf}E?Irs>M z4-KDUN+#AeuWLHKz&~$thAj>ss|cBEBqhBZjUcNk2%F1Blh2989}8^pf$ljN9hMgF zWkg5w*i`2<^fo^?9)KFg$K9~U7fe>R(a{s9H3Z~5QEi@o>(9RbzzOEz2(j=kDShEq ziY>Bld8(6~l;U6mm4U1O-VDCy`qfqa?|pc)2Zr~0hj7ET92{WzMx5I1W*Il4i$yeZ zC*uWBu1@`vTKHjzQGj>DuZqz>N?kKu4wDxGi-#Maq$u!aaRS6F;-o5}FPuVm~H2;23*;@ zIZ|VnC0$`jMk|ai?GhG2N#; zrMrs<7DED%R9+i-xuSEAl$QE(L+L}|dC^Cca;~tK5(2(C_ff#@q{dh-PV?cP{;>zX z*uaM8Mb6H;MNHInjWw4Pd#1ay*}ki>bp%1+@Z^ZanqjF{7C+B(P_8WRBbLO@Xcb%M z9#ena+(8ZHic%KyO3vJaNoLDj(4BkY1Wv)++|b|nN8li_RsIX!#hYH^BQ?`hp$x+K+zzuNTJ1Oote#7z4#{%blE1i=+ z_6X61J8LiQ6*UgT`($KBbt>Z!N<9wQX5c>ZyLUE;Z$V?*cN@=cbGPN8c8}RBV8cS01vk+>IX|k;>YDZwzU;k5MO(W%kOmn0&m(@=F za*`2m6)oZ}^K#;-Ndxv!Q>sg~7W2*rl`r`hZzZ{af(0!oocKfMpJfaHUuX*mQM>ZC z?2?4Lbti9=ggKGB@h4uba@#}E`jp+$H3aDbs$Znk`8H=jqMNd+tRe8U;o+7>=V%$` zXd~He-SvZVR{U?yiw24Fi4;WN-#;}k(O4t>ESw#qV+HGUT*$T5%HGJgseO)jB#Gcly?l*fa6aTvBgU#Mh)HQzHWw05^*WhMTwz12)7zoD=|QM3YxbAnA$ z`6eX^GQpIDfDisWVad(L2yobNI+|2?$>O4&LKbt$vp3j6m5;a2TB9OTQ_&~{q|TRw zZtZe-4_=>o6e)k~2l2bErsAa*sDLPs7~i741Ok4_q?(MZU$*YPuJLMjSR$`I7vmj1 zhlTIoR*8K&%q&}NtfAwWwJq15r)wQWd}iNiP;Ex1$Ui*VI`^cjX}VfCk_IfQSNCPQ zNcOkYn%-U~<7`dQ7Qn1tzh9UGaH96ghY| zzVDN2)BU7-p}PGX;=oMQoMwW=FTrDe7d@#$zefrPxJI9xutha=1<(5#+fOU3>pa=L z)<4#Je8yJc%TR-j;T9z#9FMz~!R~oCI6mnXP0~!V3fI8$h%^0NL8T7n3{Bo@OPzLg zqW~uU-N&?kfI5E1>-$wg3k2cD5$e=lB(y->m8zuM%a%x~b0uyFml>F0x6uJ$U1;c# zQCAy-GQN|@$%pxZJ-JT?`p)_WnI|Y#rQ>n^bD5VA_fnY>uV|Hfp9O^kcTU~J%9(GT zCkngbBr`ofm+h;UBrQ_*K+hx*Z`1^l(s_}e-XnbaV?WhU(f`V9XpK;Us&O1vhv>0- zYHL|bta#{=^EAt~&a$3S>bAfrXoB)RGIG0-hQ&j0CD+9>wjRCv>zNB*2k%RSOix$2 zx!q^lN1cU{2#y7w+-HaCFlN3SV9+rpPZ1|?Qj(t z%I{gXP56uMt-J-0XU6wtzwm4EGIYc)W({37s{V*a%V>|Y-FTgiNZ5Gu(2(Totgh@| zFSCpM^Rjpqg|V;zO8jqtI=~U#Zh^uRwT$_QQS4l9nM^e2W!F)VXo{&Ri<$7-X11)_ z`Yw&qGSf)yvaDpLqL;;NZ)v_zx%H=L`>MZB3GL=Z`$`J9QzMD+-x93@OYBd>TYZFy zg}R*Wt`%-u$~!jZHxIE57t|9t+8k_%TDqr*urtD?Oe|@h>Jiq^(-(>|%gDoqofhx02F8Z8WfqwMi?T~Q%b>aVf*0;u>P9817<9DNp*X4 z&~vKg{a2B08D&Ia@kaWbTym~AgsKg-P&98e_5&Uhh`p06@Z?>80qt)j(ym3?kScdN zwc+6j2@o(KC?5DSk?$)d}6w_C6S6~Pj?h;wVj`90K&LoWMwSlmJ7pt(E~<*)!{s5 z`brf_qlZl9t}0ekeUyQFD&9c1LkdbEk3QN%^dSkJ_wY5;6o-WYRXoeg5 zf|ZeE4CVG4vfK|?v@2w)#)Z)E@X>6}=GxlYM0d8TnwpyG>TN^+-U_UMMVklzgj~OO zx^h}7%EL1dW{ElT!$0Z5ucWh09DolnGrBo{J^5Q*LEu1QS`hxzNHybs0D+&edte{{ z6c9*-nf_Ht0W$)2w~isR?JqK@K;eb@4SCN$bpJ2}XrPmlOyF;q{|6$#xF-%0CgchK zs|tjb1h~YNg!kaT=5<3V0OUbyL;31Y^z>^OLVm#bwDH_Q{}+1r>DF(*0ihg-Px52@ zlZN~pm(~efAIQr2>*-#nz-c3IwoddsQ9w{DDuMORn}|wd03c!&2iY;}{v?DI5RkDv zz~1-(gr5Bdn2~reHWQ-xTQ|j{qH#?=4Y)A}(5F8#IzY(*U>8Go!i;a?ho1Xg@6l{o+B%KAJoZhXmOPSV^i>cABGt4=E( ziXMk92n68UBDc1-wyf;z?5wQvN=r-g^Ycqero7F~*DueHyoGfvEiEJFz31biqoMQc zH54cxMG;$>bTaSbc>@(boCB$dC$%?3c9~+Qqb;kC;~t$xjVqU%u_h#LVsJ^P0x(`gc~*qe{l zza*NTw79|N*bHi{Os0zT+~{zzo`^p$H3|*nxi|Nq^Pj4xE>P}NXn$;shJ%BnUGg*^ zf9Wb#*Z0#v44csE`$rj+87X~BUhYuW{H0!|o{9msgRk#65}=2Nhn%L}aps1$W%JeM zBPk+#lJ{cGuV96x3GktzAsCKZZhk=lS7HC=tYxa}eu9X1O+(P4fqHe3737=u+N^9C zxpb-P{dctnBnTGChL_D+^nAY*Wn3KCd3fZUs|9HfwUR>%wTsn7{ZY(9zUE>8@z8I3jaVtx>)2d&@T9wZA|5XBDQxipd_ zxc9Djgb2my{<}>B#WN1W_LR;~cYo<&AlVk(eJqznCLevAro(<#^3A7eYU$oyq@@iP zcF=Y!uyI(N*5^Xlv}<#X{k#P;3CkIocU|P{al8IZwe56{QliU9qv=X*RRTWE+cutu0DL&^u8 zB(z<(s7(y|NGBU?F9$8?x8oSiW1Z>GDiFToZRAUGlEKPvDhAS|{AKBCfl$Wa)jA04 z^$^Cc$=hY`eGvj@1yyYrxz=TA28)!YVhPqL6^L;w3puFGS!6Ev+U5^-||kg|!v)})(MRZ3(QfhCdGD;`?uv+|wm z?H1pQ?r^HYzCj*Rh|NuYq2GRr5Oj10E3VB_^ytnBB@0=_mk-4OUg)GZt+mFYT5RXw z780FS_EFgn)lsr)Y$YOK*a~;dwG0zr=33ycFAYJ}*X1$&MG|ogn;jh;bm9E>4j-_z zrhLhQ!)_*Xng+|Ii5e}!i)62MXUaPlLhZ)mn#tishAh$bbY44C#k>aDI*#>vS_B0- z!4-`}RR@EQM)C?5EK1b3TOiRAy%wthEVDOwlC0KM*$7w=Q!#6|*sq=^ zk4p_|mA};9bB@-|{6<>1gPPMT|GZQr3B47YX-`Y1%!ozzv8nOUtE*Qum$bXxILIEI z?{Mnj6+W8&Uv5x*4ruqt^(D?VK!$6%{l) z9jC8=fS}rC4E|7 zjA9_kVZ5L4Bj&2vypm}&(&?q8+0;wd{qlx19wPO(P?NkDEf;%@@5tD=dSDidW0|t^ zKF6+l0|}4LOZ}0BRKj*=3Fz5Blv)YXI`vhpQm~1DM_s|=If(&Y0}5n;z|vXES-#`c zX$4D|SB>jMNeL~}GwzVQWdj`)yHSTDYyJYPZ?$h7Os zd2Lq8!|d&pNIv8!lm-uNmlZ5zhD?qr+>rSQ&NKtL|+@%W|?Y2hK zEmJ&mG^-l`C(5+0>2b$=zs-%gLhIXZd4u)Qs3ZZjfT6U8g0yh{?y-Dn0AwgV*FL}e zIDdi%i{gOwwf4X`w%xW*|lH@go_tJ`jTDW`m)lIrvoQvIjo^S^Ev|2 zdZ&ifP{(!s@$|}ZqDJeR@#^p#qlQEof7#`Jha z4<+3QOZnb5UkfIuKikO@oz%OI(TJ-W3NQUg(McN`8d@a{Ih-V>s9{AE+!J)RJ2w01 z?V5&{M*?bWBqJ5tZj2R?Vy@l|M?GPp717Y4;Ms}B4yZ0_-9j|vh;4>`o}?&Z3;k*b zDpDMju~+F(NS?uQA2Q7w+*TI;CWRj6rLT3Ya050h0Q?|vMuWPIdo$ct>fz7P{KczqRPtr%7^FavSud!R@(oE^{LcRm1_L%?Ft!i2{` zcx?B->@o`nIY=}BVV(N-&P z_HMPI|M}pc?pWo--P2)}p!^(1?dKoiZDXw@gO0v5Y? z+tUx-{gu?N%NW`Z<*wg9GgLCm?mcM7q~XR#1gBf^HoTp+?WRaHug=-%8IS3q@o(xe zt>)V;sOeCClAh2JkB3gFjt+J0nNBAo*nJp zS11kuuTVHPsR4k1J9?tfJdaTuvX!TrboYAJ^+fi|2fuQw6!2nm)3Yv|u%i0oaRvYD zi{<&{D&zhzqie}Q4(ojfSxai#~xwFfJ65n~{U`mucUO~f-O?dLz=SaIg zv_KakV6Ahf20u42;j6{j*f^lx##07%PJaVk-_m>T`--TyI!|E+#%tYW+(?KkW}8>k zM=H)j%Bmt!lf-Db{EwY(2Q}UIU=g)~q(g&!4B0RJDK_U#HR{p?GLAw+hE$|dxGVgAS7cP#+rtwx9~B^`@hpN~ZN z9n&dc=L=A|lJZ;A%MHIho)l5)E};n8U03jx8&&!qdU-NGZ*-`vZ1TyIOu98RFapRX z7PBG;k6y5&f$sykLt1x6o(L9`s$Pqa=M28D;RLIl>9N+(ps@3XVx-Ks=sQn}X5--Y zsR=7Od{C*8`v@cVvVz{J*Fl0o7+rY#EFFbC?-#O9Jxg1`HX8Q;Af94Cf|bZJKuQ%$>(VK{!EC3KE3OS{stl} ztC82`RbOq(37cW-1Zv-e5NImSCv4*RHFPuva{2XE!)1Y3CX%u&vTEw>wUiA+`eM(= z1zSAg>jXW)Vp(JfN*VirO@#;)VM=om*SH5@2muB_H)V1A3u83yA+0qC$6}R52hm9y ztY)UHpXuq1`L9M$K9ERnPqXv@j?xLvJpYQ8tGsQ)P2#h;iya2VIAdiKNsjsFSsuGD zaoIrA7fPq|JHLPo2X`{JdV+vm@U6R7_JL3!ad(WwTb2 zR}${AKJg1R1dMvd=B+A+I~yoBYcLTYhm!yW+HeVn-4j9?+wcZm4SOd#}wNX-ny zJVo9%X{;(>e|muhNmQSQu9BZtR6eu0YZt4%s6jL9gQ!efuye9SWpfpcGlf%G?R^|a zj9X_y!aAU_q>EPtxHIHY9;dUGvVPXi{`G3I=Kk{0uesf<8X5_F+BUWF@;Qo3N>Mg9 zL`SLCVT~UaU zrrfmeF7f~=##_xi#82Yu%1bldL+!2Dn!jd@(ScU3(x{?>hkSD>J-4Z59Vsrdv~epT z=R!GA{zUsG`Z=Pt2(sHi?yzyb&t{%hyX-t7!g5`|nNoP&g`n}Bx38VWWXEVQBB4fc zMBhR=or+xebT{p;r$s{il5)=80yXhg&TN^OH)6h2bf(OQ#5i$F4f5VDz-HpJVBH*b z9y)rhTVAC$>43DGMFXm_D{n%l)?tMIpzjXzZMa%`G5I_RzJ?N*O*uw=hK1hZmy0TrIBG ze$q$@#J+?(`fKRsL7tylqz1099%UZKYF7H}+^fUj-?~0rkA*Jhk-x5)6t(VBNF!Y_zW+fT?qYvxo} zhsOgNu3X#G;2m78HvM3C%}lV=fo|#G$ofGbg+rah(dT2i^oRU8oA<4=x=*GdrOcVfa19MPBx1UC>e{6%W6h+aY^HzdP>8pNJ^{iQVh$@ zdgfy-Gb;zfF&f`RuA?sm{`@x?5W3=;`+LO?K% zEn0qk!<`sqK9htLK37;m?nW^EXhA(+H@@i<+}5|l#2uoV z^3*~T(*UK2k}K6&ah>xzRO6#H4EGHLesOGz;!d{b18DtP%PoD==cL8??gPGW^rq;Y zPS&K1z;6_{zmO??t+p5$}J`E-_=)_WzwK_%=&hu7mPEj8& zUn7@cG(R#Q)7sZ;GaqE=G`%Sgnw-8oraJ86@EfBl3N{~KwbAu@^h^&T2t;4rw<{4C zq>i?ujuGx*@{$@gfKf<3mZU*^xT@OWB-jJzwKO>M_f3E283IndK`jesWf!Gk*A-ml zYE|Lm{OZkKLQkDfw^&}~6vEjgj^Q}&Ckt4H^yPgW-D0+lDZz-35k6~g8B?58ZWqQt zrSJGA`m6+E9+&Jed>m^j(cU5tOR+M4H2w0c*}5s1;LI(Cn9=D=&@C9^V@JKNci`@u ztt(+hIa7HkLD2voVy>7Yry>5{pn=yPL%L{=A;;s$>n&T+ohbEU|LdiwizO?_?3 z?P`96qcl~Qt>XBNnLtwAT2x_L9C~tpQilkVi^vLHu zEgIt)jXPBeIyv^>#x!=iiJgoqnZ=euI0MEMx*<|dW28VXU*c`(_+qCmw5PE21ficP z78qJsT`;4#Y#S*y_KvxgIk%s9>cZX@cK$d?DJQ=6;=w0&K`SmB-@?(za_;*E8sUk< zK2&+5Pv(a{i~$DS(ElrOs%qnwc}va~)639@+4qZVTpI3xMarC+Z%n;GB}TOs@M6HJ zJPA$d7V}nmQ}QhXb_KN`1Uyhtf6lUWgHal~XyKK0zK47U)xyh0iQt*=J>ge)Gor9$`EBrX7($!ylt&V|Nef!WKw%zRbxE#>wj|Qe2^G744y-=`osjTag*v`ZpGph_yL}WYXK^ZFVM{1eyvfO3|#SO?)lVCw!&2;QWFe((?4EwBSN|*>JKsUqIOI zD)a_js)yYfgTF*q4JynfTH=xLws&Rs4cVAAR>*ni?2PGjRn7p{xs~Hul+MTca`Zq^ zbZ!!X-Ea(KYG_km>xhpXcmip=C$?Y7Izzp_iEcPGcFTMylkUzDp&Fjn_f?wfJT6<1 zf5i0MV0aJ=4n7h~Y2U@f>W`?@cA<_Z-?DhqA{DMh)r zLeaqXH0foZir*m+vP{HThJBklv%upW8?N=TV#OM4JJS>}|?McFtrIgrn);W$-a8 zf}Zw<%YeEE+E<`+Jbd{BCE=4d0SqrDDRmO^c)8rsn-fMK6IaRLw}&YXlA#fP2)G(x z`pQF+pgX)U4SV~(VokePkF=2IK9=fa(ztVj zAAh|O@3H)pX49tP{KHXrfB~&SCCYY-=Slb0bl@HPylWO%H|M^5fqC;qL2REPO3ckC zjwpsDQz8lpv0AtV`3_D{1!DPak!Dt9IN_6|HmDAy*l+l|cR;6n-j`9Yc!w`#hTn<` zDwR>(KFR9Ywut!hjhZ=rJc{qpxD_r!hCby|jnAJeD+R+jrf75@Q-ZeCeNO*Lr`rz#H}+1wY%SK2V|rweu}`WKw_baN1f6a0b@= zN1-hP048PIO@0^isIj&LWI_X@Z@QyQyF0tyI@3J&S>89i#k~*)jZ4Mr^mx!YIk0&*=jBsf2T=MSgg;~%)Q zhQ!aAr{au~?oI@!>;4>l8vuTiTI>~uehTF!Eg)--lt_}-H5u=J{yq@cMEE|$>OZFo zw8b%jHU#7(9Ho9W#t~uxq88^S4*J==_{1L&HSR5TO%Yeqha+DnT zaPwD-W7;j?vAN96`|;=HUUFR)f9g1IK$YS@D%ee=D+(#KDQhvOK*zBBg^f(U@!VPX zaI%mNkX$y7i6-;I3n{bfXMwss3ejk3DJVK7!y*TW&n;>iTWyAJ3HeuJW=pg!gArjL z@3wLui>PxrzkMiKu)7~Yoh=Ch+P02a3nNH&W!f;oddbPj6nqYY0|Ns?Lr`z!X;vX2 zgKL-}J@>wzo}MTZoAl4RvN&xe8aJ8#GV}K7NRf7K&Cgkt<1&>&C8(&VHkzB84F|lY z3xVopak0|=YEQ%_^~#L3z|Y$8kHYV7A_0Kzqu6nW{wP5Is)+wwXu=K;evR{rT?myzcBH|+e-ZtMJXRV&1w;{c@2^_?XW$4cW#b*${}}McsA#GH zm4GmrkAGGApZ&0!i|^@v^ZbWAmF>=_M9_k1Q3GF28bPOj(CTm1NNNPT6{WC(~YoOPNF)DU&LUrYn z_jQfMSxH&wBC?(%f|-W0MlJntaRB1!eKD2Uts?9=ywH@CRECyPYIHt|9N;Y`7&zUC z@P{3YnRfE#EW-*DjV{ht#c&I!-%cnaV0{q0`8+6JZOi#!L9TO!vr4KjBFiN59uXH?nTQeTo)#?7UtXiz{=z_&YVj33YC zBQGcM_Tr#p;U%^tLmZodEzk{f=2ZVVImMg?XBQ~vA`vG*U%?$gmpp=Zo1P!@N;k7w z`ahhWXgPZr#VOiCMo{`5!qL6HPo{+XGpeq%x=Cv{+E0X?*w~ zb>C;jrE-0iXaDI*Z)?Y?#V|}qxxs8I&*);FG%cKl%WpL$3>?pAVw5!IC@Xv6zvQ=D zV(!jc6U;8Mb%LzCIvX9s)7f#*2kDWtEFncO;q4YICR9>%RdEqo}a0C_2Z12KZV-keRPsN(W~Kk zI)woenS=#{`T5c%8cu2@aS9f_$fXE$qdMth!hV+bL(uF*0Q?)e8+rK@ruew<6d=GDe2nA2=ybzlq6T@ zeS2I-LrAq~iX#8i2pBMgO}iBXHa0fl{9oT>Mu>WKg^~hT=GW|4BgDn>bxQ@;@wsB< zt&?ouiEX-}O@U^yk?mu>Y<4*^z`k?w=PJavl{ENzlfw$hVX=<0m&XZO1~vsMaokXh zHf)xd?)&p5|Hp z(ixb8i(@uy^Hl6X6TG~ZX2njfb^8u7%&8TM@kU75o9y+JS@4~rfn*vTWPjGf=fVPk8Ds=ko*fvzhma7#5)+m?W zw2V~WwN;ZIpUp29?o;rI?g&b+8G*`((GCTQY2?ZJt>xKjQ;75TM=z=rM)1^yeRkgA zE~@Z)InPgVES=nnJh~Mwy7#(IqjQ+Z_oTVNP|WXetc+VWEcqaMRRw2$CKdYi(gz*R zkiB!ha?aw8roX}-^ORVba^2XC1C4s%7&SBmGg(ZW-{dIzbb?42L++Jtoo~ufU zmrd8k`n)Z{0z>iD(u9TY)}VkT>|7k1TnXx3C0JA^wOC(jH>Wy1UIdL8 z7Ja%;&g(OgVJTGYa$xJCcYcW-$%sc4lN{Zg?iJBD%8>84EgDoqvetd+pOs(D^FVSv z^UDaj>FS`M5T!@ZtR2G^{g$62b@LcN8ooY~Oo^xakSw5;owI#6EnrkLE4x_-g@n|S+UX*H;e^}G>*s!J5;4jeZl735DcBE*P!l5oxIZ{XwZ|1`ZlrFoq&1Z3Nn{9-`KPx_4ukexVF!un% z(5nHOJX`#eGWxGt+PXPJuw2?&+N(iNLEfu{-u4i%_vvf)k}z@} z79pYI+Qaf5>iGuO9qC(!+!l&krbV}BLd?T+pG?M{hHW0L)|#IG*i^HQseJv!4+7)7 zYhP-K$S3-NS21*E#uY2u-jDBPh%pgAg#2U$6Tb=AP!7*5V)UEV=U0^eD9i>TyM)YgoJNB_a zXc(U5f|seIfZdFNd9F*iqVvLM=FI;7q0`HS1xZ;9<9 zS$AeGbW~afDQ_cB7i0OFo@&)L_Gm;Au%x!OlF2``p2?z)EawzKbKA%Fz0u#_AewD& z4-|n?350gb=puwG>gsgoho*Oq^YZi2flYQUjHyFAadnMRo6S^D10W>GQJko|7$`%` z(b;}oC3L5}lZoF(^fmnVbCB(22v;adW8sdrNaV?fjL}=#&HXqS@j>qHaMg%%FWVuT z!)4dpZyYJU8MQ~@{EdScZOz%=zPK$SY{ACSaCosJGsw@>)?K%dZEy6s&H7|Bo;PIi zd%;GXdhC8$)1I4J1!7~*J4HjQef}2G#GM1>V9g|QvHFP?wt(Xcv{N{)VIwn)-1FP_ z)rdl&^-$K5QL>4mY5_FRRv0f%cy_yg8u zZG%j58=JzIQ#S)3YI5WmAipjO>XZozW=@!)G1p7rB&7|--rut%VC@_f8{T>a|iuL z8-ws-5HV};Y0Uu87m}IA$nnrw!ZB|yO_s8Fp=O+9Cft7t#x}BC@x8gL=3WDx&e_E2 z)V56|z5SQ0ZALf%;fbz|t!IAHzM&lTBJi%{IbgZ4zm^OiOZKc%?}p&8i4`Dq5GGEsf%& zOHj5^(Xdx^G>lvwGMQl5++6dyh$4um2=Eq$$jY82cZ%$t*kpQTafqFBm%cbs5$%B! z1PCP$W9kUcERZEc8E^=_(Dm)$MX)h5jSaP}hPvU%uYSQ^{$%LDEJ5 z?hMA$;G8Np$lY6xe`HDRZD&>UNW`FP!8XJ$)mEl3;_}g1WKSLvC7Ths z^7K~PV%FYp{aja19HKmV2%{=&Z;?SY`Qie zN(m?_T_PYNN;fD90s_({CEcCNk^(9zDGegsv2-a4(%rC2H!R)p&HBXqelLIc)0Lfj zX6~6YXRdRd>j)BqP0O7TJ5&_Dj_#^q;fxY(vKt`jHyu%^Xs}fr`Qd_$TO@C&^T?$A z_HC}+^M2GhtwGofjlNQusT-^HP(@5{?Pc~y$)KistChLT_Nz@MIyW=;%HEl%TJ<9j zy$)^NW0vMJn=Um%L`j(E)sA#ss?xNX7~gk&J-xGI4Sk*hn7q8{Tv$<(xK~#R%xms* z`?{bEUl|=4*=s=sMQ+FVpI{a=JM8PNCeGq*p>t%SFWe^1Ok5X+3R0ENk~g~D4!2O?I!Eje6CLM52@H&n%4Z2%8zr(+NJbQ_M^lT{6;rigKxDZjitq1l(C8* zc;(3{)H%QLzD+Ldd}IHY&*|5WZ?Hy~uyA|rRoRB>$@J!Q(&Gj;ewIF4GX{H%)6>(O z6xYedV|YO?{3hOWr!O1Z2=S}k2yKD=Ztg=lCVSpbdIsxq=Tg`2F|GbOU$HiZfa!k# zXeiZW86r@;Rayf##zVBRI$Gavx<*}CNtn^W1VgQnmD9tcXM=5rrkNgN0fEJ$3z>tyulful_|pR{%-!WyV*{kPrb>_IjT8m0cRWmhlAxef zTVWEp_zK*6xm!?QT8RED_`&|!FE+c;Ax=I_(?70ALHc~NRrDhHhKBH1D)!zHVcD2Oj3kgoW7ag_YG}_dO zYUc2h%pY~GePWxN#5pO(+{4ekZN1f?v5*jmM^7&=rAkrSdQju`mPg}pyQ!eodW8$i zwo~nOaBhkF$Wd$};cIcv6O}Z<=oWE|OsTLl5UCY5Y71TJiWlE$xdBMJVi!BDKKn@s zRcaA}U*b2zEAFE#<37`3LF-@MhXx%VF7Q4I0WcOu#lO6?w6wguE}FBE6->RZzn?3) zpXzPY=s%(mJ7q9YwV4@Zh0A&b;Qsji_Vdw_OnkKstR?cm!1nBJ*@i}S!X`^LI#sdach(sj z1K(wL2G8Oda|QUX}~&Gq&5gKEe5pC6aY zS~?!(2&Jv`ra%T>k~4!aZJ`}kfkod-D920f$Nfzf6NRd{ey`A>(Zg-w#mH07F_qgr z^<>&QI#ZLA2V&kAC*z(XjEb#ffbX@_C{6c-+aN^C)<&0!8&D;;DcxqLcGM_!V>PR~~%i$nWBAVpCn}5~8Jv2Ecz872)YvrFG!WwX)XE`7h7-hdYzrHwFi?vdelNNl~k$bq9q+?)ApM} zh8>ON--P))Pk?O1a&%-B2FL}{yZqe@pJ}XUr(6$56ex;jw(DA>P|6FY$S8*Xt2{C0 zzlf~qY2ZuSEF>L+>Jo&KbxRk0BISEw{M9x^7M8;{jRAXm!KCC?1VuyHedQ6dvQuX& zPyfZG0RAVS>C4cM1u2cw!?;;nyV9Fbkamzy2K(=98j!$q@C2t{qI~Y&e}5AN@?CHM ztB5?xaQ+{VjTQ%p&|%8({m;i&;5L5bcK$}_f7*!v2bie;eX9TY*asy-R}W>S`G1(C zudg%;sDBoP|34o$gWKWdj93i8zp&;XXs?9{T&yGLJN_qy2cWQ+EmyX5vi~bW_ZfI) z(vimhlOQx}=l9l!JYvw~eG3S>r`|pM7KQD8-ySS>j~LB^i zPasw2E=J#5fk*GCsP0J6$AV|&4X;c~8EjA2zWM)n=SLJoQTgQb7|i8#^|F=aWBwdxnY!KWA6^AKh2K8ZcuE z5ANUi|MTug4+0Qa8L_NPV1j44{YVu>kHT76?A#Yz(qcpA|Ly$%cx#z!rGGU4fL_N~ z+hMiOezViw-mM5~(%N{Ae>wpY8Qlp=?UI80QE>q=Qum#pG@k9LV{xCx60egFz(V$n zSXGY}7TC>bM5<_N3AxTkQg{fSig5`2`g?Eqb%ROj-8{-U0*y+9VnC{_^^TC|rp?y@ z!H?V#a#X3F%QNT4eA~_|N|C~=H6Q83OwQV|17;rmlh@|^&&>vvAB!E@lGG=fr-v{S z1Bt_rE(a24>|Vsx*A_V}g=>UUyAAF{4Sa_DN8R=t3Xu*xNk&Qc?9@w0OB$W0TIOUj zIT>-bF^qc$na8Jk^rH;U)@bNwM{i<~rSU^SM zrwSe1ickA;IpGOzKc$;vG~RBuKU8|~&)aKp!1g^Q`SW_H@AqE-bFbf=Z8P=E?MP^g zMxL{zUlDqFMwC-=)FvJ;6z=`D#IxPQeTt?_>H zXdJlLHdAi4=Yynk10@BNON=ep`FffsgEuZv*m3M6pk<`-X=3Ync3pqy(@;|*z{1kA za-7J|g)E)(FzRyT8?NL)Ow@Uq>B!CzyYFR6FNJYzW$=!vRW~WlQ48S6bn=>9xv!UvYXDidUKL*jqkhVG z9m!C8&0Ap(fwniR^Z$xi>!Yq3fi+hxq+h``s_81%``$rx^217p- zNDQjK>=W8}bo>zL$x5Jgo1yvDV3Kj_U1ZW-=2lg?0UH96gjs(h z$~kRr%)YilkrMv+~16NuI{a;x$LLY)752VXGbUQeSsp_y5uJV+QAyo_YUgF2zahy1m9PP{;JRM z_K8)b1q^4YmOE(qVI=ljCgoY$oAAk*qa6N0CrwDip~)XBOt!OGA1;)f}GCuws9jP3Hy)RDM zL0R8_1y9lM(4QSTF|85U=w^&pT100rr}A2jqLR8+2D2NiYhzm*9tH$!d8mCuU(95n zy&>>0J^1$8`jw3Zve5GO%NJ=$#EDPV_)CVNCbkX=u5lAo{rN9BtCqhHzU^oFRFO0# zJho|by-$UY+?3gG+9p(dJRD&-FuQ^j)30wbrI$>X+pAExmsAoFTpl&|qle@Cg8YV= z_WAZCY-2ge&ii=A4P+kzn)Vm&J`wJ-m9$1|V^rMn7N3xs6ny|Dk5fr~)&5G~^=^bu zAr?a|f4TL|y&O-z=CWX|iTSd~H+~F(8M}S3o{Uwuqta;)ru}0c*VQ$Fv^14(0_~gf z=IF{_PAeH^^n-)Qm#DcPB0J5j^n1T}l(xkzUu-Ux%h8okv=~M5JZ^c>Y$?p~y686) zDm}J(N>XK_nA4kkba`0fv&AO>GB~%v?eoWTVOTwcr3>3Q7xXcRPh>=je(R=b_m_+$ zK3>6z0ckp2W#v7v`qr(!1NJAfZlk!*`>4V~|I$zGiJPj)(QJA6`lKI?ZG`fJ&d%cU zJJ^2y#aQ!vE{9O>tI{{eNO8EW`If!xo}{0{GuzyF&K|w4Dwrz2+i!!0t(%-s#B_~dV8aY}fHZDTK!ieb+z6Jq^vBNQUWH9McGB0fBh;$cDTBuo9&Z^^g zaCCO0dq`uGM|&s5VhaKiI@cI{zzs?0a99OIs?(rH6CfXIBSK{FH!9eHBC{RcXaV#K zR8F1*n%@D5K-GrHMs{B4B=Gx1lCbE!BGJanUpyx1P9CjUjQzd!)JEyS(=3)UI!wz% z$z56Ra3^z(Vs&LWpoNxbiStR$+$peVA;Asj>>Kf? zz_mTT1)YoCDq5M*$kJZ@8dR%w3~7%~Ex&&HyCMjG5ijGGg4JTh8s_NAw+vWCCB2Se zsW$-g-T0k1a|v&@_S~S*+wyahf|WFg4Ih@;LYXd=NOMQJYpU1m+ zpuCv`dIQ?98l!dsQTPMmb>UJWvK0G+76NCwC7B1?x3v`fTT-+8XIJ+8H#wbMuHmq9 zC5Mn(;z>JFtr^j8f@0F>-tc{?X%rz-4@vU|$nALzrTs@v6v z0?=SkAkFgWJ>)e|`{zKwyI2T~!wLlcti&-x z=g-DYM|#ZH!8t)f02WEi4#!gG&mXU&7$*^v#4zYVH(j5K3*Wt}DW`^IdbGvn3ub?V zU{}VC%xBM+HL)jq8;$u1wo*hwIqvBwW+qo3cgp3d=+R6&umtj#ZW^mZDq7GY2+lL_%Qh0^B_3DRdTqd7P-}jn- z$c^oz(E?QwO!ukSGqXQcsRXk}o3o9t$f3LW%<2|UCe;NMrSgdJ7`@ezUMUbBw4Z)o z=Rx|Bw{BZa)Kiu9mAUPPmmAVAy*Ivx)G0$DHPUCqf3V19OI*8BQ8J!9v_sse)9>Tw zI1kaac%fu|q}5nN_6PObg|BMLS#tz9A+mwqj-(&9bL|j%h_Zr=OeVuJdnuS*1lyNg)fW^L)NfTyRoef& zEB(!5X`xte;?xZJi3x3V&Vk(TwlaPC%J;{J2~T&E)r|IKK?%A<)pWX|^^TThf;kcr zcEg-u>ppRjve%dJX7@Y=QjL4Ay_G)*nZWtTVKMSR6P)$73cs=K z*uJ$Jf8}+b3UAV}2W8k{$*YfYyxBbG?JRNftjJUGWu#ug7zg5`FvPgGj?Ud!rmF1~ z;o~=W?~9ezH&#=#-&B`AHxYg%yw`GNx2t$A=9FR9rxBri2?C(&n-zVw*4D$sNb*OT z>2}D(Yb}IqIIx?*u~bI9{z~2T7;{icqFF{Y9>{iZ+W#%vu}W3{Kysws^d&QzLK0aH zW8U32A6z=wat_>Wp-Q8ztB~nmZTGQ3!aZ;VZQU2b_=ly0W@f(!pxmz0rTF-`xCe8a z&%|HOWcF<~>`kH%f8EQ`h)wq1LnAm?VQ9X-L?UYz!fCs*SwdEEbVRrJT#dMH7{p$b z&edvA>j+gna(2sYk})sbgvlKhO(fY4lvpsixam6@@5X$e*}u1uq39{rf8D5D%?xz5 zR3?7BUD=S1O@6|M5E<*NLZ+}&(doMxJhYA$ut0jA#MP>~n=830XxuMUL4r^q4*R|0 z?ZgDHmcU(^^9L+wafA741x;roC6Bc$bMx4Ee1Kp?R#Vz4SRK4o2|6C!mDZ#dWF~99)*SmZ~`a%S)%aFv6?ps_t*zl1K1-%eG475zqiGW&vFMW}IDT+z03%Kd*B<Mc2qN@U^szHf48hHRna*=rdLaNnc<R-XCA&KPvl`IzO~L0lDtnJc^Bigf9&0!*UW!7Vi*r2qaeOFEMJxl z>Cy8~Q7FZTe!cHpP?`YKQM0w)nuZ^$8oCgi@1!2yN8Ct0N<)A6L4pK@toXNr5ukbw z{_yj~Sv~F6SrUhQL?4O>Ie z_5x)^)rY-no82RNqRQ?NU1{9{!ggs0LgbTo8i<^P7q3WNUT>ygdAyiviEMj|l%;%K+9gP+ej?MS)-yyP{UC#b|i-8~#?ZE-&lME`APzV0) z#?C6wc=fZZg1ZYQFm1VV^-0kc!RE`NmF@&?ZsdOP=(#vDMz`LzCxIIafBG;c@9B3O zdSBvcKW93OJHBX`^l0eV5-kgP`F4}qME!D*>sv-Y65!qxa6OCfW4ln1SeNf#N=@;s z;yJN~lsCt6^(S)QE9AURm^Xkt#m7hiNX9lMkFDR|4WVIxnma%!*UimCg3~Tano?6% zb6~cc1o^7b^VEu0t%j?{t5WJs`$E2!18%JCvlcB>ac{Tb3a)gkOP>2~zoRy3`b z{OSfcxv}Hfg9TYL_AuG19kDT~|fgTSwZ+-XN7d)^` z{+dz8yEM}E3sOVBp~5N)B0 z?%>vchCuI)4YZW1;%^E| zj;9=ZTC7G^V||jxF6H}@p4!1f+3TBiwcksegP~3JeTw>P;rMMoc;X_lS1Cim-;JD_ z!5^ZyuSJR#Y0lgCwZI z?6iO8Ud09b$j_bW&Dfobo|t*Xu@#CZ^(JLbT3aX0 zYM#wk)3>uQi!kUMVcw-(R;?o zm>3?`uADIoNtF;Qe_3}qhgP6EXOo5AXxzEuZ#ObpsJKUKZu#gF%p|I2L}p}7ey!F| zWGIeExw-SjWgDf~AqyFJ;=9ujIO>TXIQ$0lzhdb%>UUQA`rG*d#)+&5I}VX^@ZHw|uBD@HvY$bJeTp zrVS<^mOS=w8)C=mI{KAsksk=m&nz0oq=dV2^P*z7OtXdOa<8ZlYl(YvUfJZvMr^VW zH^)Z)!6s+Ux3rc0h#hxE>5PV}Ikq+O`^+w;llEmL0<$>Ayj9Z$?-gR#(DJ{a`8~dW z=hUtQk1GA)pN0R59)qZb^7^7PF(-tbP;EFSe+6mhefe#`Omw&RvhOiuHCZN}Jw;-< zU+u3KPd^c_s#jK5W>~ye{;ONqV4ZUys*7i9vei3)?4MI+q zFRkM3AYF@(XZ9vD?JgBsGlCDPoDTD@4Y}AYJ1y4>&P)j}##562>|F_9Ji`1$Jx3sV zihD=9%n=8t(jL$o7af@gfVDYXvueuRkd;?fS-cFpZ&%eP&cnOkS+`Dp3=i|r+x=cNWWh)JG+>r4 z5u4wBb-8}+c>DHa(HRbdWFx6|Gm^w-E#D|H`CO* zj*MDI_*a_c{7-rII?VMCoAY+~o)SkIpV65BA)9&2Li=ip-rQ{*q@S zzF>=185AFw-BQs%cRklBIUH4Y?*duiE+IrhHrI_Y?|o6?&${#G-S5F_x#Q>SDyZFX zbviaG=2p{SlCG;9#sBUt|3o}j+3Zgi_~m{y@@it`a7OUDM3AD9arhM(YKzk{msT4i zFYRWkJ^L(&LUse%5Uw2&+Q+Yy#>ht-PgVz|^EV25M(FgsrruxR6v?%*=H%s-XV}K- z(JGd6deroZDd3Z;dnX~Taq$DqH6}>4XTedS(BsuKK$HO)F(Y$qE z>S;B~`1UM}qm1t?Hu{#`K?iY3vwIt?qzHOc#@t*|Trh0V5MXkbj4gH*YP#mnf`CZy z_ffl;dLs>`Eg5d2b|V4y>PI+nSDL%&5;uOB^uQ-7LhA;*-Lxt?I4xGAI{7+pub{wi z?c@f2kbl;DY0pN%D!Z$gG8y<=d7L=YsstsXEcl?OVk0i=-K=}@^t0L+*am=C(aHck zhsrxU=NBulo4n3;RIZR|h#B?q@@1!cDT!y6nclJDWd^45y0Oc*;?+s%#NF3({btCy zaW+VEP!{{PUlCUI^!}hI7CQkg>F>)BB*Y;FBY)TJewO*}N_IAG57yBMDZUt2XB$8s z5)TuO9*4v(T1=d=vRzq!lfLue1_t)5fIS9CX8^Ag3)9o=_!>T~k2oCWf6js?y{Ta$ zKfyt$tD>ruTTn{@wXNSHggR;Bvok*AVnFjNzW+vF-v=b>AFq#WtmUK;6!42OXV&DK zG}Fm173CITvl}Q{QLS4G6%(;p4{q12wK`1%v%2UOH>YhW)9xU6j6@D4iObRcWPq8xbt|Ku`S3vG8!ol(@t= zq2D%R|G3u+YnwOQ>52-C7rTUHy%f+hGc(=uob7 zk8ACAp?5=Ed`u-%mN~2he^^@RhPm2W7Fv3bm97wQ8jnOe$2^;zv7_MtVza2jx1Ia| z6J<%=h|RHiF4hokk0(lVCGn{a3v`;iu8z-FmfcqMa8cIw!(G3w99^*@gUQva%*?U0 zl(@@F#R~(2-j;)ygO|G?fhCDyL`_=LpP^3hEwH#kvS{ejO7CKVQO1`VBe?UJX3S@E z7T!+7I#aRVzr1Dh8ZF#>=Y(O1g*5rzE>Qw%cIwi6$yCJh&v+Xt;co87ZV_bQWe70% z0-e*8a{Q^LRi-jA09o#cC~>~Zpo6i@%{t|oML)oIZ*Ot2mP3gfSXVH04=AUQWI z?=4e$EGs*E{ZgEcwUg}e_RLgj)Soqxf`%a=L|2SDx9r8xi*_!C>j}-(R$7+={W5Ln z-Z`+^S`VR&D)!G05*{9>AT&g(T^siUw9f$E)lCDkcjM0Mi;Oc^WE;rG~E^t+ML+*G|7 z9bQ*gH#|IiaBu+FX77(?l!VjlkJiOUSJHgo)`BrtciMenH%%y&Rubm+EYy9S*U8Z=`L3H(KwN-F;c|K!1mLEBY z92MwaS|Z_bCKd;(OF$6u3ZzfR#Vn3*@s1U0aRW_A`6RxCzEcott3-r0jV2$L@sp8~ zk;TQtyw@JfImPaAP!_tob$IGEF)`7G2ZMb-j69_7T&5lohQWA)324fHQ7`C2@lsTu zY#iLp$vII8b_@fC)|wYVa85Tz|N3l<*U)0|_h((iecq~P65@sc4MR+_KBLMK z*?KfOy4BmOUwgUL%-Q6oI8eoEYI>hHCF-t#)tkw5cmljp{9sdl@4kuWptNWBc>_zA z1*Hr_HsW&I?JO4P7j`RUN)>6tEVm|MKY;Xs*tE9{vH$lQ-aZ z$zC@D^+50i)Px7$riiL>g^f7fycKpPFC(KwqLdojG^k07&(P7+W3y5Gv&Su|aa5&u zAKBmFAJV2fEMCwId-{&CC+%p0Vr(fp*yqaz>jPIc4hshbmkVP$ zy^~sx<{Nu^RcDWpc2~=Rq!BN9E7qKDKr0&6)+jNi`MVbOJuNLoM_{R&zB#V3W|w24 zG65nBcE{fcKWNJ^>QVz(P<^!abU8^;aEWBdF7Fw1@L&uI6p{oT%UgkJA=4oQ?F0RNu2!((f;ph^d}UY(vUq4~#kXUT3+r5Z}0%wr3(cm zO_OnGwq{SsiHZu@#4WlfRuoQSyz$gWW474gwU@JPF7 zTX{u<`T8_jq|eab>a9@-^%GMQH6SX^r20zj+;1f>-46N`nCj3*aa5us#U@Rp#AUJ6 z84IYDiQL_l>a16%tC5Z1TQV))ipc`IK&?(>o$ub(qd0I*z`7>ZYCR{k+ZGkY8c$!~ zz2neoUJZjOBKXSr_$vfy8Fgds#z>ugz|t4u;!YeXX?Lpm86CAbvD=)QE!EfJDH;XvYs7oxDCp%toi?(1DWQ;VS##8)(NK1O7xq( zf4s+`t7rJG_{~4`rYR0{(L(x)`RIcc90RlQ0%=lD0+Sh%xK~H8?jZaw{v65y1ER#* zj8}H)Pw$jzUYKn^f|ofWJK0WE_-LjNkn)^RlOy@N=hG~i5s0xpVIQ{++wX!#ixiY3 zjRnSIqhE{Ex1<~5UtYw}r8gjp7_cl8?XL<+5&`F@^`ut!FM4s)I#--j z;f?*YZumlx-ueWzB|2)pqigRQ-}O(`-);u_in@()nKK{~Bz7+G ztiTQbRLp%RJ=Gwwpwppy?iOT4A*mqVA>l&v?b}D!mKe>B2h*}XpxlAb<<>NW_^kN` zzNsF}ZRu!!i?VcPRGP;*XeY9PVLzvjpV^EoIrq(}K0?2=S>%MJ4;`JH$oy(<{(1dV z3XC#_H`>}~-Q3@CUacfMDOhW#t7AKJ`c1aN^RrU(6VuC!ig$%uAyiK5~dEN?5h6Q|wn-3~lBMiXvD%(fo`UN-xnP_}N&UE4TxQ z__;YWwO+=WjiL9=4U$n5CLgHw&0eKGgX$RXWQ3)*7KCR_2sF3>W~S+7FABy)zQ1 zaL25EJ4+hb!|THfhYRPRK}kGFu)jb@r$oXVpWaWuR7!_0A{njFw^VRM_PLPSWN5hhai#2NXIk}ZqU~OITOkMj_DQc> zQ>GT6HaYwn&ifu{oZ@DF*M)+aDhrMG-0f&{!t{Pf0)YD`d z^!?}<+8!rk%~vq8+zXHix&9lUTe!vls{&)c>fO*|%TFZ)Om4BODO9&lmyK^Z-G?62 z9c{ARCaVa>H71AfT~3li)yeee~sCy5{Q&ar#) z_p;^l^QVRTrt2lBm0-D9{q;L;bRzTZo{>@=qB9am$`)djuD3igEPOLcBlx*`J;4@ixQlxNEF z#yZ$tdMta^3w;|%VKSqtenJ*>y(Y@ zPwV( zXaO{Tx|MX6LW$m{mvdHqhX-Q^0}ywPuX)zo#c+_N|AnOBYEe>CUQW@F*@b}OVn+};Kw*;mRry^G8K>f#hQ zOLlwv`gCJ82apX_?E#Ow z2AeKXK7EufM~fF7hCXJ>zw}ctNll8!>*pt6ebl)ErJ3%*-%1>#pdD%AyHZ6oDenPQ406T;OwznL0Si(~VgRhd4oG-*vY0br+Ck>R!Y~@lUN(xC~u5WB?{V+1E z$*$rUSfS4|4rMAt)JI6)JBt-AKDv^)$etKpeN8(<_9%k~FMMYK& znMJ-KT>;VFy!e`RW7I@#jgH!b5493noO};|{=D~J7i}yOE_M_}cO94eZKUm2j6;Ik?v$DGA9P z^zD1A=-aFbu0WhjkExLlQgDP4yUij-P zGB#x0j7uv%3GXi&%-dCqKE_7sB2{xpZ}bo3=@9Qd*pz=&O(U5DP8_$jZKGUY9fny} zjGmAfflkKIKjA39$IMTTOcf=gqziP`KIf~qa1}h~7uUnT^`2WS5p6p6q_jOcYWl0U zTDc~1_ZRH`tfq7>)P)}fQ-p{4hNQaDQn79aq0kL=j3(8-mI4tMLQFUDK@xp#sRM(L z(!{e)6(U-6TV$De&A0?fG*LXaWH2-tJ%&kbvw~(&&HMoun9vzpc2`NyFi~J!6ZY-; z$>`BB?$VMRMW+`UgBna}urVc@E-t0-x;`%E=9P>sLOgVu=PaicBBy4z6b1hc+(>wV z0BCzK-Re!yuAczQlPj~-7U>WJiw&k<5K{bJ3;2}=glS7`R?uF6^~C`2*60+S0*B*1pZ{A&pTjsd>+D6VEl&%Sx(ZV7=+kn!jT zbso)tv>SP;3wAimPJhn~cytZ8G>21Z^xyr1-YOWZE~Q>@Lh>m^)Of#heBR6evvJcb z3(M$-Q00-@2wGnZ0>b0*<;FiS@W$2bDoHAJ;&iUs7wz*=s+4gdZJjq;T!`IxHLyy@?m`=2K` zGoc4bxG`BL`v3LB|BO5ph@)%^9#!7{tF6C#1pa^K6~My6N-c2z?dIQK@p_8 z{;Y+6TSCVM55ctO|B=o1&yxG+J+BI46k^OcZCX(Lr>&L&%!gJF+4I?HY#7^~(SHY* zb>ePAXY&922E7vYt4noNJNX#+pdxk+(1IDq*K3pdPvamwwo_EmJTxOe2m)gugoizw zd3Rpuzq=)*VDS9veK_JL4ZsVnaZ0h(txuJO|NA%otY>iC*e474+y2*J{`?huh2Gbw zD}M5yss3kr=oL^vH3xLT0?`M2knmM*JQPt!Ghtfgh2?QgZ0XKDQ< z$GNoIcY6MPfZsi0dJJl#7F9yfH+nssM^`&HU2L8=KU;t-9Tn*{Z|(D2UkqKVl9IPs z`K*!uJHrC%7%zMyV(E`jExmeyQBP>ck&kUT6CD&j1{o37NrEGme0;|I&h_w;dAhGZ zatEadb9Qt|4!$_xfT9c)YK^DMgNfi9utx{|^$o!Bs)5uN%I)WbZVSKo*y&bv&nU@q zy|Kj3)(g6SKmh8 zF4BUr+?F4@Dqob2ixaD>bRJA^sbwMYTCW@Ko-XA|{x?|e_Y<_C@W^LiN`M()?ak2h zqxt!HAj%IAcXkEHTDUDQ509#v+Uez0XlUptS$u3PtXEY5{QUI7GNWb1e+UN!G!NjK zI*8I{p7Jtz+P-{flUm>QW$V|V=m@#`P5+D>UFzCzPELe$Vht`Eqo863fMz!`m)=T7 zXjVavx4~!fsUjY;Eq>CDG>Q4PVt`k#tfDdmeg+4rEzLu;#zTBk~Pa-aI zNm$YR6o^?lIj=!B;W;1!Vqu`G$as5S6;;E9Kp7N;v5v5m2c=%8d%)F)02CO&pUWWE z2ws;e_qg__=fi__InKs!bWIt8;ffj~>c88%W70>$vPKM#`1p7|qNAR3Cq1-;a@yqj zShQC8-#j!6M~Wj88!|E_@dUUokEZOd=gFJL>)lGq9y2p90s7e$>aO*kX)9LAYP2vx z*limWLJ9Q)WE9V{wVbSr6iYipLqi!EnOqlWVnjbROI8{Awb0srD94>uP$k5ErN)6A z(;xGGmTkjsn<(UHb!@IstAd2mbH9h56!)r)ziGE3?Q#>y$KFz@zT2;;xqttwD2F%T z&0}-K#l|l29kTClP^}-N&f{Hi_G?bv!Ceg7YdzjXTpX-MKj*fovoQPU^F8WoFkDas zTDsB@9i)4t<&)2yMsop5iqrvE4l0HKr3JV_f;7mnlqKA-h`(AyHXiHY{gnCm2jLvZ zkPKD;RT%)4-wjmp13G*^k_DZNPa+w<*Ul3N14?4(*$8o1clT?2+PtD7wHAH-<=R>T z>Ya$jlQBb|A75_x^@3B)Eu6~j8T~7X=Fsbn<|`{u_=kVmK7iuhX%;7REx!3;=|>=5 zn&&zjrhPS3Htvy@BXH`>-79FfBB=NknkO@PI7*<$^@uR#7UuZDL?dwbKP(?hN~N|u z#LL2J>bKK04<9Tj==An5*nqa?R4rmrM#pXMJ(_`SR9_d0n;4Hy%k3!&uy4k_5Rb)B z6*wsGdQ2tkT3BX+Xo3P5`4X_??gzDpQbe!bs@lH(e^}<3+`wWuz+kKdDlwZ=Z&p4| z>lXytZR(eRqvqbI7lsdqS;dDcH|=9SIhsejzY56Ozq9%r!`fjW@JU{080I&}=96EX zK?-;cv;KIP&zqBGK4b=4kKTrNc6NH~FN6AsKEi+9C`0~Qv9 z*M1MU9nOM^N!hP)ia!>$>)9aUb2Uh=y+ARWk#i&v$yaWc6*-v=kYaqSSxP@Y?e~Hg zR*Q&|)aH_q;h3gBEO?Lct&Q;)j?0%yBRFckF1Lm<+!2sMS--pV)spndzF(O;2Yows z@RHj@TYi2d26;Rk@J2{`;Y}=ckz9@MpSI+V;%K65WW?+``{6EA0c%y^5Znj%S!nKd z6ov5Gh%F*6Ccif~co;0dK?dd!)SEZ;U6-{uZs-s_ZGVAHbBfjEyrztwxjI&YkYTwg z-7H6awa6kqpt)+&6=y+q0B}H-*Iyz(Jhm$6Q=hlosptfum{q|FEk~J71g_D(ar&s*2*!J;YJeIP@P)Q25b~f-AMLCI6@jji*H=g& zSqmZ!5)p1q(n3!vk*{F&^#=y>P$>9vcnOqy|xAx{aJ)y7aQ@(e-Kggdt^B1L4*scvrQTJ~Tl{90&wXI>AMCBfH%X?m;{Z2ZHgR~# z6yq!**TU};fr*(J!m*1(`{8I0&-{dI$Cnc5mQ__Ss%2CSgJj1}HokV|k2sQwky(D( zZ7zXRIfS&)yk@T_RJtT>5c>N2L(rt+4IKk+{xQ{~J(lC$YAf``Y{}Rw7IoECUT=w3 z7cG2FoetI6nDtdOp>M4~0Yy&uHo zru{}C8#?Dbbp-Os6otPpNHyN>0{K(Tia?m<0U*rrvTo;yUxK2f_KNfxG>sK!_#5QY z?dH2jv(BTH9|NKz@V0TRBm83y>2 zIQf4(3~b%!7RA!{HkzR0+txiSlf!XjTq`isIU7>|4!^_0UaFbM9@HHfQMp;ruZGd-<9T``8FI{yad>zs zlDu6EHjG|EB)86gRK*g~;y(UL$J+wB}q5^qrh@5&eqSS9wLn_<7 z5kV=?IVK{he9B_H97qscY;pTQUZHkwA#oxpdUR#qN!9%nE-x&H8~a6Y{dObE{O7bZ zakuTaTOQfH%^t*<>C`J|8<$d{jTV&H)U0l^E5cWQz?7KHL{LYD@Oz8R^gXe)wJt=oWn8m%7pZ zx~1FB8Sc-&g&zGm(Pi$SKy`1VsH>XvX1u`qanZh)2gO#chKbma+3&2z@XffG<-<#_ z)%Vtj_PDs!WBrvgPm^JuVbO*Q<{7Sr>+v-vB1l^PXQqIVV8Oi-og|TtO5h-|{VX!` z05Rnz1>PM9h=%|$QC492$z3Hjy}%lq6O91_qjVLf7O<~q_E~_HUgZH^Mb3Z&6DA{Z ze;;0jcIasab?u#62T6&~j}J`-zsWcZW!i52emPXKot9J*IYx(oR8?ib5&V!AoA?J= z(APf`3AS>J>P*Ha30#Vg$(M~&DU(evet(K5R3%~Yh$;nJ@^<(<1dEoauk1c`i&;%O+Zie>;y8V_{C_ z6Z4s+iHC5vx=A!zA(or2c3uyyjHsFTh&KoqUN;serSNms^dBTmsW&z(-C%zF!dehiXRWUz^aCOiHcdi+c~*LZcdLAfdFd9YgIwZu+S^pK&`%A{m{-BR>;fWLOD zw5cwodOp640P+j{dNM)?K;Ll^QBB8(>~H#R!i42`4G7D0Tt6i zu+IZTVlz4UFUyD+X8cjtfMbKg(BN8jo9d(QUeVqbf&b*-3d%rVCR0pItFIuhxJ*Ru~js>Q{KU5|z?L{y{F%xkbL;Dpc6l&G=I9|9XFx_2B4b*>Ya(ZMLXuB z!CBzgh0EbmgA29g^dYG;tMf*^X(!0`0WIKl<4N0nQ zZ9Y@!l;QZK<(=?I);jlT?^_TE*c!2Bq>hCYxLjGB%(v0=_g#%RKG3jt81BH^vR_vj z@4kLAYiC~r{TDPL4ZVlf()T1-`eB&r5x1BVUkoR4Gc&ngU7)E;&q+JPH=cJe94)gJ zNVd5CPS!Y5KH9mqgPtioy|X!&xT7|I+gi2kN*=W<91BvtZw4<#B9v<$;lzJn6m_ux zV~Y12#d&O64RH!%>3-dMs{XwDq0d@K6h!V`yMIT#z|qyD`5XMxVC*^j-zmEsDD8H# z-}CCQ_RK$Y+-NQZqdR{Reu_OKhT}R8$(|kiwDFs|)zXwc(d#c*p@s304R6Wl`*^@N z!1t_(2&B+=-@46}Z>l&x0xU{pCE1GmMX|=dIjLsDdo=a1YB>ra7>+8AsKFgh&DDe# z*{Y0Ii2%fWnYSVav?J=CQcOb+!H=iumnpi4C^mGy} z$t@~ItV~lEyWjkRN{o@f?D{J=Hn;UG`QP!;N1!KLeG~mpKM*X69*ezo`lhxWBp1M; z;+B$eY$2*Dn?e@4EZpDydncY<;`K_+CPm`EFd+c|?{Q$gEBXWEaM3^UUvgvdI{gOw zH!6bsRRVe*X7AJ7-&o7v7!iQ;fS5=PPYvn+K?eN+OD-WNDShETuqV)|L%xXoL>$3_VdWtJX2OObzun#4?bfV*(6rvZbGEJCm@jma6cI5NyIl+W2VVQ@Xa8v+ zvNkkIl~HPG>2L!;VKUE((wmx^Qd7SiJbU&GOr}!&9K8Sj%(54t{BhiJ(fs>;9zijJ zPS8Vi&qdDuyX%ng!G)tGSf?ld-F0-t0Oh9B$a|zW{)HX?=|Rs+Q1cS}igW12-}CS9 z%irn1gA05j&HkSs0Q^E&aKYu$Q|)gQ^Y8m{1wKNiFJrd<4`j;m2${COI{gb9=ZQ}B>&d^|9d3=Kc6LsT{Uk1Tp&_TVC;Z$lukFwTruy4 z@sil%d@Ci9kdUCY*_5cX1_i*sni!QlxJbaHXM+!(`D@*vBhtvAsr!@_nu`Mxyxh>& zO&`@2L7M#0{rlA7jhvV)@+LW$sYUMv+5a)PJgEr}&QLwn*|CKFf|vtUkJ#Qd6S z`l4gzAvjpfS$(+;s4xmJ`KUuR&;GHxB2z#IBuuIY-C6{q>uv+yx0YZFN5g1!EG8Om z?|2S2H11@Cfo1TR=gw1iqH9d+F~IxDEiAJ0ZNKQRsu(%2c62pT0zV`bJ*~iL2^7HyMf+q$7R;K zQR3pc`g1^*@3A9VZnfV#Xa}VhE3iTGSB~^Wfez74T?|B%eX!$0o8NVRdvjfYwalK= zTIsRs1FXBegvfk%SdO9J>n5-Y2IB9FDGQp-w0w2E8qw%HyI+_7a?}%Oq_jqv8&0I1 z`nupf+IU``ofJP%JX5_NuyvB*w?`B|gyn|Pf&fkEL4}IKE(S7xE7U1}xc^}3#DETn zBGqLYF+p6T$a%CC) zBVKn<({IUlemxb#;t)K5&vp~txY-ktZ)<%9FH! zFe0VmnHlxi@o82OGnytCvNyxM+x1?!b5Y$`wqR9AFQ=-N`m@nq^Sg5%O z{%h}B)Pk1O(iO}6)-85slA}7@_&G)XVWN28W@PFOL+Fg@r2`Z!g$aP2dm;^j5Igoow{hf7(*+Y;RZGsB3Gdo3H`%X;%*Q zS8t~>MFGFd2rw_fNl{BpP0f!SBMqF&H#;RbvYsC~0(Sk$$q8V2qu>so@6AjUsXjsw zRaM>o^F)jps&+O9Ah-f*4dp4tWi~3IB7CN->7FxxX-*IQNUo%sB*g4jY5X_??;nRxxk7$|~x40Ys!8?Y-X1 zDXXa50wZ>Sw10hiu{U!uZCngFc_z60-N!VGw*b##QwG=JsN)TgumPe^v!K+$I{jCP ziQ`(N{MacxNdf`BU}k)wfh=VmR8+wHu*)W9m+gdVbR;@X?*+hG^WkTl6tsFG$cA@b zQsCdo1WP zobfr??b?vA&U;|9(XNVAI0y=3>5=mdnp*KGFAp%%x)OR`fc_|t=0?rT%mlhT->)@3 zqT9%}*iDt`=22BB?3@DeuOO(Xl|Y8b(YdBXoVi*#qnc z`Oz^Dgn5={$A{|ZjnZB`D@nrIJPJti$g#B?MfMt}3VZs`sEr%!LOh0{z8>{(Uw@LL z7?{ZM9)U;7ns5*}{1IB?c^A6ptT08^6ck2a`|3Vq>_c-kYrzn1sGw;;}^J_3aHBR zgcHKQ>_2zak}*=wVy~#GOB*os)wYVMPh4H|oO%kF`#B-IY;5yI1;$r{@B7peuN&_% zdGua(W5+EEi=Ahp$)XodHS0MKSB(>OdxB&-D3IAz5%J<;fDX`^J}lPR1G!Ydr>!;k z#GoEN4=0T^qDNenyCYRDB_UAwVF|%tx3rUXhrMEh)fMGI>8JjhNDS>#ApHPsw3U@rT6K{Y ze_FonGvR&$YoBj>C7O<-a2n4fiMFJrl}|m6m0?#ilM65Qe%n0@V0526B{{tV2&ufw z{OV@7oU$^uQ=|_uzE1(j@S zQMo_rv2uB2>h@E^!$uw3)YifV>BIsDalS@`0eStuf*8xLA06+g z2+z*I+AuX5L(CvA7xEm9k+JZ*h>}tq<{HHgp6jWkKcpno!NC2OVtO=>b%Y%0B(GQd zF%<<{y#lI@y3-bsvd9-DM8ha4*%mnjN0$@LVt&D|P{vtBWD;{OF%?1NG@urMWgI!u z{kWT?o%|N)>uA7~d}#D#jn9`_L3rj4r17vkDL*p7q?F%beP5QXTVc!Wwqc24|D_v8w*+ZCBdipoNO<{adz^rPS-N=|hc!DwOQp z1qq=T69eX!-mVLL`}@fg9!Argbe>VgT*c&AH&HvKs*gYNf7nOwj;r!m`HeA+nSHLH ztN36D$D9OcK1bA@TyVSuDZtjMxXFmd;A>%LeJ~D-Rc{nk&lr>{)py-u@>Ih{J&kfX za|GGKIPYBemaL36*Qd{4hFTdLFdw%|A}UJSgQmvY`ieUGXAX@HtuKF|8m17E+sc(k zc>pwTpSBD@IrhfnqpYOy%#)3vsab^1J*7UV9aoUMl<{&oli{ZKBa#3jIRtmP!G?GT zMA0fEl~hJSMd@-ma8)AX^+G}(Z_v;_17YUQ;Q&AUGdqM|B%UXJ*_dZH^Yb8Osy05k z+ASvNv>vdCc6a0&PI7)$%wn^8z{(-IX;X}+pjv68Zt>p38A-&>x41arc&qfTt71!#uBAoewss5xN8-<@GC!NnDl;0C9060Jeu5=&*KKK~&?&9q z+xeS=X7wFz<9BOSGIzkj)toSG+5EKpbkA`kQNJ!*03OQ#cU%@xemscWeM&dlKL%rp z>1iHO16##yWn0Ii?;QEWOPvl#yV6hP%p%>&MA)J(RHuYguIhIw;@|Op#+(hFjmwOg^JHV0r8v~c^^JG&RQ2L+Tl~VR$c`A^2$0|T z=_wNB#o(?y7p@W=&rJwepv8nf z5+R^sVcz7|P6B2ZomFXt>M>^>RZj6q2^@^>c|}o>2r8A@p&WJVVl(p{>f?8dG9c-a z?SxTJfE?=~B<{upoK??O+iiZbq4QV~9% zQlAytUdS-pSZ)m+tH#R}$nJ0pzV{VUXMDGpXOTaI=}4oPX^>r!JjE;Y>_cb9OWT9f zt1!Y!VggJN23I_OR8s-R75_xHUL7Bz5n-Ykoic&pkUno*FI3u1jyicha=?00FxljC2dsr z@zAr#XGjej&^C_$lk|6&yZLV)Gxl=)9dIpzC zX^~J)s&=v5(w8$PnZ;b^fK8zr!-yJI^W}h^F?wjSoX6a|Fa-sJDWr@$QA!*D-jj!Ez2Lm?+rD@qO6OW~-tNA2xW>kg~Nw0hB`1Zs1BX*QwWa|X zi2JnO4B(PO*-*nTx&|ThTdldxta2DtfiO)vGI1YYP&|bt$0j)w9}pHkCUz9jnT)U* zsTaYHTiJ=;FN9|>r83O1^ki+((RaAgSZj4J3~AWoo>Vtj2#msooxoFIBh{DqGGFp! z>ZwrP4|}wOA)Q4~uTk(B@ambs?p1#G!xkBiM1#N7cf-x5h&g*$W<*%Yzo)?AL#0NM zad5s)p`bCupx^6mvkC6O=;I;K+RRocLT|y^$wy$UU`~^&oR$*(N*<3TJM#Q)0LZD% z1TGI&DTU{@j-(EMRW9T4e>4jp;lrvi79DR>RhsF3`8q@)5+=hJ!7N=XQC%0ETR*1gZ1Yl&M18f$qQrfL`3HW{fz74CrR^>ytyr>SBlb+C%cQib#~sPStu?j zu+J*>g(0qQ?M+R-=7^=5neNANT0VU@%0sh9HFq-v`|2Yia|FC$7dX4K&3C+o18Ntc z1(7w8_dmZGv5zP~Pf`Z!!i2p~M)<6hOU7uh6LuKhFr1@`7>uYY-is0HGNRHyROppK zo_WzXE3e!~Tu?V!q0pbfcqJvr4<+n8Bcd9HR@zv`h~wP$m6y!aR*GDhb=+(fHPebr zIVd;*xDsm4FQ9({<79CciS!4I@uQ+F~`LBdM1$qUZRowiX zClOD=sZyx^2(f0@lHeAC3{W#s$Qyqzi^>>3)|1M6q;~X>%H{f@X*q-Q%Kg55v$WWz zrhvV%fd`VaAmeYEYYhWe6CM%7DbpZRI#aI?JAJuzHk8l`_+vwn5bDNLDO+X6UgXwN zDt?ho&4725a8{vY>Flwm@FB-4gj%vl1G^`ZFf|R-%=d~*>T5W82)D+<{oflfCMdFV zIJ1DtD9puxnI4F3WfD~db}Dbom&@rEcUnM^WfG-SUGVM+?%>hlOn zWa^%Op3#RTFBrEaedkCSO_ut-x-Ji4?@}Mpx$2A}H4fc;!p30b*)wkl>Kw()-o9w+ zS_r#own#qtl5F7I7$$k|j1ix)ZoIIQj`p(>t5cgSDsjN!p$w`qQ8sWqF?r8G@@^#H zFwSlKxya=w)ogcldDVr@?i*6(gk58@aAb_i>~Q?u;)j)ho|o<;=OZm;ceaS4N#V~k zXFpR=bVM_xsvj%1OH~M(IByH4e2m$BP9t#3wVt^Vd&{N0SZh0IL{Nb;{nqflV$gun z#%+J`3(Hl#usz?On$HjrwK#HgyobZ_@sKl=nq*>N!glgv2!si%2&xE0E5n7!KQxb- zryU-yg0V12+!LHJtdR1TBb~;s+#Ck9Fl*;&)>Mskea@Sy@hjY(El7SenYka4{iYpy za9S=y=j8VMkP)HYG_=zP&C#wHiITBkl25YTwZ#p`0fllS^1>8gRNAqGU6{JXghtVh z2PlrKJH@26qamTUjGUj-XVLXlRf&DzepZ5}=Z7aUx!%Unh|*#gRZoS=o+?orys3+F zYGi0idzeqyu~F+FqqOi+UD9@?Jl;5B^iXoBIwLDn#QFWc9qV2~5TKdbs zsU@-e_Jn>LzNXma%yLR-I%?;AQf{2A*vQE&^i_b*v?D)Wz1C`lY1EV> zB*bD+WVlI%P&zG*Xjo-B)rUlcHdS>J&Ss-s@S^u{+**JLPV|?u1`o^RfC9c~9D|^B zSY=Jc`&g+YE#qDrQ*KZD4nI+S280$Qs_|>(0GH`1hRwN`$yF>&saPh}kz}Q~S=FZ9 z!NdiV<3UE5>DBSa*WTXqlTNtBL;0^qwNUEb$p8BIDEs}M`kUTufbCRBS}@VqG-Foos_OU*jz|0x;LB$Q^$K+gzSW)?!pvC3GJGf~S=L$Zk$E3u;R*kdlL;y5~QhjIt9suOc) zW%zdN_Fh1Sg)(9jugPQ^CWbC=BNhV}mgAxhi$|WZe8E+i8(IyNwGN7voPQ-(nNrfj zDWs{d(x#(-NmqxS2_Yg6-4zx)%Jeq7-5EQm*kq_dcmhM43}6hv5|s(^JyzPw>s1o? z)F6Sqq4Y%vOQBSV{@Zla$fQ!dSz^gIp?nJ0qaW%Op1j?(^f*d&pi=swq0#c{cnwei zqu7H6e*rhht|~dVvQgK{;G@nU?3wKc`Q~fc4ysH}wo-lZ9af^N-T4MAz5<_=4EUg= zF~dzPZDIp+qoD*%b_94xi_*N}T4f5T;~a1LI3N|lW;0f?4_SGI{Q_|=rrNeEI8NIy zD?dYP5hd;VJ2@3_o$UzsAXYOyoVsJ-!V& zHdKkQp{7DBY|DQ!*=s2{f?(-djA2DltLCL%l4I0G#dO;Gx%CAr`IDt|v{W1_N3=PK zAWuU`iB|T)(&~#3^4QHCEDp=ppK>;q>gw&z(zQZLY#K@`pAKU6Q5`*SRJtFp-royw z6cpVUB$Z#gKx(26VrB77{c#?AE1LacOcYpVdS7$Cf+a_axy9tk@~|c*hY&Lo_UHPhmuNpA9lP^fB9l9m)r*d-+7lV7l{hSRpuxc;w$sfrYeK? zhg^8nb2*%vGX(xPxjaN$5x<%AQZ`Mj>hB(Ugz5nO8_7qh2-xxcSo+${XO_pOlq}t$ z-X;;18mk$t_+_OSP6eFe5;&l&pgNqS|^jY4;9N?3O96y4em# zxX3om<5O>Ib6Uhp_E1bk1*8#|RxcRC zg~M23+sinwI-1FVXt$R?TRA*>&ph^J0x8~SBTl+0N3{W=ZmdxDp&AaFMqRd@R+cNm zn?R%c-onZqr9HB_@?w*Pl!Pzo*TGFTZMp}$4)v*}nw_Dj!~io3%Vf;F>Ak=`BjkMo z$LtGsE4rdz$Z$4E9u?lph6n{dzFQUj#mjCF_ZQ9mcFg1_o!I82NyP6mY%|b#jOUVi zat`AdjoY}bd!mg-QcL3HWJZ1~AaX(8$)*csbeW)y?H)|m>8*WrGQ7YA0vei`^2RK< zNAq~yx5v@Vl&%97P*GT?;s8z)1dss4+E} zQvOwfQuJkb@@)o{2?`dYD}e8P3neogv+MZ$$kJI#!&=YzsY)_@cb$Z#nw5mR)ewch z#^)w$#;Vo`ODObrxnb#s7A8vNThuusSWxzN<0{Eg6!?~u)=cMqb*uOyBdy@6ex7t?s4yh@;KPGQ_-QaU`e z*jB}9E3*sV-DNU!GaGQphVCJ$p6Z1?S(Q^suIGx@=^Xj5jfaNJj!Ho+{QI%U<~YRR zhcW%ZT}oM5(b358R^zDhdh0&*i(5n8HSGG}UH`SW%=JA_)typTue&kzUkUFePn9i} z%QdMv=PWH%LY#0G-wGGLi!qU4?(RjmZ6$g+!`dJ?zev9vf~U4eF>^Ci<~M-v?B-_H zRLl2+`+#)*s3$A>M_#~mp0^7ea~RjE`paMj^mLFgse&_bT?7f7woQCNti5$|Ln8!X zc@>*szA*eBZ)oJci^q;61YlGZ2??JGVHwsG)Mn%)LU$lk3OXC)l9;X%;-WZ3k5`Ta zC}Igx)MmcTVh~g=m9UPMiOPE7w@2C=YS;>o{m#T9RyB^wqPEjlb4_~BCUbZfp-mWt zg1A6hbcSh=?DR=!mu&Vvy0t$9oWIqOMm87d2bg(cuW56beohSccHXD|tpzYp*=qVJ znCM+?en$%$&p5KHt$3om>F`EWzDMkQLA_3*NO+!n$+be=(ysHHg4}Uqiz<7KEOTQ) z9?O1#=EOvUs&mKdl>qWaE9r3GjqBbWtm;Ve#XVkW&NBFL-1Rlwju~_JtuFl9i_Hfk z-qX8-x#+~W#OQh92gU=c8J@+9%`ws?UH}P^IHAWA41Iw{5iZy34?_-9gLIY`qJLYq zRT}+T$|l-8#ymN6mSZ9vCW}&mMM_cn!IDT+SKu&D3>T}K5?2e!2bob^p;a6=E+&sQ z9f6cp-Y{4~ai5fnO?D62T+>EB|>6!>JFB7d>qX9Dzx@w3vDL(cr#kc<Wpq8^tOPCL;)?AISuwbxb> z*Qy5A!Pbx6XFY7V-wXJFWZZg*^1$s(Y6;FpViRk#d5&ZZj!-+@7A;LLbr)M*Ww-pj zWs>D)#!@)WiXvv2(~Ob!fVTSWWI;j|(CX4*tNI21-&A3n)l7O3Uqx0kfWQB>PAJ4@~?y?0`@dp)MJtuY8e@Uncvme z^;6G86_pEYRVBz|xd@m*X0dchsPON0RK&ML+#|_f77A)u850lmT82b}WPRXG=PNB| z?hV(_M7`J0lZJR5cz-xs7I>aJ#CptwIhz=oRyG{LwcFeR>U52|*cqlb)vTQ%umLY{ zwO{!DD;@C275nM{jrA4Nl2W$~d)&lB^sPEt&Gx~{-cz6_G=0d)6zSbtu3!SLPCnD)KyZxDqQ-YBBq>G z8JTPWYS}E%M$=S`RK=m!cuRYqdotSEomD}6h+ubdreQuxxH8>UCsx}jQuI~FUg?#` zejpeshf#P|r++p2SMhO*Dh!%votaeUH;C*Wc_OoqPK3yhNp+Fn??kWZ(;XDonMa0) zMa_aE29#?|QjhdbE|ivWvCY&zM$6Mnzq_a1>7% z&qsb068dRVD|-o2HzG7v4Ceda`g0yF6QzFf$C8m!wx{Ez{z8FD7n?(uK6fLqT4H){m_M} zJ!qo3{!;)?GV6T{jHhz@I}(1FxVSi(@_C++&XC~Q+1U@OLyO+4t2)gK#rB38gkJsi zW>F~P^*>wbwuvBr*Uf6cZC<3^t8-*t8T{y0k6US>c(EgolVhGQGP^E}mt&5FYlO15 zOF~?YDwBY=lMoHL`ekdA{m3Mt!)!8JsH=3I==8stoEd#X#fEBE_l9LUP6 zLIzmLo&#kf0?qsxBHMmv0%QrL16H67f{lfBTJ-7z^rZrUHO=26A84k;#R%FU&UO?^ z{UJ?L6KRYU=S5r--OgkhmJ$AaNLY5lRf`H3P#Dm9{ToEW zQ8C_4^)@!by$sNvNT5>N9twl+FPh{W4_7l5kyHP{SHs19!w9*@$7%!rguNu5-I*z5 zuMUPd(uM_`gQ5vDx&J|WsZ|jAp^(g86DfRBp`+k3H7q!!Z z9)QR3jUjt~&-lMSqyp?Aci-!4lmFve{&}ZQ0O_p%-|hd`IUx7*zmdcBziIWjKKy@Q z6q@6Mx}g;(Pw99XqdtYKU@)B%qKXnk*(ZBRe1WAK19o`7tY z77k)F68N$4aiMFwV?DFSqK`#lXeXf@o`qOLL&M)1pB(Xkzz#Kt-R-Zijo&TeK_lFv zoY{udm!kcHw0c}nTpa!K{N^fk(nlp11P|X|Km(Tki%&y9C#JJ|>t;>L5gUBfvX#3Ihu?$<`B3w8{kZ|54ljSVrC(+0cOq6zqs`ZsTO2p0LFy(womZA}0nQHzIb zA?KxcFrw0*TLI9bM!#khtiphv3PYg<6J_*I3jRNZaQ{V%FMZ!q5~uJpQw=nQ1VoU| z-&jq-f=@@7Qpi@9x1X1n*Y5YqUe+}A3=jfMrz^ICKF!1Re~vnBa=n>%yV=vuO+~$E z0u|toMb4y^P4=LU2{=tm1LXrH_4nB?;CC0A-A~s0RTVT7WFWG3g^K0t0DGFPw>OVb z1)2{kz-}oRm#yu1#xaj`9zeTBTO*l!l;#1ZlO?+pD+(d1s6hVUcDCH z7B>eq&q|Ff1+;(Ewh1)gFU{)rm|Xq-BQhe-u5hic{`uHa#&UOe;9iWhL*sh9+Wq=Pj0k(=DE`gUSD#a865jFndTUMKU)ndF z;Krtt)sJS2he!zu&~!IXqt>L`OFo`8xI&PJ0-u;XL6_MatFt zOMdwG0B*x~%C!Z$Ajxjv0{ZW|jU;C!VbREgnd)Mh0Df9?#T2p9ZG@H0i{iZO_>+R9>N=174wwjsgoPF-?sYi3PgXz#x6?;49TyjkNbUfE4i+XKi zqgpn_wh1j4C*#&~rL(fh$O`RV_I=tlgvS%8Of2lgzj9rQIhNXeJiLNdn~4bv3AtUi z<8K#B;npt)Q1m5dPoqD8kXzc<{!iG33~~qfPaj{thWjeM{i*a@k(3Qe)N3J!&tA_F zM@WUX=Bv`09;NJL%ziodA8L@n-rWbkb*eiZ5hf9h%WwucGNGVVl~OIA=exJt^37p+ z5v!pQh=t=XZ{=16n!|Wb4jN9x$o7VCGS6<9Wx~BxO?PCZ9v-UgH%jYd*KoUJb&!h) za$fQ!yv5-PLVcgJ=q*)2x~95QD(sxVQ=>W063rd&z9_Vbb^H&$U~g zSaL}qKZ$)U!3A?&dy(4P;pTUeWWC_L?|5*v$2ninc&=nzd^hKuT<*T(I9@)TYFt$7 z3ss~_Ua=-jTRGhQTrqR&{9Yo{H67n+Gd2CHe8YmPux+=ckvO^0N0vg+&4JrSVcFv; z(j>4g-FVM;=srjW zvUJ~C-KB@burRA!g;>-5S@A=C^9|b|{n}Mb*XOLYa_ovzf?=4$h>vURUD&nHy+{>4 zePw$%;e3!TbbPz-c2R%BCS6~rYTq?uYV}3eDPCy7s6Ryvr85hTC1$%)x4||KDf26Z<2v?>+H9M;*=6JF! zrk?XO_0^S~JIFZoVM{+*Cv!_eDR$_f_#Wzh1AMn3NZQtpxah0egCB`bz>37-Trd7W z!oDRa9bkIKlMK4mKPrqJg1H?XJB(V%;;Q|X!N97UDB41hF%HNw!;$YdjVcpMWWQ0E zl04@hY|rL6%Y6m3IJ~1gv5~gpj(M*#Q7GQ25;EfT_RyV{wnUJz2+$K z0yxL{2IQ@6H9sd-yXuj5I|WX@(e78#Aug+gXSe0>&cj*qFz!O0_6;kzU^+VVW9#vpc$c-sU=tey;!qSV@#xtRBgV*cCy?_pWR@_>|-6nAqZh!{yslIzxmR9Qcnd72dlmpnB<*?tt)D+KgGX< z`oUnUS+%6*yqAfQK3+MYwj(@8gKwg~`bclVEK2+urx|LV?dgU2tb20*EEeZ?cRYuF zVP^Md`Mb(uSJAE-()vie_+v6!Ep8Of4VFPg#bkCU;#n-|6r)*6R3TB#HJ^PlBI53h zNXGiBO11N{Xgbd0-)Al(4v%E`sp*WMAjJw*_zn{k7(6-}y5gNe{7sQ5TrLu%c+-B~$BLegaAl3g-7#LnW!awurG?ttzrHxk=CN13xnL}k9>+iO zTa?8*k=oKi2m|K{c1Mkt<_Syh%n8tsT2Ck4MSeO^vR>ng?%IG7WOG)t;@+bzBwO~d zZQ6yzaj9WhetYdP9^uk+k`XRu^z1o2)5w>B)4AiKjM1z&>C0k3iCrB6!%~_`d%`>= zae_TnU*mM0++y)e{tIudG+N~oq?L~q3|=I=&lR%*eG3+OdkfnHqQjv>ptRWw;?4*#smY`* zoQt;UtoPOs=KGpY>ZrqxP0gL-Q!$~P3$zM$j_oeqB?7VQx%a-=Fz0;0NE9Fj>LQIX-Li7XJ1vjr8kkkRG;uLY64OR>!g+k5i)5Q-z3OId zuDSlBh41AUxdFZKYa0>030CT6CG68nVfAY%tjV{ni$l9}7-F&CmTy#{QS@JTRL&*^ zO~uDK3%(L}WVHw;eSk}cqHM!8XX#YEJMM=!d?zhAyiSv5TqbckuqY#Q4%Y4pX5B% zrigUQu(ua6^%OZFA9;}on#ymsy=*YOVPnS@*WonysiLl$M6D@B19z>*#o}%^A?CIP zChH@MA$WfF}X5D{Ta;r8As)k6!*$WiV z2sr+JrS|?#=lwo~kaIs`p8}4Qdyt7ZlqdnhoVBIkUd?}wo ze4O5AB4B%CU$|3ohVyO-_3$x;EsMI*Lf0zNoJZ5{GO&wYQ2#iJ~u%aKL)1KuHz#%1OS@Dz}h2m!sr+l@e8}((g2go$@zqdV$%h85HWg z%o}fx)3SudVcyv$u1p_jy*0g!5282etBrgiHwow&{#WA#ukO~a9mH7)TRBT=&xqzC z97R>f3x3N&9Db-kl8qI&fZkgOnJoIaM*PnoP61ISzvg1op*T+Z=o_EyWh8x~o-(z6 z=3lGwCO;Q`MCDhbF?ZJ#E@eRe%*&FxPrsu_pOZaLyeKjpRfyauSuJ`OIMJ}qsD^1L z&!acuMP8Yo7R_Dwg#c=f9h4g;AytiwL$t4V_db!1{ui5+5OI@&T#tXcTy-7ZQ zLDFo2lJ!#v_@EH#v^P9qWSC70U$Pgq6(e-CN3A2)tAAhkVv8O(^XhE8%cj_E&H4#0 zMH)F=_`zlC@bHdNb`twi)?{7-?O1$qPCnUv+8uIx3g_o2Sdr(?qAHao9gj?Y21Lm= zq7&P1^;xCp`b7*nN4+$Arqq_coO%YsgueyO9VNJPw=nkh;$Mxk^$;EUgw)1=f7>X$ zx|VCYNLzD=g385McTnZeRVx1MhHjLMnU2IK%(tESLHqoQuw-tvA4|fY5F^%S9S-wSYJyk35VuVaUM7f=@%vz`0`#}27BPIk?I#1BZ<;S9sxb({U?u2mD{5z55q z6G^Cr+R{HJXmQMa>Z1f*BPVFMkgokMYH_!a_1o;BH$ztpkGb*_J{yW*^{%RvlKywr zSn*IL^ES$Mh?dOw>=+ z)RQ_!HqItPtX_1Mu$}p2=4-FhP6&P7GgzyhV~i|`LJteC8(XLs)Ya?Lho=nDEy|-J z9SgL*(Hc+G+od8RYj*mbVL~_4q>y>_W{7)#cQQSFEjoUMjJ&s;4(<6!-hQhwQk|8< ztylo`9_hyF$f|WS^+)G|c#rB7Q3_f%`yNU~qHgCE!u8E71$CJq#)FK5R_?^sHjEJxbW$;T>%9i* z{v?+AW0w(8Vn3L(ycg!eoGEj3hu3a;n&d)gkWu5eu<*o6sVlH7wsJ<~sPcV8bls~w z`O@dSndjAMs^93CMIl8|h$DtQjz3a)lIZk{p`X5i-;RE6CQ9`Bajn>iZAs=(8K?Jo zTza|S&^)in_cq3@>?ylCq`aD3A2p=pmxu^yWzl_Z#AhVO&o$Mpu{jNB{sVUxndUYs{yNA+nl;}%{Kg0Th zl8XF52Iud`BG zZ|NIRe#-U!tFtEG{Bf5j4MXHiXSu>};S?kFUE>|kT1(SXbs=R(KVa9QO3p9>Q$k&u z?~K`aNz8OpzlwP|*~QU$p3kaoPx8Lx%#YyHO*ORu@6$fn*i~(*j$`aUofp6bb68h5 zS33#zFkRF`I@R&QkIC0BP0Ol2xdoe6%jAR%2gK2nuCzx4>JXDB8wte5lMm7HoGP5< zzdXw)k;BU?-ez)#Y`EPlUnqWNj#kBA%UtcXnMFXBM8C+lO~(a_$mlIG5!Sg$5iE6i zXr#e$u6o9z->hf}D+Z^w4dYsEdIaZapVTLdvgS^ny3~DNDEJ61QCV9vnlHlIb2P6b z<76dME}BMs zKr8aVtNWo~cHZfr%)j317oIiGmQ+CXE4|JZV^+FiJ##<%JVa}TGd2tm)}mru^9<-N zw&d$bd7Y)ryyJn#;ZM5T)&H?bG^%ar+Z^LKAe{e$x1Z2!sFKTObbfb_@m1UE_px2#S?`{x+USvOT6kixbNL(XyTTC`NFNSY#+PzV1 zl`WOS!G+WFnx)}u*=m64)I-pq7#cVFWEP4E z&J7IOP0sS25X@d0#XdqKp}gF-{ie5Kqmb#a!n0?6R@uurLy8zHu@knn$W`hgbytB)*bZi; zr&#owDYkXJJy(9V^_tU8=4nJH9%6XDPDhs6p2RRw$y&*Wi~BSymy2jRi;CRwwsE0Q zM5^~k2Kdk4d{okK3MR|aiRpcl7Ap~}=8%u0D__8p^D_0rbCb|eBreM8jRb_YB7OsMwC-6eTiqVQ= ztktCO#)zSsZ{`lDSIE&#hd**a_)`MKWnU1#>pjYD;dgL$U-Mdd6J-=SyLQn!xr2U= zrzq1c>3MV=?Q_V4)nPQ(Jbhn6Yw3&{q^0n|f9c}!Jcg>6s9-BuHguT1RR~nrrZ3)* zs$BgES0V2}hq5MEM}cW?>J3%QofN)dGVQ`%y;w&{Xn^@X8D3V3b&3CKaW{>psR#!CJa&U|JVbuJHl6c>H&U`)!xI3DqNg zHQU^E2>XRbo>m4{1-~}wqO!|a%q-TF5_~YQxv;dWT99h?pzSHlLQ1%^`b=I(dseCS zM1}lkTgA~BBX)wo(~t|vgJruqs-GH3XE0dCZ*Nc}Z`StJtfDeKt1?5}-8Ef#Lf8&RSF%#Q* z+d<<-clIJ>|L0~QHcCMaf1*)$K8JdoJ^Y@kwuK634Ymc!(V{G!+%iU&n*?3`V^~^Z z2G0vsJfoca;*}?IjK7D6!mK|FrcxH)v*hxWjC~nf zBBE?V_AquyT4WhSNR53M`)){-oe^S;CE3@kA$rf~_g=s1+xrK+*EN@Gl6mGl&-t9s z`7HOjZvq+9eBapa)v+Y;swAkM$_v%(YFE_rXS-_9S>u92QTSsdm2@;aM`6C}1K!ei zD_xv1f(sf<5{#nfa0?48-@jWWe0#_}TqF@#ZN{rL4n1CBjf@Sk_Bo_gZm;Zdy)SmX z{-g`5Wv}&Eu`Kh5S<`K_)&+;yjATMhFs6Ettsu3}jb`p}$#H#$Y=r1alGv9$G9`13 zqCx0mPZ0RWw7L*qM5SJDtg~9-d9~n8ijQc&y&;UFh#vSX);e{>@$Tckl6x5!N&Pyr zW$yAP*n4mL7~~#s=r@1d9Z7jsnAU;Mz`_l-4jXZs$dFmEMTSjj@aiqwbVzf_7`p7| z?Hs3p2{I17yKxn)X*9RRRkP@-4h4L;Mx$Dj(e1g$vk8NqLAS8n@F#3&RI{uMVwaHl z)|y}Wp~=>RnQ%Fwz_F^KivN8EXZPqNI zj^=8)-m-!f{a7FCcv+ZF;BpVVx@2_R_&iFKZ6M4?x(9LoMg1KaB^@5?OFAQOU3zj& z#g%)r8*Sr_h`A;s+rbqvt&aY+6A6^9^}VB^j$KmtIh)rUCru{IE>45f+ZSK=e*4hU zhNn6U%S4WM-nn_@j%#?h`r+P8)Kty4WgQZ|&;L#f8Clcq zd*aHKVZj#T1l|jcLOs`pDTgmuQDtk#x(q(1$@lQ7V*D!Af*sU{M6lhz1*f=HdDbA^ z%8KXeAzx6>{U|%;IG;;hn4RbSJ}#y(w`-3HC!G^?h0oERp$w+MO+|jhsYf1CAkY2W zq0UgwwSW7L%{Rg;ig3VT?<^-HqSw|HoVN46r&}aMgo|%v)?YlQF`Y=fSS&))z1Q<3 z?T|gj4^>jcQwY0o6fnwuRqvAu*KCLp;mMkT4Jj|WP+8l5Hnb(2CJNrsG>MCl=!v$3 z%+hgTr>e8sIUM?~D~tB)>J(;bP)`TeUop~yo~Vtl&O8>Ue&E|j&Jq>|RSivX1dE~$ zzG|pALckKs9XjBGaHw&>wqNYNP55MUqT&5yCr|Xj67{D{<*{s{S)`GIg{tn4`FGrv zokDAmh=sv-UEC#CJ_{dbEzm!H$0{){v$Jo=I2w)A)a~vf(#}g@Tl-<|n7#28oc=JU? zyt&(tI6DW@?FZy-tC9zpIMN9vd<8`Gwh&0$C*Igq4Xw%U1cSN^zNsd)qNi%kD)S|@ zGR>rdK4m+{rzXvVXXwov{8Y^oLpHLG+CNzdwc%$slPb-@GJn%RVx7(cdEWoDOT1KD0q z9adU9_N_P2k8D==3Ec{N^ZgQ;)9mz0Q=f|k-))a(nicsnvKaeb7w2~prkux$kM%9r zH*ju8+*{-@9`Jt12AEwuK6qJT1 zF+LL!45J?pHMT#LIhRBt7oO+>xtS@G_|A}R`2m&h`yz%XCK6#2Ae?%@UT=Y#zW_@Z z?>`&jo7ajlF%>dz&b^d-ZIDl~z2ExndIIs_W8UksmdX$NJw9YqSV-5F8vA83q|6`E zo3A-|7ok<`uquSKq82bzzvrEl*P*0jA1N}OS<2@rU0B$CUOOkp=XH0fSKABGnXBbR zm#pUELzdi!;&7YlO`VLZZJJR-?~E_D9>R z2kx0`{FcN|ky-0){bw<=-XEUY*=yXNuW?g;cs$Px-Wun8W+=Ym(e)~MrX#!EvWzis z554U+*#*-1#5$OkmzFMm4>Eo2gqQhBR9t*W{%SBqH!5*Igp#K_L^squa?jjpkDKG< z`kYf4wl;0HHQi6;MVI|;zt`TIA5xDJ%lN)x9zk7x1P(lph3RPvy3Uje$1yb*#xuB3 zdN1Rr(v$H*zER?MA?D>w2Zp%k_WSdzL4};Ef%5*W`TXZ+7whr4+~H%_FOjDm`Vgci z5K^w2#Qp-hH+kcQj@o`zeZNXk=S^NEoAJdZwGK7Wi6TB;l_h?1pZK|X#nRLHv#l-(h;`0@uPhZ0?1k3xB{>J({n(e_ zc8gxN0<*N(GAOrD+#?LmN4vicG^7zN>4Q-6MwqaB3c2y3_V1sXhk11=Xt>-B4Bz5l zpn)0fw~u;SC3}QhVW%W9*QAS;=_2a^>Bd)ZWVuXvI~o~M=R*I@-nD`8;%(J86E5&A z@?^*dEW~X~WStcUBe`m!L)TW$v^?8RneNvKTIQJ}O7{bX_)E9TvqV~U)#OwF$bg8u zb*=5#>)TPRZfeq?gpNz4LTFJOC@Y$*ss!&UKV3ocGV7J{>Fn5(U?<-T)*o6_f0EaxV|?VU5(* znP~n1O&S0Zs7?KW;^X%x0u%?Fja;y-TCBA97oQAK&5&M^!u=84WHft zF_KlFLp{ru@1^8)Oa@2^cS?;vM(mLxFYx@YR9QbNI#ExtKDu)G0MS6E@_5hb;~>6# zRfYt10Out^&;2)e9)B@Qi?tFL}DD?Vu1#{|4`zGSo zu}-^yn?w$9sBmqTzc=7r<^OTZR+fNLy4huD=58=Ck85D6nlFU=7Zb?f8ZyO52NvDj zQ8m1t=L3*O0Eom#I_6Q#JJB)();ztF_c~cu^N|hj$T;Vw)O2caDG^y!+fqn)hMKpM zfNzZeC2HXXP#%B8RF$L$8i8bS1p8Sai{lUGi{ugg>(lu8{w@_%iz503c9TVa0`zC= z&<&K|PIP+Fd~KE=6c@j@BZ0_@3cGliDiX;A?NnLzCfPue3@%7)Z+ZIosB0(hc=z$GrXD%a5^i+yP4uO z053xIlHCA*F2Q*R2#tC(2jz6{G;8@?(8e=$FXWbB+gezLsuO~II;SX~bmtfigzLio zyD^a6BklpwT46aei&7obWjdc+vSN0v~jS?G7O}N^^e6& z6bNiAqNrkpew+AmWdp5^B_qV-HVU4tr|SrK{?H)SwY9!3U)kMYWI;pBOEyjktiv*R zX3sT~9<9@_Mz&Mc_SY<8a!`8~5Lf@Z)H6=L;~X^{l0+}LQ{L2!>1w}E9oPpNsGzzO zu1ZjCweoQwxa2oqQ&6`AP+<4JrZsVzUC=w11tl1dfzr`<%a0Lt$we&z9|!xL#VV(X zo=>^;*ft>NL_`m_y#p|zj=drFBRmo96^*-gTdKd=$iBTC4QzA`ti_H1%K)${VVXkv zIw;=pNIVgRvA}Ql=GRV21A#JZ3Sv&LFDVTzR>YlYm-u(V77jZLs|Y@UCA zQp^N}PD(^phIhKcRCk5><4wo9Lzw#xtnXy1$UIu%5=uz=)#3eoPf0c?JuKt~3S^*H zX5K{8H1)O$S2;wA;U<_ocCL5tM&W_8))LVJ>{CGuBGowB_bS_fsBXrUW(K-oTrg}> zB|W$HG~2u=^|c@Em!e++fuF`3O(z&n9Uo6t4H2pRX4!A0K?3fB>=|tiLTwY2Czgj) zoS=Ee%#(X9o&LZdM5CN0pS=6PskD|C`kNk#EE z(N+jjT7pqbg=#>+JcC+EVXW2V60e(q}sG#_1rm`ZuS{%>SC?iTJzuv5Og9V zH7~s@oKZ`m#mB619vgtdV&JU z%!Fj}${!2*42}~#9tLr-dDrCCB%TA8oytfaKm$FR>_h5>j`}6s@`xPp5c)wxGkH9^ zb7lO*E=w)e2rdP8bw?)Xv8ZsN)}CHol)}{AU<#*|jPoE||JNF}qJpYZy!Id28Quh< zTv?cBG_2lpO_90i22?%6O;KCp(`*&O&jd`R+lp$W4XlJ4{KeHWMQ<=(B1?OE`=6#|g;Nt zBOi2hYdIfB@bd)JWkAZ!i+QRCX+r-;kYq*0xg0<{-5q-D<~KmT;M-NgA4bD-(cg2! zl?398FWW%ULp|GJ@BXpMP`c5rA=&!0@ZoML!m6zd_E+ZWY|6Fq>3qlN}6VrABAhdX4em%Umd3;w*qE;=7!$abgyDmGgyOE)a9 zdZh)#jE2G*N{Q;d0Fy@7Ul@ecPh9K(cp(mDr;}1otS}m3O_I}Z0I1NLByn&k4Y!zE z2|3hx{7JdyrIC?x`OdHmT0mAOd#>l#-L38tue^4*gJjeUZDOe@vI66hS*s>E*Ckk+ zF!jLh}mAFLbM@1c%B7}J?v&>;wKd^b2 z7uHwW&EXbnHiy@?l#ZFb;e#i?upmP)W(Xh@=(&*BVrsa)+;CXM%9_*#iP!`obWSBD zb!w*l_>mPpiD|-+NYFqfZoV@HufTY!t0QW6#D&_Da@fV8$K?-8S)q&9$%ND>UN&>@ zH6Z9GX#^qPyhQtCvL_AG4@O-+?`UK7<@P~D#904PbI1w+1mkAOLUq?=S4!5ttkJyZ z%k$%gc&#T) z0p+6KLYQ&-nDjKS#5j?!%4>tx)CPAd!ffuDOt!qw|d@ zs+bpWVmnwWN!{*yyIc^^Wx~yufnY z?!C3ZE4gLjx?m%Yh|71muMHE!aG)51#dFG& z*iRonX<{54tc-BH2=VIGo>KhdS;1K?Ox|RjliAQ}GJ?dbiQ6LDV^5iE$Be7R%)}E# z2L_%#GYD9iUQ-;6#ojE$qB)Al%?m@!URz zB@d!k*cC|-(e72L_jHM{Zp zNFDVfcB79c!Ex3_5elq3YD$$g#&6^1n~b!T8r;koOQR8%%`~(VWs6`KeG@4~e`F6s zPboBv(8_7>wCbnub9wXHC7RW-T=`vyBEbf$mh-lJ=futLayE)fs~Vq4&8~kjE^q;z zP;gtIVpwtV(=|7dl-sY~oHO!X2f0(fpN2y%X??^m>`a_KBA)5k2emtG857*Y4&7Fr zn8zCB{Bi0FJ#|+v_KTBvfCTGN1{^fiDyT0gk(;xGC(CFu&S81uSaIwosH7BCxL(^d za+rX1%={5Z^oa&ukep3b%rc+3AO^;3mP2hT_0=Stf`V>CRZ&5oUcXS=(s}*DPSPL` zJ2`zYq7SDZgp|%QS&GaZnt{h0yftwyaImey>Tq`?EtfA&^uwV3%>-K6<-iU?inqAN zO_sWk=?{$+^XUj@%fuw%^PgO8(XUL)o}B(9b#-nEND^iCMUmpEW)%*5M<#Pnr;e@n zq{_D|Cx5i4u&dGyJtLIu0lTxNX2>`7^Z}RRHq8fK(KvhD%#B@y`7Y8=L`!On>TA_M z0ftkx9TS2e`FoM^)^fnrR_MuZtEH}wNB$r|BBrCTQsF~Fh`KL2gyUl`ufG+ybMb&` zSaCW|WI;hs&x4Ull>KtV1p=mjdnMRar&Y82=;eq4C!bQw=NMBgKPeva$nW+dr%#O@ zy8|D|C4A+Oh@mrn(PUTc^yDLlzT1_I+wvoW*lpMO6ze#0F0Mq;d1MBSY_Zp~k8zw~ z#q9pJ6B|)OS_w&5xtQAnDW;k!U6XHY_)ZC&B3>-KUD?N+ZHcNhIGGgD9iN5HwTema ztPqCr_qoAD&SN;D^R5dyYNnmX)bCa2MDp6GupwV=N>bi%@mx$R!%GTx(Fh#6a?<0| zEtDU@K04sC(r(lDD#;~HZ?@kYoE{ysIa+obKU#=)W3{Htx;|@pMY0M_e*72|dpEgs0}tT4HSw9x(73?)&5a(5KeiPCW{WNsE_*i3;yz6swes~x9~NBqZc_^9 z-@{w)t>@e&evCt6&+fx>{_s@xf%2+1@Q(HhI7ub^;5R+!DDg82mN8QE?qy67R9NI^ zVf43&qNAiztAsv^D%NvBmW{@`V5jf4*Zr#|13n@FPhE2;nn$vfb2!#G63+!BPCJIn z78xMjfA+wBziOa_QmSfy!Q0XcPONnvc6gq_SpB^>By`F?F#b0S1|?1Qz!{SgTcVo& z*b|6Xu6Ca0rc#a(tlG{nmNoR(62{N>bpi4sIzBUl$6zE_#=#NZfE>5j=+e8oDSAq# zPZdWj{}P0r!M&hTLrcaE3YNs`MU`&rN&By--toru4Fu3M~3>w+8CB7*%-W zC8h9wkGymv7zp2^DTYDyY3G0*!98e$*fH1<_kVn_MhNZZj^>Ay$|-ILRChzjg+(#O z{`3$mlorzCHvQ{G>Wmf!?jQ!89NW{0io*5qjoHqo`*o!t)V+aP?_I!A zwHBI(oh^3BxJ+txTD_mwR$7(ygc$y%rTMRedM?kehaX%s4cE=*v7OLQTVSc~4SWK% zAc8`|Y*}p8{T>8{XHI(;=}_w$y(oq$`rF)}q>rIn&uLfd$qrTzyWXFE%e2xicGO@} zl0npT5pa9>0eVTb%<;0)K3k}Tvv9z^mvVJNn(CT`}~N_p24OTB;JDr1fI!!uA$pZB_6GGLvNgVDp^`SB46HI_b7G} z_)G1U*(geWrYju=cR{3A#x9$3tiu$V(TC9A$eOKh(H|B%OLVF>c7V=v10bCKEDoq2 zUd9H6-%KnvPjJ*z(lpl@>_&^-EuG-|o3rSlS2v6c$(zOa%^7O4)?xf6mcRj$%L~PN z{%Nl$Jmx`MN&-djMPD~3)sZLMUeOeJBR6R_NwY+a{r1rxfyyL2RULa^#=gt}&lAjl zOB$4DWk3Nvu72P>odr(|lVCT)=*?X0b?4l_U z79TfEOPUxg-Wxp;saYeCQyOtm6#Q!+uUX;+m!v`um<}pR4?A`7&nhGs8A3iatCd$0 zrZ8crQCXVqw@|qI$lH22f2-Q6Q%Z-zFrptsihvV>N27;{$~kK&u&-n;*Qzt7?Jx5h z|FSN#PO!$9Q051xFsyAv?ZkJ!<})$Lx0`-ppI2~|#a@&#m)5Z-$xULA<@MBc&A%+B z+<5sJ(_YwEnIn~G&e8)$j_V^l)-wc1xqCNDJvpZ*tI-a3mtHb|nJNd^hP!5Y|2Ahu z%*SB@aEopCk4?=3rJz2r2%G}Z3BN1YE>Luc@`_ai{~z6*{jlFYvhD{Z5A62YYqFm} R1Vllhs-&rychw^3e*l|3Q-J^g literal 0 HcmV?d00001 diff --git a/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step3.png b/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step3.png new file mode 100644 index 0000000000000000000000000000000000000000..336e1e293d96fb58cbe70860c5ae1f06c4e9a0fe GIT binary patch literal 69230 zcmeFZgJWGEOeh(Dk*_DfO{Me7TR?X#*asUzaTW~pD+J-MFEZG zpL=yQw%^YHG=r=`m_MHff%Jjvk3S~B&yRP0T`|(o{-v0P{^w~dP#VUcd-NgTX^_vO z2XBB2?#st|P9PA`mmfc9F`Ps+0Nxe(c!!B& ziqD9H*Tdr9LorKwHHV*Jgyn-13}5Bo;6TbgKyz%tGADZ!VaQ1HjvF_E}+^_2DXl;Z7Tr1vqON^Y!1?tpvpz*C=%B+-~r1Y+y^Ll7D!iS)m}7)P=8 z%%CQoo}L;S8elM3eo!&8+(<-3q^G4mv2e5Os3S#$o%BCm`0;9lH1t6>?mZ=v0?bnh zXhudxR8*9|zyD3dt2084n6$>m#`N^`q9Se~AswqS9=Gm_`yCol1phrT$ZrzMcGiaS zz~0!<&@dn%z{0|Uf`S4Sn2o}0#`8MfMnImkGs!lop_^zpGi!)EhhF#c5D=7$$U3tzyJS-+1VDW z1IeP&!wz{f$ITGdF~SDxx7`C@>Q)-yWF@43uMrcAe47J%zZh$DgN70VocY{ji%#T~ zxuc`wH!4qDUm6-3x2@@H5}jeF7)a0&I=7`bWi7(1QS~2u{p4_=)|c?D(i+5#3^%CG z*V72k_i*Hhlu7P|cVP+oJM3oUIil1o=!5q-*tX7F>89}_`prL0gRh0gUW zvX1(**HJo4!fYNI!>G*;9|!)o_8XkrVqrB@kxh0qn{|e{ z%oD%qN^VIwRB*4&G^W9jmuuQ4f#03MN+A5dB72$!zO6*#kWix|l89v|WhKa=PV<*WVmy3j3Lo0}s5lf`ez4o%xvVXVo*PXOKx*$>Itz2LQJmu|o zYgDofk^SQ^|7QVpM^lf_F-*3G_Pr6PhV3Dp8P7f9Q1=P{ez|U`A>C8gf$? z4T|gQR0(TnFvQ5*vZsT_Y4`ij#dOpkdq^J3mG_DLm-+*gQgCWA)huo@KN`Q=V0#bz z_8C?^o~Mf~8_3?qrc7)$ztT3Crq-ky9OoX-UC;EFxEG3={c89Qpkh`qq!A9YD9N=M zRUK0(5doo}y`7x^dqz{@C|f9HAIu}R-+!w8|4iOZcJDzRp<>%}b(kT(DNjHHhgKf8 zF0nk`OsIbcZe6x{tEF=l{M+dGA&M?qip`Zl7$lmC*WJk72DwTX@92{V&!_ zLN-PEtXutUozUn0He-+dd>?AkBqd;xRUpS$)&%fb>IDplt~IlyxiFZF#%IjXDp4o$ zVo@xS|1I)&xRVQGT%XgltSbuQwx%i#p8Bg2mxx)L=d8R=>|-HP?B@7exa?&2TJqh6 z93gXU1NPOQ&C@}X$zxow@KWs0p{9`SPh5WsG=kc%WP$(Hxa}~^Jzo=0R}D$KO03FW z=Tm($%KtRsfJ$Ako5<+U^c5tzPI0|MH#r|+Dc3Tvl`2E@qW{fDBgMVlLb2;rq{Zy} zmmp!Ti&o+~nex1xhb!L!tl07wkAT>)Pu)<9|-?pRE zf;IAM{~tzX64rz*+)^)0``7=WrQ(x-&g=cy=>L0WUm%T4H~+M18q0fU&E^a6-fui;Gq9Dp9`TNv=X7PZ~lJ`I{{Qug-F_E{&m?u?!L?U zf+lI?Up@G1xE7!?Nb`R2FX#4}NgA{n%2fN8TRC|KP+28>wENdq)sr%UjyfWr{WbjN zEW#QH%2A z)}9LT;ON^6DnP8`Q-z=XA2o_JDZdHCz_wLVhrri$(Ib&FYY+Gos!&6;jmoi3Ls(p# za0ACY8-(^!TTm=U?w?=?r$XATy4>wU`T2aF&28ih!tdYI<*s=AUSk@g;h{L6QW%Otw9EmH`ze*)FoTp}amRQ;?yB3XvMM zo<(w9<(H#QRTr&yT<{cSg5s2&u7!%-k>JEJ4+RaP4J72`WcFQ(TtrB*7oB(s9ID#N^w8rI2)+zJL> zDX`F&36XGac_znlr$=ogzI_UDB5D%fR^N!A`zRAcoJiZFKt-w9k&KJYNb(X`sIL%X z`@dRMf{vV=J$&?nJ9pB0dUexiYjXpay0=rL%}SJcIlcLHui?OZzT}>z%?I8_c~EP^ z%96+|%ut%K(pF-N_Zs=W zmmYx89co5`lLqKCh$i&*zscnzyiGmirZ-1Lt!r-ia{CaU!xqFfIWslsnbqv>KtyCZ zqn>9_iplZ0)7NoUx4Z5y!Ey9Q8^V07n?s&p@Ffj+g`?3eDG)72#4VBU_{&7HtSlWf zmY87S%?49hL~r<(?#q_R%)y%?X#^rTe!jqYz%FcIyZcYlB+Y}av~euolxXZ?BNV_Q zm$G-?Y79NMV%1$L)F6pYG0APmMUcnwNat7DCq^%(3OJ9dpwj4w&6O-E{HF?V7Amwu4NpBAZNCcE70gY#HY?9^*rp zLEME`BJ2ePMZbO#gxk@9cJh^-3JZ%&;#=k1=xHrVk}gePm!I0{Ap`!Xvw5gAqBngi zg$A620Lu)X$L}+_DKdG7z!T`8sFEtWX z4LaYcbTW&Esk*zLGmVE45jQr>JhiP_c{U!DGG+(@4&F^P;!Avs5G;y+Q%9T>=~0-n zRrKhyRq{|r%QqZ3dz|stH%RF6+EB;SI-{X^Upy9q3_~tB8JNjzS0%P#_*+`83^ z!`X_ILQ0H ze%toBU+r?e14irG)@_~l6p}O4l-o&_r%aW@QgX<;;72ZmK3At=bav@}Em}+)2aPWK zC9lEPMa^48>E(I|3y|{i^Ifrt#5J9E z2JzRvqfCiZG5ueAavLf)cIS(AR;*d4bfT-KK99aeR+MY_a*?KC3oBA5*xdeIsXW51 z7#B|5!IN0GI;XZeHb`IXq$F~k9y6ZaB1qi|x%*NO)k1Z}o_lY*ar8jlUe9oczR*B- ztay)Ks8d3sOymo#y-k*wL(J)yHtgJL|F1%z+!B7#!|#st;@0c|vMZW6Us$DyU%oNQ z`o~B5-x?af*3p9vM!Av zW)lj)QNQz9duznd6aMEAf8j_oFh9QQLJi+arKxo;&~SgaVE>*FVk_?n7vtIK2kY^f z*>)y6D1mQ!q?el-=q9{{AQ;$ZoQ}V;Ru#H$#|$h|jfUs<@!&}BJXU{Ok!9(vDv4Bw zHVj?8dEq_wc5za#!KsU`x)BP;@Q`qUvQNF2mIh*9jOmx;e?0YdvOG^jzIWd`m$S`; zNWZ{9n(08v(NMWR;E1j<#OO{~i`6ye9JVA*jP2Na4tl~XL%Td(*YgpG@nOa2XeZ=> zS&Gec>2`q6jmCqnD*Q&l%;&ml2Qnj#LiugZKZN6aC}6!2p|X^3_T?58IxcYLSaG_( z{yx?u(tci$^AbGw4F=H|kZ66wbds{H~g^W3thVvput{NJLJSBtfk})Q)k}pVj zRR7c2A6BH$E1@FXE(10D#2>u-?*|7N90_wfh`2ySVmgnb!v|gAJNA{#Bjh@cU)~`^ z(hsnWdO8PH{6lFo-drK@i9She`*=!d=RURqB{l922C9s+g2K+Dzo4}MB{w2v12y$u z_9lv}#CkNRKT#+#I)p*WyY(IJ6G_v8a^r>y%k{RIInSOAu`0US&35aseFX>R&n}y| zSWMN8m$QJEdGn&Lk1X&owj+GzHdC8ljQZ(`C`kE|@73rGcH&(524D0MI>Dw>uNux5 zwxO%!TlXy=@%}b)I2FwqvL9^^z_Ap>*mZKI0rRDeyWR74s?I`LWu-Ye;QNEOeq-dEgU#3O={JbuPF?Qr$Qi52_zI=(mg zdU)8SL6l8qhzqwo@-Of{po~9yE3P%lR?UAAY-tWF3(9@fVmQXk^W}c@{-O*SZ2yVH zshPMAUdAYS(rpEcQBPM2t}U$h{%vPtOWT(3TdphMwoAtBvjPE-mQZ zlR#ZJ9k;eQl!~o_+$D=j@a9Lzbb^yKOScC5KWOPf754dG|;3mYj_yMAu2;|aD>Y`Z*< zCF6?PQ?tjljLK@G*_ejGqCOqB*t*~)uPu1s#ZiMb)LPak@U*U)ePY+xr#?!P04D71 zq|GO)UzXYdbF2koZ`6=s1$XdY3pzJ-c$<>!#2B)*IXe%d$u_|rbrwrE^Sat|oh+!X zb$=gM*3!)`rgk$-MUM6)>!zu2;q8SFKwXnq!y z8l*(%NJD?W%)3wR*}q=CEAh|E(iJo&;v#7AzYj6cv#_0|amOfO-|v*VbeyM@EcR#K z3y_R7TjDUk{h@(7tGbq19a>_hpa#ifN0^t4m{LQGV(+6brHd%>nxu~X6$VooD@rJw%ZI6>T3t5lC7Y<0ArI=L7Ef+ zc29Z>RDvU8>zqbH0PPw$Vaf6TWpEGJO+0MX^)}W)%phFPTXSTd?AB4o^}o$7HH&c9v+&XaU<^0^wqeFa>fxs);mB z?4R4*%d@>R7OY;RBSf$;J0_V)J=Q{ekO{r%%j z9y+)8Q!=k8>wkyRjOCb=-1=Evg0XXk%=l+*5miNYd2?f92nu=e$mm23fpqh}iZW?_ zPaPe%!;YMZFWXVv9No|a-G|sFjQU9rZ~OWhNc$CO!|EQ!bsWr1OFlw0=Ys;L;ge)zs;bq-%SjYZ8V~1z3L`^g3DhO2R}cc>O=ep$lLzAU5^grrnEII zna%%(+CT3w0G~nsp1Sk5TK|~`QyDn!FO}r{{-m zLP7}ozd7Phf&`CzL9ZEYXy5*!`-guZ)oB)>!ultY{Kf6f_aL2IaG29~o$oS5CDUbpy_TwT`J@DM`7qZgJ&bR*#^qStnEkE4^WYA@a z=G9VnFrv6qzpvb|5wlAke54e^*O;S8^6P|<UT7ZtV?`0@XaOzw zH|~yoc<-l?*3bS+SY0514#A5@1wWQxUg=v)_PKbW@RCH;{ntd#$rwRl{U0=c5|pGY zF>FcT4W2V$5F(fT)d*zlNAE*VUb(h#kb>{*;vygnc9fRrh;r?CrO;bPOq&(O`E-CX6@8(Wfr{ylINRjL3vhX&^bdF!vZ1Rp3==%`1L1?A1z(J zFSGyv7J=Tn2>|-gnhU@ zHZU9Z1|Ly0$)|W7tPLON%a^@|$)-bjvp6RWv|jz*L<~SX`O-+si}CO@#`9}D4sFJt z^-^O)c1t~ULJHUvX@ZF83NZ0r2wYax*FW3wV1MAup1sMb_9n%8=ETaM{Vq?4W)dRaP@6v_8Xe zASNcJE>~x_05+&i7s3LXc&{g0U*Ih>2tGgBihY>=%D6O0EMAGBFu)MzYByJ}cP!5N z9R7NTOhx?u;c_*i9+{Ua=2pz^o3Yr1bCe!LfA)n{au{4<=9b%d zH|ahGCi%6&L?M;7bqd@>`>zCfD?5t(VUERE-L6dLJzI;FM3+n6J5DFF0aN@=U;w=A>IDV z9KQruhY9Es`ASdPHq*zRAl&8;?^WNi$SziEzv3u(_^AC`FbIM6GD?}+tSZ=A;Iv$+ zFhZsm@52skM{9!3gehFStfJyz$L9)Gl^1kb4BM{z-ip6f)H z4#g+KV~QDZMW30}HACC%Z)R!8un=3Dwtcb$le?W)mILD8LwiX*^DwuHDdRBdj+d~; z5KSQ(dtJEUmCEtfjFA1Tu##nQ{>jM4=BeMF#255=P`VjY@-SZ|e^s?Kf-uMyYm77R z{bNXQ&Vk{my%0XN95|CJAQF$U(h%Q6Q$@Fj4T2oem9*^x~}xof`WO4yp+Ox2Wa zj8Vg~rItiq0(H-}!bofRpd_UDRbs6|_PMv#wOfgy@(f-TxHz)aCN2KVim@+X_U{|c zw_)?=bH_H&bsn$1Zq_Y2c-=YNRue0S#{DM#1J$ z_LHF$XW`KAGZINH4w9dK2VK9TyMYS4f_+t<0I0`xbz_$eDl|F|xv=x>ypFl3f}4A( zR~NaS4x7D#1h9*}-g9bRy792xr%r zk*u+xQ=&8+eNzrWn!E|pl`u?z_oNu&`<%2f%reI{(R-Pbt&s{$L#FC0W$7$DOrn6= z=-O?Ik&Q2EN6qG3Yay;I-0h{?Al=aCVo9rmsSd9BYPVrdT66(l5^SGU4v7zTv$48+ z+%|#vM;|BMLE+V{B!CYhhptpZ6^4QoZWQqQ_({%DA6;!Z_{>BHN8tW;U(Th0CBNqAXh3%l-q4BkpLqP3Svx4b=X5}9hfBYrix!gT z(iuW!wm%HF;06zTlR~YlxPC7|fGIy+7M2h2(q*BD<^-t%mlwVk;8|mH&)C`>KvrH4 zdd_7w>do{P@FS-twqSMsY43+=GrS!Pu}U7m`dMwi0=pcVBz6|pbYY=eR-^4w+j6b> zxei}`>8H1pwg(Wc8OL1H)kw4wkxWU4aPhsaRmtPpMoaQ=h{+x06yNnrB{`jvj_<}^ zhvR1#I$Bz&HF>Erv#0#hOPA+6a4RPFnA=vr!-F*)Ur>$l?dvD{(xeN!oeU1ad47c~ z&xXIW;#K#_5sFuph8BHzSjOYEuBfk79YPmxdn{)>#GQX82cflHpp$4oKr#|Qx(4JJ zVB0Lw*ZoE@LlUH)^d=SzXiyxJl3Nv{{H5gzFw1^wx@?|NAgU0*y4bHQe9`Z0A2 zq=Wj#QP6FC%#aXpr-8FxYVkc&tB0>vRt%?3!GtL6Ozlve_A-FyP0OdRM>7sO1cep| ziu~;|)3(55rvVZ5ab%s#M#d+NuNI#gFlMs5wA;%&kvaWs#8S^73H8(VnuYd3L!1rh z-U2}>RoNXQxyo=x+h!<BrcWA9T7KC2Y(?#VSBoVpj;yMvA6IGmro zg^(QKmr}?p%gN%WhT}2#hHpn5lu@^P#zYf5*6pcLbi+G$QK2`RC7Ar$h8sA}D2mRI zb9r2~{P|Kcb<`WF?6)wEStt%UbJO{D}DkIQq;S%L6$-dwTePa5R>NNcTOp8I)X$9;T*V?xUN z*&1H$+rhbM-$Vm6DWKCc0-l7v6bI9iK3=kyNS!9+RdMn{Z5ZG-06a^FNdytefvDy>G{3s7~ z)ENpmU7kA)$H?uQO1(Ek0v@Rkru6i=OdVWZ?p$r!lAaml+!v_3Yvxm%U3QOMIL7-I zkM%rTECEd(fM6f({AHGRE)y~?O_ltYqc5r2>k_&YBEJH2qJTkKcr}( zIywoObVYQ&c8$u`kw*L8FEn2Wnv90|_a|>r^DJD>C(Pk>c)J>+)P%E)xe{`v-FTq7 zr4O0F#Wjn;^A{^ZM?!X+wOF^a?O`XUgFeL{)BV`&ZUNPE=qMk*wXkA8LAb0bU5)Hq!oEfuJ$NPRLmSRW#q0fcE{8?=L#|QVLykSS@5vS(lOcw4)p7)pLZA(9kLAJ^MOL+RjY#*Pa%B-mC-H&5e5bJg~EDO1L{V}ZA(MYiM*Hd$0t@^-^L;{pf~QX zJS{W9W&4o;e(=K#dxHX2ArV-*&^BE?o?pMlrhi#6T0D--M1jUX2gMrd}T8)?E}xW!@*6e4@8; z$NP3AG{ehHvJNv%es?`8$bPbW45S-pAuyFk&05Cs7pz zIS95&Tup}~3Z;i{hItF9Ym)h{Cj{KaJdRi|xn4%ih&DqZUP3y2#<+gV?o7(wewbLb zUEsArsvQb1LBg{txM=TcH{558_1PeIypl8OXj6$}_T9_9+ zY46qebnTBI7_Id|Z%HKjd5_(SQ-LE96QiN>hhucb)aqvgPp|1@zC5&$t!_f1S6yRHahr3GG@hO{wjWcYQp{J6=pa&ZJN6?! zsmBvI3=r59SeTu~uFHJp1+T^^YU_}iL9XctdymVd3x+8rJQhr(^d3myQro@cRN5b;nYZ~QYvnKzI zXT&?a=QAd9SFmY_AK})aSpW@Y>w;fdpq|6WU-Aw57_(zEpc>D&HBNbI7SK^V6IW|u z+cj;HFYgZ>W_-ta%_&V>Ql0WG77vZEn{Tx5z=hXk^(aAFs6P-jH9GHo*b=y$37UBn ze)ilPLo<VRQu zH+JlBXbl$C7?*|P^Rt#`xVbHkaVbvDZ58g$eMsfsPD-gAPSByx(x3q?t;FFfVS|ha0*jC+iP7jIB%%JMrYz`3OW7 z$_cGSzFNs4INrJ26qB?^@N{9&=d!%@Au70B=$wFyA(bJ!DJx;w;6{r;^Mk86XdSYk z;zHo5U(}g*++D+h>2;1;H!iM2+cur*gH0LbeyQammT^Z=b!| zr1xG?Hj|M@O--cUr+e;Ebx5eylEqSh`!Tuoc3*CaM-QdUt%?*)a~+K;6RWMLMArH9 z@ng<+C3`bd1+7(`u!0tdt8<;i1cI+Hw9e9$m37ozz_X-a#rraPWNqZ_-O8H8wuVB? zDhJha=bSxjtt$1Y`n1*A7s_V1v(BSR39LEBcmXYr60A>{1nlc~<&>zh9a0k%kT;r6 z0xokotN0nNC!euOojz$|RZ>KELdZ99I=d>h74L2M-PaeSYu$!Cv}UJgJ=B)=N8&-KCTcy)EOk5q!C}5gSkJ{rzpXu-f!KzuT3~v#8n-i-M#hbu|#? z8#cghwaKMF9>(O?)Ma|yKi*&BT!%Y)P@k^>7Z(uX^3(2bxbJ$i+}?q&{AsNQ6709W zqBWFp_rdX9?!?Z&~WaHIGM=x#KeOW(OJYe}GR{f|UZPpIbHRb|yA?l#DMPi3YS@cI7%x zKPaHi&M*JmgxXDcydJRLXp_cmx|CWe%TQE3=h16aUV^_pYMpu4uBJM#Nryqvji+=9 z(lMf$eT1 zQc;@FFXaMDI)2*atj&jvX+NB%T@4p^ zRUm0X*wEUcPPLu+V3EW(sjGddaj73c&WJs(LCLL9xx6FM1^2J50ixgbQ?Kg9bP0R2 z;Nv9+mS4&moqp^gd6v30?27G1}d{EmjQhMUpPVySw$tiB9} z?t&APXr2}fYL|!CeI!3G<1JOI2pbVrZ#WZu=5(~i@iqst#rfc13(xIv7RQjew7a!K z3JgmLJWD6`TI*&_bym3vi5-`v@=AQOmL7CayulJV+*qu=Mje{+!h#R(zi7T}hE9C! z8XL{}vYR<>?JTmj_O++q=KD3>l<2Jc*+}e(Wg4%oi5nV-r=;#pGJ8njlB9kL(HLz+ zQ^L)4uwVN&=;F9p0zC#z!)d>Z;cedubmU56@3^ zn$BXj9R}+`y5YhTl^iIg;Q*8VHdb{M#RM#n%UTOQ>heJxYaj{TPZ}uCV`s#(ee?F~ zG9ARoRFf3!2~Cj+>CUiveq2C&Z!@1@I-7G;(9cjcE^^&P%_q}-e%zG|W5d7rGArL| zJZmiNk?i{R*GlBlINZalXgF26eOzD5HfN_YKXtAB)BGomW!w!^mB8VfTP-Z3eKRP^ zCpYN$sVW<_*1C16ql(x&m8=?xG+$(GS$4h9l?*PHIw}p%H$31q)^H|&Ap2PzPg_6~ zLEuL}SxbHm{KfBsUnQ*sS-_Nstjbbq1Ir`$Yc-cEq~OX@xRjB_=+13Jo+0#9t*Z*n z+sta1ueU1U$lLjsOc#S!7b?)E!t34htDOn!jULI&UsjoSs$W|!%6W0(oF}AvJ2V{F zmZKR{Zy%2VSeaiX{<6ZmC8F0|oD}^rgK;ScQ-DfsgDa#XrS^8}Bw1&ox>pL4n_oVE zf8t=nW}Kj~60M}_U>XWBXR)YXJ-pX&Ee;83Dt-bUDl_t0ruEXSYD~&k)b^V8qNA^s zh`f)CtUauhO-3R|M>)Ct@MtY;m<3{=6OdWHqwBKq#kAFV`7GM(w8$^kDDtV~F5*o;M;bWvuDBY3@;XLHwcUnHi~<`q=A_ z=`ib7WcCyrzwQ$|ua$VKKtjroLQW%0e_Cl&Ms!%sb&=!I3P&!{C9LtRDIYwQ2Y^5o ze5vKqv!D;xYhr{ybLs~w*<;l{ZG=K!TukqCL8J?9iXZgv zWpCS`A7G=d4Xs_HknBjj!cbE(c#Nok#u<+L+pp+=3Kexc>@z&YdT|(~L48&rG%36n zvt^5N=PykVG04XIkZHHYl$yb1Pxh+%w(6^aUY$@ z)4PBBQLvO1XmS`ov^$f90vQ{Dr)cqtl+FR92YHvE5(@Jp7oGygZEyJ*38v2V@-#hD z*;m)FnSptQF5NFwUi#pdoY1G4^S_Lvgs1z|@;Acll=Sh|DQp=lOaMvxAN15Wz7Y3YVqN zZp`Clb6(@U--8p20ukHK&s^>TB0ARZ#Ehp8RKk8_H1f;4r z3llzrCb_I0x945^RaJaI2{hchisXNlqTg^y>y(7k1ZDjEoSS(*(7}uq6d(WDl6)NM z+q#hG(5OfJ+*H`vrdi}S5?u`l=rR{d@PF|^jCnxI)ZX0l9_BAGbO--e0q;5hyjBK< z%ls;Lzt?LHdp|)tj3fy=6~6u(s^@D3;2xK$n5llwe4@R8Ep^fYfM%ZV+JM28C9iiE zodzY~wzWX1bWA8Qp_P+hLifdfp$T{HN>F1@sw;zV~aIWlnd0F?mOt_#OksPf7LNII$n>)j8xxG>Nlz z>x4cu{V8-_BVYQ35Qpa+?|*b47c|i7{Jo~{k^fvdqZ0tm;H$RFr}`mRX%e8vveE7o zF0;n8NUPJgrCDhj$h4HXEdy4aaQ)V(8s4gg@S$yXhpxb%!v5enT+R2yZ0K1 zCkk@y306d&#Lv%-{cNQ&(g8%)RacXKHAD|?0i8n6gQPorQrZmplYctF7^EhYYd{M) zq#FHA_x?omEU!^B=3^@3em07}leuB{{T1s+72ZPAxnx#vVlw%KU_kySjed4Wkv5_} zwh6Qu2L@b^b&5`(uR@Q~_;F4Qq?_)SeTxfhlt)_JCAEAKgF+2wI^!@7h5LG297g*i za89Hdb73&g(Htc#EUeShQ;)$7_vMupq;Uh}-HHS&YbSPgHqI}e&EtGK9&U&+BQB@L zkzq#k-rq`clQbna}mINgw2OA{XQapK5u{y1k1F|a+zJfSonw7E@0O>9Xd$=8b$*| zKxC4sX8ubr{t)v!0d{yvXbI=9VGsc9!!8+Ss(t`1M&ubey$4CJjQrxZU1?>X^Zsq4sFc{+J2>LRcy$X8b>-q}4x*|@p6$$(`h z-S=PZE8r=%3YPtBtN#>n#1GVU)?SqRHN0ouSE=yrISTDkPpSR((zikH>@@S_`9u(SC3*^Ba(Bmwu6Z(W#k_Il+`{nZSY+PW!D9euTzvpetX%qe~u zl^;ry4+!GPGV{JI>5Huw;e|*9lpD66Ej!)#qOGpy0kajopcrpxwSg#%cs=1raG!it zy8(KJFx+TW7Pa}l_N0v^z%Hb0dSaNmwmQFPy`$4-Y-H5%;%H!8s%h;8@Vy+7-Qu$& zQ*?9OJEkkCTaDN7ukE!_xsihDY!c-c<3eZ#=1&F(sjq5GGDJ z%GIE2+2L#qq_u9I2v;(c4Lq!lF0_cyqJ25CdvRKQMib^ZDCg8QMyk;p+%p?=WA%m- zt?01rly-GdLwQ^;^Y#JPxoTQnokv?Bp+>GUouKV!;*YEDk$KS7tj3;*qeCsH*G8Xh z!N!AV-S?KTJWeN4c3l8O)FV7+7dNSD`~I*q(e}_b1qCUuP5&-_Qv-6J%e~f*ZGv?= zgDC=5yMd|)m}mxr|NjR=@ccRr03sw6VM|)(7f! zJ}rMmUnS(JeYG*xYZbm--eib|e3wopxHWYr;*@Na^(M;I#pQCNsB=fp)y>Hh$h3{- zMUH)l7@QtEsZiZN>|{WFnDQL=G8W1egw&rKF91EaR&-A(!XuDcBS0&ZmVLT6BZ7*rM_{}ss$Qk#@vrx>hL+8qSDoUzp?|El> zv_tKv`P^PI;fiB;%k7EhutVx_B6n$V?gPS6=K%2vvo#j%=Yme zvarMQ<{YKO;`+Uwy1Kf|Fi|7`3^6Qy-u{uN@pU_e4p233$}hn4lH}Wz^|GFSdiHqV zePDCYiN#AhUojXk4jKdl>mw{B5YzdToQ{N?j%9;t@p9M0iMr`%R+Vy~fgPNEzuWcm zL9=qY;~4yMza=f)Y8$dmXGljaK07h2t@ez5f=sOVRg$;!LQ0233U>2p?P;g7Xcbhw zy#`f{zw`Q{sMGA}(=uPY3!t>Db>P{G%X$Gc-+Fp`H`dmKfr13^-6@9)Z;8EkKN=If z$j%^7JR50Llce{XXLN;1Xr!D^NLL3%6!3ZM^sjQ`jNf>|2fT{D4t5RSl@NNo`+bY&uCvtCTIkk{ zq|YFO(m<-%(U<7lr}$!u2jgyFSFfx5&ejoHKKn(VWM`Yn4qa_Ue0++E#7}iQ)s4*c zl%?iT3H%@7+}`HxQhl|`Ngj)}@<~>)arLlo48073y*K(`s*$UKh>cWz)Il(JuiF?y zy^=nv!D`^LS;T2@c;7Edb77w65|xnb5M*%~bL^#cOb}!jVo_O{$sKE+gWCPu-lw3yetuL}beFK^nUEb2rnZlE@{u z0F!0yfR0C!>YL-HU+Dx3yAJ@Q3>%!Bw&;t9c(^s(wmB1p1S=|S4%?i<#$bVj^JQ!E z2z}85ft=^xamjL@(hG_UvAUUU(MP8%jZ9Wukx#3CVmQ8vj6FIkA&3bNG?=+q(!VKX zzLvX^nl_asCA|NgQ$-RL?Q^*`w~|xy>apHpdbSLFTOX_pA4gbVEM)K8EuHBkz(t7E zHh6cEk$UAO9G$B0%xTT5Uv)1@#b@z4KxS+dpR1%_TES@Ht+~o4W4$zS6e|=)9wXV6 z{go+h8l74c1lCbCV>ha4@XIMT2ySz#}nSGSbPOisO7^T&^0cn4`3xi_cHI^W?)cCu&>dV2o9wFlfU~P@7VzE%4vmMrR(Veb+ z4qMQy4mE>4yr_ukIHGG+x=|Y9O9ZUghCP?8?dz zU#imJ%WGQj^u`LF)YSz%^O>vpg(`C93X98(AIA3*wXs^0m$E0&SdNjUn6P<1Qb&M)^ z8@!H^{C5AXd~V^*B(^PdnV`UJYq+>B@)7}{EZ~ht0vcIoXCB-HG?ZOeS!$a3JX1?l@gzQjynz=l`^phZ6CfdG(6v--FAWM(2O&AjayrJ#3cIciR?;)X1AQB>KDIG zgbiiQdQ3&O0bS=|2R*fT`)eupoc--#sBSVTLH5zhONP!j;#(GJAa9bJl5c8)OTAGZ zfaDVdHcpmg%9GcpI97~cefYir(;GooZbmVFgHDz{DGg^y5h|`gnqg%H!-h>PVeh2c z=the`Sn4=+2jx7#t49*pDRE%do;rti2+LWrmH@I)&k29;Rc`oA5tqsLT|qSL91~74 z{N)wAV=^IqTP+@GkyV=H4D}KMlP}rXQ56O2Og(a&Hj~-qQ=RCrPN|(de2Mm+`Y>uK zhmn&1$J$$mMY*-@!+?rN39P*7V;Q2Pc(Z1c3Q5Lh6>a@DU zyH<*qyMk%UJCNE1C1#EC47KaV8mkEZo}8?2bCU+aOG)lZ*g>_~ZGvo{H1vS>d= z{w}_wZ|yG&`8VtzAU^`4pH__RnXkz7?(nwQOkU)g=yZ9nOOOp}L?a`yF*M}17K>uF z^;&w~pcmrHu?uzw9&CW=pe%Q-KS`LmV82x{#EdOpI=0C~p=Yzg_f}A6yl^3bi;9qq zeLd(E{VSb6Xl7f3Cb^VcZ`JWD8l=l`D(+;NagyO$**$TWB3He)__EpkF-=$^tS-c%;AZO zW*1jySC~6gE0bR#Ir^D5jAFPh^U2^d!g3B=i7toK#!a@tqN?LAgWr13TxY6o375-Q z9kCoY1;rK)jZc-@Jo_=uxS2_Z**-&F)xQKEfh$0uU|IwXS2a5sw-V-nNu1sOr$w z8_`45T(-QUR!9SbH*GM#rn3UI61JlC9z5bOzBQcwLlJAtHa3f{5pO|iXGX8~Dz1_- zpSwM^w=>KF-Uh|yl;X}`2inOYMSYK{-kiRXjFv=-AeU>#93}HilxnmBXQ!#q$k;x2 zk5SuA?U@~qt;hnXA1cNmh4ztBz=ABJiFm$EEK^47N6X5tkGty>NIc@Pf3yNHH=nECI(#5xSZi z$-V(OC{Q^__qS?(Ii~&Dhk%H6+bKpj((OseT{R#@`GT;yT$BB6D!o z^w?BW$_k(H8RMs76>rM2QSrv~KYQ_w?_!!A>DA4Np^g;lR{*MLx6-rWoqGZ$gVa!S zO85mW*FB%eeic8TOD(3Vot>LCio;-`xy{}00^tX-o8R8uvh8KpB4DbGGeA&ENJy-? zF6XF!>a_~jGuHOL0|KFUhBd{smeog|BJ>SM(xcEO@7|kb28&l37cvmV5uy1Cqsm81F&*BR?*s~RnDrT#5Kb`rhXAtlWp}lBpkVKP$8-1 zW&N9smep%1v+moEvUHqlTz#&e=9*Hlvc<;xF_l-SMNz0|wHE80$>tx;W(K!(m5?GB z(JzQs7(PuOBCWsDw)8%PR+4mgciRl4r)^c8@P}?}X4F1>pl*k-Er{2d* zJd2i_d&72yNG+NJw>1Ux{`JYxP!WfeSJBjvEUr3>d&s5e5i*l3*Z=aZ9>1E`QIWVm zfw6zc&s&RpG(xg6ww3!v4+{fx6qVv=#YcD7Kjoz}7?zh4$UC`q8;-r;Fs@C`w#h3Y zNG1OlmT4uwNt~i{uY=O5a5iu@J2ceR*BdH1$^oB9bN7KZ{=y@rCu-axDw!C)V2!`i z%-#c7I_AogFH6;S_Q%_`SpEMa zaZ1BzNq(5cS@I%5`AGzp6;p`5%}LUo}f^qUJ|L5+9uYJ8Oo3Fce^ZXN|!( zIQ}mz7p4YwB+k1ZssHQOaBv{U%-;D-^KXm)fQqd=0BDHi?)tB!oK^%V1?$44@!ukJ zalqqH`(XWFDL?bBcwrV8ZSH@I&^-icI6Mz4|4p#?Puq53pcMFYG&8-e(d?PYw`Z8go;Mpd!!IuR#@S-^TO=kk3iLOQ=-+p~A^TE3VU zz69MM#pkcP^OKqpogdy^2NpZ zqlF+kU6V5Z#87kpZ#_Wk*w!*FHstwK=n5mlMHWf9KnE?YzaeGr3&egjZln^|{^UmJ z3hwS|DcQ}zcHp?f*7-|3jg5`4(XwenSW87Pc8gt7e32b zieLB7fE|)=O*XukRN_oJm^lqAU?Y@L+-27qa};~ zk+`)7IJP#=oqRldb1phKHab(EKfb=vUCf$pYL>ImQsrc?t=%8xuUWf2$6$=!|Jps0 z2XQ<9F23(^aP7GAA5w+xAry>T07e$C4?3?FUbd2e4&-z>SsiiSDe-M+R8j5i5lNS} z9ejBz>DnS$khAr%Pt7^F$~$lyjlL=?l3VZmLpJx#@<>O&(#AK)jT<5;tN0uA|4g)b z8RFI;%0EU$bIC3^aJC<6hN!@NkPl4jFH1-?TFb9+S4kc*4?W=O=1@~AGs<2(_dGN2 zd4NUWnEH|yruwh_TNE=P=c}c}giLl|!C_p)az6U5xc0H%Lx2rmrB3m~W~!G_M+$GW zZYP@OKBAfIru_FwTKr6`zL1OW_H&>KG&0?=ail8HGdS}cb6-94TfF*b;;ny;e3-YD zZq2L9bM7L;+S;0(gG2YgfS7z^CN=5U3T_RZ&+?N{n7KXY&EN0^jPps2|f>h_o z_##)lH&Livd66+K^JXL^vcpP^`MuE2@Dn>PhQF!{Ejq!e z*mZC^YCr!8Fuv%MRJm+oi@@!gjPoUa;2tS}V0Y|sC{8+igQ>TBCir+3h6o9?o)l|2 zSr96#yXGNpdVR3csUC?!h24|BC}wJbZuzA6vzyHL#^FC9(*VVt^`0;XaOBb5tdF{{ z**ojeCzVxsQ=HhxL4E^8c@X&c=g$YZk|Px_>N8}T2BP8}kG%AyYjSJC<}%Jp1b*ae zZSk&z+MAj_AZ?YN^%{Y-ph_Qh7DHRm9c-@eXE+Z{QY+!q=d$QgU`bt%}auId_r`qQB9|J55#FBjwV(hSiY<3A02TV~w-LZ-oPs46glnc>ZfOBNV%; zg!A|IHk=tdU$5uoLz_dU_lA#ZvmY>VN=e|9Y~r`#cPY&N`49Cr|u^Y1o0)k&`~LDJhJDrb+zEMS=I_)7{~=G z!n0~oOd`WsAnoWZQP_C+*IKg9`44sUrFz+Z|B-va@Gl#3WkHo?70qW!Pqvlr1eeza z`31P!khQvQyFh9&AGqB%Ti#QL8BKVdpZti(2~7|3d`Tr*hbOfUwA?>hd;k_Pd{_1^ zAN~mWyjVA%fl^;ze@%X`^UhI4k7(o8q-_Q-fp(T+{=NlweruHyrR*D$xkRxVStbSs zFCQQ6q?^AtA-D!P1JS14_FkGW?^kQ(ce4s_?F2Te4ndhj(-nVy&|yzi@_gDtls{Ul zywQKOf{Z~xdt0B)71A#~!fz;U#0qC+^r?b+y2MQkN#eHsuW0Ycc&$Aa69-wK!`DBU zK3N${_UzGY?iP$bH3O7?il|zg8*9&pN$@;gm>78W5h z?w$Co5ljdtQ5yI32HRT4*WLBYU?opfSXCtm*C)4Iu{gHZmihe}Z!ZCK9|L)M3^UzQ zY!BE1lj21aLRK#hWGEH6>3AlllO9gi^HOg2&gC$*@+ZfN44m!R zGa4y8VcxO&FsknI6=HCoYDDk=f|wNlJj2iE$!LWh#6WNlJ}+@dH_WGmPQX*^4=gbY z%y2h@S=qC#DzFt|7oN+H)d$-03EP?mi)Zx}sC1Lu3*Gj^?{~@kH$7J#%~qd+#F*c; z(=KQDq<}F4BYW=6;ats~`qlK_;u0gtZZ7k9`R}jp=zs;XNs#e0=Rsz{Dq)3)VsLnd z;$H-dmk(S-vR+`ydt8 ztT_M>As9zYw4_s<<9vA~oPjU{C8(vlTDZCHF4fp*{3b;e&Kms!sh>TVo+D$c@>OB@UffX(xL z9dTPbWEwcA5<<>Sd;C-|*x&{+Xrwng10T%powX-OQNJD!r;MmE`IwPy>Z}&OOfAx1 zDA+eLV3x0J#N;F8gS1<7*ljymllH0k(t^D8g5?T9&7S(X#lU`k zPOjjc-uS{&`andNo9=k$nBntVJ-nmaMORMG3bJf}iv;5W9YW4GZ%{m+%zgA+{Q)b4 zB8f(s+IxH!?k!?YTWW8(=Otb)V7)@pS~PL;PGPJ^;Xu9Eswdz<__6*|qPzw7=3;*T z<+4}w8j{{#j&gPwh9x;3&4UN=D={~^N`+}Kz3eLzwY~xu4Kv)HvU5(b&_61X9e6~; zaiP%VCBM$M-#)|L!W3a(Z?Hdk=-zFKj|`}qy&*|N{#~rxWmb+u!h1u%kDLQHH0g04 zTTt}1zRJX?jMeRpcinB4-QVmDLq70Lw&Ao4ziH`o_L41mp8njY8Kh|{>=y< zlSN}@kg3Y*G1InQ+rwPbv;N3PD2kd4803yq%ewKif3BVE*;fr z`8xPG!_8`fJxxEteYE{O^tAM$>Ci^U{nQYM~c3>la%dq5W)&jfx z_e(Ecq$}?C>7b~@G=Zr!VM}-hBf(D0b<^nqK{z+n+6I|!qRezM#iJ7KUJEn{h|=plnQ-3zHt za9t&oDi zt6uVEAt5bu2LTrK_4R0@ouUYTX}J7LU7_x`%H6~gNk=#g9A~Qyt64QLHkJ(zup`X` z>O$rrr^^CD6#O939i*KiPCBF+NOe*q?E6z?(bTopT^~Ur_CV4tl>^Jkib15x(^{h0 zZH_MkIJ3@i4j;`es#7bq=+HG^SrQfxZ&Vy`J3z$bBNPNct0KK=^1dKD-(+yBl2wbY zu{TVwW%VnrmqVWa_*X2gOyn=xRQYSTjuBF3E;-x1mfA90NRf-VGkV44T{F?Ge0J0H zGmM@f9}N=o3cWeO>P_X#kXB%Lr)ZEwK3IBspf&%KQ5%frRNUx0PKt(Y0yklMG&4z<)JuVrHg< zzg6sMxiZ2=5?6)Q01011I*O>zSY0()slOU4OIz+b@pPvW5n}d7jT;*7!3X~q`N(wL zO1WsTlfoEJAVcEZCUDzW`Re$3e#V?5HA?jqs18U+#`~;7M}< zN#qBs0~6mq018{r?I<(j>+cHNVrcy{f|>&K@0u63X;NVqF#bPXLk_-xobKX@%5`RWVK2#WL6-v5N7Pxjpzvl^eFwC-8Ga(!Cqx;^*)U5M5zrd4@}Y~I$A`-`nu%KIAu@WEeVSyyYDg1oQs6^ZGI6-|?~ z#_5j*{2XIzl*8`CN0bV@4>7=w61U35aRwjeiUr#beN`0^$vW*VE-E@aHVI8ct#V&_ zfZ+V%j6yl9JT8|2Ye@d33_YH(FbJ_%jZR#71Z?eP2Be$egNGCRd}>9tXF+f)^HU5* zaF25PN?rd)=eK(71Z)=3{bSz(rN+F+VH`~m3BBdc#8s{0-jSTXVh6dV4HvCJL9xz4 zL249!sbF%*T^$>)z#0B#ofS-NrI6s6Sh{-J01XV@4i_lD{L`5rGEbO5>ct2o+@RQS zouZYMP$F!Y6p42%=UM+E;x_WYHAXsHf?8h=-q|+ z*3-w9#B^86mERM(?fAkko|Lfv3BCIOSmPIJh@cvsiJgkET52&ys& z9di1e`@^ zNUZhx(_}=)-$8wUOCs_%E}ho*|NJ-;_W~r-^?mM_v+@i)j!Dm_EcX1BA?+qkEH}w@ zabJzwk>mq%)R=Ps+YU}ZiDkbr4b6@}?rb+0(p(*8Y z@-_FinoPxF`5%DO(!GmY&9`N*(cJ)nXv&hzZ&d%FA>hB~baoCqSlpyMtqiP4+|}{A)((>M zIKrLmRhz^}AQdq^clqm&9Y`BrJ?^fMtJZpx*S~00QhXu4NB-gQyJalo z-2U@m7y&W&hXmprS6(mFzTXsj*;|`Y^iYE$R(KD55l_kIgflYDnRD4axddZRIWCSH z;3MBc1$b{T4eVYZqFdtyg^9Jfv;A8Ww;SQzc`@C2wY10n?>t`Wy^(xiQfx~tFuY;h zR@mN9G&JetR7Ol*(Tq|cy zOJ@V_i3v%Ay3lY~XvnVQi!j?z&k6nCDK=z6&l)%4wXD4o7Zr$arxOs#EGiKZ;9G_7 z8SX_2yU3ir8p7rv?)W@3)-Za!tbIze zu<{mK|M#tMDCG1EEi=9G?fC>vy>^ZH=~;d7Q`=MnYEdqU0w=fBH|CGGnG48XH6RX#Vt5s@8l#+oYO~P~Bb%VZYCX%~?(5s+X`)2afex?=eDu*}Pt*yxwfd z+|Gp4&crjdM6^eB$$0o;$C={g*Ch3-WVbJ$xR&wzR@(0S3>=xKo989pQt6;Mx>&ZW zU$S4~a3sNBzJ0-9r*Pc?txSr$M?h4_A4Z@Z%w0-S@^uWd-vTa+pJ;0v-H|?=~#&4;&h&bLm0>3L>w0vcU-m19)zeyp*T+PfN z!Tf&zebGSifdmPgaJLZqn1tZ;C-W53;d$qWFR23^J`R#=fjOoK)^Q|@n1%{45I5=Y z@!ACCKvN6t@EIqnYV2ACG-!K4{!%Lv>DBZ~IfL$JcQ>oa(#YoLOZ5g?cX^x2ikwZ& z&mvO7EF7NOW%(+@tENrHkJLRQOYUEW-73BX=YdN+wxJZdiYo41DnO{1Jj9zXFR&+p zHtlTum=1qd)>2t+hGHLmo!CSv9$;T>`jk1@m5KH5pih(X9#}B}1GFLTK4_|nSXs+8 zZrdz+rQxTiWvLc?HXokYJ$$5Wkm>B4>1`fp8F-Hxv5Jsy^8OyCJaLie(_@=^Y`Grb zg?$uW(W>g6LbquDjP6GjWzT~I=YX@832d>*pp(2c*&O>LMgl8%?X%5fOWV9JvY&eqt^^M=!Vue#Mi?lv8VBj zI)-CArH+)^kU-b%^F{vDy2=W>4! zW|MNdqq=Z*w>MK9zL>C)GSI#?xHN&E*7vUt4srgSE(izL*l|x+K%sa^W0X??PSmiqYeST~@ z5rf$dbYLrP@tzcOX1NS{YU*-o{-O5?}po2CoGOg=^k zAXPXDs}UaQS?oG`5eqD9eRh*A)2Md!4DoeVMIw(DWxqD$Rc(x!#!=^+ml4pxzCu}FJ7`oK8R z;OlXr>aijg*<9p4eljA2j+VPWW(HcRt6_;M8}o(lMMj@Vb<|$#e_kQ9K2luDy2{SJ zF|&puRcX_0Uu>GYJpEe9_IUzKU+-mZva`+B{b*R;*htZBjqjpe24LdR0}xR>D{&8u zaG0_%QAh@^WIeHu-E%lp>8zBxv2i?zrZKCqW~czNtivH&T$p>;^OYlOQ4G7!=*iXJ zgt_H`vTq28KR%jImqw<;G@cEsu)8zK-xr~1NR7|_ks&X)A;I-)(MaNC{sVnR=tGww z+rV;R!-Rat&Fm85NsSu1u&_^{3TdOJzwnzEw>vSKkrsWl3~xDj8%mW%z+O|2&YPO% ztgEebft4jN9l{s85!5%HNtP58{CutamAtyP7DSLFFE20a+&Q(jX6thniA#=?b!Uod zxz|MI{HkEKxn>8x*6g6itL!7ghZ<`ihi;Wjts+!TbLSrB!7~je;>(XRn-A8?;x=s{ z$8&kAvMs4>It>*K6=JMu)GwP<;9G;}U2J-5SWxaf@|( z0-g#BD<ecZ=GdxcYKUO$}_;=Zmw+dI$zRsRvd8;5N{y6A-d@y=KREda|mr zkb=@)f3wM$KLR@&e&h#qzFks{lXjVi3i--Iaad_YkoYL2$Q#C#kzc=R5UzGC!~ZEo z1C*liN`g2*otusvPnsZFf`NL4uz+lx`7^QN#oBM&gX81lgM;(OXF%HXxjwyZ#TCiA z_2InP;sXy#KI)9qz1m#)V%Ya}lV=urL8FY;rp*D~#>SJsDy*6(s9=6{s*;ipZZyj=K?C}!N2 ztK6 zdhP^-;M4>MPG~3u6nPA(GGh*uaGr(lhVQjqibh_&hOdp?w8*`r?~J6 zE`2n<6D=54rC^+>jTx=Z8o4(sYk$}Uf4U3FlFhmL;R$eVSjJ{$raOs!KrR4Yfd+6| zugGZcEWQ^%RIvthm>j0tsO77@31#}Bd!YV=A|d8Arq4CR%8F-}@qN`(Us$7j=z0eX zi|%lR8MyrQ+vcuer#eZvc*F|zw$a)w)_lMRQzv-16npT#rAv=t;+_00K^>STWeJ{I zL>njqY;kX~Qw&5d0gwmChyBGF7<}*r@krY7?0Y&8@YbB07#(Hz9(}TJA__m~m&cvN z;!YZYUL3&3f`{*$80Z=s+mE_xYsWm#-!LxJh+ZC|`#3y2Tz$I+`o6HZm`(rA&tR-z zyMJ2_6)7Xff?mq%z*(eKnOGm~&P+mW0!+Gj*672h$Fb5q@`0liAkhPCWSuU^)VHgw zD4{n^Ssp6J7V8Be*@-)AQikk}Qf2pAIY74|*VLLGQWH`qd^+T~ZT<7OHSiXPqYBtobG9rL46!A$(#)enB0Z*xw z@_PaGrF6iGb|CS&ycrT0Z-hsWNLlLEE)dT#*zi5tnq>X;)iT?L^2S25?Lc}bSLk^Z zDB4F;#$%&q9`meT=gmW!p~hCPdIjI?$HZna8h zWsXR2eezVb|41z2JT86zCuZO2i$ktz0 z-u9ZP1TsWuW({VZ|Eh!k?W-TI;*E58s4kg~2u)H#2tzZz=v3LLK+CSC9z+7o5kN3L zeE3P9H9Ac0YfDQAa;6@vj|YHco=PK#sd`$?WR`A(sK%Y}^eY5*KrA&u5UqLsk;3@EXA`cCs#8u-J@Fo={6FO<OP!7x3`AkFB-2XZcQh z%!+aq|NLHc01?M_2co0)#-@!rdkUL@FeX$0S%#oru^H7<)#8oc&Q(iLXsQpW(w62p z`C64YtW@EGFrkv5v}8D5|S}5 zC$z88q0gTB2sz~V7Cz@cop&?M*G?-UAG<1z`>QJH+q1)t^M!NW=B+d}{YOeQ2luI- zi=40hAQ8%FvNH3uf3~VDzy9@Qro^zC!pZQHr-Owdis`{)>8VZwdANPA>cEIp#-^on zo-L^9yN6s=K!-A!!!C6P!LM)vNHs~)r;fn@UovLCPJ=sNRdN(Q1SVSYqzzc@X=_GZ z+i(7lbsZO~+&&+|)3c2;h9Lx+)nJX3lo8JAqIid>^@bNT?iu|EGHy|hl&E+W+h_14 z!383h{k$VZ(Gly~rwO^Umr&4uD7Ln9?2SRrnlwvJCwflRjY!-u85%;xZ_^Bh|VrapVlHV$_Sqb#s~q4FZc2iyAf?ijTJNQkM}Sr8;P(*X`OTC}6hn*M2( zO>~Oaq}UH@l`a3kNPT!-w@ehODcE-|%lxHSM({}fJD5Kkvwtmee`(3D`<+648ddL& zC3)+x{9J>mUE(yO=bs0hmru7v8v&7dnEVD zUErt6h1Oj!3vh@~j<_^K#A`JP3rw9pN&D5_UPb>*H>qYb1wFG|%+U`=&GM5!5@ zEK)bFkV)P4+i@6Zd6AACnN^+-qcypD`a6cM7glNB+ThUZW_7-prr6-Uma&)?JBV7-!Zx`&KyGH$tr=}gLWf8r81 zEaY2uQDkwgxr`Ut7cJnAX@QGMyI7;Ww<5cO1XFYp0%Xt#P)U99~r>mI+1^ zX|n4mpJgi?^omn*&-hF`$M}+xaa)p)`iQP3Y#$$rQEQl0;hdd=-H6H$%Vozj?#UTU zy^k4>HsN(#*Jidw|!D5Zk$SljgqHI@ShN24TuBjQtkl7AT5r?LI(0>@E zx;}e9;@ghSn`E)V!An`Zq2_h& zS$!op+{b>#%o7cz2}wHrf?+F`LlJs#wH|hd%4!P_f4_CNO&JPuYgXX7=b>bG9nJpWs{ZVz zno_wB^1x}DL)eCv@kZHx$!SfWo7cURUA#2k$|DEs$S9b!xSnziz#`!uhVl)`79*rq z8g{X_)O?@k?AdHUM_VXx-NCA_82Kbq+RUPxdg^79S6r=i#v+`&tP`82*eYKhw)XR8 zUvu@ciy45 zhEJwbN^dzx+v!cZb>Rk`8!e#-ZSis%Y-dlOth3XHLl^$)EKo!47&4^&&^dg*SbUT& zNZP?T%E|z~G?Xqxqr`xUN1N9=O@0>8L)Ig3{@ zd8#_}L|pn;Fww93_jCOuU4KM`E<>{x$I0DRPS{Qbpw2}V$*-cwzIegUL!|TqiKeUN zFH1&`dE`Lu`WMKXP?_XAhf9fI7*ljuiIPD)YpQH;bpG3GyMVH@&(g;foy;A#_;#?^ zUN2(Vc%!UAUTxQy-5+rvcYHs7F#a;qOtGMcec92gnpDYA={kprsHTJe>Rjgaafhd0 zIM)PMdPCo%$LK}1ws1f_F)m|gCG72eS9f=JZ*LTkCwD$M2C2dZ1ZIvTv8>Cryq>)a z7AvppVMJ1TX_Ayvy>EP0I;9%MkVh~Xn4jhE)_Ec8uh4-_@#Lw^hmXg%&25M5C<;PI zFr!2Cq}t`lQ=&ifX5$Jr7+UGcIP8sDKSHw4qQ}pUrna}IDq}7Xv8acsmfxiYTFnV)0QqEyYPG+#fiJJRalY z%UfA}C&6N6@X(T@)Va;RRjz36z|Vz}kKOMf28itf*$W1$F|Il(LZ3`}#hbGH{2VCMJxO_ow-;A4AM=CSvTg=W;J3DGM<56sQe_yRZeO@@Lm zS^Ay@cASXk8@+}$@Ee@x(!)z=%qxStMuak^3}Bi)caq0qtAZz`Iee?eV?=wz#JPr* z16%J#-CN%oDnQTYOY@&t_J3KHR7t5hty)2|=Fd%@AB|qsqw=H_OAqSu+V-UY(|(b9nhyR8J(}}5vB4*NZP9hzYWrgovC~g<*kuDGb=qe?U!heox0Pv=Y0o$ z;0I1k3%0fvj<26v_@rCpy?>r%!dFx)D>FNLsdI4Q-Io~L_bOGAYIw+$v~`fT6$*+t ziQjv!wjFa^D?I$5Q(Vwy?S=WYVHFBR4V?z3`JWO*1gW4_WO`tA#R>G84ldoSARA2Cq8kkjd3f zhw_r+s@LXv@`FtRi8K!#-dDdvmQ7<;BBbw3VV=NXxrjFr2?3Z`CU3I`zlIH=CBeL6 z@(_1Ry{88u6Fa83-lBZx&oY?QZ`a7Lrz$64L#km}q#btt%FUah@*Dx@_H$z`#=ATl zh^OtP*#J2Kc+5kQaOEi5w%qiOTbYc*4L(=x(lijrvP0r2#aCzk`v))IcBjlgwG)3j zl{7oEBD*L1X{p(lnBo1{#U(b^rV3x!F;bM8blmk7$iHoL6?D?)4)`oUEsQGWQhGvF zCoKL3JLiX6r97Y{i}x+4G_S|PuTgWfU;i_J0JplGKyYdP{dNF*Bp$<( z4u4`4xKHXAadG!>=lz-n+F_~cNX`Li!Y5d_38kTqHY9!;xpKh~cRe&Nf8lUmfb2gM zA>J5h!K^u4BP+p!VINkA`EcLqWy;ADYVYVa84b;U?B5QpksclgAkH6R(uGc7F zFY0K!r!>Ct|A%d}W&m!&18?bm0Nl~7iG;VurDsV8NDLW85|@MRhNNVx-(OK z$*r|Yk%0_pw88S2vv+U{acRPD<$UhbG{SU}o2#-| zwe%v!bgj_cYA4j_nE$?fVg&N%-csx_yL(l>-%AyRwnzUC2MElNS!`gB{6jJmhVK`q z??aC6D(e5w8$Lb^yE>@;T1)Sr3jb*~zSHHv2mpNP554lg|M>74Xwl_oBMkoz);~?f zzs-Bgp5@=h|6SI{hrlnm|4V}8-*)}ozlA)lf`f%tIz9iE&;5;jRC;Vea@;m)q>c)7 zA0DJ-^(M{xr&ZD_KGYF-_TleF{duo4~s{i|PUQ@2bp3huD8k9^+$PlcyIh4Y;!EdM7Q;qPY-@zDX0 z;T%!?%W1AXt`9w~)Y<{Ss{BuzM}V&Q`xTKYt)Vi~o??G~mPu^w5lZ)S7|MuDFbqPT z^pMfW`B=jD`cWi3f3G57rYHasx>M!tZG3~x9?nP8LOvMjkvo28qYTNax|u*bDq4QDm2 zd)epA=~5e8^~M&QA8i(FFJ%1vYEeEAdEvggz=|)!g(9fVyP|CQ)RXVxm9qPo>CHz& zF?ohN=QZ006E5ug4}0wC$;|eWvGg`xrc{8{{Cg0>vhgtfnE4f8bIO^qSB_;@}*pE-+0(`Dc_8tIVaE)+$RD9Y(ZsZ{PxBOqEd>)36Aw?%1MqKu-)zG!(ey+U*YJg4?(X~t;@b;K z`zpGb1jGSamH@d2o`P$7DE3sD+fuNwpx^|A41=q8dMzfpqMP~fNe&s9En5(oD~CB8 z!K?eKje7#^>*;jLW;95oT#t0Mi`g8Z?(^QSxXHif9jIlM1mEW1m@6q6?2T!#L1=Qp zS`POj;*M+5@AwbSIQBevL@qMleQ4_1i~F`T>Nj$D;ayL5opPJEL$IxM6;h5yoCHSnsL zBbbB)1-m6sLdT}lCWq;ZR9LaWxfv!(>o#n#5~BX7i=s2qX3DE`#Q_&gCB)?yi0FNm z?is`7y{>fPIeWS_55n8ehGIjO4o>^!HcKk3I?F7YO#+_kG#>KfrV-e2B6qz7pcXp* zExS4JI6Yva5a>W20rzFEol2}1C^5v)a&e{#AuF>hD~AgpibFSqg-7S(R#t3C&QUKc zOg7pHarhD3(z)XMf7-g1xCh5>!bKr1=OB|t;sOu+$$4*{t7(e&uJU}*E{WR?Z|anz6xF&J;GSef0d~FP<2;hF5V%cuX%5tq6j?z{OMcm|Q!L&dj^7B-l+}LI#Mg z^~lTRXJ>yaLd5Z?1vFbkQ_le>Jyo~mjQA#Y7#@Rt^}~<+@G4<#-I_|0e*9bF91r;J z8?o2gR0`k*GU9viFXETf-UW?coI|70^`5KPWwB7%)1^;()^EP0i1box2bMV(O*BHA z1{dW|5N)v$IxPjlK`X0VM^17TH@eS{Vm2P&s)SNp(0dU&kY;pfNL}c$VPyE)+|1V2 z7O>N?A_yb9oY&fWmNv->)UV>Rv(y09y@5!skf-CJtMZ6Aw7~+n!$kV>9uO9flJjjX zEc2NXk-}1@*#gs|G4_T zG3!m1knn@omy+bXH-M?kfuOtBHS_K7A`@cGD&2n{O*QX}^sM%=3_)L5-JG4>JBZV( ztfG0h^z%7~<1u=orFK;Zx|1RalIXj#Me90NarZ4{_nG)tlE|}!aIMVXdcJ227szN7 zuqX3tzHfov%5Eep6$aGXk?9cJ&>-=mn}OfQ*jLK*ZE@Rx;K=A`&Z=E)d6|QjaNx#E zVlWV*Cs!7Uh|#Z=OCaLG!vyil+}Y%Ir@26`)KoNdLocKIMyDRq>3qXzk)XzKT=973 z0D;3GVik{mm62CKdoaD}7%&W4U&23)ltNfCG$%o$3X88iNxf zM@Z;#BO)H&J+bQ=>1$bc%6Beg+egQ#g)yL@KJDKy!}gg`!#lfH!5or1J&4+-1Kafd zKRf7)+k}`$qt(i)k?*3STpPEgL>RwxNeoZ=b|ryivd~&4EYcU=2slC=x%KN`K3#n) zz!MXnA0BgL&?w!{?sI=AmB%6DDf`A*|9BqP&QniJSe%rnvtDq$QX^ccjO2s-(fApc zsc!znQk(Nc#;yJLLI)o2oDBLkn!uGx0nfpuF{IR&$FH1$e3Z>HGm@!?W(UGuuS0cY zml;amzL262-~VOI!8~muUNBt{OplGfoKU6P-e6tUQPyT`6s?AD(57cp4c$Xx!fUN> zGd0p!)Ra>k)`xTX5(+YYy0)C!_@4FIu%}3PK8-sC34aTCC2a}Qr^OEjQY1BBErbC0 zY+{V$n7F$j8#2`T!1YwXl#2smnLn#UkY=UTSt6WKkktry#xqDK;%H^)l> zhH=dP+;5?&UYh+&&g2zLy;!qG=F6Kiu<(^T?KI1VaMC01feaT#o zqH4Ee7TeS;RY*-roKG?9P0O%^Q{!^WyI)ftEe5PUqI4lcwH=McQ`cyGh8(|*Zj_)q zT+t~b3D$!D&SyHN_42xs5mMub;fbGC!JM5(%89{=_E+riQd?@m@Y(GUnkfdM|&Y$l0 z^(a&~CRx_L?5Ln!Ld=N?J)BBTj~y#CKApT6E@d<_^M*4y5`O3!d*>!YL|7DX^vq}K z>k^!ZSQ-66u9!eW0qzCJAS*smuN28_=Da+3O?pa7T{=0pC%_N37?q*0-t;o{t8(Dk zr{cNK;;Q-VaiNb+kZM`lD}K!ow%c?}!h{Es>IjY0~6jgI|tRqP&prK$d0_Hsilyy|?T-{d?uwDHX818w*t5W3f2&Dn`~ zP5&W}_13kz^g*zWChd0E&6iUookk-~clkJPf@s9mfhG-Jh&kL(<%QeZ>~M&Dr;#Du zK(pY%B=(z_fKr{QZu-RW$R)jZ!5L3Yp3lS6w}r&)931`K_B0z_^k%*l@01J7C~{wX z85LQ-4>kRw&{YGxB!Taxw*L5>PHoBEK;R;1Bo%Aat=zoyM8M)ro%4y&kmX{gDNieY zuU*l(qQ*Vw2>c3*?8OJ<_xUq3m~CpV9H__!T=f`<2)kxp`2x5m_Et}Hsp~CQ#$IU% zG3=u;jXKh>8Z;RLZOh`vk|v$Mo>~p&twc=MmTbkjW9}&B?=H2{EY8E$`}BQf|bAWDb0-uhR81z0XJ#Ufs0mF0~)yN**ng z*}8Lo9-nCh8-C-+UHmNhynNcYUhDIWlW z$3;tqdyZ+&p;O2UN*dB1w>;jqBxK3p?#5RPOg)D)<Io6DyYpviVQ!5dN&o zj0*zd+g|WlHFsj`n^I9xDbKJOY1x2^I*Yp0hSv4A+)pBH#vU0q zh~+rVZ9#GU?taB^u%r2=&Mvp7lX{)iLtm;ir{Tmzg1*}N)eXDyTt2?K?0rvq@TSDR z4H|lYXRE+E`lh4HPaqU`JHvTQU6E%mDOPb_eo12xMs-kge$ej*T>AleY)h2tz!j_{ zb5hTh;f#R9D*Sh^DxuK4{FCHJ@1%&RbZplJgx(8)5SO&Pugj}S#dq8$yb^m>O>{9? zBt&707Si0YQRul^?cR_2*Pi-g5C;kzaq2Py`{bj>rdmG^d66A1BzfAC#&Md90B%Ln zAd_%J1RCNrE}>KOjsa|Nbsp#sv+6M25^Nb4n_X^W>9qHs^#lt1F+4J8+1lWEU+5@L z+I3lQeY!8=i*_YbUzFc`*-L!8RrG2m`)=@wVPdTN`w*mqHMR*xDSG`S%mM6{7p{P1k0Z>zb0QOYNG@)d}?ybCG7z9^ROE^ zqKkr@$EMoFA2ll_YxRW!FZ7ikFX3MQToeNt>cfXlj;Hg!i^Phb-Qzi6=vv#eTRJ}k zd`hx!`#sC{V8WE~C>75=H(F_Jd;*E-*8ECLd0jV9eO`}<95SlrXrDhG-{`zEevCEwP^wS_jx!p8|DvM>j0Ga_(KQf z)6e&QI?h|;Vk!F$d2GJ+>Z@OL(4hZ6AxjA|^f&tn)a;uTbY`pnh7JVj0hAH$F^=+RfNlOH=jfPP|WF$fe0t z{{q!T!q)JKu0ujq`q!1Wd)haR58B;>Y>(RoT}W{_8ZwdpMvsJ^U?XxftS5!Q3L114 z?8Y3PP)98H0;DZG$lRsTv^UyiZyvfRUUPhA4rjNsHQXPbFGm#DVPT-`)BDXA({7vp z;5ndKcK~jbn0)Uk_k==>dAbAQlJ4&^$6l-=v?lY%;$4V~7!{I7Fi$W|O=hYI&k4%6 z!0~qZz<=Ms7f?L8&rd zKT)m~SwQ5?xx}&k>z@CDmHywq!Zimgc#-nzZ{kF7%0D~&N0#3}L_Sid%Jp3+( zl;DB@J$>{z zCji4K#I+USf8I@y7Cf~llnm#8UI1{=zhCFMJQzR!(RYBWPvHLcRqXD+fW3cyjDQn# z^>_w;DfE9{@Cdt~bI-&3pU*qy1-{^fKvDchBR=(A8-Qlv zUx;4(ujj=BU+jz`EB+fd|L0nG06vP-YW`oB{I?;=^M76P|M6LJ2Vb1>uK=;l=>Z}f zjcTpr6~*YSa_w6%-+2DO2jZ|<{{HpGXe%>9T-e#T66lAXk|?M+HUA_lP=#9sHLT5f z31!$|2u6Hi&Ruky%K>4<@{ivqVdN{V4Ov)^X0S3HQiM-D|2wp7GeDc$;^m9C5L|GK z5kU%V$)A?g&}Y!nE#b|O%U!hg1oW3AXo=4TWF2b3l=T$}z5OSSxrnspYq)Y_P7SDwVAcqC*(W zeg}t3Lfn^8Ka-$(!Vv}##F9_a8UGGE1Zgk>;R=^q+Pw4WK(_S^8vcW3%kKd@Wl!Za zWw>+CXMCwn|Gqgqp=JhgnRbhaZPiju`j>a4VJIA^N;^_s75cCGYH_#= ztVw4Z)T1{jYD(_rUxvhgWoJcgL6RZQ7R6&%+StOg(!I0GThGo9LY$(J{I0#xg|`@c z+74(dcuLH|dWr0Z6S-a6PNFiNgU*^rLBYyCqHshhApG477(U5bK#BrumN9Q^jET=G#NWrI-;bI0z-0+&P9&cKs;r${(@Xl#Or2S{D8TrOD zikdaRA0H<9P}#WJywTQL!VxSUBx`kF>4}l1scP$}(LLUOdNO*2?{I#vxiSswE7fil z5{J!!_StRm4t8&=p;ety7QiV`tdutP^A{AJr9$PjNM$C9UJ9K-n{j2e?5ohn`owUWg!)whSJ|H!v zujso*{m*E4ur;w=ltjz6CoxUzdFq7}d`Yv*(7HUSVw1d-sAbKU%3CmE%;dr6jQ))ik^=Z&p0w6`4C3PKzrRYMAGk zwwCeXXeS@Re1K}b7yUFcItmnp5y!_Nh4R3=^fAo5~$dJ4*&QA) z(7S+y9z^hbZ<>bxXa2#c06r%u4GxMw*s|g$V~AP6|7*u_PUW%mxOAYxk&6U44e%Zz zU~34fAxe*f!Xr;EWxy@jN#95I95Td5f35Mbp%524oawkr>f5z-8Azp6hZ(ds0ApM~^}oJ%GwRE1ks#&#~(VfF<6e!}`)R zUoANqwO8XsgCj8iK6^)BHk_p7?!AX3cy=?bT1RnF7NGS6O;h!>QixXbC`(L&-iQ zXdFnKa>{i46z(0=S@VX#TW9(4;c9M9**Nz17HC^W*_qAXf{FHj{IHo#W}00Z`bhcn zJ-R_Uhj9Z~$bH9uOL|uzH4=7VGG8s$l<^W}GrUWxKu4!+hAq5J+7Ds2?|oB3J8D@` zQQKq`ke(1pHJLASk!e{8;rn5r=VyqeD(bGQ1%~@I5Y-iD6rV}LaQr~h6(96MY|IJl ziQ^*Wn2wQ~ooIHJ*fUO0#+p+tlMwkz6HI!k#>&O=p@Cw}BPz&-%FOCBpX#OVO)%~c zlOJZi#q~-3c9ciO0Z~U+kPuAQf#dQwY|Z;78sfTq8~#ZY%<^Wxeu&JeV@ZG3*$vHu z^fWi^6hWekA1OZ+*+le*ErmVzU`-xGb8j*v zLx#8W6HU?;(vrS~zkk3oUMgzGUf>zki9gwu0k1pCOKa62+8FtXAbCaqG@lO??~IHp z&aLy`toeIW`~Xfn!168@I^aDGF!6~ITUTO}$rfb;s*ZuI@bK`NqJr44Z6mcdX@(xv zG))c@#U?|TQqSD*^SU}UN1mH7D2E;W`CD6^2I$UsZj*g@7jEFXjGn^62FYY%VewTb zPV#~2iiEhj>KQsLwGnP?1r0XXJMXcm7ulqsp*<3QTN)PKg-`_S2P-gF6&9NW4A@zV zlhl5$cRq)Wpx)QOz){Jhna6;BBC#6lxmFXE{Jz}DJ(Utb=79KZgwQNwiZIWg8ix0r zY$P%to-2LScc}1mYozjso}L!K$6UvX8TbCI#~K47bDfLt1_~=BwDN$EWxp(;7&?|; z6khe|0NO~83dc3GwN4>SRAn65M==|qT_a02%!)s?iNVD!k}o`@ukmI!vOF&~%wL+*HA~M zsjLI=%IPZ$&b^d~ zJT%dmKaYd@6wMGh)gaP?ySmC`bp;n~VGm38fSbpFslm3_Z0QG9mJiSfg+Z*|{Nk{nEbsFxwgGRd13Es+y{ZnUR)6NlWsDhdJRhT^1tOW&gADu26aJR6D-KCMC!~!CEiKL5*jl^l$`_zmQuW?cmxIkbGPHt!UJy2LW9+tlv>V2>uwP-IqXiE;|Dq znS1>JO;7@QE4Y_o_iFKgYB^qGerc6#sMm4VUlYUh)O|ni`5nb2-d%&Z#Cu26$)pq{uimU}GRXzzO4_hGpN@F7dE`LVY z@zpYs=*CD%$~aDC#Xam!@fp`Sp3&0D%!fr-j5F7OWWwU*&3(TkVf;le;M;tk$3R6MkZaY~q{;}_7nV;uhAeIR9 zYfz{H+6ha6qBF%lx@!Yk2bbHj9Y|=%)Bdv%`yM+R-&euPUYC*d5Y-bSPWsUPU4p^o zXov8<-Ru!Tn+Ys>+OxYKZKFqjF30V5GhJ;G88%n$X3n}die^+|JQOWv$zaaon0D=e zqG^lU(QIgjE#ig5;gEgQ=lnmUYDK2`F4j94{P(-EUcMK*hsBHo!~L&c*M|)vPEpNB z);*k6w2UN$75}K?c-iawZ^+N$GgKa&ACBoc){ftYRuY*oiNXD<0y-gD5$;$K?e1VE zcrlbH=)gjvheSzOJjWQ507i+QeD7mkQ!UpRb}Cmbd0Da-R+v`_L1Y;0r_?q@I?r#=U%RENCSx7Xw}9&y%`;eQ@sx;o+w`PR`wyi{or zmzh*Y#Fj64_muk>317@YPnw$R&;PR4#rJ%!x^aKheQTucApVe|w}^U*Ux}Q=4ZT%? zHwC6auS&d%Hn!+%1Iy3i1YNUA1!Q1Dzb)O8jYp+#>WlYvsxY{8?fBb_tD~HBi1_*J z)BNR2Ir=(m0PNUT^wat$>(^c~Q<~!>c16@`B zWZ6Wb5p7Ukm8tgmW3){`y+pl&n)&VMRLSe8@W}lb6T~3kueVW{lv`*}TKDZ%6JAh< z6Lr|XaL(xVv)1>Prd(#(`}lcSu_{Ay8wq0>(c7Qz)qlB61MM^h1dYIU^}KG5hMV<5 zp?hpm?V|?`UP{iu;)RFW#liRs-IxtshlFtI z^Hjyw4qsR(xWDmYN}}M*muPPkNAZdH1XE%&)!JLwXisM6i^adf+1@@0KF_XRZ#l|a zK3fnNtxg1Fa!wI#3|ra-$>&9)3eJ};s+N5l-j}TxRr#fbVTiSZIp?~f zXiCSZ)Y^b?EP7w+4@1gLpmo@bL*9&ugLLv+OgGvAr*Ti~N!@+|I%$N8DN{;S7~Gx3px~#~@<&!9R-p1Z1aqtr=1A(q8YPy;IhM_z#RMOtu}BtKHWlHjRADaEau| zn)Ek;!!EPdWKwpg5lK;ROgTNY;@rI6Zg}Res=g_UYxs|EvMO~mWuvVN7kfVOZ%9{T z;Zl7eR?3SD+_ z7f6tocGgFc^Q5g#c!&}-|Js;qd+Y!qvAXk9+y~NGQ%XROfyy&7VJgI z%&y1wD3)-DgnS#qi%^hSzs-qRXgszdO?hQq_visvD%DctG37Jmx03=G0bgCJJKFscYyofVDLO0W8NF)=6m|iHSzDk`?8f_2BK2wNfu`( zkmDHHVTb2(Y{qKy_8Z^g=lvcj)!LRT`i$n7Z!Mc-(y!=|<6}}drY`q^k%19cIa^{( zdIvN`lIot6w`50-O^lAZUAeH7yhy7!v7F*qR|g+cvd!1$`9&yIaU_INMtmEW;hu^7 z&-w#O3lN}1vW#0RTx?U@jnsTm!@GCsiP27?^YTYP{isrXl@`=|r4w5*U*@J%_fpEy zrqjj4B{b!S<8j-*%6wGy&Ka?!wDQIV`AYffx=aCP`DYXY68iOfd58JoIZ`pEh*aju zV&0pqABx~g$|~42+gsYQtRL(1%c>S-YzbHs5BA8{;6nb;oq8n|wP;9dtPIa7zO{GL z7g6!ux=v%O8KUkx(G|mtb?M5*F@?pE;CxLU;Xy5MkW_w0AAxUx{JCP+5XoDfw*f}Y z>Rg}cU46Rs%t*L@5tB;#(%#6xE9==b&Y!crgdnZ(T3^|N2kXrUxfsH{PnRBFP=rqy zPk43MyKvrWFHQ6Zq;YH#cA#m0UBr-FnZ!>KW$-h-Nh$#E{yc}fDklscmf>*~|ACOc zMSu<971@or(F+;#a;!etV;I^6PJ9JH@rH=oVcuf+dc1AY9LWo^Swk)xQ3@ti;mu$n z;Co)#ALh8c=?iZNuU!mEf}d9f-f7S? zLzXqV-#?*|Prqph(lfuXp zxSRA)i5}E;m*nOszr=y)ED5m=#e&YGv@TvNnOr>A#a#?-G+zVup3#=QP-jA>bCyUx zOOf%r(ibV%7fc$m#926u(~Rfuekva1`7q8`o9Kn0`kTz(^m5kIjsD!QUtf@d=Y4zf zZbtI;6vn&&(2tEL>QGS^31&y;%*@fdBuK8PooHzX@@n+^ys=(AMC_hIQb2z z9|gUwF`{q}S`RjIUhOUH{+9L7E!Cp3-D(jxZNu1`X2vqJ8oeIC2afBR@OjBS7$okl<2U<=`MuB1Xdwv@*%b zHjcMX0u$E2q$9zm1~SOa+Bph4DL@!eiT#XjNt@S*dB zE4p$J_%oEEVJNFnEgG%KEYQ&Bw6Vf|0KnI?}Ql z%<6YYoYwwy1^{CSF{k)szbBYwvFJ`G=hg^!ZL&bYPXCenew>@kjO>0kfYa&2UgUa( z)0q)k**nj05=KYqoHXOWxYVY+FIGMmq(rjMjUDu(rajfA+m&zotfSKx(lWAbO7Prf z6b-h!8pC{W>APQxDF}785`VpTBd1J_bE>YwBa>vR49|%O$2oX>UR}>)r6N$=LNRlD z>>CvSs!-b`&RkHRz?#qIg$dHc4dF<1&t^wyHbM*%{QG9(2Fs-k5Mu=`Bh`yJNmZzc@IFIQ0v{}@bnFmEw+zs#0aL-oU*uyqb`nl=cHxY zE+W6;qpIYfm8-!jKS$u*I+>&YLBb1LI#J>|Gv}eSR>0J^63RrWTq($RB1dUWMHjU9 z`aTx7994!Vz{7C02IJ>NEWyMC2J!eoczVNG4Py!e3G21enMaE1A``>q=|@j6;_S6t zkdv8WRmL9g0n$JI@&+GLTDH;pS%21fGPC$7PWNT?S8+FiKQ*+^)q)Z96&9%@+y;*0 z(x9(n->6}unB>_eJR9PP$*|>&L!kvj%NZ5q)v2LTzxksKf$SD#WKKTj0AfB8jG%dGhm0Sv?&#Wf&vM>CwL{hzX3e+cEf?sf{Z;Td^0m!pL}I;?oVE%T08uON{ed zqOISd+^>f0aW93=N;RlC;l`a;$D48(*}bB|WOr@#7RgqHTxQqI&CLg6y6yuZuv{we zaM^mm&DM!;dB1^%M2+`7kh1@1Zpcwl=;w(tqQ-1iqw1Pv{1x0LFhPT!A>;g+gd#`) zWeDFnj;4r+G;Wxn(Ru%H^!PAr;+sK__>NNYvly@7o*;$KJHybv++%-qigsSBz8Isi zw9*85*|8o)B%Vn^xeVdVZWFYL{Ua=;3QMA<%G=-<3%jLb6w=}XO)SK58j!J=b^D2I zrn9M64Dx}p4M+l185Tze;gSF41@L{QUQ8;0OlJ5!(V48e&cJ26lt>?jn%p#LF19?+ zEwV9H-c~UH2w748Cm5s!#9fc4Hu&!SX+2;M^}3H z*OkQ@V|~b#b_+K{50I(t6rHW~`Q?Q&Q#Vb~I5*k6$H$~|zs@G&>lV%yTYZ;l9nCEW zhF;yCTC>IJx912iy&YTz2a4q|)HPaGs%%PT3x-{<_b#v;1&bqh*X{xRhr$s?}#8UZ_*rB!x(s zN7%3wK2b6yU9Ql9I;L07Kv{uo2aiy2b1b3#s=P|Z2wj{-0_t@Rn~kshBtik931zV@ zxit42K|WbwYY9YcSN3vS$TzC0eAl0%c*Me*bF-9b?~R{qfp>1Qh9Z2C{Dg77Kt~|c zmeTx3B?eG{Za+^bx5P%(vX>Zmrt8dP&Ex>^Ve(-z%|oN8FS}|s8Awg6GPnw)j7LVy zPRgJa@q0i`?wQjNN0BbQhf6OAB>GZ>Wc{lDDE?(b74xxg)&*j9Wo<}r^t}g7v!qMM zxhI3b#%uC@+mRa|TkGBq7m*gx zvMHlE%1mY}Of{;`N*W#spM6ymv4HP5d~F`havgZx+CgDgyO30|&X+0xlm7U9TMGTj zpfORLXYV;T$hGi43XU1DCN<2W!V%}evjrK6E-9BArl)@qnubEj>HOGXzZoJ@cI*BT|s9n8s8L(OWdb5IhFmo4RXzFmOW~RPt%#Z=!IN33=1wt zE)-Z7Sn~V0(%UK;i}I7!D>W6^i&I<@EgEB~3WjD=vC@wZ(U(X}2=)!5lh3*8<~O)o z?3SSf)*uQUUPlgBhkq9Sl;2t2b5}LlCdfdyOf`wgVYfVvxy_98k*Kbm61rppKVPs} z4)#UWC0Mb9Mir^hAC5)|SD);v2t)XC!d_&UIg^yFVT1!th_dg3bfRF2Ui1l53aJXo z%)V*Nm+|eQ67h}YBm6q}R{mJVmn55fAQ#$yXm(DNm$x5|ZIB{b4vVXZH4`gbFB&gG z4v;QiHxNIUxCqqn@~;|+#QOjj+#?vD&&MaVed7Hg*YR^&4r+LqEll}!!bvOEVX*-y z$y!kfA5C6%g-{OphNU=968GcCk?VU2$o33`SYcU|=vFk2yqPojyfY**`>Ct8%SwSH zX12OTtz>{ua(toecM#z+24^hT?h0tTow+vEH|+M^fn8A4{Vy3`J^~TsCPo>N7Lq}8 z;d|GyH2nBYHMxY8_MOk8hZc8R>VWABb$>7)n(7@S{_}%01TQ?K{Rato!LQ=oFvw&~ zFJTKEnyEadisx_*yEPlLk_~S<6e+z5xNF={JG-uTV*@)H+YaxB_Zlq<@26=}&pPln z#xt_D(=blc7352mfQwCG-(+!Ls&Wo=P~hd3tojqU#ALX)qW!>S7$poS8#ySU>g8HI zv(ADy6GR6hBn`!YvQvm?-J|_HQ=Z zS5!@MS)#X=|Ik1F&Ugdf39z~a?#iVyKc3RMPi{b})<`|m6B?ZZ7sRthFkVd1=;r=fw8Bri0fV}mI_Wv(F%2#00+@xV$6e22U!W8zJf0Jm+ zGjkemM^i_9EJ)bIOA0qpny#}_c^bo5Vg9%B)O3}KMxk9ARp22D`}u^fp`FuK`B0yt z9q>Y%zhHg+H$g`5Jv76MI@MJH&#nzYlHy)NgJOneQM~vcP8UIx*ck4e;|0DINFMCr4-zg33pOtCU>cjwfnw88=6%&8d;Qm}r3FS;9y*b3TY*8%367EJ zI)J+aGI7tupULKX|DEYPCxZ&Y4Ptk$l)x5J(zQIW8gEDEqRuJgMjf}9P?wj%A~NZQ zZ{T~bLbE=-pj{mJn=RkD68BSL6uQHr1KSkhjm>VYOw!h~clsD$&+22tdiWRbfd}g? z2AgGO;e4ZmRb~)%@GsjfWxJa&3!IqoB;&;mj-fU|FYQ`Auhw|>2Ir2`#mRHH$m?L0dq($82Qf2y6rvB-QA8#a| z9Y73Wuc*v_6;xUw^pe%Rn^ac~_V4tIqjOX`wB{Gy;(|P6bX5(Vbd>p^aCo*ZXm8H- z*k-mNKo4UARoundM;bO30leN4+S?kF+4RIWANjpEtPniVP>&Y@1A&Q&i7RDz?;_rg z7-YK#8YpGvnsY`Z<3YrK8f~@7JVHXfV8>%*Yz(~CuyzVzR;9ek?U*W$3e7_rcWZI# ztmgOsoONw>mg2-{xBmE@W$&mIEJz`a$F6I>2ufL9v9_0B@b|^2h~IaQlO*G11X3GT z9&fv57{wdqV;5-0h&~paZiJJ?4CJzsk4^aEISR)4KLZ>OZ$5EPIBV-*p%Ev*u2>!|x<#rWzJ(g&<&tWXA)qhi!j3VI59mylZuPX5l-&#t@r_;O7ZeMt(9jNu!4R2j z1ZNgIAj3cl@2LqG8yl;rBrIKce0|oCH-7Rr>m0aS?N0v|{cfz;ed`B|*WC-?F)k?H zXSu>Z*4Gz|_tde%(gtW|Nu_8Z{MGh6JS}L%E}d-;&ff4(f6#wZs-Xd4=DTMvIHGS; ze%IFLt==W@7tzN8)k6}ha5;>95|#e$m$mf!?-Vff;qFRj4y)}yNz#s}aWRFUbH?KV zy{f3;<=+f)0tWbPg^aHyH@IX-iE${$Simis6pFKy1Js3gFxeV8v}&3Kyd^uj*MGBp zKEiL;WW3IBt%~~Pm&-c#HV_OEdN23p@^#WN76!-pHuVqgxC;)kf6w>gPp!{*|8+4B z{gZzIz2P`;9slp#S-|!U1ka?qZTMt$`!9aN)5q{9>bU4!<^OVKAOrF&aZE?7e@5uP zc)cDvAbfm^U^e|_W5U0{p*Ja;=pODH&aaIUw`?B1r|yV0%l}dj@I8S zs((M1APEQr|MwdIt-b!QH~w=x^ZehS`v2b)Z%rBKGI-jx@SMU|R#oMVW@l&f5QN@M z{l#Q;mw~%>ZU#F2g$30djKHL)+2i;HB=Qgq7NDU3DUrv@sv@&Qd!SSDDcGxr6RQFdtN?`K#~(xFfS zaB~=sEM;)>C>V^(L}5rJ@U5x+`D{7MY)9x0Jnm}{TJd$hAb=&}CDh<8@#S7{@oz-% z|5Y&OYWe096LUO!Pp<_5b}+T(4$`2&OQ<8XH>LFQOU0*M#83BTDNMM7=%L@p;3D9; zGq43htpz%iCwlR&c;dgAX)eGPd?AN557g!WxBF`c67a0UXkjwN{<(+f6MlRy@$7-G zRTrkxMnJ=1Vq~P$e7U(fpQ(3oF%H{(d*C?yI>$$S;r59-0*Dvs zXbwkRLD(9AoCho}_|-q;Aj9v2;(`ZrRW6EJ@v@T&uE5pf`%JR_&XJfo?sggM8PL?U!15dR@;1y;d>6xyeUfPfwju4l!)<>l6(U0E%6%KdBy zAVf9|yDqvp7u%FL71@+rG#)t}T__)2DH~@5G0$0aAnhq=dLc3AYV3H=oA*caz0|?F;}caB}^8Jmulsw0c(0B;LOfWDv}nE~Yp( ztqT>)+8=+(*Wz6Q^OUX-*YgY@e0=PSqEK;$UAh^vOV_6b9Mxb9)%X~h{G+F;uH=Iv zh9Z?HAuD3$j&FyS>Z^V`kmxMjU%4J#1@7zv2Me!rqwBZaCvTsu>K#C@Ol1MP?W1Ye zgK4w6jpzF63;~Muv zhkn8QcupKjm-A%4oQi4HB;bZ|5zelR0Rxh$mYal@S?v(c3msw94 zSM9g5nNn={w@%Bq>TVXcv$`>O4aj#$=JsBjvE{cDIb1fpgcVKeH*;_8tLwtKV0Z~& zrr@F6$`_{1-7wq=`6S406G|es!V`QzpV&fMmjpK`@0v9Oqz+$YF^BWOkiE_EtQ*g% zQ0Ix0tBrp<6CuuSX!3{467Z$6uz`K^DahfkDY$MZKtwWDk)2nbKc3P&wufHa0wr;V zv<&2+dR|=r3MjOne7yzD1EWX2yKdRJZtWJ7nB`O;Y&Qe#X&MzG;tW6t*U)kiX&OsA z-=cFIRKD#mjU8lJZfub?Og@dw`F!{z#CT^5g@F43Qu3(W^(AUka$;<-oj5AI5dc7= zuCx8>*@PfXGgx_q8g~_4)2QpJGI+59%KjdwinVaBdcTFVcjm4}tC-Dx2-++?#PTqt z^J;Cc*<*2CV*#!Y2pNJUE(gXhqvCcWhpcz}B%J@q!t<|@Nh1s1x!xYII97d+b%X87 z*YJV{&CtzN137+~GbXd2a{c>LFqmQNO<)YTT`0IsD&&2qaU2cUJ+s!hV_C9a6OLF$ zf%8CtgM#_seuf1`zjrGJ^m!&1T?WNB0mWdUli2Qkz)EW*-C4>jYzE9tk|+*l^SYNZ zuOZ9YViKPKhxf}%T%scp^;6X3M3DK0_f2~8sK_Q4dM#Z|Q;pN27m7IzxR&Nqn=4B&FribUC-z3q1H61>fAcow2x`0f)$W09de*MZWb=5k-lf)WI@b znty2A;tZMk9(rzx^~`a|Dw$zRWk96i+Ff#VoL#% z*d8xAzT)FP=aY1TA}3}!s%bhfeepgKA;osi5dCQSSzH-I3l#}yU=I|{mQWoEG<#Uqp)rM;=%+9_LB8?4}1$V+9>) z8>~E3{&Wyfh+U(Tm`;J@rHqwqoL_??1!s^ev?H^IgRc0wF=XiaMaVeu*OxJ_``xZ4 zvMghU{#ijzX&gO@Ic8!mbQ@*5bapL!YDZ&sZa4iJL)bFVn`ExZ&?SzO1;0YMZ!5UR zjeX1N8iNr|%k1J%Zd{hHT=oy@pKzjzPedwPF-suf_Lhp+DW86Vh9QH+NNl$&)L>*C zaKbmkjTqC|U{PcjTt3B!rizQx))edqE&00Lb-Outb1Z!^z^q+OEhaUO)F3Z|ZfBBS zt186c<4!Os7h1GdpSu8chds0wu7~jN+VC;R^TR(qV#Qv2;9q-OfsS-L7jQd&?{@P3 z;-Lcmc7#M9#*Nog1Cph|4%yByuC_=i48oDB=2{*7@&ON@igP#^wbf@8ZQIYvp84V# zg>usbP3$AcTBx65+(BLyz+~t_^Pi-4JF##hJr za1!!qUD2{~8(7$uVLH0q#JCA{?wk*o1_5X6T^v~r8Z((H694hKTP`r5Twte)ECu=_3DL-Ghi7G>YWEgJ+;yklNeT1o~K8)E!#{jcd( zURecm#7J#L8Dti8tGN*clgafpPtOlT5GJ-kc3J zp&T)B{>~()!#c-F7`8G(x--^U$usQPhFZtn@yDK!JHe>6{Je>Q{yQ7}s_!G~MwF)o zFczG~ddFCBl3K`DF`#P)|1h51I<;Im(IxUx{34a`PiJn`QuT?}JN8~A==UBCWm_<) zR=VD{lwNtGS?DWQp-$96tczfFB-f-jr(V1=c*mh}@PTakZj%K&p0Ko=yvr!2q`^ED zf>8_V8(7VC@YddtUym#)!$?&()-!A?>vS9}%v_xKxOP!s%5y8UOFXPg(Ek|D%*}*@A@4elxS(aU>WL##IqNA`PD{<o%Qr;!%GNcV!^|sn6Tf$&-xOe_8PEj~rS`r`;3suFXnXeHuh8iEf|;MTmhsF; zMLf>)&#f9Xsy>B3MO_m%1C7y=jH5%}xQEr(8vzVhP0YtG!97q%sO@cxIb`;#c|f1i5X#Nh!xArOc9IyO~dmgQsIw?%d+VigqOF< zu#1aGaigvAehkykV`RhgStPE3LC2AZ2q-xGEgc8RsH)dbp5qVT&7}_sWlM|@fid68AB677Njs^-v6mO%EeFt#jyEV+_1A= z10yqS6q8U~eUHmK*L?9C>EPms-?iNfY5@>IzV&D6DU15}-u(ohe`UUy{`!o;sBGX$ zH(uNDyAFPZefzpViEF`kU558B+|GAjQ4BXLkonO?aQ_tBWq#M{XGcOj;>~cb8Y^UY z7ORIh-?P2gy>w07$chI|z@w-C1CjJ%mNuk~m`pv=?SIi0sws&9NDMZ{k z&@fyeqh94)7&sUCj5%{mK`f(u?^ZraOOX8G$|xOP7{Q+X9J^xrs#w~{WKiYd@aVxT zgK!G>*cKDxPsXT31_oNf_+P8s(9qto^ zlPU0B>0d^x&dc!XI4#MvT((i`O*k^Eeh;utIZ!9a^&5ZCIPzDbu9733efe%Qx&T%Y zQK@py5Ya+=^Vht%3QvY|Epjd&P5&&m*zbe4vNO`Hg$w(MnTBprjWs(z@ELq*>6ZN` zM^lPIu4L_qu3s|OaAo<@Yskc)8*k*3Hx{W*E*s0Q8kaOhlgYiC()0}ra16+%`aK>e z0Obs5;Shv5u_klX^7$Pc=(90ud6~DFb-!+>C~>!}OUa6j7+Uj}2-HQfGcjbBYO+PO z<(4|a8+>Th&*G=Ab!OC%;^2|XC-MTh5b9naLggkKwnlXU3CEc^=H;#BD^pMu3lEc$ zXF2*~FQWRfCuB`qr%&?N#AvOCx2$aql>sq25oCS z_-~BQ#OUEi>VICt{0{g0MQ;6XX`)uo3F8nh1;mD*Lhc8ZqTRCo%?-p>%Ae27nKDDH zqc#;(l6KVZ{kNdm<-bmT5QUt06=WP3)g(K#^8Ju??3XO5OD|5p4!GodI=|U$up3jRrziiGer4P*;eqRQNMR3Iu*8&TwU2*^DOM?xG_J;glu7VC zH03^BxUL&SX}=s_V%~X6tsH0vA4;x;H?JF+KiRjZAYxjD!EALlmSkB=`g4{#2npl; zN9#Ndh5IO4t=O~~>D`<}%RZpZGg{O$e^X=IFLpAD5qXUKc#P~G+hjhG?Nf~5QKgJz zoJadY<*i+s6EK?NoTS$~ui)!FKn}LaFPi>21FHQPiC<7-=B=s>ZOtyfLr%)EW=luj zjreOo!~ORz5SasuZSXI#XDl+G+IkVCkgH%U4#LME8QO%=EBi(4@9I|J4TI@?m#Uxp zPZkwEuvf$Q;Znzq&%n@568gXjZ6%oK!Idx5s6_v?IPa9(p)z*}5ZNf6eXnvcI;j^- z9bm0C85A9G5U{j2UZJ?F9&!*lPwTUoCqb!-&fr^T%WI@S(VRq`@7`1xnwT$qUy(pQ z_-aiFx?el4z(@tn$j9MU=Q2repNg2Q*{CPegkhdb>F3S+5#HqenSzWRee5+V^8N}s zv=?K2hn>c+>%ZIIiB}<179>W9L<~DM^z4_JK{tCc7q#w#>JU0vL`Oj(u^9#ViDa*Q z-&DS6aVDczM(KaK?e)Pu$m|Z3aH{1jel^ZmgNUAsH2KgnDR5F27w0aw0Cs9_QmHx}l5;444 z?%{jQmN`7;Yb;K`K{7Vhw^7;4qhl!H{>?0O-%NDJl?cWe?(rt}myexIcMq>a$I0~~ z2MZVV{7Z1toG0$n9jm6?tl@nu@}(kJx5{ux{at0IWrw zd~mixB)aweu5*n;pt#U$uBCm=w}nRGR2^OlMnyF+L-DU*Xb;2qKMw$ipeEjLZha)3 zw~um1k~jeGt>HjRRb6pxp1eo$vaN%8QIKlG&!BKJT|q_Z_JK7nf90?F5<3)Ux;M0S z3Z>hYBiqYX{0idzII&8O#?N{X#<&zI%x~6O-^cfvt|?Kj6H7*!9~j?yI9QlgXb7t~ z#?1H&sF90Hmu{a0{`1tSKY0>Lw=ZQ2+F$Nqr6q2fqNaL%uTFp z`5hQ#-mT>yP@ozM@$9g$p$Tb?q1nfE7s=1^%nTx`nog`t>auuk_>CtMzhL3Y(1*w6 zI_InDn&tlt#!TTAQZATjDZ^K_I^i zIPhBdVWSwy z2lNnQUQjaefOCEsd)-xEp^o)RyXU*TONKA!o+>U7PSv2Dx3*HKt=z5-&mhsXKVy71 zJ|MY^TAVxU~_;I5%dR}3~+ifDQI}Xuz7!ZP=Vph*N50si*H-QetVEDoWv~#QN zbWF`|wk$~w_*dp=s%o2DR5FcukQR{_#mvhVdncZp6+y^AbeL!@#^OEtHs4Dxp6ts+o`FCe%Bjzq6d z6v>L02BPdZ*M{R=J?}5(YRs`q!UpuMv?h*e#tslFvPrafx7_Jh-N_E`&RY_DS#Xxc zHPM}FXqzbo2tu1Jh*4Hm3G(^q+<~>)1-OVHMbB<;Vne^+f$w;xK~x|`;nvFjO&qp_ zqw6uB#~4CK{w6!;VT0W7r!I#6a3nRGC{5jksT(i!rCyFce(rX#aG=~0t4>F_!q7wn zYg0>M_x*~yyCgZnjYp@xE^dgwN*uI2@R|ZlmXRACojCI|D)=pvPJhD`=Rb~k<7|c{@ zxx4d?7;TybSOYVUWk_r-h*#?l68bSG1ZpU8oC!1ImimtZ0*@Y0iL{3m7C?r(4SBgR zuQN_(((=7evR1o~Zh)oG#xn?ALF~BMh+G-xa{it5vYT~Izyct&1 z;zudVxN@d3W>;&3n}g5E@}&+#DzDgH1E9(F;;Ch8cZwI3*hBmFrH)s8+oz&43ix^M zx1zZU5n-4gg@MzP-07x*C-XA~16Myo1?2oU99x(Z^zurLKHr+$kY6$%b*hwt;>*VsZZO`se8#?UbPTb|`s7)!Mw+dMWNrpVZ9u?K&9gr& z^|{TQ&{H&DiAFqU!oK8s1{$4%>L|W)3TtK9%Z~3?z16q6z+|TCXWZtYcO8zJ9ERYHNiKbP!JEzq*mDu-r{HE%Cq4j>f8t;;LquYn*?KN(e{Ln^*L}&~! zDUZ9~AN2-9oB-yVL)?Nbw3YsvkFuvF}DWfN504w-Hu8+}XZ zMnzGGO#Na*k!~Xf+S~n#Z=~It87F zG~PP5yKYQEWP-6v;#c}RFQ{wnp=lZR-TbXhuIJ&_iq5q$MWnfgf5pZb$lrN=(#Qw2 zdFOL>COZf2j+`?#cW-}ZX!E1sQ)gOPSFh!O0%C25(?r};l-=Zg{WCQDXh#N8OGjVr zsZ)QV=!obtN0d&rz56jfFbd1iGEg+p4L956>}%KL<|64c#d)}Mt!8-PU?Ru2{sX@I z%a+c{1shMRlLKk58)^@{OYf>9J?nt+iP!6AYJM0zgDdvzkl4C!T#2^YgW~6~vhfsr z++n2H<2IHOmIpCV@6Y@^4K`fKSpeC=>#2l4Fv}*mw?)#X{8afIDcSPJR zfd!pTOAzQQEoDV8Jac__wC!wNR2nv{Fh9Rx1D!1huV_5r&{=k>OLb(t1(W^LD=E-O zKo2!BUW1Z#SCkjkcyw%H>Fo~8ci7nU499sn&0eDi<6q1-G@8Yau%TCULge{1UUgxZ zF}X+qr!t*5D@=c}g$(jc?HQ#~E?mgXvKCd3V>x%vscO`_P*=Afio=9kW~+ucls}V3 zn+p%BYsHvozu)Y!?@T4sHQPnmQ1&I5!XgQAL0p{p5J}soQ<1HazCAX_6pZ}()f`mq z-CmZmo{HG%8@?}@{}B7=uw7(wIjJmOsxg=^_8mt%r@T~v#hEQc-6&@`cD7{FH8^Sz1(`58b<~4(#uSho){!39Zb2iO7F+p|5s+o!_6v;`+6E zKrio<%1eSp-@iFhWOy&}O~EaR2SZXfttL-uU?VDC#!J(1aQeMIKHlVj*w}d!r`_H& z;Em~2uabYzd?6Co;xEYspAS45MR}AeshTwMI)4DiZxwJ=UKbrlcX!-?xrm*L`TG?l<`xb9c%d_XAP&DMc6p#rSmdbDxWyy$Avu7 zODpS_&Z(&xnr?^rr1eKw?U}Fn8U-%rfjz8E$UkEZ^Sz_+Zw*rtKz zN@s0bE&*;ot)Rh^o=Gc>i}LhJ zj>m7@X1yZ&aaI>2lxQJdz_k2ilUC=mXR?c@JMvx+q*VoPJdHK)m8>*=JTBGF?D?X- zQ}pdPJlJ~RYAch!D>YoZ+S)YtJ#ooGzXG~_@3Fm~TOtqq_{js=d&Yc$;8@9`@Z(% z;?+EkHA#%=c-Pi>)A%){%ln#hQ5}$Fb|IB;-^=yMg_%qus+Q?=#dm5=lMAtTzb7j( zE^9Bcy}^~9)>}?4R#h7wcye-}Ct{PZyYPrb%q2-(_@x)lARJ9AYa^4 zD@e{*m^P?JSaQ?&7z{qJ%VPZLPE(pP>pr&d{yfJ0%$b9I_J!U^8`!0!1s*oEiIsdh z3FR`(KSmC#gD<*kcJZGh7chQZsA|@S5Q`!$c_I_}*Y@i=%2Lw(kwF)x`ty+(fdxc- zlFm*5UwEDuCD+IqeDi}qEBOW}Q|NZ&C83wRnpdS>^BZ|F0 zk(o`ZRZn?lS9Ft-6K>CwPAW7rYZZ zDSo9O_pB2eJpjp%#w%@MAy%l(y2gbukEOg^;h3GQW2Z%t(C>s%&>l6+ipGupDj~C= zcC!OC1Zbmv-cx|mu8MiRrGp|TptyFm=G8>@CiIT31Ibm-QlJY?Istftan^4ml0=kS zdHKkagLm zjV$0x?FzYt)ZjnGD0FDu+pT>-Wp2L+)^bh?+(C@R_+l3a$0Qi#Op1Kbk*5T<(v2tM zeB$>}H@^k6Gg*Sau1BoK+VtxSM7HuI+7(dMbPcw^GxK}W|8G0V&y>Ut5>h5pYSZL5 zzhBvEz{2LqSNF^OM=AcN9flyEax)Ab{;MAIm(KkBbAkZ4VfN09?%E%>18+?ZJ%l`G zFXH{1O8xaA<9u?ScPSQ8|Ia(xT?6Ydtcv-SLjUg?$ZD2HR?>VeVbVYE2p)`d6>=H$ zUX$S$g{BStY=Qoy(thOA9kTxa{c5^*$YP7&VT~Pxn`D8=|r~#>LrvjM8 zCnIk1n0E5O**lG&E{Uv@(~dJShzg^Ef>yX;;Si{rdjmdxC1yE~SPdzYc}aDuzsuz3 zUiPpW=R?1N=`%k*sK{?(i~TWY{eJ9!4MdYRhPwj1KvZ=J%;;QrE#C~V0D_;uqWwX8 z^tS(C`E#t>Y@2+uHB;cKd%6rjS71wvd5^AC%>kD5y3IB+PGQ8cl32`l;%7JljL3hl zZW=+=pIK<~Xa{FU&f`^}nWq8x?S>_TVgdkWm_h|M0A#@_mt1T@`aGou&vaIQb8E!e z5^Fz@QP0*pulu{S0DR&C6yeNNve-aIN=!;SHf+xU2mu@Q*7yNt*%G715JAMU2auKj z$_?(L7@yHkT$LkEwT?B5IsR~UIx>OAZqi?syuVB7Sg?wOL8R#ng%e)F#j=%5Yu32G zRlwQMF`wo=4hVm?>%Wfg0)`vR_2<2BUaU6-CLQ^5(tA_~t)4LkCY-}Zf5?BE4Ebo` zhmo#S8wW@n9KpUI4V2D-9NAZ6t9LNY{sqW5=DU6qV1%k{9L=IAZ{QXQ>=7 zpgs-|2(8@Tny9Z&uxVawuCp8-AcFd5BbgMY@+A$btTB!6zry|R3m|1V3UzxNfnGbl zFjL45z@g*;wl`d>+$vQW7Os)uh&G)BEa*;F&~E@bWG;gVqJ7Q1zo4=>qbSS1bMHtU z3aA-+gIwvfwaD$ii2U+nuj@xe0Tj>>Gy^b1OzE*&EveJ$o@5}vVW0C<&Q(+Q!Kefu zk)XznC}(y4(}{nV4yKBv~w3NG@6PktqMmo7qJ{l`}%;VSL)tZI8btCA%sqTNID;JzS5AA48?)u!6oqRH>&SRZLejy_S zAA%J$yqzoOMvR#G0+@T&U<*%GxT*mnNniXe?h6#uNj05!CDXo@yk=ti0WdBvF+3et z3l+k?9);sy?11`NPTTdscA?4&%B`CO_b$1S#ZhR*S5V*_SF}j5^1j=`_FVK~*H7>L zo3&6lWbDgb(~z4UP$Tz*U?A|dwIDh;`D2?d`&y;Q6Z4NPMnbiS z?0~lw3g)nCWSKmQt1P*xcBL#be`_Oz7QdgJ{D9~qaoqt|4l852I2ojPSM#41@K`YH zTu}8Hr1la(Zw>i;ib5iGAk0<`%|Nluxurj^CjtsCTvoXm-(c@E&m@P@3H_NNYoj7F zlMM-}+b)RL&edV_Cs22|X7;x#8yz(_A6(v>GO4yPxQR65N%d%>`xtEA>GB(|fblB+!>A}h5eATtnbvetR3EKO z<^WwA{-yR|JHH4I!RT9Tu1Gw?f%39UT|z8X2{vzMx9oKB3y7V87k?iM1YK`GkdNCv z#JE?HD~Fj#&ujE-`JWB^rAQ^uXl2z>h0Iy_zH+(a#m=UA^k&eKTSX6wotGva>CXQW zx26Byh6LY5MdmcyBhTk`g-VQ;lWo3y^Q#b#K*^~;9hxzYG9im9Da&T;A+eLq3)hXVTCn1S82{P)^}02U4V7joeE4?v-i8 z#`dSuwX5#G?1g}K%UB1$Wb#Ca37`)9 z@D`OF&>^vOV=3H&3jBnbW5 zA;0b+c}$Y{CcxXgh>CpEe(OdSZarQy;RkWzyCazPRFYi6_@Gc`c2HuyJbmB6kLfEF zZao3L$%Y9u=5T5nrz~)9Wm!kHh*XUs@&dK9EWWH82RzVmy=o{M@WeF zANK7itFbj;n@}e675Hi5afMa&rz_1_FYc|s))*z;Nb(9F!Py+f?u8AgWJ`n4kG$80 zLyxR4*xOmz-9&z8^4bSLJa_hsOW_zwF2Z2Osh|!G;*7ec&<1HyE3hWJkTIEn z=DrA8zConeC(ri}9iRz5Q}V_vBiV}FM6b0xTic_2Ty9>v>%~Fp=;6(FhVqaPUiP~a zIL{I!nXz%7qd~}oz6#;)>GYHh`kp&*Z?r!T)?l$!mKbF~AbJg^49)-(R`{0pKj!a$ zdfd$h9@uz5$afEbAzB7iLTKv;1Y$gA+v3R>JSN%oriL^jL7tDZI=1lpcW_6N6Z$U! zrqAsw%v)l*hZk#qeBW){j4uY@OgUt$KPugSYf-C%ALs78ZS-~HJoD#1yEMaXosD9b z=i_^ox%Qj*wV1g7efwX(A}+89;cgP|!vS*rS1WL*k36nvxHAKL1b4P4W|R(JUe>y;0; z%yGh>DVV>m4`kO|u*ZG^*wgaS#5?WxAznjP)RG@xeP8`~aQVC10^l2~aSXI7MUh0T z+TnB@s1P%I8m{}k;v6i)oZ64r$$MmDB}uhyIOl~Q5kHgU_%4h2&nS0R9kp1kATpmD zkNYz&1c}JxCRQAztGc6=^UohC&8Yv^F(Y%&_<=w?{FyY%@f=*5?G^|O1F}bG6YVU% zGw2D=)Y}<9KrQ$w!h{*n=fUt*-U<9)@mC!Ti*oq3p0+CH>iGEmOpVg1YL6) zma05!FO38t%>ee;8!%dCBOFuh zj~gulKYRxaV9XS&wC4|gyGw8R?cU14!$B{LqEsHH=PPJ@uA{NrZ7I8~_x+%xzJhe$ zU?36pkjjTxs?qw#9+&f=lwUA>VI_LJKj}@Q=gLG;^VBPam@VMq2D^n=eN&Mf@-+6G zvG9FMy}Ps@_P4BY7N1p2Odvi!dKUlCR zeQ)M0+KrvdFOp+BzW8p~_fasEG9R}JKSGbAbT2sQUNq3cKLkt@{?xVb5o;NPV2|Pv zV$E`q`XH*!b26CPv?_AbnPm1&fpWd&eAmf8f|z$~{2JFbq(m&IqZT~d=mfePL^|G} zlP`!Y7OZd9+Vvi`yGlY1B7*ZdP2_I^uwR?i%`GGa)Hn;eIM6QAST{Yg1{ixy^}O;l zZcrT44BM3TCx2&39u!+>w0#!A6ZFDO!u^V5j(|p;S(!Uiv3R_m!cxpck5@cI__B$3 z0~Oz>ErswkhzMk6^vv&}EC9ICm59WKGb;r400f(}%0JKFsAk;1V#lV5De9A3w&KwWej#4g&Op zMPrZNZEhL}Guxy2^fN`QSnSx3=pCR80hu`$?GN|Tlp!LVOQZaMti~6r9x1_|{r-C^ zCjz1u&ezc^f@c7lK89NdYj?K;xim43DDM!9>v^6M5I zj;Ap_Wx8sdsb$eN2Ewo1s-MEsS&Jyy@_Q0RX^iCFuO5-Cgi_Cn452C4@4e`Z`opc} zoK=`5q#9j{i5@DCh?z(sU8+QQBBa!v-r;lYUP_%`2_;B;y}{JW78Bq2We7Y9EUV+b z(2e3CGgApE=SacbeB0PX^(`0vJzRm1XjS6dgbv|@9RwrsUSbEUyrbm1djiLDb_Ght zDxiHF7q2%uY4HSo6#6kL_qz}#btrGOlu59fLuWnBuDCiS=6g(I_I!jKC1V|Z)TQ}| zf^;26QdGn=G<64AwGB`mu2}JehY^{LkJI%bQ^@KPRW-g5K83kr3~%8+Pb_G zdOA@*A&IH-Nqk$9R{OQO;g|2n5yu|P=13$Ha)X7&%=$cXG4G7E1)J1@9@CoAaz&6{ zIjX)Dzw8(s?bLlq9(j=hKiA1PcDwX3^4Z8A&O*={YU`k{7_8gaVr%zoB-_4#5lyRK zrj6?Med@2C#u`O^(&I_mlcWqs6nqUuw2eRQAFhiqu2##Ca$g-nNX$@BpFkb8u~if4 z0G@AIcpszkdIm$)giAaukZ^?0vaZVXcg;5fwYm8Bo%DL4k?h9UY=HyHvTf7%u5bfh}B_MARgOrD2><%fzXg3_xP(o}-0i-_`# z9MiJLIw|t=HG{|sR@Q1ER@PqnQ-l8898ECqdGjw@j-Q~8 z87)n(byQ9cygyrak=94I-x=Q+`FSzR@Hcxb8VuR6UCsn^G+$;Ps?>{W9YdLVE2 zNp>PzwGO4q(dqq54-Ke=Nir;mf+KD8h?_S zRFRJqEUo`>yaeHK$rDPo1gUs}AeLP(2DhXq7EOafavQ`=DApt>D%@}TWYiYsfLvw& zGm->*b1mqU<8%d7<0YN`kaPJ8U2OfPybZ-F)w&Y;$HY_&YAmCL=;0Eb@T>LX4_N?B zQHxDDliERuljEhyWD_gVqZrcz7n#LUD~( zx77BYQ27AQhdNfIL>HV;BtZ0Fxs@9d$DaPHspt2|IY|Vn;Und03ajjs*A7Pv$qRaL z-+i(YKYN;j+@<7P*2+h`l3vr&TQO}~6^(uhmHrpSKsC@j8hGw+6ShCt<4~h9!&+xk zydx{YJX8wO=rU_{<>nkHlda{LAYG|Z7p2CI`RX$xi%a5MuoyUB;~zRrek&SHsg}HB zkORE4uaGJ{2TF{a#L&swP%xl8Ddfn6AS zKrsuEb-S-b$gEmzUF~xTRg#c5NVV33t`4f14Wvqe1bqo-Zpb$QO`EZXY^}fNrvLML zE=)m08WAdh3)c>gX~=AhTmJ;QFj&jWWCer5$O%f2Wu6D94rSHRgqH@D)5}uqtIlSv z%I?^ezbehYpVSL!;+7l;<(roKgWa`A89WC-q zW#dOB0`@W>(BCBw5Uw1)l^L~iSpCbT(ESL92ZGi+k6MawNs6UH^1tiHFC|SjWK7wu|sjr!Sh#h&}dq(t=J49O-OH9M) zzpgcaHZITelz&8qCTLd+QV}5aOYP0g6O_#_Q=%Xe(HkJPt)N1 z*dzhDf7Lc>sD`;zsR@MusLXmwke1_!AM=go_EWlKwjS%zO$uzRf_d1WO(zsh2!;s9 z%XHUhj(M)6u^#;w6L9{lsDLQKIpGUtg0+d=*?l6QW4`COMzImQ2S|qupAMHyI%pW; z#jjaK=NR1M_p842$4_cbJ>v^zg?)X537RzE4Mv@0tyPqH_)?7K--?_hH)aM?b4{Kk zzS6YB+5=MyvlB0Wo(xYPx zM*WWUzVEl6??3Q+JZ^lj&Ewg1&g*(z`Fvi-2u*b*a#A`{5C}xB{NTPe2t)vU#O)`> z1>R(tAL0OSIBwcXa-fnvhE?D%FRTreZPe629KdH{5CP6*5Z;edfIlD{x}RVE@s)=- z^#6R;#$o$?4j?n!4ut>nd=SV8c>nQZ4*dT3&aZd8Oq~Bn%*6e3HUTIT@6Tu4e&B3S z;Qf1kz#GZa2ZnAS5XF}tzc}%n6!gG-gOu;z)Ahz#N0Ov6D4d?3&p;}{6=K2|MmgeO z%1zMo%*;m|1%3oErP*hv0ryU@dfTru3(iYfIZ(6&i@I9)mniSQIqo9(cHzy_TMJE< zV#1KEzRfKuq1U@3iOoTb2u$7F-s8Wzh3Hi zG*i6^B*ev&Q&OfTC&eMAk*R17Ha1O)C8U*gLbb?Fk8;F+e*yS@5I*(-hgy)`bIsdq zn?CXJ@p`(t@7}$;wW8nel30XKPEH;b6}7Rk5gHmQ9_L@Dk{3mi{?_WR3lrbsF`w73 z%=9`&p->#$+{HyjMTLc{B3v7+ApI4%Wd{Yd$~AUu6cs2Xn=`2^)?4Vo-}fP#fm@D~ z7a`)b4<#UI@lD#OuC68xIqvCloOWp_dfaDR0$ zZWp@^r<4WZ;Q!alOJ4IF6?(^Hf>5})X7|zv)%ihtOYDRDjh{R`Jm{E_woCp*Eycyf zDIzYTIFSK4$R>!XRF?C;8ojFLol&7J|MZ}!zt{MIE%PZ@EYXWQ(bP=i0xT*rxY}#s zYwY6UV!7@5O^~y_{baodRPK>N^CoEgOQUZsuXX<7xa!37zcs#vD||sXt8_-~7Q+Va zrVl%4RAox(_0_DSa!G=1!WPsqvdS(lB4EV9u|7cUs^h#V)&8Wqc}kANRpGV@xWQjb zm1VgXd^nIRXR>H%X--9$@)5jMpE1VOae%i)1>!_v(Q6)e6dED}oG_&{Ra%!yXL?mw zhn}<7)3{5+?d8ni$>!(>VSnG^rR^{AcNPTc<7LTgCb-y`MxQ@7og#6xoGZ8NsNW)* zS?)9M_04UV9w?0pw#+f}dg+j`f!q7H7}-eNJN-hjc~K&2)6mr3=QSzPk5gJwhgCNu zZF;U6hc)7qrF(x6%aCGhwjr+asm-rSs;5;gcWmPNw_OvnyvYuVl6U%ANZ>Tw?O0B> zNoAMd>xM320cVx?3v0W8V2r*qrSRr4Myx$V<7|K*VN`aM9CZNOG!Z5z`+FlV7_=}x zOSX=SzoA)K<&C&gq5$&lx@_#~ls z)ko>yRwmnGhC}6@Ea^m96g2F}A6qy>{ka|i~P-i=lEH2WojOO1%VI+0}SnM;2e#kN8VV-&|{(d$K?tJWcg^a~twdgD;b z9Gs_aZk5Py>c!SN)8hni6L?Rz<2S~Cxeg#Hkl^4;+CU&uY)TI!CpeJKHy&0`3h^*^ zCw4bBPEo#ej!adwt1_P%sdS|2;rTxoe~=MGBIjg)icsG0Rk7ZL zP`J@)ym3-*`9I&gMydJIK18R|{JDPGBz~jKrx&$0-7}zZ{HjenM{ynZlpJMC9h*#x zRNTMU8WTjW-us-yC|(ijJCSYzZ@cyK00fU~x3Q0|?R2!zc&PukZoei5k*N2=BaG3T zVsNCzTLnCGFYU^8>2@Y_VPAbDyP@J=R>C5Hn?5J#IpR0~_bJo`jGjHCX7c%@b-iCP z$=}BL=}K7^l-x#79qvezjZ@}t7dKJFUBf)FX^r#Ood>s&CVG!Dl1vegy{i)e9efyS zI&~s()MLN;Z>7pMk$jY&9uIhgpmDia_UW1J8~@53#Q6Fw!cVGkU?zN_ zv-BW`@C|n1V@YK^V`I|4-8u+g@-giBBh33JAjeUufl^!QhyNXWf0*KP-Z>JR-XanR zrjBTw@fuOH{yWOQ?_U)6&e!TJXBQ9y{f((BuDcfhC!Ymh*5sq7|CW4N5zidaBukyv z`u~UvBoGKBNT=u3=fC%f_#+7CX$$?!@7Mpm(8HUup!JxWb$`jie-s1~og=@$^IvJ? z$5Ku%v49RbVjul=7?(#D#IQ}6?fqADIU}Y79nU}D`RgzaEenXk8&6B}FHHsF69=8T z-lF~Y;ou-LN)S;)FdNgq_cBNZ7r*69^ex%H56iX?5C`GS$X>blZymQVo-x7YolL!~yGdZa8LP>qC4@Vq1aPuUTChCf*&HkJM~dFCK*ZF*xR|H#C4Do%aT z-S!K+`sz~`y3tEtRe(C9!O@c=>YuWN(mr_8*|tN@c1kCDXv8?Cl_MWMz8ZNSO!g}E zxq3d7zaF;z(XLM5q)~n2QGO;%eJCxi50@+`wIfPnZB2~^|RLmza;s+)zPISPlUr$2bgo2Kf3VLEQJI0XF!!e_5FVe zq_5<`*nCLN>?pDiju4HFev-eL7?)O_FDif0vrFl|!$kea#93J78Hw%ZxPk4%Vxly5 zR`JD=0X&b7Ve)%&EM|aa_o^2QMgFP(WLNNwM#ZL<6;zNj!}O=o(gR7S9{JDKrS%4i z0@^c{1dF{)PwOgolAe6G+wyqGnG+6HcO#6NVwK_~UWfuAaQag6^0NO^tY*>l&n%QTR91AbORot|MUiI8R3JzwV(ZuAXpXP{V z7>y}0heN@x)jHCnh3VJ)*_^$Z399Cgo|x=s<`j3Q;Yly5N^;gGRsaeK|2LIoV{ z_pd8=J)~=({3-+3(20Ya{p&wAv`{9yXK&jruYO$PUo>7mG=SI?Nge#Aj$Bo5`{I8P zW|3i zWf@b9nIlT5-%0es)0|wy;gIL;DA&I2Yw8Lff_$c4DFWXG^ADatG~dwDp{Cw*7tB`V z`LwQx*dij|?{^ZgI=W;mav6%fvG95%T(ysmLztFYnw@r>2+`XPs}&^roNS}Aa`8H& zEGdbz08=8SkGgU5pH}7zsiWC~BZRBoXW7OU={8_HGdb2r9UVYQn&#ReDzB&*Vzt5O z3qvyZgeEW{8@6ZACcfjP^F4`W%3DrOZKcirFTX=N>=-nY$V5cL8OCbrV6e(;tD&OX zIbiMybgO*IQo6vwwvG}7n>>Xaixto^TzBA$eDz1KEdAlKdtEatUK=WXeL@2fvH5<} z*M!Cj*L`L*hQw`$r6s znCyq%3HLzvy>^k*G>0@inYC2k1c6f>^dugGHs)- zaAu-L65EhrXyzG-tuNH0uT~*>ryAk$lH$U`TCW1yW}%YZ#Eh;Dw(cW0!|275h{bJ+LS8Y8TS5MY*7gy}&+~>fo)bh+5|1rsLR8IoN zA_OJ4;i(n;eft`$x&gO!EEu!#JiNC};cHVEcHMjKOkC$?$2L=QVuk0+H-p>T0MB5?xWWOX((da&TxX~LF!jS>;zSk!bb2JH}j z+c<9Jx*K?9`{4c8U0N?#fd5$d`C8C`4)kqr2WH=i`x130#__mUlXax9n~)lv@)n=? z1vuz>|2HRzhtYp5cb~Pt{IlUPk?s6!>U(tbrpE?bM3aM|ag_-~{xI@>J#&*?ZbX3I z$)^w^QK^a~tqFz_Kk> za%iZi3=Z8)k?;AIr&zzIpQIt_{nV3lr4dIoBzK@NJZA5kvdT5CyZLe~VmQ7*aeFs} z{1DhY?<~yZS?q0K7-o?HJiMzL8qDzACab3>0V zR~Nh8Fg>5)&+B0%pix#;ZoRU?7U1P$qT;rSiobce=a%|i>ulJ&tP1q4S5noiI8fpj zCN0NV7a{&^GJkaRrXa{U7Fqt)&S*eF;-UZ(~SAG^<5j=v+x12yw8&c65IsSgTunp^D( zy$H$O!aVM0Q0sct3mldWSM*H|kfWV2rhA^BYYs3CUJA!Avsr1=hLMf81MEmi)y3z& zb|i}wdmj`IE;Elm?HkJ*yTJJ>@WCi1&m++pP|ll@ENKVET|-Ha=c2ddSrmX&cm+p{ z;kRkZv6UUG+3v~rCDW$v^7kDIq(#Q6@7dUHG^dV|C$mhjb9oinz9`f0%c+HF??c~$ z^=dbruMQ)TfnFtpY`KGLY>ZF_Y2S(9dnFJh;}6ARQHZa4OX3R3hJ!lxN1=JrC-0HI z`g)$G;?<;-k-?w|obRx2KTxkgdTqQvI(R6HfjieD3^8v7PWSp=2plkp@2J}LhV(_w zN)UvqFJ8&#TkSxt_j1syY!t_mchmPKb$3r08>?LA3f1OlRE9CU)3EayZEX(dW{|k= z^BH?xy~j%ZbZN-gjFEnCkUcD#=}M)TJiP^HViwhwSQM$IM1>!M?)=6Z2Oxpfjb;xM z{ZkG)Yk0*)3ODZvfHzJf#2b%Fhc0 z3zESMD{VF=tCw!gcf@4yJ(f$>NL*c?eWcsGaz?%-wG*#KZ^IkloIKw!4kn=xM~%OY zT{H8+CPiAx2->ulf5twf`r=d``>o|+p&u?IDF6Ig-NZ&n@8$$LsnZvS@Stvy4gd0$ zmGizg_yRYs{kHfQM&@THN5WX4?#OU?lbagfM2a1K_eb?2H^YX7N5;UAgL?2sdnqQy^ zMOo3a^D#1K^s|RyG&EcW=E7l3zEac%=t&-(*oun2no;kNxhk-oss%)M`_lAdV{mOw z2EG~c_>SjtoH<33>igjD)^k@}zf=U^+*l)c8Sx9om3tN%%xAxB`rS1=!$-si zuL$DhztovnPd6dx_Qm-=S4I{*4CG%Oky}K+tWl23T<#Ha(*YZR6dIr*`0NYzPQ~Nj zZgGxf9M^Jgf6Wb_f8U#eO`P@)yuEYFbGfovzT%@4UR!qy`*)Dd>CFT*?w3!GX>kk5 z`0&L&!*Jm4S{c7678tuTCg6r=ujdPZw_;8N*DSx9`l?kE%W zN&T_VvyoE2!Rw>Q5#+}lOoZC%*V%YbBb))Ar*~ckdQ%<X_q)Ov}ie>%up^~~-d zJX_o<(+hOlyTof55+~q!h2Ot+~&|Cyt9sOs(V=FokST3ys4AeNweJVKIs{fWjou?vFz z*ogcAKbC`Tit>S85YwD4ff&;-XWb@!(P$>(dZc^D^Woz77nMYdhk@|I(_ZejqPmMN zP6x}UsX4W4ZMkCv-o4vNz5RS-?+2xJ52ODhfoFa-r1RDna*-;iFSHfPYy~-Ho2iGQ`&Hq8d&y6XA1(xpsXjZq`)wOGY&zG z(FQ8glx_%e>oWhA(F*XhzCH{;dqYhLf{z&d#p)5K$v(&7!+^p|tl9bg)EU1@=oVQZ z1rTMo^ZzMuepjAZRDk!I#A%uTHs4-dYmdZ7+DtGk+F@Sm3Yw-~}3vLH!Hq3FM9nlzxLTuFY7`=6Hbr@#yMfl{5t zo+0E+G*?!3kyoVhDl) zX+Tkws=;DzOZb)M06xSi3vy)tHPApD9F#q)!c%t(pCI=ouwvPC5;kn(!KA|;Y*1Ik z5AcC+4gwEK~3f+-rquNr=Pu=L3jC(Djer^iJ9qey!>Umg!mtcMDf1<;(A*& zaHz0FoCc4}GLGKB_G?SqK8;Pzc`8`>}JoMlI7Ms^F)lfy4!pzbqnx zgc8KMbf4pgz+e#Bc!{~%H3gTaLG+Q?zeve9);Ko7eb4Zo`py#6*Vh{v83Cg^Dapy6 zK*8X)tll^_GQtR1FZWM}!r^el#_^Pare;~QiK7Va4gZy_aK&f_pB5aV;o)I}_kDeR zkj=XMq9Xgl?Qc}9>bkgaD1xARUH~B};DH+MjANyxrmh>}GOBvg{hqt6t*r~Sk`v>( zWg(V9aD|W)D}+<62zvDBYq(3 zT51{@EOcX%>+?Unqya!{s(g}rg5TBE^`6@}k7DMF3QW4Q?G3Rg&<3~r)piim@mV*_7Y5eI~`pd9CW~C5^Dbb zjfQ&C8dB(YkTIZc8|Gf0d0T9)a2z>niIDc)f8twgX?TLn?gw8Rq+T!OsQbN(ct1jP z|4Sb*L1-aL*Ke;^M9+=efdSUufimGQ%I0!n5O*)Q%$7NFIE6ylH?HO4@8rIF7HCID zM@Jkqzc$eb1((hMLqCR6yI*fQ)oxTGq<;)erL4z;O_QYku^*J<)@cx}^ezoquIn~A zbo)ok8JJsqQ1Jw*w2ACD5-d1nVY3b^_Z-X7hBrXb);s>MR?l)d$Gr^h{sJFahH+0{ z;AfT4ha!}NY0dn!=2IOU96U5izw3Er6pLR~1Hmp;MtbX=wx^G^N6~YK$}OI@F>P82 z`+uaVaYojAZWtl=U@uMP8)GCkKlUWddhQkFoV}5!;3v@3WuX5w<2CD_ak9XAd_i9= zrcunZ^#w`2aJkb|P0v`Uz|s%j0et?6Xyc8?L)@u&@!6!mV;Y}15LR%8T+Kc41&_6w zLoz(P3hM3&*1WuZ`}Q$eP`+wXw)*_Ie3gQUkHaWz)eTG|Qes@IE+VOsBHa7_USpnc zpDxC33C}!q_JGGsfiX8Xmm_8J`wV1DC{#Cm`*;Q;O>Q)rHbpmnGS+;Ceon@7^{&;+ z;^&6_$X3tS`&vqjxt?O%EqG3Jdegc(^04?6q%)0T6y4&)+hV==3DZn1a9Mn(zR&JK z1C|Y?0Dd|X04fmu<=oDI71k1VU#HyeAzZhNnDxh)FuSSf=%eWLPD-;m?(3?S5cdNG z`|1Z08)%|_oXpf0Qn4!7n?skt;}p-Njr??9)cQmvCnx9m8K&9cmG6oRe51w??)(`I*KEsKg0qG_4A;5f)>R1;Ft`<|=G;ns5w2hz;DR3O`O zZg+c$8%7A_!Egb*#!qx_vX95n=kAzIeex1>4dS?dH1cs;uuI9u@+i!gwMK;^ya6DOXI$Ijg&j^@oR7kot!*(D` zK2;Msd`(Xh?r^lbco1jf#hwo4-S3L$qz>S`3$_!YGb^u`Hjji|!AyQOBu) zmis(ch;USLWfrg)NfG@OnWk13%+s*UBTc4X$>5nOkX@HkM8`M^`s5q0G@+Cq97Bx} z#49TT&!%}5gbv>8w!Jw&S(ZUUsLW~}k1NocR5f28Dz9VbsNB2CDH};1B(z$d)@y(u zx{Fh`xjHiU*h_b{)9_y20SC4sX$rrd##{syz2Z*Pf?6GH&_7r=I{eIhHpf$Uf?m0q zA{$dMVA{J>%#C6j!i&R)%=j#`$4RqBR}ke{-m|R|uw#$*AZoF{GWE^x%O#~m6Z?1L zn~(IAu9x4T2XhM|+{C+-)0>FLOXl{aSe2Wf^E974^S`w(qnmSiKWi|Qqt`4mrZ(&} z@OWzWAoqsVpFoR82~-*&x^R3I**NYbT=oTlQ*RsXCzVyy}9Mu>EvY>@nx?c;v?9){h?0!JEw*f>T>M@%w7uFUEFkQ$L)b4aaP zSUQ8>BfaYznbs-AQ2`?iTRsTLu!sZ1FkVq=s@@+Aro6(REI+(e3LwQk(ZG|HqjDLl%yZ0&C!;0EC1S{fdat2C3e10Q46B% z^mcgp3nVaAL+V2!>U+65{%xC)wwVj!p}N$ilT$*+7lNqwC#3`Z70yuLX4Kh@TowA} z(+w9V+FGMvkf?xD0@m;fVTxGx2)$!{s7}w&^qbO;TC~O5Jsh0jYqZ_e;73{T>D~$I zo2Q2zGM*Q;Iv*Ae`f8tI8;_P!w(8e-kc>HYM;7^S9;{6WB-QbxoXwt}W~6b;1$GPO zDUMmM^VX`;+`(HORHbS@Y~ywG_fUDR$Yd83MsxS+0avK#+|zknw3?##$V%er zey;uyV;kbunJGrp%76BmhS@qvklMvE`1l=B-LuV7b`GKElB7RH(GhG=1odhiJP+01 zIJB))0ZiVMFHBmNBbk9r$2QwGtP2eH~5Vnlo6M9hBM;W8&#XTDA6r(0tl3?>r#lCW3v zbZ+iQk1>MEo~CyX$&2dO?4#2#UlJTU@q-^-2DcYA97Nnujd2@mZTp2X#)jhZm)QAYbp-*p z3gyDN+4iG@aW0-G(QH!tY&D=wJE$e5-C1kg5!-L!W%r7q=WtLOL*bqC-MH7ni2@8? z+skCpk1d|~&U6mrMm3J%k8wmqc@z1WQe21*V!TvL-0x0y5$1^XEynA$h6OLaY0EhS z!qLiJX)t{~#4t9bJ+UBMvspoCXDXp-KbMP!i!p!ne8&3!m7Hr&3l#Ac|;e3n;>*2xd5Oh?XMUCD%?&p`RE+YKoXjkxkP#p{qGZL`bn zR82PmPY-Gm8U%C_Rnrv)N-gGqtZXE*YPH*tN)}8B^gZDPk&TV?N@j>ET^rbjrQF7{rqmHa?*0<811hKpM_tHH-q)Ow} zjCXyKIx}Pz^Z2l`rtn-Wj_VOFss9MIqs3fs)XUL=0RK0G`}fp1^A!7^NPDJzugo}w zUGvBWbv_~-1fzKUzL9ydJv4e1Ti%1FSN3vRA&e^pQcit9KGEA{PY{g%h%SHr44TC) zbAE*IIMNGLq+%$GD<4%py>D-(aJ=c~P9hugedzW@WLX>(r;MlDzCvmQo5Ors0N>=g z85Cp;AYjFxo;|@A?A2R0<3(24v|dJMqG}8@tgrC+)_eq`$z?C@y1EQh$mujn(i1fl zsoSZESbR5y1R}pgQ1!Aqu2@S%Ux`TzJUdDzu17d4hOX5o*tLXqE(D`A@8%ETz0r{= z%_ggEVr4cHM|re;s(FrUjyd^OgLiLvWIh==P^uVdkIQFLsrh1#>Fx&mb?EiWWhS5S z=r0vte9JVLcRs&Fb4scAD5MrWF+m6PSH;tPsV1>aa>?TPgiJ?{ZLy@a*G{o2bRVxu#`OHAy=ylKMN5w=JEh65X=^+Cn=B5X7KJ=YV2F$we@ zx_7jTE_^7LA42-i;YpXusrf%fHxQ<#9LsvWgKl9xGplV+DvP6|M-<4TQgw4ax>$~S zPT30w9%l{Gc+FO*waE43J6?5Uiht+5(?g^@=D&Wr9l@M}d}qTD*(9D4v-bI-3~W~S z86V!`*71B~X?|G1m$Mrin1WR0OeU8-GPNQ9W!2mR@FjvHY%|Pky`Ii_>1fuhOVj0i zYW6H|H^Z$@3~L5E9}q#h$NA4$6s#$HyR;1sVloeu>g1-vrF(0emozckIh;0Si;EOo zJF3iYK1GF~7ji&T_hYaYR(QH8cP2iK9LRFBeXN#Gt1dF&sSy}r1VJkY7)ss-u3Iwe zoWLl-mp)4Ldn~))IXVqvT+nt7;yNP=V`PG?mDyfi-+Q=J#%DHQJ2QnDf(IOmD4%48 zD_>Ae4ntxt%m%q8n>z?2529K+nZhYz0%%eR64_icH1)oB=3NOl9jG)VmD*+{E>Eln z_ij`Yx4kr1fr?Hqa;v9Zm{1+ST&QXsg2Dm_?6Mh8DF*f$_tkVWsE}4ApZ36}KDlW| zyibOgMlZVY7Ssm{#ljRtR$`wm7;K~SM&TamX zyl71ONvP%Cc>-_a{vhl`s3+(xHh`Af=)>eHN1b{;nL#rU;bnGXW5(f40-A*C!Ln=s z7$-lVdgU224*KhAH&lXxA5M))?$}R$2}^SmhImS3KgbLO~)NTlGGn!<(^nQK#xcw4Y)yf_-SjG3@tdodap zxo+Q1vko(Pwzmjr#s-AWu{Wm`cU4MF=#iUN=L~I>Z>B$X8Wtjx5KHQVPK4+QVfHfK zDJC$v7q%f8Hmz@$`G+Lu3dCtZE8@GB38a=4-RM;2k{wD0B>7LZ*)FJ~#FXg@UC|?8 zhb}h%<6&LxX6#7mzKltE?$Mi-VVUCz1XjvU|8nA*G}a`m(t*x1-t6M*fr?d1Pc!PS zhdZK$&A2GCMxaKI&)@XcSlmi(&V8ulOzKa4E!h~nuVhx+WucS>`gD*msX`=cW>zcRJ^K=8-@(;8szZBu>Lz=`k2G+LlOdMtd^X4# z8=ItO`NLzlL%q;U1(WBSCPu<(*VCg&N_^WvDNH7(Srx%0yRJknOP%DKJ!nQFUjL*~ zl|6$Zw@!cFs`l?3Mmk&Xk0~o*Ds!mLh(>(Ml)(<}pO~Lj5#|LtCc-C5P3)+=Jv$lcyQ#^d5Gf8x>9) zu?F>!h+b2n_jw%E%nvZvx$>1;`uI!E??vFZa41#)-SfGHl)r9PDGz8h`aK{@McDQ#0>wSchG{(262QH4~eiLQ2ru{<~ zFWQw0z3z8^(O%kI?_>#a-nf3g!+f4~_RZ`ZJLdVy_EVT3UDo7s(DDx~@(ti2dzRJ-xO>*Lb;fF7Y57~YJ(Q_DF$;&#(L{>z=c z*MXy+QsiiPW^6x@?NRA``rOc0cd^1g0QRwJGNxvc<8M;jFGMFb&}CCX(AL@2fzPzL z!IiHu2LLbq=$oN#od>uRmj)svS(Pmxj$DLwi(n^0gIewByzELH3^ZsnGsFs<0If- zqBdO+jUa;<(a~y-Q?-Fm^WM*_+Nac`@Gvery|E#W2uZxYRNP=q0B#S@PGYkvm0d4e z4jy1*reADFT>oUXBTx<*AzJa_2=QF)s!}tj9k#aeH5r~QnDVMderc)Cgv!{V429T* z*tE3jA>PGUZ*;Y(nZrJ-rfao6b;>HlX^GXBe^F#n*GIB(wkEHM_rthVvJZUr#%gxR zot)8HQmJby@)bghA6I+Ef=Q#QkQc52(R>9v&wT!IQh|3*NI~h6uJ^mUWkmz4HpKfa zeYt!FHz`yvz!nda*m_1e1Cr;~g|64PzbZNg*YYz1eJMvu0PU)8FwG&1Fh5tDJ>o_S zXURu7T4glzH~8$Sb4D{vyHjSaBYCw6n|k}!-0iJeZ=3Im{+!!>y_(InroJ^3LA>M` zi2Y@j-vGa%DXjrxM19?D1>Ck5k%Sszi|?f2dQ^xo-HAPHqG^Z}8wj z{iP30tY4+^@*N6gzhJ^qXht3DrP%8>J8NpJneihE@ha?X8l%q*z64znny^@LrqgH? zqz@(JEG*WL*bqdRms<(2O5uv#0q}fF^5ppuEf9>}(7$lu%=z-c?Y&_ES48I#3R-G^ zrsvVT0)JI6A`~9pYZm)>ohoKyriZs++*3BKF#r$wUt*DO9Qd5 zeO?xXaZLNJ6?0S)Zex6kxxbENIWpm#64agScGL2j!kSK*&$h#rO$_#}s3Y}yg$fLt zhLKJgXPL7gI%H-mtM^_YTag`EVafrz0&3LXdqNuIvHB39=I&y94F||~M;BNq@8d)n z>rX#nrhvhe5lmakVB-&++t$V2Sv~lZ5Z_W z1z&ZzX;Gm~@L(z(X~dq@cYinkRA}dte4hlab1-f%;fLd;5&9f6BAcqLGiU*QM+0Z! zdo)vl=^ZZu14Wiv0{1#2440j$K6gQ0{U<$MHMU&LbctxSJc~c>kLm|2rn7}}@J{n{ z7Wyu-HQl5TR#d5u;~A!8=+GqPFVb)2QofgzuT2}Rz_N#n#VcCHP@md3^0fDi#R}XA z3cAEMw3vLPi7uhX>4DR@U($?`9)bJDtl%u+g~(2Yw^qXT0 ziU9}?X^XeizmV@}iyu@Ni5x=t#)T!k;9sm;CjM1i#)73xzMs&(*o{m+10jB#GQ+;u zNGv#xWH?IY)ehpprR|2!+DW=gPk702PL$lXub2>F+n!a6&(gnL>)0 zzO+7F=?QY9`6b~y5W72m3|;)Zh&1tvm||pb{3S1f<4k3mU!1}R7QoGMnZj@UBxB!{ z{b-f}u;zN|w(G4wli0KPiTK?j5AVwTp!SGiEinQ6IY)cTDc?7y>vYJ-Xt+ezV3j@Z zyLp=*`Bx^fH?*2Zg$g4`N1xfKcNzYsI9rJO@p@Q)4yv$V2nzJIL4W=)03%~#pukKh zbW&34NGF#7m^%SwbNKUXm(r9y`$B(eD+ri@TeAFhhsU_HYd!=;>3PQq%eM%MwAe2E zSe2!7scD^DgUarstg<0bZ=6LJ5P#Fu|;IdJn=2oWhF8tCDuPm-+cn5h)H zpF*tU0U`9L)L*lW!KcJ*7e<++SUY=Ssxx#Q{~)Eo#C&A?00G=?hSRigw5IGV#%qrI zA+^*dm2}B(Yb908P4I?MGyTR;gj-XWYjTPSzyROCImMD{*|FOP{Ft`2g%d$X{mRMo zA)0jf&$%;aulnq&$j{zH-%TLi`_T32Y(+~PpK?V>wz&eU{K**~mqoK5iEOm#O@90K z?f35%jkM?nQZh1h>5QJ-o~foLWze8D{x5UcRgtyKLZ`iTK0mSdu>U!`{H7rY*O+m5 zWW;*rX6(!N=n}A!l9H`${tT;XA}|8cIrI$wmvQ~FCSW6o0W4U4{K*v{vi{!sA66sF z;sY4$La*pwPZYDf09@s|qhRDe6X?Hu=*hf*&uGa)1;{VLFAsc=ZQA zcF;#4`rjh~|F1#tpW<)v0!H6(P3gD4{t^E|6Oi&EN-_4YnSi&zu!{Q3>zn_*9$-;& zY{0}pt76XAKWqOjr<3doYap=_%bPLWC?}`b5+0Fzk00OjFsy^9)VwzaSYe5q>6PI3 z@zk_SXSpSP50q4tp!u6254&EydMHUR&^(DmBF9Hp#9Xu5!ZJM;t z;leonnX?aMZ)&#b+&Ep*HY&+u#Bg@U(Te(??PVzjw$N_z_MYFRCu{33i&2ccyiJQo zM*t{15;=K~I-gNgKsc(|_y+RWUF|rq|8LeG@$?MfN)@C=B{8>%@e~kM2e?~ z_d%G0$Jwq=py3IvMv_LNXDe9!h!@d5J`yW*v-?#}$e@yL%u6KPbxe`!)P5|_Bqe#0 z4%!8^jJsWJZM)YuG6hbN&x=kp($im*ZfN-E_`N^EraR1pUi^mtBOcd$TQR_MY8(x# z?MH?Sb;8N%X>SAR<<{X{?h36Js%Wg8` zRN%qzO0IY>%9aGu(Z~MC7%X|{g_t=pJVnIuB#Ze106nQD>xHy3 zn`63Q2L7FwS8`@cO1?I~m-bu!E@32|oxrq@?K7N}^6ruDm9qXaoDr~YYqOObcw8@5 zZMZ0Ph`ZHvya_<}<<3ZPondyJyb}PtXA-bc-Byt1CdgYR=TQELJOn_;fO? zRW$eG5$)zEwybkpQZHs7ht&cUU1;nbO!R|=E~!iN{8X%g?W;1N&02$e2EO@{daY8J z8D#xS=VuPg>g9EeffsI6>DlS`(7=G5x;}r_*=PBkjf**V7)`(?`MyHcB1pllJAs^s z^KOB=yU7XNkt()PQn%t$J>0g`_M9K-HFLzPiVAseYpJ~F z^4Y9g1|}-L&z^32d3X#UY@-_scsEj1nC+wyb*Dakxd;xA&N3h%g?$n;8O5&=-kh14 z)}5W+M`1lQxB}fyOVU_X({1qP1gTXl-)``&2B*o*7nB|@H$G-YG&I+Hd3xAyKZV#r z)O@P*Sv?g_WW9Bmx9JuaUvs1%)Y zn9-53Fp>Dx-rBD4xo4a<=2W=K*CF=7oh0Q-^S0+i?@vCbu6-=*Nb^{ku<`i@Cp+Ik zo!T6wiY;^OKB{(Fv~-bfvO+>r=niJ?6raru*#yk>$oCEJ1rO@Z&GH_ePNmH5TAi6~ z9T#n0J8)FBsxGdSinaPcQu<@i1{kR=*$q6X%+TA1hW?;BqoM{_jidnT&g`(2wx@kt zzhCDWO6AyB`f`@(;$G@h#dGGv-Q@SrPd+%(Kp;o^C$|eNrQ`GNx{O!P zy#JCecZ%>7D-6|OK3=8O=jGw%Xxu&|^Qu|jZ%h#{Y<(ljhzm?CIXF8r-L?AKlQdx5 zHFA2~A#nKcVEQCKW?1PWdYm~&4RG4=smRm3=>(LFT*;pQBeKD3i`9dUArttX>M zduW{f1Yrxs2fvDmM%O{n*Irb+7F4& z5FBBEUTPCiiaGA>#Q(JPbS_3bO>`&S5Y>Bwu zAdz-kC>Ke`tU}c#pfwgQ*Qvz#KHE_+_+(m`yJ;h7rly%TK5hbYjq+Z-?K{=;9TeA~ zi#P8uINR8bqVDBT-0i;8ds;d(HcqqgSfvA+*PwWNSi{iO5OoAj00$oTKm%N7cw^H2 zR8e)Y@*sj5ml)GW^isa=k00;5ECQf(!{g6kWV{Ve*T+{n4EUddx>Fy<{&4@UW!5Jh zyFPsz2)BM|mv^j>aFsF6p-|s8`Bee_$-y;&JQ-S zN46B}A&@}8zt&5eF}fSrX~t%7e)*kG*fmbxbXQGx+*?uaMm3CWmpjK!rx_q$4kMZm zw|#7UKwCpfuN$^nU%Aiqyt3yHItl-B9^=F@O;^T1KCt>KYj_8TLfNRpGB z@PNZf?3k~%Sn@5gK8C(Ky)84%e!I7qS`KTsn#!=D*mfuXbc^lGPaT_up_f&er0Wn? zjy>eTPE*;;A<4<<8g{nJKe}!Ci&FdBpD&ieV8fn!XmJsHd;5F-c4pFuZ`5CF&zrwF zu|H(O2Z}Y8DqQOaaGOI1=L)V zB?23v@mdircAku0V&pCMQFeP&*zKjM>;h*q)G^+YLQE9Raqqdz)MS9XySCXkjjW(A zQbh!C;a*CmNZtL{4c-f}O2cW-Cae?eUae9JLj-i*_N0aaaW(1agY^=ZU%K1zUTTuY zCEHFtU_pgAP2F3gOVsR#`Z>urwS028?ZW zin9LC9++>>t+2g5@H|b9C^+^Sg&F!@ep^#jbwA_C(KpPkpq>Jq>whE}4rZA>I}fYP zW%Ho~fQ*oS(s$Rzf=T4Vl0kuj$L7Q&@g3C(amFiN zl&W864s^=^THJvhgrOfJU-ss-{L*?UD9%27!v*}NR!QleMv}wQb5oOMqZm=68rLND zGRAY*2z`3p<4Pg6MWcG3`2#6LcQuvusZ*!Q z9jsJ=^YAJWqqerLcgxFIo$#oa{vTtrEu_H80xF#HQeVDKGr@@H>{Qf_W~OvLlPAOu zYCt8-?~lAJjxzQTJ~lAN94@QDvo*@^mVRzWBndjceLp*M;<(cW-}*fah>9Sf0-`iZihxQtinMe|4v6H?T|){eNJvVf zbazM%sDN~L3?ZEg!@$76yT|kKe9t+*>;1<|$C>BZz3;v5wbtI-_B4bc$5vect+eTi zbQFz{)6(~M8(fkh2YD@cc`%-{0!75evFO^-nh-35#;N}*80J2fD+WbbUjB1hR2pQ& z{&V`gm#cMaMjn>U8cEo%h-l@QahK?4U1hgAuAqDOv*qebQ+L+UaW2Gd;+eyTIiv`u zx$IBh=BFCWz;QzYGuSzsglcTE_QntsWvEq>OkldoYz_R{CHBT-#Qct>#0J=Q1?W>2 zqPQP6U~<-at)-wvx*oRQ8c65&IXEE>$B(^~W0Yv#1b@+p8X0*%%j_8CM`y(Mbyu%x z^MpF1`6=m;VUu;m-0}C6C=QsK$==!PewadZ1cfog{oFO! zdw9}xnfuISaj#EuD}LtlrRBaY6p?)iTZA}d^K&d0oo4d)c9OZ)vtS9|`PaiYH`e?5 z9ZP+WSs(Z~n*4I5G|(~7d{tL-cgm80Kdxa`i|5+D+=kBjWfi!7zUcz1cki_us{JV^t@02-?Dxo946IzY%NP zUB2jbCniB(LiyuYnYQcX+BK!2x;nu;gCZr4w ztfmf6^dzb(n~es!ky)V0QLf>*bDRN|~&pNfGG{Gx)j&gKU;6 zI5wRUMVeP5g$J)xZ*Id}lH4C`gM*DXUbu!uy4M-NQhr00Rv_IsvG_uN& z@0kY}-I7bWTh=b9e=^;~;m8oD*PR`{nZ_tzCg)BGHzAhxKuG+0ry>f-Yg7k5UCfF3 zJt#oU)ou1d);liu3$3N@Yd=u#D&XFsUR%D|VEAW+rD-5#x=a!&Y2py}O#X(Ps$`X9 zVm%#%*L**Eh_Cr(nbV-0IrX^j;zzRn6xeemPfwfvfT84{}=?)}pu z(9w^dzqVYY*i1E_A|tktS&SpQK12)G?~Z++sGvTb{rsiPNEfaxapblx9-HI6 z9CuSgcKm&{1sGK0GOl&LJELWdWR}ZGp{XZX&c33>PlSsGZ<0kGy6mLG)~8tM9Af%Z ziE`R=kV?p1&fS$2{RHTX8ecd5JqLU4DL+4(0F6v>efK5bL^6rzuy%Vpz1)-5!>mvV zWD3uf8ndyRW7u3O`M>0+mVN#eTGvaA}Yu|Tf)3}%HOmv|N+ zocbp+yM#2!dbD_pZGCswsLEVPUS8hOv0^^l%gF1;r*pn2Vcxwj0^=M8JI}4IGGq?* zud=xkBvK^0xJe5)Hjtxy*B1SrH+)t>6mW6)9V=d}bb0D1_7u`0m3C~+q}v-3>q{Ft zVi$Po5U*qu8tV|}1p_KnRB7C-jb)0vCq3cb*0zdHm`?G6fLS*_1|lk+rb6Gbgpj^( zg&|6Jgomy_8!yWmLl#|$*Y@}doa311M>DTNYDnpLJ$!cRohGt^;ulV|20E3qaz2e- zV9v|Tn!Cfi;RjJgqU|5~%oA8}4s2fYmE!^LOtij`)iBe+%Y_#S8fgR0tG!?LzY_6y z4nG_hh#jk5``nUnkCoCAk!qv;AXu{4zeV2KRA&I?)4atVvT8^K9RJeN(!8*`K_@Y~ ziaZk{K=(R6l0qfjEmuzS9jsFk8=3tzGd?AhU;o_wq;N8sMYwi9=qhot7~6(JMbX-Y zzKz}8`CAwj`j|vvQ{79th-m@A-*HS2!F)HdV((6>ZZ$FOl=yPS zNqe+D)06IrO8Z0+pFB2&k;{(i6Ti3dg4Sa9dqJ$Ju=iM!sDH7K#Lpj&WT9HyX*{|o z9ND8{!z~$X{&%g94}WPyp~!p5jJ=P{ZL|!>E6h?JTg!A1w!Zs6nCd-_ROSiO&*IjG zG?7xD{k=V#nPCfJ*Ur88y!4?pE`BU<*>>|s_l@-h&5W3Tkhe_mBeFO5bo&X{P2>bp zQ5#nin~%^P#Ep9TF8X_!F~f`T@ySwK)f+rJSuswv=bG354Fv&E?1;jc;IM_RcqKtn z*YTH7Xz!4UomTkterx^?i#4zHchgum4x_V*_?g<2uru|Mh*72Mi`wxyt_)6xB*PmWRu3{C>8O zpl{a|?xik|?{ljj40c%eJDX<=``xRN%w>ECx6|4iaq;&^aX~iFi9Q^T*7^SNkfEKy!LiGR95T5J+xAmOb)};EcR``FdK>l>u&m8as zBYu(i{{~!ts{o|*Vmg)TzvWP#>;RJcPoI)MuLw9J1JMzjKd%D|)OX`4()k@1Rpk}? z?!?DA-}@_emSk`S?VVJ$`U3kcI|3WSs?CHljI5e`SF~D|bC%5LU+DPvbN@A*HF+rw zB`_RX1p-dXPSkpK=PodL-dCdTOaZXlZY_+sQJdd&n~UQ?SrMIp)%yFG&*}(e8JWLG zs=pt2*|}~E8EBd4t9UaLW%_4P-sa6H%+-F_z(Enc-|sYU_i7u*iN<1&LC08ekM6^$ zTUQ!&{|egz7_cDACki&K%J&9Uz(j`ci{=(tzwKoKZuI$Y+p_kA#*!+tsZ~l+dcWbf zg^w$||Iga?HC8Daew>Q!0)hrf9cRVYr@)`5DBsS*#4{=Ew0}t&;GP2%oU`S&N;Vrs zJ=QAX3#q0o>3W(k$r}?n|JFj%4GUvz4NL*RI)H)DrKXyH*3)lUg5KWGB4=cdmh+XG z$w{hi&?=qUMaS?{3N6ZFF&hlEEwy{8%^meYJ>5woBUN2JG288%p|GFf%<=J0i*K3z z8DZGzM-z(kQ3V$}c3Q>pZ0$=X&mcOkcC%~!My$hQ=&!dhynH2vP7Tzh7^#}&^-iR< zQ>vd`lkKk`7pI8I4Gn`C-M<+m{J?vyrqO}Jx8g2gs~jk~mdi;E_ec^zyT)&CKP8h8 zwTyv&6Au@5Sf=FCVmY-j`?m=rK?oRukER3E|Ev!;@-I$^4IAA$*?R`RI$^%cXMT&9 zPsQKL zNu&q|KT*?`<~t?%pe%+cUd_CiLrWOx>gsC7^&EIB<+BUJ6sCF?nOE8`N9-6X*13w8 zH@P}H>uB$cxc<(PNqk?lo5iQI`@X3%=f0supHT}Zc*gFMoFsNN%;8G-E# zmF^6z!N9{M=grhx9w#dRdaueTF=|Htj8(xhnh~jm`hpzJoj|zZu^drh1_WRS+yIiC z+_Cu}xwWL@T2@9%-eBVKt0;a?q%E`-oO|b`E?6mBp`5I9FdQJCcEF$7VqQKq=;&eC{rX8dzN8jTBG#b6> zVl!!rx)y0GMtv1#n<+CpDW>!~g|T-*qxK3qC!|umEH`dL0XkbqNC*IQz^U!&!D?U6 zRAETfD^`$hhhLn+L#wuR*<2^=`hIQ?^mG=Vv8(?6;^N|x-)IBW|H6lW#C!2Im?$OL z=Rm$i&`M(IQf!aS;py~vq)1||&uVnohEx`UEQI8NsH4vYS<%dh(Oa9;g+6qDhWNzC z2p)^I)U{OSeyj#9&DmnPcvJwJ4GfSZ+NND4Uw@@8h#&YsRYN0d?>iueb^Pgx}cmg3LveYi%(at28CXu#wyYV;fN`TEepZ?$^< zOJ74{aS~p6@?snPghS-Y89(|Ul;gMk}Fc}8vF`qF_V+M z0jdB7vq7|C3bK-7f*4hFUEF*{=fjogs1T3q3;B=eo@5ti)|aDp5jDg^C#B+^`{Gqo zr0m>-=cRAHO%@IZl6A!2M*b{FN?0p_B4|hNyaX`9oBJF@Gxz3t+@GKC>mBINhXrI^roGj9S^kHyDm5G^}JPk1a zHl4hG5Y0l$^<)T~MZQ)~FX`~RGtr38jF?+4`lsusxmMn!kd}=C5~P9f*q`-lJ3pWOu zhxXMX58g_o`9O5(*i<)4^tYTCzA~ABIBM1zEH0}+d#7(m?6co?)ejWwJ4=56=cqv4 zjKEd#&zg+-c!xL^is=XU{3X3EM{amdIa@x+>hD(;8(2L*-Mi?u5I3!>TDZw3vhJS{ z%)=B^!*80uF}qc>3>pWi$Q(|?UiLxDp1P+2%-eV&TLcGvaR|LmQK&tSb0TRcSVa^K$CV1 zb2B{MG)zMe5x1(8+4{ZGnxWPgak=M;Gof{xR$a$w^XKp7&Uob_&cZOyV}~VXl)%cC zzPF#y`-1LpZaFX2R2-n0&64YabQ!P8!2UP@23*Io7)%_LLCC2~R!&C6j?*9O3A<`+ zcyf*!y*RA9Bsd6r`^2%VlEx<>lFLE|IZAHu z#%8BVHZqY}pGNOe>SAmC{^%O(*ub`kjd!-dkG_zoMezsdAi@et|s+AB)ygW2`mlrMhJ~uyH50hEZ^6E);0wS<9?D#EZt}aMEe% zxczu)TgFMfFYj9a zBUN796dR>;e)UJwui3fSTE@~^%j`CMT(oA+Tx~hrT=;9<<8M>EFJsqcKnHZ0$wodH ziuj(haZ>nsDqX8HW#Ep}_xzZpXR<@bLL)sAkC?p%4l2=+vs<`!P-r8dck+@3VoK_v z6I@~&dPu%YK^iuFnY79<=SEo2k!O}DIOh-UJ)JxKG~^GI=U&{Ko)oulZ$g@jo~$-! zGCYdq4VexKce+~V)!5UiW5}VwwNwwv@#+pnf z*$_%Gp!QV}{q@Gtk zeSa;+uaYO{DkgUK#zxoi$)2D7GTd^ru)BU3OL=^Bwj)S<2(&gCu%jtXPEN0T=-DsM z$Q|VaiLJ8n#ZdZlKq=qg)=4$}Rv-Rnb^XO7pTxsp%(d_8D(x4rC3fo_xQ1e#x{*a{ ztkGPW8K+-T-yzK4w(48gZuY!GT`8Da%Q<@Mbhf?YIzcgr$pDGmbEjrvR7Q9%no%lJ zaEU;7tr=9eC=!Sqdee!9{h+7tdIJe6>F43#%7q3y2SiOCS z7)siuc09FJy}axiKP0}`5cZ5Zsm|?a{((W6e(-7EJv%(5Zaz=Bip?)hYw7IeOSveR zMF*a-h7(5Hpw=cS{YKHv$1M}g{&&?8SZe$;1|_hrSOqfcW7$F6`1g<^<7s}5uMfuY zq58Z|-|OLLxt1?WiL+mnqc;>2PH+TauJVs9kP4)<-_UOoXwZA^?$!2T+@6g zbQ?VV2L!~OXSUwfSoeqTlx%6dzT31D_7^KkN>O54E^Y}g(p5b-y|2)EzlpTgJ$yBi zHQ-ikuuM+5RREWQqk~_eeV3t?6FLDQN4p_AY4-N+-Qj6!Vk#8cYy10Q_b|OwYL$w0 zgaR_GedqC|t1hley6hCvF1 zylrDY8I@(Ty`7yWRy*F>-RbYw)cJB>NeM(K0S;K>rr90Yjepott(?GYO6hG*#zd{# zrpedj$BLx4OSw2{bPTiU=N>dE#hN2U6$?0*#JRW=sHv#5OAQJNmD5F&SU;~x$7;yq z-*^k0St1!t{;|m-`1!`4jB`YaBCxn&U;^Wc(TXkX@0~f9C@go^ecdgd zOL%SCSF6-lIy{#)vGqK(=y;S@AU4ff+!*X$z3RkN+yWuPD5w$XrHOeTNlFW0VIqPe z5IMfok97J6=#Z?$^U~gLn7fG~zbzORKyj@|puV!&q0rYle8#27LBj=Y8Ieh{TddnP zo^<%9rMUpOM0q9rU#-pnEG)z9Db_bT9fd;^P_qvVbo1pro+%9|m&C>dVVji-goTA7GhqOsrrcr&1RD(*q2h__cLz_Vzml+IyPuGWHbX zcdYu-`3e=PETq9dj3Oen^NSrvKb6++9*@D`Sf38-HsaP#b_#^Rx2OOB7xh_1WCyzD9HwABP{R1en{7W;Q==k5<$P%(I2m=Do!$j`5JF@Iqa2zF4C z$074#C8wS-zFtr5us5e&;f3+jk)`2oEE-c7SE%9{@Hwgy2gq*{3f6BahVWE*2TCg_ zt{fMLgU^72OCZ_GgJ1n@xVG{N?2NsorEf$Q)u-IoL*Ofex~@4Pph9mrlur>KqebYQ z%kD^ycf()u4lcgKof~e9*dOCyKOhPSVL|NvmVv4Jw502{$ybd&)eC<>+(&x74>cyh zzXMGd_kx(2HtUBFjvU?a$HAXf!ol6%J!%*E*8_0yg#-NWHD4{FLh3qawuFWduAxs? zvR5s#Z?Bvnt!%7^86(KXEf+V@%JnqB?gsnLK+w=+_zBj3OX0v8P&u7jDTUl3IjO|s zlnSPx*^S+quy~bGtxinAMXH>f^J@2yz6@JsCLU0OCv|kSQrJgS>M3rhY86QZhYI+tbxY{D>LA%h5f&(r-Py*%qPx6>Z79NJ?s*w&?W^!6d_ZlCIG_rW-+gnPTe<>682%hat{WK;u;$wAkSL}fX2btqgzUwY` zM6wbK7*qkJV1Dlkj~iSAFZVyVHaa%J&pU3BYtb``4r$1c_*G&L@u9?&qHLfbh$pNlNWsI@5!3&>qc4XT{bL$esNg=)rHzfv>jvqT%gl~N>V=ibmA|P8 zdu!sN(OTv@g^1^k@)&Fm$Cgz*(aW~MGn)@9wnYZTkrpp^YJGx8ntS{D0jM47%+1cgKOiq>; zX>gwwbdy{mxn4*;+xRJmyIG#?KqGiE05tLfE}m=y!lPB!;0<;*1@IGCK3wW9+3STJ z2q1}(+}s&?!}>>lHvK{dT9vOcTchsf_uqmaZp6X+XG>{?6iobd*Z=m5@A(w8WU<54hjj&zG)1r0h(4z9?Ag zB$d2W;K$D7iCb^aiMI6I9k2iC@>jQ_akMYiCUhhW4IDF@gar-unu!lo?uebfbKmto z!i4rGyCe-i=2&0=;Xx-gM?b6mQtGJf__z@<-kFB?n9nW(YDFy$J9|J-cD@fEOYWh&ua?LI1=t$l6Z z6`7lx>w7tfg;dPM17XVKzw?ebf><3s-TZ1*iRw<1-9)!vi!(%5>lccJ z{a@)(MO#51#%ABoet^l;n){@->xncspD0CK-GUx^3NSP0>0XP{ET5pilN~kS?q&K7 zH-fCR$Y?T7oH1HlNNnpXdeq33&%Vr19qTxJSZx@&lb>DYTbnlLf8_jo|Jmv0vwPZcLwIU5%=2zs)H?X6Ol1vh(vEI><7(2=g@bS)Dbz`|x#Ls?S z_@1ApplW(gXxz+Q4dd0|-5G+fWn67@I<6#Y-=^nr*vvmC`$||vgqwGBvf6KXu57*; z?&D|kZRz3d#+WxWO7asw2b7dns7SYo*rg?De ztbrnkz$40UV2hx0V?AjCbv5zgyfp8e#|6h>lt@Q2aSlCL6WWd1u=I5+6Dd8@*rV$W zp`f5hGE~R}zJR?v0}mfQ^g(ared9`vPd7A}s{9$%fk~K%3#E9}=y*Ha%kCM^2#*R; zwyfvK=%^-QRC8+d+FCSnkUdMkYR!1T7#jK`6uT&0#v?H1u6`o^3~X<2rH#Ms zj?vo~x$N(!kH-1udf90Jk#@CthkArqC83|-T2wybrGwCTFg{gET|OmEw8R3B2DkTF zkY|-l#*sY_+@YsQzq_#Ko??gY!{6VZuZ>Pv4_2}i7R8Kf*{|hxbg$aGdfTC%Ya`h! zy!=_GZ*F4CFt`Io$lN^s9c+-0IYTfiFnX~9k${~2;@GKd&4;A@9&@(eOM#4*IAR_j4a(2Yu#rM3t3Qq#jBFXKSYQFY zMkCnd{Sr}e$(v+GzrlGXfLLGeE)EUy>{n2(3YC-3sMPCctpWJF8BQsiI_90ay#XYz zZee?i;5$ev(vUKDVj+R~SAZ%a%Z@b)}0btSIhud4ap``Dt!?X3@i z%iVyTnYs%sI`AdcOOiuPTib1`q)zWr-0oCW-W)UTpGb%f6z}Mm*#8teYkiUzPYRe( zQ0=v#P!!*H14b6|gr$y8joFNYocJX4;!T!Ap9dA84~hpj@RYd|iv1(|k6?hGyeAdE=U5UHsSm4v}yW@cs#H-7Z@M>-m%w!a!($d%GZ zmGrzApcAk}cZ%q2vrp0GL4}fxzKl_*8PwSsK7HEu_U3(c4UOfWv7_xrpf+Yb2_Npu z9&L{GEJm*DpxP-RyfWkC<3?T^ImwsN0K9MH6JcfT%*|rSj;`65egFP_adB~PZ*P8k z3@;D&eS384MnQq^6vR*mb2Q$rM*FVn<*0L7f>0r9{cACvP@{E}IZ1d6G4MPqMCAt- z<8hug!Czf;8*&tF6=hMn`jk$bx%vbd8{Yz(1nblLCZA)k$w7f5ieVVPrcT`%LdfTM zA4`qQlC66@5Eh%JU<7V4&_CLOT-NQwPjG#1uo}swKU=se z;e&!1@Zy3lU+{(&PTsLPSK4us z{3VuPJ>TEu8orGS0-uEBKO7~)<%ihrB$0!Tmmhj@XG0aw8?}o862dAVbMwWZ0hL8x z#Sowp6T5XSNX`g(js5SR+!}|9fdI*SuX*B1pq}&N{#PYwySrx;`t#?VUnRChtA#>R zrr4EOEd+Y$qJ=Uols8N1Puw<4o^&Si4GV0`Kz8@NAuM< zbV?0Cj>9xcEO)}FZq7~riPvau-%PD-Y!f#pXPU5UVN*g6KnzdNfLxO&fhn*@wAiMn zzrQ?s_qlmqULJ>PaYFnXZ3vw!W7oidii~@4Uzmf@qus!kFSgZssqHp3kxJRQ+5I$L zOThAvH&oZ={=i-DQZ=~xj}%f{t{=Vh)6SEHb0r;NrW50avdydZ?vASscFbVfLwlBh zDVM5+kUZiXbBZlBKdqhpmz9LAGPu#rA8Q$YMo;QF-Y-yzBbM63_nHorks4aF=Ck#V zVbjt%V3g#qHHc^gApDw;65O1A4|7wLAlrKq>@`mm4gI~>&}RiLn^X|32=1~rovLBf zP@NC^C7)%#&7Muls+iZpg2t^lO6iBqd$2u_W8`;(9}X)vj2uqDfZbxFQ~5WE^IxO| z2&`R+Y52XOTcKeddkFYN7i-2@w@ypUiwMfT@w^0ylio0k-VeW%onj}oYj@VdeRC%g z6JLD!Ia{&DP9^HMbtWBt`+@9bn~JC-AD3^lTk%=1^i(EOwJjEpH36jV)upAANJI@N4ncB@!!=)PcF|BXgFt9CjipA%4<8oE z3!Qi-wSa5Mcj9Jdv=)Mei^?P0J&b1-nb9R=X>Tst9pzIxJsuASqCQQ zFPj6$Dx{YKBwPaz+{kOk6vM{16%OjI2S?~-wtO<$8**WbPCks1nF*^aoveAKFcgP8 zow$HviXb8ztNC1=x(_$MDEAoDI#1Ou*c*s@6 z9u7;m6y?jlQ#|fVVYKmI&$hQ+K(0#y?i}OQ8Q1SI~%wd$q#; z3XlESi~0W1!kiXbZ$8ywo#?F|OYNJ4Cr5ibMI|HerP|Q(f;88&PN=A0GrPZ5R1VFM zq`M8!Sjp*T?>Zl+6#_r9i%|~IkyF%4B4m6ULbX$D)e4;E7>HB1C7_8ajRy4IiiJ>j zK;KIp!?8EpYz~>$kxCno6Iwp}lm`qbMGE}5$O#O;K-&|RI4h7KQh&(H$~vtHf70Tv z?4W({jW5(*^t=OCEUcHhbg6KOa$Fp~G04K|O=*=faTdWONrEr@BJ-pz=bMG|J$#0J zD?BQ3R4-54V{?!_k0F@M!Vjm3R^e5 z<*B=+@%~z_5nr`~Wu?bWcp4bJzE9!~1#$2FE3AmUX4(HzOj-#r4EmEtx+pV+30xir zaH3AtH3FXraI>=cJ0~Z~ah4s@J~)uyy1iQS+`sCH&!y8V%dIk3=wGgh)DKTUohYeO z279bRe#+9rG!q@Rp2&rg(kHl^i^GAsTzV0)=Zk9WejGSRqr|qzRXTfUDlpI9Qc?uP zHQAxs*y=XgHGDB?m8g+nAtIC@L&QBP2+wYK?F0Ws`M&Yd0%`9ZpR;ilb1{eH722v! z=(*ebTNa+NI5&;$YekG$-}sNqVijiOlJ&@OeKIFpKwO+4V}qvmb8;?h#nZ>tn`34w z!;Ejd^qSnZX1u%ez}8E)N3JiXW=V%L=uPAND!zeQnn?DQ8kv5qTb@b9wfbi$9jn{t zLk;!n{)89Sy-nKBVsw)a6%i~?W-;1s>6guwx%(!Jzrv^a@Lw1Yp78j((p@VXB~N8& z@LUb-yDA`7v6*qoaRi-~AWj&UCAhLe>U5W&>}0EQ^;_DTTOhn4=Z^P&>u*kX4oDGY zLUP#b?{%u5wHLBtqtW$LGMNRlc^+lY8mHgSYJIk7QDH>T3ANFPsqPA`&wSy2*=zS8 zT4!7T!aL=|=ryaaAT&Frx?h-1%Vjj*d);$P^d;d%w{FS0xto_ zT9_&9r0kc?>Ky9?K(apc4*jV5A#a37tVFz0(E|*tbrX}x*cFy0Ci&Z@{6!%t&8}{RDmNRd((Ry z*N&oWr?P!dk-w3aMm91WjAH`$5`CZI7wXh{l4**5YC)v>Sl5`g;9+P5OsT;aC*aWQ zKI<5|@i?Wp|9iFmi1^0wYF`S)`Iy%;m0k_+^?FTKM(UK(zBGzsqJok^I-jv2U9Be=3d=ytUKeTyPjEt_fYIs<% zB6c;9nU8vWJX9}?c}+qFvq_wI<>=z54&K-8q~*_6ti#j z0GbSH;Za)+{G_ht>-aoq9V?zu-R8``m+Oz7%U7o)ewbj*Q`wK5=}g{HL8kQfN*);d z28%6om_y}`o;z0I(9NLkh+Ik#r5bns*y0|)Gi*K3wQKwly8-pW+!#TZu(Vl6TtB(- zHr#)7kJNYn5=I9NoUhLSTK|5FhVUuFJA&-1Wp^N3f4u;Fg3u=qihPp#is)%e_u~^z z6-ZC796^)z6f|C2`96g8<}S8PZ*B&C^RJEf-G{^S4`y6dZ_8QpocGb zEzL5tYhz;r8zlhKIhD)aePu~97g$fm!lkc7+}A0t@8plRN%OsuexPI6VwUAuqh@4O z+BQEO&#!z$u)nQlq0B{?5v=TlDq@8xk#Heu&ZhXw4&}^sbLDQ;z{$4ht z!yyQqWe1|EniN}~hgKWCrAt5jXqisVN#dG9Bkof_$NWJM&(+n{ZRv4@;b?FIH^jGx zvySZ(g*aejm^*I>hBNo~XBP`)e#p40?|H7$S!ie-cr2Z>l-rY}`6~1I|&t z;l?ITAdSJQ<84w>MPPYEUI3ZOmd&q$PK4am0j>lI-|PYj)=#u(`J_0wxw!a+%<=S{ z+4^67LPj4^^dIeb9{)J(r$Xy%%mhj+;!@I+kf6sN8E+U>$={(ovXn1+EqyDhV1$}% z`9}ybwVo*YX9lp?^@(vAgI~IGy&h1a{0_6|OI}ty+tg80RkF@(qupfOejp*?D6_}K zyocJJGM&*gad_(Rv>N`|vo=qK#ukYH+m~01DT(H|_By4wwM1}vL5UG!-^*MtVW~B@ zyaQt=T)*+9W?-7F!qg|s>@C=g+XGE+SV{ztV~i|qrCjlc--M#aFWjaN(IY-Z&v+JP zM-g#HkVqcv5nqy_fPi-tecNDj>%(ncJ-rEK{1U^B8Ca+faYT!+7)&>*?fa);V~FL5 z?zD%l#-7N=<=CH%rugLV#|VB>+tq(E&2!&==z3+ms;%7QTMUfNCUtj9E0?!?w2aPo zNi?S_Y;J#n()o!q`u5>gF(Iwdz^YhE>O$FV{N1Fnk;J7LXOF1#H?$->>wQ}ae)bZl zp|bXm_#`t;@q06>bblBk!ZefYnGf&V`)-C_zxxco-qb^g2n$mR5abuVhR6D7DF!*Y z?=eD*+{#6&?;?sK2dU_a$Nc-wUanBIgcUAWNYX=Os*G=cT8%evdfKgJw7R+)NiQ+4 z#ntX2^BBkWak#^v7C!m$9L%Gx$RKRvJ!rpcK|we6`Bc>LuKts_f#P@g4(MbojcjXg z9u(g#?=2owfNDqzp2p&~Mud@``fONnZt5_H*Z4lXgsb;WLHrMV0$`IlNqQj2uHaj(x4N=Q zmfrc6^JZ9c^8JRGmd%smnn-KC+;@bf z=prS4GW)%@OeX#m5_!(3HU=X(8*SO0m4e;(L9#KvX?iNFV$S@=#N z$5&<-MkFD&w_3seaJPIoxpT;J5n`)%y{q?TiDkL4guv_L$S-^!2mY@dGyq=O{+Qc( zi6I zd`SQ&_5&JaQABPYwJA%jVh6BhqM*A}7?EQtM6tx_6R)XdraxmI+YJ6v zz!RlfiVE+~G+??G(3{>u=DW*bv@?UIgzS(&Jil-kd0MaMxO|GVY?dHWpTOwPi# zzdZRnO#apg`~=^D^6RSl-w*!jF>K-uz=WOY9Dg%$*r|)Hh9o73oBn@Z;;)<87$v5I zEg|;ntq1rI3OTxq9*9Qa;$UCU0IuSw50`Yl`*&b4+>n>ttZTB?c-^A!?DWKWc9Y8Y zDGr`I7%!>heQ!SeEmeJVYu&(EDN(a^PtG-Ivgrq39kGvi2ZSBUe-o{L{}22Gq{m#cH3|N=0hO5m z(($UunC$;)Kw;2;dG~Yvz@UGNi)|=}d${=JWcrs={yhNxDP-md$ajd)vQuDZ=syGN zPs{p~f&WOGd@zaszi&@d2Hm$ISAa#a|6Nt=UsL74fc~!C#)54PL&=4~MHojN7s$(Q z2SOdM{kxSH@bPe2?*tJ4`Sl6!SpRmP?p2Hk6IqV-N$58?3+yHAm{MzB^b$*n;QSFqh#JK2E5q< z!2)cmOETuhq@%;gr{9_5VkY|@FrAjFf{o^#095TBW-Eo5^*XVE2YRq-Ej#;n46l4L z8PDI2dix)V@C7bB`rVf-p-(PtWOLYt(cbW-r@=;ulSY`F$P)<(l z72Z_XZ1jLp;)hsz?GVF%w_CIU7${w1MlZKiv5^X}IbrAGqWhz%|0Gp%7@~_rBID!Z z10)$lL_~&{D=RC1yjVG0AG&Yd63&B4xI(e#PuF}n_fkU#p2yozr6i8%o;!~p*hP`vCh4z^o4^ zCMM2}`TY6wj2)np9&XT<#*&G=FrKP31AfkBb--q${C07zq{w@MFnl(9;H-#y$AxNozg!*I4d);%r` zu8*0kvxfkwU%q>Da@`<}K+dnkBi9kkd+rNEk3kyQA~>Z+F#rW@0M)`R7p23-*G zj*o`o={g)Q%z5*RilN^OzDTi~2KK0p$AeOPG^yf)?)}%KxG{B-AtDG`>@%>2&KTbd zz{v-<+Pno4MtRaL+r;yq7r;FHC4-%U54gz2%hLrw23UpccGZAUy~Cm+cQY?^{GpJz zrK&@5eNU}aD9K%9kpPvDkkFu>KCt!JpRoAIq4Qd&GOjKH-Q^M2yti4+`ImPQ(f)Hi!hGM9UBE= z9XVsx2ykzRDf#}7Pra*?_O7tBGo?NSA|yIzxEZp^!^Fs64!FTvCmp+r3sK--YlUwmU>4A{Y-u3|3SX_IPp3?j_E5&D= zU_LA?42ZwZzzJPqB~GKED~zepMp?#T(plFvxCIj|B!2PXg2Y$c@=W^j;(dvWO1~}n zFKQii0`Z!c(;CNFc+xt`MW_H z0}^KImnV)dWe`4;O^H>_rR2NTnnip;Y3aB(ul`F}KGuXe*Qa=3@KMfo)Pv2E@y3=_t_GJR z^NoFSTqXD+$RlWR`K|I~{EAzi+qD4#4Ga#S&N%3D6jsLr$gBJM4XE z*o_>{hhY@hBDzYx@1g!I8Mz#v`j`~hSR9@~WbtxsSPA_Q&N zw_Gn%$nSAiI5@OlwLBK;YKnqa+do0TCrJYOnI!S+d92hE6R)j5R{MrWLEGKa(+JiH zKCR+JE5RxEu8&xv{PnPg0gYe1Fr{*)+x>c1LcD;w=>KXYM760M>-WEuIz##wkM2&h)<+I~XGm1WR*hp!l&Ba zLoL$&0!lQ0G16ejA>qm<@5Tf}+9~=?SGt2cguo_a6mw@@oTnz{i}E0cC`{sTQ0lkl zUPv5&Oc#wUB%+_I1Yx_CcI_g$tyP9CIjjFD6E?ocy))Ay5;Ap+52W8qj`ybVY6x|F$q`3q34DQ9GX^3Ag5~q9oFeH&mJrELhY7V`GOlp zqw^Jo>*sZs1ACsr%As?gx}Pu;!uJb;8%J**=Bf-_hA9m^&09fQTUtL6$Jp)Vc9QH6 zBKH(ibVNNE1mD@MCwPz7SNn>A_=3C4^CT|{ah9jEdtEG#TQcqoR|paV0_D zrOvYZ7s$HYXMbp)_#yeZ;R>I+CZTGUpNjAXQ^`d(4Uf|>jjc{F3AYUkdiWy&+4)#$ z(#e!d0KY8l?z6_;Z;*-3xHC9tJIYb7a1mce!NU!IC@bDu4z#{%P!(v z@Y-R@l((z4iBW!{kyvxEoz3mrU4KwH;!I6Nonh1{*Z(tJm}P5ne=O=*+6xsWjN66U z`RdJ>ZV9eDhH4QPS5v9I>O*Srk|Yi9*GCPG3547;HNN^?;>_mng$)s1^?=-9=zFrF zckrSE4xDAX)}Ps=oT5mTF(vaHSOna+?ZvT;qJ?d_s@Lk4#xI*?NCr!RHyh@3F~O?vBfm$^Lf{ z>n+#pmvCd#K6sxW7HR&DQ0n2QCscMEU7J}uCaPQ((lHZ9KnIzFWwjN%Um!26_uq(K z*v#LFtW-3YG89yKf)f>}pmX`|+Wz>zl#JM2U3z&uI`NQ7nx&Hi7$3 zbQ(90o52lv%)M&N231LK^U2Ge&GMfWS(ZBvmfXbdJBA0Wz~FHG5eO?nhaB(We3i5M`6)eqL%yz&6aR&!Uw4^G_yUya? z@?EhlDQI#%O?JCGaiCb`qd dmn8#o1OajCe{sHk6M5hBu>>*(^NRF9!L=S zeoRJ=ZTD6ISCQiGzV7ar56Rcds;f1xSxKX${jtH{D&(H_)6A`F&prDPNErud0w#7` zdG0Nm_2On^q0g^(1mn_(*rF zWh-;D_cBJ<7&N$S01AVE>!zlw#zbH#$dwP7Vr=W?%vogRxynR(rX7QFA&yIltIe1- z_u7&^rGbXs$%9HDglkyagt?3BG_LqJcMn{l)tKzYliNHx8!_DK-yF(Q(Nks2d!2*C z%%2FcMyY#kL`I!g9ttscT>t2^A`hvlVU~X8{z3a_9FqmNb6t<-1>R(JqA-aqnfgsy25O>vZQ7}MLC?8h5sBk#<(=v zcW^J$o`+r9f!e#?y0QNd0B^fn?5ca0`9uO+klh+7zO=n{^&dk$P|awhP3*GA+O5IE zM__BJOSm?s5oeuns3rSU?4eh!qV3vJL{;tdw4Ty7O-(t~Qk36WmQeQmh8|}99c#KO ztIEoHr;^H%j-l(=ltl(D(qQlPrbOVW5gdA+oT)D6x&A%H+<<)lG{t&Lk_vCPH zvPy|7IYoG2tLYQxC+ZlXyr>=5kzz^2g1WH1jR8L@D3P}-VO$^GwgR7sUd~?=7ShmY zL;%#fcEX<}FqsdUz~rdYADwMhl&6$p&Vty)(@t*Q17cm@3k(X0x+lO#d3Zlcbg2cm zH6h!#fzPu2XzBlB@2lUc+S;xGK}tdpP)bxnT3WiKL%O?RgLJnNf`oKS=cZwUbO}gz zry^a_(%;;A&Uv2iynn&>UGH`AgKk)JuQlUdbB=r5V~p5-PE~y|PiUtJ-_axh-M>5j z3P>KGMoMpPS(lR66fC_VWBZNNUL&=Wh{6cLBG0h=vGInNQplq?wEF@7rv9+!^Sd>V z8}<^5ddO5MSc7xeA3sTahge%R?!y`gi@o)piyt#w%=XJ%uAivXH-FCB*<20Q*Bi5Y zF{#u;%$e#rJ55gXy5}K&Dj~F{dC%XptbeImXVafQ_;&el>EtVu3Xc@mwWaM$rP!9b zs7b?OAQNG0mvCUiM3U*%uYbN^z2#WO~;$*}VH! z)X)`9r)%&&z=j{K*YVx97H&Pir(Gs{$!*@WcCN_R8|?HS3nepU-zr)O%<487Netv& zpLj}>^g)t|iIuBHubUj$#Los&6=J#VGhD)jx1_JXhE8E~mN9zmyw}v{UcWWXZLkr@ z9F$n{2w!;Wr_B}oo975l_rZAYl@ZwPPG{~0Yr;{4`cJLq5x3u`vwChXd(!}Y?%MI> zt@>1p@9D;hJPAwg3wmZpohSho*-l6U`HGXah`(V@E45T!mQ;jBN)Ky+`S z4B9Qmje|{SElGx+b$Dwh?9WnQS$pyN9PdD1H`lpFK7%6j&YFk4`x}m817GsV(ux!g z7vVnk??2w5e|^zgr(EXc+__>~>RwZQI4p5lyJt)mrj1?*dO&D+)7djsZh#_%k1B{9=H@(+RiiSJop>F`x~C}Hvu#s`ALwe_-DNS z0s}3Gz_T~lDL(wCH+}Dc;0} zd*^?Ay9-F4)M63^{<|p(O?2ew3A-Ja`~L7l1k>SNbvU)2Yu-+@z3kf#rrVZ;F+e%2 zuwK{g>@?tP14K{*`PXAQLb3>MplLGwXm{C$-mUz|<#NTl0p0EDK;-Q(M3C4)Q@g?T zfjs<%hhfNhaw(HAq90bc8)*KqfR7BkOM#gv3KF9CWlAledtO6rRV|y9m6Ujt_}Bk> z_}>rur}OY+h1%8BEv&8CmjDbSy{)aS3o5t0o0>stq5a`9a_yaoe~$y8-vShH1Y6@j zWBONv=+s~sUKkX?S)6~PvA_Rm4v2cZ*<|JaL4yCf6FgafqZ+iT;QWP&{d3_U5Sefh z=QTk2S4(J=iD2N`{0hncJ)-{?g#Q~y{OcAZ;ajPqi4H z)W~=8JpMfg?0_;&Iom8E{r~-VAZ64BW)1pX*^_@awG5PX49u(VsQtU^h28@65C2}` z*1u;>>m$(Apap92-!t-`S&j_CE6u|f|2rW6%o5!H9gzR8R|qnmj^IDjeb^oe;4s>a zZWV*-St~HR-{nGI@PRS%S3q=jP6O_w+z09_7`Pi0V8Nrd*%SX+@D!QIVaH-;)DKwW zKqyO-6lzglqk&wW?$V?=MIYPIsL8A1ZZLoqKloIN|IhmMmw#4X;I8cK@pjxXN6JZS zR|TH&0L|w3P+zYIiypFYZx(IA4iw*XL=| z>j+E9@<%wur6arTB8F&`pc?n3y>;QEKS6rkeShr zQ-$#WR9((du1C&gHp1}VGYNU;NyTDA7bu z(Bq=z6@Nbcj%mpAw!(cM^-jy)bQs?aIB{C)wDn7^;%L>spa;F)x z-48@e@K=%Wv+qd1`~P!*UlJp=RRQHgH0P)b7#4MY-l^ONV${=&zUU8`TdttC4k<1r z&e)zxCabY1t1{2a_CUb6Je63`Km=mHye3p2GhSvxv3YLDA6!F-dK%HUHWlVxz7sCo zjt+PCKriw8lLdoUge3~M*gTLP>?o<5)bergIH!ebI}Y5tP*n_Bll@b@?_b0%XW6fUStgT~q`s_;Xq?wjJ@L4m8tzzl(eV*D3=P$>B5$r1b{=1)- z^76i}^duNJebf&<7Ux+OPI&g@;~oDuU`x0s{!54WuUE1^h5zK!ET%mCj#ZyP?}Z}&YW4$wv>J}~E z3?%K4nTO|tVSvbmwP3?5INa_SlK-^~Xs0f1)T_i?en#|Lss|1-lWu10-lpcP_-Gry zS*XAcim$#-7iH6piP+K|mvwE}A>Oz9LWZ1(GE27mo(RMUf3R5!#3X{_>(k|i{zBcp zxTpI|g1$FbirM1ft6$MK{iY4x~!r?@ifh4*3ze?A>3@Wv_yvw+&h1>|U?VyMXm2Jlz# zNdJ=%PLs)48EQy_pu)hBWI+V(W9o6aTkE_DQd&ntg1$S`6=ZCpbRNn_)Q_(AZ*TT% z0P8;*9I>Z%J~{N#vW)ABiHV7dzT;;Z44?}ekFozW=Wo>Dy6>|Zr%rj5L;m{Zw*YYS zotiyBO^S(+t`U@yKuYVV!#cVj`(90I+in(U8Fqy|qDl917`OWH;RC!1d~3_fav7lR zzTYtQhc8j2z83`A` zU*hlm(@UT{<`sZKi{=*Y*WLoOdRX;jq;qp}B$qq=D8hN{mRms)N;zxjduhz1{>DZ= z+UMFrhOblf8eD8NQWe^Z)ywqqT0xp<&fgh{fTQVj7P3o)m;CV%URf2sfya}5_PsHS zj{`{~fQUiE4x}Wf@l_{)(vh*t1O`Z+!GcBb!Nwa1_q?}$k%UoCXK)F$cc3-HMp2tU zrG*%=IEZajCH&2j(?%$;_4OZAdMYTW-?g*QFr_kDYfCcrGZ9}^<2eIXh@-Y>4om$j z8VrLbj^S;BCd846PC4CsPs=2+|4e73&&+!=-nJQYnP>V-8C#XBbXNatZF08PL!ChmeB)7L$-GFC3%el=dA8<;)093bl(di%Mp zd)4%KS{tO$_EcG9F>>xkkU8ySylvaOsRqT5nZ2UWv*JYu3&${JQqKZkz1xs$nyq+m z^X)kT7~`$xE3IwnglA39P?78bm!um#)Ckv(1W}CD=9hY<3&`va@t^6W$V=M>&|+D# z;m?iNDOD*c*r-K2p^Tqtcqg8aS=G6KL2d&gZVab(Iwvm@)W$>9G(t1$pVAXrSydfZ zL&;Y{1L*D7lFZG_%~Rk$0iHnUMD~X!H$JuOOQ6>~kO|WGUz+PfABB`c={2l)lIR~; zH()_I!V%PPRig!-WsBiF7i6VizP8q0I<;2M&{zo$=g(XMXl>Z{BWej(El{Mfp*V_7 z`GRlB_bll=FiE05Hl6F~(RW4XCGYHhGNl(`?zi*fL*bBfdQ9G&g{j)fQeKtEN$Ib+ z-5gjk9jrwi~Rf>&!Fy-j=W_~1}ao5L3g|uygi?czqwVY!6kA4s* zkcp79MJIs0g8LkH-xE6(WcH+(AMu4E$#m>^^!(PVNwgWCYHN;cEQU1D=`7il#gn(P zJ}@cqjY)H?ZZpjBYN=P$_od=_D8^*lVu`};i%pn)@7mJ5hLw2_#7A9t0Q`CPjR-z5 zJyiWpXmT?2y{ws8-sUR(KDqa?V#q}KN4En0@5PIKncUK#ceC0I&*S6Bbc+$|%bG8W z81pDf>e*!q=?v%t`I_CUB#bF3{Wz4S9u!zD%Fc?%B{1n0k#4{56+HhDV@Lq^0bxbb zh}P3h8&E8?4}2?l^;7gji13kTD8u-%mYF&}WafD-qfomdG6tKnkEfcCkN8hS4^jK; zPoY8nbvNHo{EaKKqXb|b3)?N{t;iE#NNXr zWB1ejT{^GW0(=h-7p>D7dL)0jik#oY|1mPGPA`{Eg=plL~PB=Y4--y-P|v zWcMIB_mJG-h2cJKO2!d-X|>HBx{zDA!K2|QPylQ{ei}cUAT+_yyPoFyNyQ^yyOcYg zKtPp7pFal}kRxvaQK4Kx)5NX=J+Fh#G*7XCkqNUvjH0im-q0N6UqA?GujSTfY&bNy z)}03>Rcd+uTax&7a|87B@~Dp(u+lbZhji}W%MRa9GW7-w(R8 z{aE*=V3V*^!AhJ$l=x+3Bb#Gs>S0}lcXyX%3a?JYm}aOj1t;i5YXQZK}TI(F#r~+Uw;}Tm>$~ zO83Vo=ki2hpMK7AT*KNs;cm`!Q)nig6CI^lW^wiLQ!b>R^Ld@16P()#@|SCR|Lo4E}t!Vm$OXsqR9k z4z+?ZyD!Ny$-YS6eX6uwJ}=XAvS7)kb58GSBT)01S@@;X(!Roqs)N;t%4)1?!hkl* z5O%2Ax}mp-3=dz)T1;T2ds`O@C=9$9?PsnvEB|Yz4>N&Xr^~mj7s%0{{8R#7MG$&~ zAHTY95L5UyU;K1-CXq5R_i=yb{b%Bd0e9a^BZ{lx`N|g{;qc4Uh$ZK@XU50`$M4Tq z8ILUXST~l_~ z+Si~7GVErcEKU|<^KCd_R&1e#YLVSCDJPAalDkOZ3Qmd{jxNb1NSmDRD-yo$61_89 z)3~D;ygod*OT0(lQ$jr{qDe>XOVFw=;`%nNN~coc$gEy7GfroO5KNx(sFqd_O;%iEN<~b<%wM! zAf&UZpJCFeA6sByf^u6`Xv%-a~x)yD0hgUIaOrwY5H z_R3_|@cr;_pCLLC-<}0i{3r5o;SPV)qVH(GQdE)4nsR!pYCVQw&`DRK0UIN;JRxJt zR??tysr9|Ri588H;*>4gs|+QF>^-N&)h@2Ws~;XwhyKhc#v*WIeSKfoQt$*X;efid zK(^`G$G1Eux&%(%2c1N6 z%?u-U5UBe+poyb78l29BYVZFnEQdQg5M z(1cC6c(i}<;Z?9HxZudgOtuJu%IR9Q{{WDY=8BujRCxdA4<&pMzUot64dVj^#yPY*7-vVjMUu zb1t(ISdTpQKJE&d@?1X-dE)0R(tyERw?I_QdZe5)&{Xx#`w~_%(_khPi71K@L}#X_ zq}m8`gWkQv#SZfn^HA2F@`VUg5ebAg?}#L_l-hImGlS|77O*HH>{l4jDCrT$KEOD( z3D((@r9GtB6q6oB<`Bxdhug;$b65@#m)Mt=Ok{gu-6iL=PBf}hLx7f*(Wy**FOZ(F zlF=1Wni(&S=G@@&Jhk@Z?o79VQ7fnAie^XEm^9#}U(T=Rd>FT%sxJ;{V6TnYd`b?Z zLzi@BoRl?ClF0Kj%GaW$t1LHa=j42TZ7nV>BRfkN_3%vzwERawW0fswP8ZcnD!*rC zqgoN|o$=+#;zUZXv;wJ?VGrYYN@d1^Fbt$Tnx0qm31}h~6RNz@x{5P~0_(yZBTrfyP2gDjE2IKd{~eTR75Sw}g6*5^%wER?3s%EA_R zT0G8JpXpKL>6OXadi0jT{(7~_(!{{GrIxZ1;wgFx#MI=4tYq4o(S;0y(Yaq{q-{xf|)T5q-og*Q{?EU=@@)k zg|r+NhEl(z77M0IdAfrJ^Gz`&1uTorHb#Y&;zm8qcA$)A_-YH58pYzH$zHVd&-`+m zZ1S+BLN%b)SQ65KmUk^ub9R_{? zcg>*!2}E^3R60$1uj3UO9f7;2oY)k&9DBQk|A= zEC&@Vk1IyW8yHWih*wBA<>1cImo*;36Soc`TyqWDx_7E)M2m!Xvq(gbj7f~@Kk80S z6OOS;9uDsG=49V*tydH^S8D*~a3(cY@z35Gmp$ajOrQ-arzr53eN@tu!^_v@8D5}T z_p)k==KgXAtnFq9*ej8Txs%``-rYWBcATQ74 zb=ybcePmKnS4F{rd4#XwV`pdF55r^3Q!Wikp(C|YCy6fL1uC|peUS6VgT5qe1aL_N7ejCmscz_#E#iQOl2VM_-kG*HHlhjn(p${7h*;L|d zT?hu^Y_K=u!U9(S#sB}dA zr_n&{Id%?-Y&dFDMs_7h{CIUs)W!pX4!qklwv=fTVBFmkN87zBv*K>IFA^S^Y zMC2D|2*`XALRIXv)+EF!SYi}d&-^KEo-tAeTJyHiU6JH*$vRbox^gXZK1<#(HM$WD zG0!Y&{)>q+A5+(7_!wq4b1m1}CO!=gz+6?%eU7nCHy>(uPcB!WOYJO3jf7*2aF`xF zMW8p}ZElTVU7>XN<;8Z{oE4QfcuUmZwTOELYaYPCAI0lRX6dU*(95%63lv2uuWoAU z9uJT_2d`9`u8_>FMC< z<030B7g3nQmLU2?yh?4XCDTSwSx{caSDcd{uo8}^Kbw-;)IPBQQIXy*Rm-`K^BInz z;#vG1*6fnTJRQ-zvSb(7=$#|fdRYX}Lu+cQm`Ec4!OM1hLWIObjmxRV*00=&4#O@F zd8%~+%}BqH4~`3~hrUetQuN@12Lt`{qSg%HyPgyKt}8j{Qx-0CK{dh5lQCtcG~nr+ zirjVPS4?>M?@Mu0{sJGEwv41F~1Ds4@T>^k)WH+8S+huL}v zHIRz6C8Zx@dNOH?a%Yw)#`r~T$tAKOyDyq-WNNq!J8R^}RcAfxY0G#ikH36%Zxo3X zm-`^Kp{1kPr{jBl((@91GV}G9lDl{GS$Td1pGU&tK;dkkoGtp*EUO}!tK1r@r*vnyWr+BSQjsPlbC`Pf*cLcq`n?M5-ZpA{D z=U#mfxNn_qQkOGrkm}9Lp)>Ynn|)qI*qCHdrCM(GGY5}S3;OQF{H${W#VrsQ4QZ4f zXK9Vs%FJb3x5%w5(|#p( z`Tf%Ot@xj1qhO=uY@q5LgSjToDbRJZs6X?zo$hvel{+~MSsK)*x+DuSE~Jy!|18tK zZCx!!)s^q!#ZPGfw!EMF6da$RQDPw$5<9QLW>sAvEBdhQA!6T?G2CW!lfAa?Ln>?z+#mRgR)oWL%wfp)@XR=)sG&QY1fu= zT(%s9+dh9odp+aN;rVns_D2=usEDY7`fW5dyN6Okrej!AMpL1%anK1JrOF+Mhk4vo zfSyu&@?|f3eAawsQ=W6Fp6|4V#YTH$R1hr5xFHXkW3-HyRWBA7k7@G>f@x7uB}bp> zikR6z{(7ImwxH0q7p)rE^YL}L^gDl8`17A zLBAJJq+fTu{0;}MO5<{v`A}ZEP0wY<6o>DpkcGJ}C;-WOXs&txxf}sc*yILjT7eZo zX*I{RSISCjK>KV|T6)KhN^4u_w{p?6W~k2;XBTwaDx<6^An&q<>}vwEJ~@wnlAgIm zs-4A|GZw{E9yqd6#ho0A%qoetoy5Q53Mo;>m>EPW!A8kkakEK!i4O$Dw$z2KOInL> zCbmq6&9LB=vg%$0m}xSy*Y%>dRh;xJDD00CseL{uYLuai$b~D8S_VH@NNZ_YOa0b1 zI7EzdbzC1GRgn;hJ{Xj{n=7~LHf11h5A>*;En_b4lAo6soyOP?R-PBxGq$kN3u@xc zoJYLUn&ztA&q0ilYhH7~V0__uymN;DHLl8R$ZBk*RyR--D!q8Tc-YkS%NSJ-hoH3^ z`;DfbfoI4ZaygiJi4X(+F1g zfjj{3f^#PGvoJr~vOo-12xMQgN9iUooDu0I4Wp@5OLf|6`1-xR&vq&bc=Ru?F4St# zmk*lmxTcMW;tLFsCm$?t6*1NM`yKy?Jwzb+z;^so$XQO1_Q$Z#IHz*2rs*;Hm#(Z1hLC!C<3Ugw0`dm)gDjf<cn%h%%Da378DB(W`C_GQ?Jgr)mI-;|Fb5xYxp zpW$F7rhfz=__&6Vmj8psYQ@*s!S^o&qUllJ=Y1X*ZIHMb*pbddgN)4Ds|BbcL}BX9K?RZq*N@I<{p_cIYXK1HTc1ntcUC3& z{>sS8syT=r(d4hv>T2724*As=8j3OpS?D&o{1#N!sN!(_V}~u+?0CR3>8B=}ei^sI z?aySoCTvNo1OU~~Rkt*9zPwDUaUtN$e|e-m%b|pzt$ba`SYqw)0M};&2uqTAzESjv z>KVxy*8Wi2+!>r7$q^*9E9ndf{&wq1JdEEnFGOv7%ENmHL0pln- z*>4Ti2jGukH&HnwR;AH-UC%Z{F_}IK9U6aY55zd2RZdnOX_xfnP%X=es_#n{4V|S` zC{{?R4$ku!6%iPHl4)+KM?a95`>8rwJJe7*rkW6w!(7VL@}b>yXr)sUVQM*f``rZ1 z+!-YGvFR7ZvnuqugKL5>RWy}5g=hPl`RS#y z3-^lTe!x1pxju8OSl~rBB;S)|F>xMAaA(znb&_kBBAK~W4Mw{<$Q9|eZDq4`EUXXe z>gEo$HjdwL-^fv&bWWu6@ivUulpGzeuf|>cFweO+ghN_tGp|$RS4*O^pjUaFUv@(& z|9V#S*f#mj=(f%Yhs(>7I*iY*m5rsy1M`Bu+{BH?cs3bg(Z2+(sAXH@c|LukPt26j6juTWuEz8H3o+P={IVQtIAcBzvx>;CgaQla*Ywzh zn7~&vvZv@^r8_w7dz)_4ih9109D8m8+S_NW(Ro(G3J*N^Gtr6w;#d-3lC$g2aFZuk zUf)2U%wybjkMV7;^iJK|Vwb49la*3=OAU*W)7hEmC`T)#cjpaf7?`UA_1Hf-9qFDH z3X~qCuCB6zaIv*QhLp=8*B8V0V>_*D%jq@F6zOFC7}VJz5ag752K;S6+Vrcke67x> z51`a_K}-m-+SYqpdu( z4x+byRcMlEGcPh2AaX=7%j)(<&QWN4=s=BoC`JIS4T{?eH*~Hqz{7*R>%9xAU-v^5 z($_XN@gYS%kzI0w|EWQif7Yt-JRgR3`aei%B51gX@@T-(4^FWH+KL$!8(SaCGFw8E zwi_Dym+Sv9ED+INegynP|M0g`zdr!Xoa7DY3XLP2lJlEaP)aRCWTDQ*JP4=D`uWkL z$?g(OZCv~~;TUcEn)>5}sc1C{TmE^nfpok&jA4}I3wc^dnA0e9|}%t0htSHXJCCfizAyXmJ58 zUFI*Ykbsy0C_S8A0xl5w4k@lWr1xzGV3UW>zESGRLrP~T%DVdT~!;ZmZ|uoPJ~^)uaw_C?=X@hBTgYu(eoo#vmitCm>V_ zxib5ff&ydsNCWYgr&&{Sy`LJ2SW8EHlqKb1g!6#6GGI8$eYBWvPF!AhWLJFC(K9;e z7?EcDZZ3+G{wWXO!eDX^MDoQpavw_iz|2})brzew8z}C9f?$pZKSV(d#IOC&D-olk zN-az2$;npDagtI}EK97CmX`}Br}kohNR)oAXrS)c_+)DW=uhI)$cG*COw)h1on7qh z;wK1?l89lF*qi5J+i$#|#KHi&7B6%KRraMqv+wP}i}V5!-D z1MHc2wFH1Ue_`6Z%066;Mo*BXO(vdGiTr%^^;zWmcTZc9{f(D!NOxZ$$qjhp9h22_ z_*Aa0ujg<0@zCV_VaUqdA$eroY#JI{-W+~!?^R#H-=t}ONhA-w@!LKcF54&xxxrhG zZh!S1sCIL0DU3B@AR#Z#X6ueMp8f2maKd`iZQ^qALX*otDTlq1gFFlVzH06IL;t0M z$-$+EVRvllXX;mn;|owoQ)q?h=f+WAiZ62&9Wc4LD;`mAdQPaSi=Cv&qB$`kqHC)@ zFN8D$h5p#d+M4?M`ilz>@9m&h2(C;na4s2qX^|2j7V6`(g_QV5ZV{)#IXr-Q8g?-u zmg9qdOZ|{@q81lUgGVQHz4vD+whAL3=RDt}f48%e{VuHI0emp*amxkuFE~W6yAZ>k z>Pq`bmDfMuBKqqas--AY5VpCM8j?kVg1>UsDT0Z!p@(EPxZe4JC?&wBq1w4jN{0+7 z3QXlF*{^@r2AUHaB0q=paq_yPcvhhh}R~Yp^{5XU4yH$0Dha-GlIX|;nz5kA@bn5 zkU>iSs1J}+0PzHJ3zI+c*?(sEAIfaWAtv&lN%!Z%-%-ww2n5ll;oe{WLpkA?7vC*)e~$=k^K2?W0-*gnvD-+dJ?%$!=J}KX>@&hWK6s1oT`R z;kP)(e_hik3_cHc>5}AfL9Y4*G!MEK90 z|G80HKp6ku?nnOL?*Ba>|KFeexIcqZ;IY9i0VpsxrIwYJzOk_Q$%MiOk8Q(*h~Ui` zA0HnTHKHaQ0gs)MpaOT9@(rIX`28#;Fe`kvIcLU#@h6fk;#rN^nwgnVvnF050__ue zu-3J;30o+ZoslrE@J>!HcV%bg}^ z09cry=y~G{2APto`8mwvWGg}62QU(J4ShhJjm^RLPN3C>)fefg?%N1VUB*XqWmXt$ zL%o6EM*SOOV`GT<=0Td^V%ni+aOOiY1AxjuGY2%-1~_m z4_dUzcDs zW_1M&QIJsoj%`y{S9c-Uu2xF3qix#+oY3{j1bwbr{qNW`OI_WZwGi<)g#nRg@##$Y znYb-zNlBbknv!<3S{C(Z;CA{&(^dEF%WhF|^;?jF$#v!F2*SdoXU2pB%8iG;^h7## z{@m3-NgHmq30U{=R9Oec+=Be^HmE9{Yji&nk0j?e`huq2f(LkY^N)dW_s-4^;7s=3 z5b1j<`qp^tpfln>V_buqBI@Za+IN!?Wot}l0`BDz5VCzJ;05OdwPPm%dnHwkNTd)*FE66W~?&0C~Bo&f*=2V&7?hZw&e`JdKS=TcEK)$5wMVJkYrfo* zEG#?(b8xa_4;9b^L4mTQ?W;)x$31X}1oEmg3oca0zXWV%**!y_8mz9a&Jq+*$;3~4 zg@gkAYI@eLz_32Yrlb^H8%%@ZcgSuDsrGn=*5^n|?akLIR|6iK z6?dik(e2V=6Y8zn#L43`JR)uhG!-Ut#rq1v!z^PDUmS#*<{D*%C97DO%4v>GSnWhI z@yIDB4LZ4|S~6TbgY^ydNKuO0+Y#Rv4!}izCA5e3JmYOkH@B@8)9Hr zbNZ3@uAo5G>Fjn*y667BYt!Uq(`3Cvr)x;RXE*O+#dO1(haC`3aUafGYB|nj-o+%! z4g>^g%P4V24cGA2>O^6K>;3IB-`g|hx*vGilD6)B+IPc~T%^tGkEgo@T(sWW^NDRC zh6t`D8h5GF2c4@MWLV^0$GFtxLzR)LU56u;3DCp4aZ!32`ha-X2|`dQnYSYiaKczTJ**4C~y@PMwp@{nEj5DIJ;r;uDIRF z)3>nwVq?gC(6Q^_sM}7cFQ&WP5%Xg&EmLd+f`A0~A=MfN`6pb<1>8H+)F z3|-F;U2m1$6rCjDX1M*G5PmD$XzUrO?GNI6SqA(Rt5LI zAMqZ|8R4P%P~! zc-e^9T_@6CJ8*fJAulu1rS9R@YfH?dh|(9`#-B=Cj%8YM^!qjsrhIm%j`wL7Y52aO z7Z);KQ)JAQ2`fX7OZ1hN3Z*>mk}e|Q=n+%o7$v1bsNa+^)1Y#;d&HR@Rx8_ zuE}eTpbJy8WAR~PZ$lgQccRh9XV52I;+jS-e*&wAA-Ie& zI^!KK(s6>-@Ix6YHKEVqR*mrl#vA5<&0Z=^9V`MAWr|k;9lA^npr_42|9&+Hn2qik1`$+wJ!lUg(-v&M* zfzN#zxn)4*XKcV)br}{$z3;%ly|+<4CeJ3iZ+m-cTlg~VhI{|uc=ppuD(A8qsf-*Kb#d|5k*BK_KvB4Cys*u&>~eGuuZ)A9Ma zXRKbzA?*tz5xkb~1Z5i9DNP8}^Tf&-R(uTj(sG8YfqRTp)IB*cf`+3ypYJq2Z_lG7 zO3z}tE-!(LUIb$?bGhU6M9#n1Qd{$dLWM5BID6FHLhOEUVKVddS11Dx0>izY@I z(l7Gm+-;1SF;e4;1Dd4TIT+K0Ow+mKoq*G>x;S%?)DF2$ni`U?UGIMOQ_pX+K_9o2 zuC=ZQYZcy5o0&9%cbqkTnBZYlByQ`QfRKivXgjn3A@GB2w+LSWxDWA`>t*XIN+#F^ zk4Q@FJKOVk`{s@+`7N3MC>r<~0#eEx(v`$TxpgwmWk@FN^}R@w z^LCZh65A%fjX1_;1n}qiP;pz{GyEl1=q06hQlu`W@4kMX+r<$?`_NsFt_1|5ow1`4 zd-e0}mKzAw)pzEbr}$@lVpMYQF198|y5E|zuDLD`!ui_M){m+&K7IPJiSEGio_T%f zs(%QL?X$i7sX>BGmsX8<~KF0dF@9HE!b*#QS^}SaOSEx?J z!n5nUmm>b$W48_ACWx!i|1PKai_h%7RgC$v!Uxw(h#^%3opJMv;pZQ;+@zUbyYtvZ zJikTD25pl@muRa^Ji;I!TavAW>Fk zto1>qX50Mp3p_pxgYcB<$fho}m(x>p8&+H<;&&S4QtQY+=R~PAxwq%1I!NnlwS#B? ze$ZYataJR1-l+G7Z6~#D6elAK_gKGm7th8w83x&4lv91WKirva7Hyx*#{qp5eD8Dax{4#PSyl9(nI;=s8ksz+9t_*j2T! zDs9yN(lgt7Ou6@Zd*qiER_?h=@yRM3&Uu?ePMYi& zZ*jx)x=gN0_ZMw*4|Q^#syJ-VgTa`D2Whyr&Y2p#j}#XSH&U>MPdpI=HvIPpi~X+! z&&ZLUyr48Np7SqWZPo}$gd}p6u&hwnZkRxC`d<)Yg_h<Sd+I8&^phkx#&-tSOYLfs=!{=PWP9n*+AEB{gYB#f(RQ%>NKi@K;;`#T0_y6i}i z(l!xT$Vwq)+_Cx>t<3>5_!fJiMyD?wX2ecz|Exc=GVg{)(1xCt_i)cxpX4m zj8v_>X2RYrOxc(4J(&mv(#w5#_(bCq^6M#}9&Gy6$^7OL-9Ep11XCLWN(;fkQcEG% zw2db(?X1YVUwmUxl_o9ju0|A3t-^4kN-)xKMbMXD!8{=Fe8_Y0=nL-MdkW4WxsrY; zg;V6-rG#pMQhvMDvfV17%?(tJT6{;g3zCmF=?2PRJY!841VboJS?`O)5|39<)w|Co ziarT0XTGFsGA(e`Q&d;H61#j7B<$(NfRlI+uXw9kZMCziM5VfDhmoaD#ytKGBX5q| zVwH)V%rKH30domg`MKm{@8gl(twGe!4W6P}+#HZrh}vPLIANphz*OybZz9I724u;& z^43dM=?SgX6KHOY*1eQE_f2_jo!EBk1nRqCAMV$WmqBl&6JPWzSks;;gHvW+%A3kJ zRkiQ68&>mvS=^KUmG65&BhO1GVoco@66w4%WEbQq(3}63Ga1m2oe{Yb# zWfrzW(i}*-%6U&h0-=U$FQ()rcda^W$FK;>69}TmlE(!bEG^f<%+VPlO-qYx6brbI zPJ4EE2d1^9mAqgdrDn`(yelz|m@@+TWs z%&HFd)Pbh;PQUrVBgXfucaDYc zw~BN5d?WH{BUcZyOr8t8J04_pYul~pWo<@nKybEnv%5rCkV(q981h)lv`{~}rsSEB ze2R~?+lSh!sUEas?ykE<_r6WM{1AK-dHXApH5rx}m}k)MHgu_lhYjB*@P86Q{z-oO z-tiq60VsvRov5!Pi^03;7OTDvO31n zb*SO8P1*d*y#5x<7UuND$F~QMNoF3^&d)7UpTEWjFPjQV`q~LE-Eb4RljX);c0of> z4{APnxF&mWH$*nNmMn@fBg}@|t0jPC3DLIsKD*L<-~K)VcZs3+O#iG+-mlsFy%Gz| zMd}bNjkwvRP#4Uu=J%8>66p>HMJeIrKo`(2fi)auI1l#(Q} z#}AJlp+YCpmvyjtvOfEY@r>qu3BijI>BjFEvTC3}KEs{DHOp+K{Q%Mb__Wl)tP_0)*uQ%zBOwV7`sh4j!3~wwsimS`_ zktS+-KoGuSj0$Mb+n%qq5+(QAtY|X)pi+pl-GyBGyW81R=}T*SBus~PGZA1|B5A}- zPk4t3c;Tm9>mFh^0Nsp-Ws{At8_1`%eWv zxtTt&FN2~J&^Jfw&S%<@`n7Hf3Cd}QN{6OAZ_wWtXq`bhvUQ7i$uJFMd+StWdP#LP zF&;<^2vo*V3vnmjpY7f`eKBCbvs!JI=(jLLofdEH!l(PiR*e`_0ao|9s!Fz@_h8XP z#lxX^CeL(6_h7L$I=Kc*y8;Wdk=ab+L(iiz!=vM^2z@bQ_E|@8-1po&CH$Vkr{_6i zfLqaf*7n0!?Tvnu?aUWlmPzs>p-rXGTK)bo|0yRUy@6)zB^8o2i!Ilu$mEFB495xM zrpvAQCQ9$Fv*xHo(MiSifT5Lkn@rBs=sjP^T491(EgnbYJ&~!qBXXWqwNrd(G0%Fb zm!xHW2T4vgf3)R+t7FG~uyX0~%PZ}vSWVTZoAa*l)CW&L2A6+fW!n67sFs!(7)#~H zIAf($faIJEA|UJBJjk zy`tE$6+<^LRI?D(YQDZ8-(?a1czNvhXk2Ey%>SETH{TkIiKD*Q?+s~r5Be(v+e0FX z+Q&#w%9W9)RVBMa@|L&U@ev$n`DUzuv5422QC|M!0U3LQ^9B$%CbVoLaeRQ1GZjfR zY;0Cz^e)nLap_$)7QZs_u-4AbUx_ ztJPkyBw&Gnf@C)I%L?bcHzuhDE@8BJCW&*kuh=qKc@N&y^_FTlDm)UAI<6pUoz)W4 zh$lfpU9kK=)xCvRl-v703?r?QB8n10dfD(ct3J6GdNJzsF zL&pe+(nxoRbl1@G+k@ZFIY-ZV|A3b@i}ei5JhAt^?|tWW@of$%B*goNqINnDIwZ@? z6L#jG1@+4req8GEW6bOL!W=e8{K~9?vmdbES$5=AGeQkH46NJtX*C(q%@#W-rS9e5aP+@16qneL+z~F4w2);CiyA@&ZcTmEuMG%F8~b5H+Id% zpWWCuuYHFjAD^pdy?>?_7gg0Tbv7Q%4SDTzOR07nUW;BoA|%r~349s30}DT&Vz!4G zjOQE535~Cg8ygI7#vI{#k!fa6-mp&lMR!kR_VZS+XoUe)fHrsgOYZ^&&uQbG+ zL3CezwPLgQEzsTPfLg~Wu6^p5<6==r`D=CS%?umadCH<6Vb%RVWa1CJ)bav_HS}PQ zuCI~R9v`jFnq_($Jn3389*PmGQ&gRf&%P&B5NVTX(bQbW%`Zjbz$z?aTJ7k=TGFu5 z$(YK$tVAw+mR}w<8&7?BXcE2q_Uc%XuvgV6tIAXfzk~k>5B@OrLc&_lyQTLz*4D3X z{$OorZmF|;!CX`Mu8BL*bU;i+GR>73{tPW{ES)=l2Fqk+{h%Oo=+jr%_V)(~J5}jY z$<22fD+wR*Hk*|rWJw>S_F+smpdaG*N22`OOtCyDywJfui}M{UL!B3L7bleMD)O^? z?5sV7gtD5jg#*L`nY9Z$OkyXjoiZaY3(XDV21muLG=f$AsjAjvvEO-3W~VF zT*4=~>bc|jCJGL`%mEDo#SCi)ErzXFbf$LG1^>&{QPt~C29cFc~5|g}?G;lG}#So@59V$9v|2jx)A;WJnZjbH7 z2HUxuymB)WuEOJ9`&s2di~0NtY1rz(1swqdKdsJ}>NrcTi)~48m51sI!JG>)ZA6rl ze`AIo`sP_0>sSrY(HsRUAKIXcO}L17Z@{7M&N}I7YcBk{e{`Ian2uE`a%wg7R>9wd6n3 zGBLew_OQK6DA;mTfaeFp1ec&3^Jxu^k8O|FEKBo;DTA^{vXr1GonVR*0{&DkRWmUW zO=Dj#3b%<<&1_Wx@#xn%S(!EKv}xR2#kHqw+6$RG?hlXEEi2QUce-QH^5zeHmQmQ^;lfquF zn`u~$EV+x6^)Z&TA?`4J9XcnZxThKt^L&!rA(Dk1PE)H8N;GJG)3K{UFDE+TT2WHq zbP7676#eW~8+{YWjDb}+x0M>?xe-4_Uzn^9A1mL&XP$1niXjuO0%&HFZ=-@I`+|zr zC8U~cj#(-{yfr2_GPn3MQH)}YHn+)c!|v<5svk<4zTpKGo>#Q$a5mm!6xv?YkMD+m z;A?1|RISyE52$h>YrJbi+c-EdX)?*PG`;0#^-i?Aa(0c~oy0(176;dY2u-c|Vef+8 z=~Y+@aiI;Ep&Xv!K2w#FsTr(UEI|ATVp98av9{d?w!YsG_MqikuXApjVwu#NPon-M zb?(B9T(d7O1`u2%j2hoxe$;CP596;rlbE_%jTN! zm9sJn%r#vO@f<@LcJ*ILa^w7LkmB}1$FK3y8|*W#)@#rQ$SaSQKO{%T-YuknH0Sj~cEb6cT+CL3Ed?hc%FcsrMmeUvm(_@?U~_R)CYAc54g zI@g%}ZMxf+nlT4}T6gT4TL2R`iO)Wk}zqKb4EJ z=Kb@y4{}j|HSD3Kf=UD7a-bSG>u;pMpOl*}N2zNFz~3~*_pP#)Pmjzn%6S$f zi);+muW*&p;J&-4h}qD(ew-(`sZIxOEt@o_6i;fiRWMf$k$7J;;xn7`cDFu!55QWC z;W}OHgw}@eSo`IeF{TgUQ?}lXsb|4SuAov0jDPBP5l5Q8xlPi_jLJK}OLSc+#~O-F zdefK-u}6xK;0x@6p2S-K21Zogwd)2O=>QL-apxeTUjUg z(rXhR))sM0TY;enq1L{If2I>uqDGwSp0N>8*^bGrzG-*Mx!Qt~-*wsZ6nST~N!x(n z3cE8E?&_6qJ!?bZ$ji|?mHs5@17IctZW|p*z_mTc!PbcIc2&LvCPL&Aeth}XtjMrT z$C1xTT<=5zqA(S@o?m*|%voq$My43)=;i_rJ-bT&7KanL(4cyYXF!{G?Ic3-+yAsR zfj!o;|D$W=w?F^?=34_Y1Z&lxc4KGMIe(!&QfwUncg#ja?w{}4{qupeEg<7X?b|#k zF~6On#G@qusP5LSB#I+DR~Q3eUKB(2m!JIzR&o1IL#N#@$A!bk*~cZp8>sF9-1h4$ zTa%y-Mb7%s(1`@&@Y?eGTAT)f#^(0E{HnIBJAB8Kw&E02T3H zY4(4t+fqtx1^m{ZdmzQJo=(VoHSEO7@#{G1ZXJZ$~dUp(lu(r$Gd^wWUt;i_U!7R{=>7zf6>qt6S`?Jy2!?XjUcm=TdO6 z1)A4X=5Yz{I=56;vZ|z54kwVp9V;}0y+GGV1OW0zPQifi*3SK}_oOyDWhf*;pFFXg zmKKkqh7NYnf~jt}^+&$jG?ze3^x+%`Nr97Z+%aFU+KjB(O6zN1JsMl>jY7u!wL1Pf zzz9KF*gWsI{WQ;LPM6xwF-4Kg-C(Y9|7+fZO z*pl$ZrQ`k=M|5pB-?E7VP981BCm4L5pDav!BxVNbJvZQ6AU}OvoDIQEr?tUmj4yqq zN{MmQ!H#pyW_-lEJ%BjxuXqyN_IzUcN~;=gxt} z({>a@=b||fFR?OA`<<5Q@vJo+H&|M)$LqE=og;GY0h0&NG*O4h9bP$uG z(F$Mg1(AY1u%*~Fkq_E$aaQXq{Og*y3FaU~Ife7EZ-Xj~*i&n}5e)+qC(&)twzA-> zrS%SQ?9>iLJdM~*dPQEbeN4{V3A!`cBEYUu;+J0F8oJb9gGKw!&-z}DeK8G+nJQj-W{vkdg$eqrCwuV zj4UxWcE!GgYkJ#?=@^H)5HGqYaX+KXOs+Z0$~dfbKSo-Oxcq;~i@(*|l}-}wY`nZn zMr0y{{f~6at)=y!d?Gw&UMcz=gynq8c+A@|Mv0SyG}*yq=&nJnYc$yg~1|`Z3aHRuUTtP zwOyHPl#o7et%R=JA?_6Zl)9F}=i6YhIwBz}iR;t-Qf(Zq&jMhvkypf&XMg z4m_$>teydV@c{iWft$$^zJa~|>dB^J#1c^Pc3Mo^8TZg@S&$OunPCa3hMnmrr4b*4 z?*Cur?!-3xB~?iz+|h(lQ9^BKLWc6FN!8WII5B1l_CaEQJ=YotAdhcDf_ipnhRkm%+`sii2TSqtK zSjLBXt<)(A%%$Q!DGZNl(zs1#kl}Tk7UM5BbZYquvMoS)&yZfu7bqL+KW^*f=CJ$_ z7^~;9%~Jq#Sz>{)Y4IDr;oS4184J}yWX?S^kp@najZVa`x zGEMj?hKbEI-Ki81?+7hak&zJJkNS^oAIai1?dUZYfKqk(ebs>#h}odlHa@+!aYv=) z4|_ob;)N?Y1?+&O`#~3jH%q_xWFjo00q$yr;XJZ2ZLcL!aQbs(w#{eH)#XdIh=0Zlw zbGz9Z@0;~FIv2E{S~vbZ5aZR@%bLx+3&_~JfZ)Rg?$vSouqJUGrFTkEd_%hTWo5!_L{t@ z#bU6|*#DlHx6|5F(#VK11k%)Be&rAT1S56iWrYhfz``BV00(LD5Q3chRYOc5yK+l8Oa4D=Imad4?v6d`%zEG zgC6uc$$B^0Q|kn}Pd0|ySc~TIBp)Jni_h5A3W5T>Rk0F|n}5u%6G1C`n3q9vylv&O z$Rp17E4M|vl(VbV3EOQ65KZw>B98nH0)#D}*R1{c@SlPD@aCUseD1zKSPWq*?YvLE zYt)4=@4RIRM9oAHJVg)d6x>eU8SL;t(tiEEpQ+;R$J#&Pt5N(FEYH_J-8MoK^>Q`x z^|%M9b8;`XmTR6)F-qLdvACGZZV$Ifr2^7KKD=c~gPn?RENLFRW+xiLZM0 zCqA<}33~mlo1zqxglInxSGoC$Ciio2Wk_UK3k7ux*co(bO-?(vrE;{sT`0|og!J+k zy0t+D0@&Zg584ru7zLB1YT*nlb+-d|!0V@@hRjql!MBYgn|}5j*xV&T-4Jw+)*c&- z#q)#1$jm!)MJ}Z(y+TLwe1UMa+Y2G%>+a!ryti}&Y6#7H3dtkE&EY#_{nNU{8JPZf zA7HOltyOP*rwU@Okmno1S*$G8eNQsOmi3%&-Jr_ldO4MsqHkHrgpS%fwQ}dX$b2az z&3ttJ%A3mq>{-+q>xY{ik^KRl6UArGp6S|2&zZ*A*=0yU`j!^0a*2$z=x^_8yrrWw z^c2rhp5e{S#YkUSMAG9|*=DFc{B7OgqJvTAEjbl}jzX_T;C-Z4iV1f_L~7oXJ^zy%(D%2o5>2E53iADNkDTK~hm)y<04 zO0d5Dv6ADh8M~`W5ApEG=QUGh&!!}{$>Kcn@tx_nDFrqIQ`62ogz4vIrM=VJJTf{bJ0Qgfh5S(un9 zD>H6ElO}bCYTs7h$TAJxO;U)+K+c0#aB7IRw|eM(#BJzlLLz6r!k|pG@L|f?I{Uu9PH*MY2m5U}V!LuPR{|Hc8fU<-lIn1}f{t?@ zdp1TEYsQyad=8`q{}&(t2bGa{9P<4S#Imi3u2@hDtm{zQzSoLA4DiL+7j*M-C>cvG zOFtA;K1;w)pDjy2*=6m%FRkBG$Qfg7alNn9f~%zPX^U{mMZw$Of?7vuR-NU%a=Y~D z*gMu72zM4Ud-L&(SH3<{`AD2Zv6b81T#LaDBKJZQ4^0WyAi*mO$!;R(dxLp-?9(7p z_P2SA5u?-&pRrl~aINL%i1EIsKsv4Xvg-Um&3oE<^qU9J#d@BpBwjg_kGh3{4`ZZe zE4b6pAK}lvF1AG>?>TTd(g$25#BS%wL`5-hjb!cMnE_yn+^VCZC(Zwj7iTu#04Z04gfMl4Qx?i@}C2a8}qk# z#711^>p!6@yLKH#Yumh5P5bBIcS_4|iym7fbW9VTeQobW%#9hkR)2ds&;(gC-u2~{hGWz!q6P_p|jmNFoyau~7=5yu>-*G?_E!_r5q8{*FwQQ7h# z{?N0yxA+7K73cjziUo}sizmaM^330`nE2C?!r5Ft&;bdb&@Bg-XYNExSytxs7_#(N`L|EYOwbpAvB@1OMi6qY zBk?63vTLOrq=x4}4N5B&Wd9g=+*^L;PFRmY1-40H$1k&if@DHLf8HAr)AL2W>pV+m zWYqtacmG2gZE~nc1+ss6 zWh7o#Z@(9@*Y#q3NXWTVs)KypWbmvM?&*qOCAoaez)Px0NNbz^N=!HAKWBsvpjYX> zRHua+!1a|1G~JF5^zo}Yqz<;|%)PT;Vu!5|+rR3h{&>#Hx3}&5q58c7G8k;+R|8*g zMbHj+>*J(YD&^z(tZ&N1UPv){o%LUJLVxIIS*pi%MZVpf>;E&Xa|*@!`7LD^{)TtO%YPFn=NQXdvHBiKwry^ zA}dyVSxS!@g)9vK*1n^Xr#s6xIMGJ>S>wx}oo&0^Q=bI>@l?3C+RR?mA3VZ!9_cz! z&Qs413E7<1%A>C8k}$$IZ;@QXoC3XZn)uHS`@Ye)LAV>|nQQ!Lbb3p-{!>KnkZgq% z-OVtLC81@X7yp^&O(8ayL|s%U!R+~(aqsKBF%ZBV-|s6`_+D}@x183nSG_F1EF{1Y zz4V`_%OVL|uo)`a5Cadli|8sH^dP)!ek`(?Uo}pPDPH^p+Qh}+S+ z#9jTCgB~y4Z1&x`IGKH$mc5yqL#U6uX_JmE7 z7w4&=m2)_P|KnWzx4a54+|@c=r5-Rmt*SQn1L4Fkw7EH2v`@bOM zE+a#D*)#b5JFY|GB}~Ka&gDvYSn>+be0#LQiEF+xnOk26uatG@>h7iS-9`|rZV?}P zxUE6xF3iI%-4gqk9{tBbtBg@{2DjIIzXmaz#TOh!2aaDHB(XNnakX(0ti?fDb$-)% gztpnLZOAc`%0s9t)win+c;M&0tn%HwJ4P@6ALM&%?*IS* literal 0 HcmV?d00001 diff --git a/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step5.png b/chapter_dynamic_programming/knapsack_problem.assets/knapsack_dp_comp_step5.png new file mode 100644 index 0000000000000000000000000000000000000000..b84cd23b52f075e5b9b49092ebcc6956e7a82b32 GIT binary patch literal 66215 zcmeEuXIK;67A{>x0hJ~lMX4exy`u;S7&?R^AYD3ArGrRSKtZJUo)C(mC?$XjC@r+m zL+B;+5=v-yEZ_Og`Of`&f87(FWQKWW_Uyg(T5GTMzH25*Q(c*of{B8FfPhlv(L-$l z0#aa?sF$1w*vPZ15&|1ScWvbd1Z6!e%fLUctPNCbKp+A>V4s|Tl<*1x@y}C$KLSFg zUl;$pM3s>FU;EmGJb%ss+KjLxAo+DZ0f7;){rPzce1Bf^drO>6_;-uhM1P%4N{~(b z*FI4%a5h2E!~6cg=Dh18Lw5oKs&7BPgo*rA%s{^hR36^feL=X2KA+B_h(6nz^mE6g z9mzg=Ky;ok-j#x)Lx4m2CBt=DS+|1&o&&Jv;Ba1XWHB{S2kb@glb7UfW}i8poL7GI z=zBm`{pf1IaT7dIS5l@PE>tfggqFz)oEKRx13td{tLw*w5a>>7S{f@o$t8IL!hie8 zy{(z-PrS0S0);^A9UPdA=oNd6wzs!+U;2nBj(V6tv9!mBCL3T$&m7f81X&ob9qTHqH8j$8#kABU;W?u z{c9LTbdl6|Kn-?zW-uOtA&dT_-Q~PHpHR7vEz-6i;Sw&R9~B!Bq@<*xc+YdqlBQ{r zF?;rR`4>vZ9=ob~UHf}LzxLi5G%F>EjtKKpczb3Kb@yL6TaO80W@bh)Khf7O-0FkT zPx=x((b0kX?AWkoFNqDU)w|f8_p}sLz0;avFx*h%OLXP(e+u`~Qhx9klf}BT<66FNdb=xB!Jr8!xV~5%*3O_x@lRnPE{xn6 zd4=#5dFNVs;ghu%L8;<0G1HrXX!a@_ z=2zAI1_i^Htc>pJ4M+Fbih-6K#1b6YM*pElm+lZ{e(Haf&ZWgzWiiwpJ!1~zS*p%x z7A0eo^e8?GoFZPoI8+Me>tsC{yDBb=_l)@Dg)VdbuVw&=o%aCc+s%S#_90gAT#o^6 zN1^0|)RK&di;Na46>Fv;i<>R9-PnofN ztEiC5YrWOg)m+LmV0WhZzGxpkSd#?TNHkJPy^}; zw%EDRB9M+{eJ&`WJ%2@*tJQHk`aeBUuzVfq<4AiV?Tk*=f>e+WVZwT#-U!rWl3?V` zhdo*=q`CU+`H$V_|ND-_L=;+1uJ-zmdm%b$*~6vDn?w1l)WIP671`zmA-&q0snVjh zWK3MxJWsd zGYY!6D%xuGnzVVn)CL{!MyC0OuP9o=Q2C!12oYsY2?q}3FQINhP{3FQHA-r7y)tf_ zIehttXOO=`>bPVdYKR&rOG`8$aAdfdqFXQ$XjT8;_CUn{bjHy8h9l<_oyA7N3hu1E z>clG}h1z%jQxig-(6~l?+B*4^N$Wyg(c;EnK7{7|Os4Ah58p<_e~GcuyF@jj?cTTY+)|D6j*^2ymp5&M~R}iHoVTG+(HbmX} zpOg2~yZG(z5qN*RMfG1&pCTe@CTx<|rT))$o7oA8o(8e-lK$smj{lwhKgs-03FQB$ z1piZl|0#j||CHc=O7K4=xCAIc$b+Eh9NK>mD!)g|qt%PH1B&dcb!t&|rO|ph*-#zn z=4HwLW!X{Q^@K##=h`GMdF2UMx>g>*{w-*GrjhNH24C~BMc#8kgrj3m^#bM`{aThq zTNqt)btW)r5l$p-+Q9_k+=5auO8<&G$*bkv;c#~Zd`+?4ZfrdEcwOB+XJGK^c$c8- zHdA_G$4K3mS((m^Q+9WJm)G{T1@A5Lxe5Y8S1L3A)YyNe7?+en7L;rU@_N?y_lFwG zei%w&O2)8Gs>6gmo1^NLTqkzXp0vZxZe$trb967@l`-_-*C{)S7;88k5) zZOZ8#)D~;c9KG}rbGjS$jUbAsB-sUz<+V4mlf2(wlXi83dZyf$&pBtj9-O$uLC_Pk z@|gech&%UM#YsV9pG&v%VqQ{48{VL3{Vh#;4!2|gwU_F-Z$^f2)aX_w{M5(x>xIhA z7t1s~$+PMGY?Z{GpTZU39B-jOX#3fB-cIaa*{J*y31nD8YT=u`Nv*BRuGE3}Vy}x7 zI-ISS)(MN*OvsQ~$xv)^6GgCeO0Qe0Kd&8`Mz=|rgtNZQq9k!s3SJ~w>?$bC`For> z^u1Gy0vg|pvJ*whrD7<1u3@}l>-v~a>c)%j7%;vErFceC4dy=Rik9MDA)gBer0x8~ z)+p+4X?ulq+*DutV8B@uu%x@;L3NfRbow%XEZ?!CRKX*go-Q}n;L>*$!f^S;IqRan zzwRvmigd8C#P;l2Pf;qavVGc~cXj$bm8b9HSQ%QehB%_JL!(jw-`>8V`JMx$6sJa< z6W3=R_d~bEye6rWwl=o-{M`JOm-6dBXKqrYsP*64h3HEeXL?|3SbM+WO=3^i0+GDy z8}Bn_JXn32Cp8@hH$3czTJ7uHV+K5v-8Tz(hm9f}QyHWhFiQqnd=XnO{#(}sdkt0`eQ$ke*pjgwHd@Snou0tv%H=<~EHT8(6Lm5WEt*(t(vj_G zcQH(PSw6?vIM)BH$v(CZ6 z$N?M;x)W9`Pv=05TK3Sc|3N{dMaMx9i%l#r`AdlM&g8-(;!&&t+dy}wsS%$LZJcCPbmzS)9bnj3fK|2|TF zk2INt|7bXv>2Gu~pRdAQr~75QXsJ(=#h%t{>Xe0sy*=~tp0ufnef|FVC!#yW*2+Dm zl?PFzJBxZT$rJ8FDGK^h$ZOAJ;Ao!yxB#YxfW@%Pn{3hC?TBz$~R)p?R)`1#JZ(x{oS7Q~^QWFJ;M zPsRdEYHoOaU!{|5D64yZ$SehJG7cOHies0q@q^4CWH!~*Y3Xe@j(A-dRPn^d-zHsK zxIs)bP4|@C<3%~E$iIMsF`8n`a_LDPeCvLZZ)2XH$;xYsawBkbf3?^vHA~AK9q&@% zzH$f#3O-k)EGop8_Ek?kz-d6oyG(c5kbAUI#sPlUc5BekwungLx;(XO?2~7yfBC=9 z+?us$LqC^Q6*=Z4iSfxe8M&HINjgQR{We8sR<^G1;0tj*n}Xu4y!#n8Np9$Kbn7j3 ztv7KH4NuRG0nRjquIWO>x*h$nIO$JdFXp92Lh;b*zLF#AooN*{#?pI54>&{!eS>v& zuK#NdxI`6F=xh6xSUsQszH1h%gJn4}UR*uwVMMl>R+rHhIFxVm*cN|SW452b3GhVX zMn#45rCoNtjl|ldUd(m z*Gll4d9?-nNO`e!Iv%idA>+QX6mguJ?59pcpC+LNab<}kcJ}L$Ra65YhiR4Twf)2s z4g1bc=w9@pb#}6b2hZzE^8cORRFK``%D)jLjX%t#oe44Gk zA6v7=5<);EUv5tY3`8`_46^^FZjbckYrn5QnOZfh^md+ew#<-)Kk!y4Evlr%f!GlU zc(%>xs~pOWM!B+&8xlPb%1r4au{7iwVipk^2Ql31XO1bLX=9%(s^B4M9OK zJbdK<+k0@f(#Lq7TGf2k4_!uRiq|YWEQY$c;C86ys^-;Y^jvhq_T1E9dnYNBUEb7k{?S(+fXreu&^Ky z<&co%eSGjW3pQ3YlgRMe%+LSpC?>TRZl*G$@DU*f`|2@Z^djDaZX7;zEvr?BJ4!*6 zL=p(ZCRF7A0?0xxQLLUej~?g47d(qRB885Hi&o@|?BVb|jCwa2=R*^!=np-fV}T-MmsiL6k@aY&>Kdig z8NRUO3JJjvL6}^(m&Z3ofV)zo#r(|BI;Tx2OV4OWvgT3dGslBLf1;`P7!{UFY=lHp zn@{-de`DP8siZ@thH$SnBlq~}uMu%q?_Tfb?(*1TzcqrGs-$F>T3-V(Q<)rK#oK-I zs+k^3TIlyR<6N3{-G&WqsZzwDW)40tlj+$Ry-GjuarY9s;UFgYh$TNeNy$~N1*wzgk=#W89G^+NQY{>f6xNJ<1 zu1roKV9Hstet=j#-}`~z4xgkVzL&f8TP22U-Xbn~6r6zbNrsz-kch87$ld`W&kV7A zlHd8j1HtS_@axo^SWI4=EM5w87^}a~oo0JQ0n_k-tkkz?LO4G^Kc|MPk3$?V;e=5( zI)4)Hk)grWC9`<7osbmt{0+kgTyX1bUq*37qRP@B!gn}|8_nhnL>UZ^cE_7`3UrA6 zOcRNjyiB8LuO4^B+By9?dX(9<=bOfT&Xnld?=@+oHcKN~dz+8TK0zhV5@~JRGQFSL zcuS;|2E{LS$a|!Q=-dEeb!1tzbtBPv*O4&!MN;x$DVIm*f7|3MgXUMC>d?cxCw0cT zw4`5s^WHO``v4mN6fNq2>rZs0R(&OJfRPg20C5k`G|f%NVD-``gOvmLl`&1 za(N;auWP+NKrrh)`S!|hB`2>8UfYPFeojZ1))*ZEN@tnz!d|N#-&c=+0yfh$%POXp z_e*RS9%7?At~nBMLjfYcHZ(E#)=gH>w{T_%c3iu-U}U<>^a`XGwh7 z_BVRInv#6s2MlV`LAm>PQn6GT{KbeAa4R=2wu_1Kl@*JLoOGhwoku75@58FWK99** z3oL1@sJ@p07Mt49xG&)SQI$`*!1Lo<2@vh1<&o#&wz z-I)Ea@#5Tmm&x;haOi3|ljrZa6F><)Hz8?YwV9c*?zab<9ILQFxcgpMWlW5ppEr%J zV^F3I5K~Pt1VzB`#;I{H7@5=4D`|XOiSEsCaG*A)Dx!h~0aZfW(f$*)Ow&KBWgm%s zPh}@1wG)C6YPWd<{W8ph#`OC?{BH0h_1DGQCCBD%9_>u9gy;Z?ly6`qIw0BM4gfIQ-T!<*CBeiRq#NOH? z*+m5BOI(m`rc=EiX1A}!zCWaT%X{d0keXCQVa6ENM1QPgHbGV#=;7eArB`eLWY)zo znL6VjJ$Y0`S}Dh+yMR-PG|@~A{T2JlQxN^wC*aN!3LCJ!$kV`@Dpbc3Elupw0JhA7dtq@gW4rn;+Lx~2=%G_$xek;F zqGZhYNSR}mXo8q!kN?rhZ|(a2N`AyIxSe&#w%d=6HG#>`z{?`%y}i^)e#pv)^~nY| z4OQjpRG+COXFeFp#8BhbZ2|H9j!;vdr)cbc$BM&olSxgnaL`;!W2}seftJdVPFRsj z9_H=stfQ@#4Mca8#Q8>~dUGzz+bIMC3~u+rDblOV|u~p!|}2INjcd1Ubzpx^64KgHRRd(t_&&gJEwb%^&Rwk5!ajy zOg~$L^xSnQwjP=WnvO=6DYSV~w_3Z{+fRP1gVeZt|M=h`aEo~~k*Znq$#Xs+hI1w) ztr>H)j}8zi9Byk<+JPkE@%Zp?vPLYmAsbr&Y*y1!Gi zUy;pHBw#wqgW_a=MIpcA9S#pbNpk9~bg%yB@oY}uIIGs|)&IJ~C4z|ccrEjP9FHZx zMBx1pNDcoLM*Z4*zzsNvIyI%Rzfp%@$0e@=(ZFSq*nbeunj7PWrCESQxSo0ZRiI&q9qXwytO`ev(oG!is{Yzlb>w5~Aq9@=r6-mvrCr{VZ6K z;L2XwFHCdps8kx5*aWish>cV&OF$MJNu;}kD>JVJ5i4c&yO5jPw9$K@1Wt@LQ{-F~KMZ=Yjj7;Kmg_a_u@ z+#mwVXizAWo}QlH-YT!4Ag0?>3%FhRbYF%$T^m_eKw@3i@h!wkbbBRbqL}X{7 z8(Lmo4uuXtP5KJy*J50n`wP_Qlqs*?X@)~9E4RQ+_(blzEjEFHr-@+G6b|D08x@YD zAH9t~EN1$xzr1k!s;vJfbDy)5eK5?(dPqZh>zfzx)9$VMV(aDpf==rs6E1)EQd4lL zL1o~HFYIxq^eETIm%${o`}SHFOE34?{9XCfOmT_8m_q*nr>3^{N2?!KJU6+_EO9Eg zpM2UJ)X@OBM*4D%W6^rU1-_g#ES47Vxkb7(6-Gv=KTdfS}`ohO4xz6U^sx?D&Yy5$6az|4wq07~=wp(I~ zIpq}j{3B+1s=tYH4kEyQFMo=Z7$ylT!Rj7Qg;Ec?!5mmh+uE^ucOkjAZUXIs8EJd% zkDcP!WOgBnj91inZH;~S`1r_!X;wxXq2Th#D!YVH>Eq3>PPJ?NzeMN&;03LM@NrQ1h>Co%u}B|PYL=J82%!Q}4+#vWYa2VkTFTF#S7Hktw!U-5C96Ud4rbZUK9LEm z^f*yEo*rS&0;j_m_SVtkqAjIM&gN1|_(;y!UQdao>`+HGkqKcPg;Kp#kQjd`8Rd4q zm4l9FjahFo|44@wR93>zw2`j5mo6RdLcsJJT@E%5CK!RW_>z-I!wHWJW`SV%dwYMs z?)y3=x-%$*;xb(VKat0|W;;o{_*T9`)ErV2{(uU`e`VewIhSN(7tlvs!2L z=~i42kD|=({-ocA#A3Ro3!Q%MgZ;&8SK*Y;g)^PE ziERJEL5QAC5?`Y^_`-n5&Qxf9Aj?|!`?>?YI=cF^-W()lX@nOf6?Pd z??WZYM{lrsjhJ(B(4tpI;fu(uau4(+KTN=Wz^Wztfa!o4zDo!>B30(OU#_A{9Md3x zVL9qo6a1{B`)He?m95x(ye3{U4`S;rV1}4L46NR6-NA;Les(MP;-a_!fho=M>kl(O zewR37>L<>BpGI>6TJ}@`Vqj$Xv^(?^(3cT+%do&ry1@DfA(HM0{BK1&{WZ^#x?CehIHg1`f|oPfq{%k>TpM< z34hS>EIV2xsfjJXN@(e_$QX3AAu%`hp3EJU3P3Rs6n367Jbq6l^*xlN5-Aj@1GYGV-|EQ_7bd{m$AdJ;Amyn5CX7}wR z^Rff&&A&jedcY~IU3qt{mnfRRwkO>VQXEps^`!6H7mC`)xnykPBW*PguZ_+pnU#S& zJ4fFA@N$`u*U zc2tJmxOL};WuAcKOd;DIL)(l3Peq49cVp|ZtvW0>SPyhqch*~)zpOWXaxH>oW|>ew zanfgzwc5xQEC-=)oi;ncfdE+(I*&;oOodjL{D3;;T6jvmGCic*Tya~nGp;l69bT3qXWrz0HI!IDGty7$ zbrWRLQr>i0mm;LmWaYTaPZD5Pu!RgkZR$R5V#W!?1+wAb%cFECP=I^R#?)?_PE%y= zn9{&)G$^l$6OS%vy3G2?_J)Ue9aym`7iEfOPUJKLVUceU0H5^!ZH^yObNV}g2rHjn zQEmu{mYJXtOc}!N4IezDeh92GiMiB1*eKkpYC}mRkb#LCADgm1Bjy9j?aCBe)$yy= z)LQDW`dN7yibu#ny4I3(@+n6_xoF_o5oY|hvk58DRM(UGo-9T4iWE+x#+E|kPYKqk%-eKgST(3RePZHQ z-0OKPw0y{9ePK2FUEt|XR((XVRe`>pOiADgFnbx)(wo8Sv3*tkTRM;OajPk}o|vVb zV$M#oG6Wt3bqW_MVLC_5oKs34ll2s&a0Q;Mn2qmYKaj*o_Y+XUtN;L0I?VaL@4!CsatI}CSaW%OWt|FB%N^vYdC)>tYV;x7j1ufgGPj6 zjFliQZuUxFt8O4P+>4MqZaFOkA*wXz2Avq14!d1; zt)KDOkEf`f;>v>>aYM$qn!!X=088DcRG<4|1uGlxt{3#YXX&!cdNC6PqzURlGICo{ z8cTjG0p6uA49sF3y7~laC$4l!9%=KNXjYqG7DjwD{CPh-xD*)=Bu4McY;M;mK92;% z=EmIo%=IHkmuDsR#`wMdw|6L??5?7UnhzdoWX_492`kvFt(3bUBrvGP2)(tN{agiW z1FSLVg ztGcc~)UdRBx^CT9jwe#j%0YW#HY+iA(huYcOC5=<{kk~cJVicoxi6Xk2&3Aug>l1@ zz5X$%$}1mV3}R0A#2U{;Tot?p`Q3W2(%>#(Ym2!8kH&@(A6laeW(V<%XQ%s3E(;c# zZ}Kr^YqD0dtf=n>DfkPya*UY69`9IL{4@9dAVA&a4u5C}-^^d|7_4n(gsRx6=NGNp z-Sa~8ST!LABS_G%%D8L0%5uu%Ym4wG^tza9DN*l|X+5r2PW@z1nPjA2Zp<5u=jcMqy zv`iH+KhM?3XP)U0c0I)s4Q+1}mbiE*Q38>mV>TU;`}#*9VUBSa4b4sXFxoxy@#&47 zrLS3JK4b6Pe5pt5p(OhUwvF?H?{ec9R6abeaXrto3xxJjAyT5(jq>m~CDj3D5m++Q z{k+2w&*;~;>%PB{)C#sjko7BL)oy?OMy06YAaYOtQy}MZ872DZ_L@tOF$S|7GS^2H zEW%|r>oE}|@_rj?cA{$5ghf7!C-%DME|{&gs9f6PtGX`n(*ObJ zkG(;xJ5pvC?KN~+zF9z?V0GxJj4)B^0^ezQlCzPJd6ciFr*Ub~T?z&n-gQs_QE#aK z1g{bJ8KgB~WE$kD>QHx?W`FExMJblWPTgEVaQu1xi&Y?->V4P?^Ah5FGTiu>wd^q$ zwQfYbL%#NiHKT`y|?{j8dDBjR}l_&#BU->K_(NcIl)dH4Dfm{U=NFc|^Yd^N>( zZ>{87+O;bcCM~SYmtkV@Z2g5<&)r`KMlzMPjQhH*fP3eDBd7-c zjg4y!zSrz^XCS72Yt{I@fuxOvD_J}oerSL;%kR_1+vr=;YqJ*7yj}3fX0l1v-ktpM z;lci`;us2?XZ>4{6_Dyn6CGwciC!j01YcL`eo}xSM_M?P1ztqnc7j|Omi;okpj+qr zMCNq{oeqDd$(V;mfK{i8jbg#*)DW2w*8fiXm{jisX8|r}K&go-3`)&kccN$Z)IoEG z(Kh|^Sx)!Kv=VFLk3l|SpVeLuZ}t}vNT{X`yMc^agql11Qmh-!5JOe9_a>;OyRQ5hi3NOH;mi(@bVZsefN~FYsV4uG?8%h)R(? zqk9|&IN*L4K8fUzJLanc&ciwiax+2iYW#Gq?iII>A3qrIz`V25HKx>hTb^ZoVliAh zZIgLc;}$M?(sGVn_XH_>AO#|TN65S1+5FY%T6mcE~?;YvgQ>o z+b~AoB0DO0@MB9GXzB%iZ2#cAv$u^UGjY)%UFrF85qX8Cm%@Fqfe}lGruZ9>{V3;q z!&8S2WuM#BVGP3d?kf?W8FbN2sgLyYn0FsZxa));d4qfH)dM8CHj|o;70q)3TSwCZ zB=yysC60BJNyqF27>cC#v z#>K%99ga&BK-jMNgp1E7IM{b?cN)bSk&KS^>$ZHcJ8srV*k0xMjk*Oz<2}J?A`|fp ze5?>ZMjh|?%+jVJCkHIXb6UKpSg|^07nskrtbD1<&=c4Xp z1?(9bQ-<~WYQIBwjYI=Ujx)l49gHczB#o-qHSpa-Y)q$JU7E9;Er6nuc zO}`N{_~{hCB4YhsHo%^*1aSAYcondt-Q*2P9HrPq@45;+zQund4?`zeUuX_K!8Kon zrBRvh*FG>TVdKVu#Lk46{Mq7oPFDX0K?=Y=VM4&$MbW|3qL5= zzu*{+Udujq=#&0{ed30>0IU2IJv3N9FZ-=@h2Ir7_brI@Qr18_Eoi3N|Ike*JA5RC zdZ^?>Ih&)&^COwCFQ*4Z9%J_*Oa&$I^>J0jToV-ab8FEz`vlU~=TiHS>!qP7$pe5KvFFbz} zf=S&~+w0mTZ9q@ontoz|)r4P8jqUxm2CjATN5W))oN}dUQ0}-MbJA1zLf`RG-em}k z>IIN9#8Mr0fDtoP`Z8fvb<00oly0pk6iE?yQCsR`vS*wq@lLohExTtO`aois5V)O? zHJIfkv(REfSPk~3vOSOr1i|e>RdA#Bds^O!@n+y_qR~TIOv;b~*jEwTID~btN41mn zqx3RYFfyjM_J(Mh?J>L`Cp;*Gl-^0P&3vwL70M44wBEJb?o(z*K@nn6aeFwYFhpJ- z4a{-cNGrY6?eWyKx~)tA5rgTA9E6;>JZBnmV2)(X(~=PF06lpSKYE41Vjn`?yQ53C)-N_78fv1I^lsTxD=5zy zqgNLdf(t7<%`dS<%?)NhR9s%4uEx%y+I<5``?BO)QDeG}g~;lZtfkgL%%Bc#Yd3KG z10K=;L#;5Sb{rOu@aVM~#Ovm8TiQLUDP$4LIF!`PdDq|lR!c)7x(q3`A#K7SuwWS- zd%bv*$NH&G-az-IrPvHL^9m)FYj_vXKgd+THN;Lel3xGF}Ie@$~*VgBF+Um!>C^ zy$rTSg8@!_rS;sqP;Jb|RwTpKV|p!xxZqqf+zB?-Ubo3vQ)9*KF0zF9qn%Bo`mw>c zU|kWE{gN}2M!hg|7#aVvQI`MqkIWW}5_ChOkJM|Lv_v6v;4 z2c4{NRieu5KuOjeeUk^gyovGKGzGLY5&pt90Lyq!e+k2mH?7yLXyYyV zbaZ({MTWC&4z;~&1Ms1N#LqlJ`LgD^&vKp>!t$NRhAqUZbJGs#2v(r{GGcuhJ(5HK zU$Mg`&k;$)R9pI~PnN~NyP(eHv51Ji?JN+mFcoVQK;ny(Yf(#JoQueMHTc1O!RO_u zr>zVj5$u5nh@*9P22TJxIomCg^DN4*V{0s%LI&=^1siwc+oT=d)XIuER7BZ04-Av| zmOg`s=X^gqn2r1CB?}=7`>;9$sA&mE*61sXX6;Yi6`Sl!*T3}QJk2;r)%WVvynTy@ zyX*G>v<@}U;K3`hF$}=mxz|pj1J9Ey1X!0y0~G)q1nUjgZgRY(Cj`wEOOy#~Ir%SB z_=@C&aXpJ+f61Jc0#qfx4Gy%T-D08u-LhSGz$q%3yQ(5pw&(Nh_#O@f*x<)^ zk34yk;=*YQJW|K`w6IfTCkLx1MwASOEvd=N;C0KfGP&oH#V!(s(Dz2p9JEi8)wStT zg*>6F;LN4_nCd*esrsyc`$qzIMD6Jy3n72D;PV$k#+3S(mLvFHft#ku=-LT&K)q{6jz(2D44wF_XdmLeBvvtnMYT|MOy|pAE>z0=HOP3M(4@ zb#BUK&8z+cAhG59w=PEXTokwc&0BJW0CC9Anu}khX;Y!WnU?ui4M?AZE2vXxLw@sq z+&?kPkoG?{%{K&|DnE3HedbBd9l*l)hZJoVBjjq?j){Nyi-I8H--KtKI0Cq3p>Ms{ zT7O4_&yKp>iATaua|8##Q%V}XJdFtxMN$m;+NgIJ{^2c}$$N>rqzHeqgdF(WrpwKL zy$j$cQwBts7DmQqW=(?TV0U+SuJexIJDj;fKYBj@zWObJOmhR}ul-!X&Djq(HP?jE zP14-FnhaqbztGo*rhu@%n_2v(GcRfW6qc`f@h|QYD9Hngnf#pW7=MZGDhVx_!?c8o z`6vCs%fI@w3!s-dO|OK1)vyAzK1HyPy7q|Rb=)uTpF9kZBF}A_#9t3*Mk0d0Oi8#e z6lc`kO)60)?mFWyc9f7r@_O^F z-k-E}5b43&MIdugG;Jn%^v=cRC7BY!+XKJ5%yb!siP>BJ_RhuaWoP$ol1t?8f&&A> zifeR5#DHPA@6?@xqWmi^4B@gx$w`dmo9?>JC56i>+ut?2yznR?^kgW{i=3l3HaO6_ zIw2HIP9o1y{`s>{cdCe@qGDKBn6JPy!|k0NT$8Du!cJ;Z5<~EqHtBEn(oId?G6$Cu z?|e2E`ojH)V{chso7tQ_j_ zY&X5pSVhG=aJLC-|Ev@^wpG=rre$Y-{~h<;oiw0+Be6O$bBBJ8!9!pjt71j-M|}X> zfdRT|H(#(xPVofaYxcoq9BVg1Y=$4}HEooMqDW*n*ljyX6J?RL7?jW{VxXgaaqj?e zh;C;${N$BbG-52b^I0L{;?4~vm!0lv7xdmTjy%F*Il630IpWDi)jjzl>z>63UQ$+dvy2`W`loRhRe=WR7*L7s$-O3*LW( z+7Dw{q21Rk&%IeQ?MDZ^urvSh9l;6e2;Zi}#lZYcW_pmm1f(?pnQxDR0F?xLE5m}p zsGUaDs$uVbdAjd7%E9@~#(ST=0s_wEH-EoSwOt~!?Nk^-{_TRWh*(pV#b{x)TWico zo!8OKW%%sQ3vJq<)AjTV(oyT(I^3X?TE3wO!?#KMKcddndwV_&=pTk~mThki0me$!>igTN;RZ{Q{KhAk*hcr4KC~N)&ziDA56klOTY$-;Iu*$c zKfS;+5#YOg6zaRHd(}?0+{LlM#_XH8``nVC%Put!DtRH>PStoP3D2w>v2Puu+p}4D_R)%r+svR`YnNkZA(=Nn8fjxqgMmzc{ z=3DP`bfk1XIW0zNiS%63{ViJUZd7Q z2VdDcIxIGjkpfjoE6fG8!oT{Tw`*D;z(@}~opeXMs57;Br~@qp*>fx(gFic|TTBY7 zK==>e&1)Ye5C@_9ldYs4*I5?ziesT?xJM)HLba&9@Io?5wlas}6~nQw12j^l$U7$k zAyRv*n=%Bqy9A{z1}{6I_=R(alH&IcVQLHBM39Y*HGG79y$fB2Th?u5Y)*k>J-!X_Ap$q)5!$xpzW%XqUA$*JIJQY&bncZPf5|{+3A6pC$i$_ZN|du ztUkWuZul8BAZ7N`u|a)`47_+$N7LEKMn}g84J;E#!hWLCH^OT}GE)ak%gPr8E@MHH z$i6_i2zZHrcUN6a)MfCl&16Xd@can6S|RPE-V^M-&g=jN525UL z26U)U&~Qr3z5P}j80)8t%cBlM-|yxf;gHBqvqpyq-d>wgo!Wg`RbJl5hqn2Jh)JF7 zjy_ih<3LsZuYUU{U)~s|g77*t)QFj68o9G1DQ|snf$oF0d;4oc-iJf;3m%|ukSKxa zW}knWY`*tu&yN_ow9MEQDO43QC=fRpof?fq9{ZQZ@rD}#x%)-j<*u5L~0*yl|Ud^bp8I^k4obX)X7ni+=+^1PSS0Yx`3lS z6coG?IOUNpW^CPWTINzS>9;H{_)E_RwlZ_NCN6(RW)$}KKO^Vi5T-0~aMV0oYb-}L z?P21UF`#m3Oz+6ETr=ydL(sT7{~)=In=~g_#s+Rsvd3<5<$^&BPp0&<&Cd}mg$FfN z6`Dw%ez_H$huePd_YPX@hEMUw7#0a%n?<#}R|FKRBbSfdoHnOg z4i77v5;bISgW;6Sw-E<3ISQ68maWZ%rS7(nhkl@Ma$`HbJ4F-fglu_J&}D1GtkWxND(eGJwAHw7Am8Wn&}d96Nq3E+gN> z46(uH$>eaQw5jQ&tt7#MhQltu>J%j^obK9L-c>0S9(Rm0Tuj4JHK<>`v)T7r5EW7n zfvPv26xOI3G`iPk^h?Mhwes4i)<-+@85rovMSNvTeQ7)K(0Nlzre{HpIw5f@2_t-Fn^;L9`2Xyq&oABlur^?#<$V`Ea{_t=) zp!6n!#qGO!1S?SGBvz0XH{AEE#Ex`r9={z(RW4t4avZ*|sEE9%(!9oPPd4Q zeJsD{p%?F%pUHbCjkQU?OgO5!eqr)6EQa94<1b0Ti7%lh(TAZPu8( zcXse`>!f~PtS^u8(DBGqbC%2W{lNzXC3Rr&7ZKChCDesq})i%UAPaFR8oD>%cd$h|7Hn2Ow#;X zv~=KIY9FKyVXC9hXiGfx!Mva5995=-#V*uApD&Kh*8?TT?vb@n^`mkNY51g#nn%hO zyIaF|2cpqb<3%bVBTC`hR>A_=lsP4V4Ibi&}wxnHdB`al$UD8nKMrdex%@r&%}@6hULU`s=fURR@6--Td1!C^6SIqyJJ$gfTZHfr#2|H;ZC>QP_) z)-vo35+tBE>-Y_jO1?omZLCL`NeiclVb1EWw>_C%ECXfyTWt z2PRq>*Qw@roXUmuM2xqy_9)U<(x$yl@apYH5@Q>SyN&KV5)1$u;6BZ#42!`MdWqRR zLK+aPs`t_M+6<_}8{FFWB>3f4O`mKp)QR}MjSpvEdG~1DFIdFO`(c)BXKm{jQ5m)6 z(t2k`R4pdLxRyoIYD5ru{HA`>YZ$yX5;%UGln9&j4BD1r9yU8ESAJi-NS!sh{bt6i zvy$<27yq&eaR?bLEiSpszu{=GUosvD#a*CID_OiT+I8O`MP`pjru(}I&HlmSDf--q z$pvtu*Aaj1^0m4PV1cSAWYA{%HpARTM%H5;rTw)rZ)H8lx{Ye7|HIyUN7J>1Z=-6o zAW>4ZNI?)p^ezzuLG)fCdXLTynG9TV1*k-jVjF$Qj(zLj7B z8+THF!t_E{7oep~V*7m(Ya59oLcIRfTj%wsTeam56Bb#k(NZhg5mikfl8is5B`d72 zOHM<1k&M=Unz$qnc+}<^?~!F3DG=}J*dc7)IS-6*7vlBdN#h%XFgTrG`azr@a&#pB zQNWTK>9v}zAw{ba^LV%OvSt|MP^L7;>u2J6UYV%tFkwrfir${Z_?b;X{zUO+3+3r4 zmwbf{ghk#69)*`(=4_p}T7GcAyH$R3|2@;^6^q*LuolYTw}&{5E@!2d&V7K$DW2Ce zV;1F>)udnls+_5a-(pwh2<`>9=tt_UcX!Mm1P=QoD0A^3wR<&pX^gT~k!!^l<1I3` zG1LVLL`$k_M0q8`qgt03y_%^$ZZ`nzGuacc`M8%(KC%|Rh(=7>!RtK^)?POIC3+r? z&)KNHMF(YeZ!4u&Ty}naU7hTnvCz=?oc&PsjiV1s_crxmcMxeXe+HX*2iXdCQ+q4* zXtL-qJhm+Xx)PL>k+H*I)taxfT)&_oW-~AZ>k^*OD>+oS$(>FQKe+Xot=1J@`?~(z z;(J+b!ngHY{rSr~5S|_9H%58Panu`|2UZ8YGbGcN6;4bcHy^qf_WB&66_c{6uIrTw z$O@r%WQy?~6TLm>?r_^;Enm|h{;9!oNb7Ag8oqNkofP$tM1hrv@hyedFpk+0-B-B> zb{fM9t{zI0sA6}8B&79p?4+LJ`IQ4oXpu$0684~&$XU*Gc>5I^pRe!mWCje`Y%Pt#5cC)*f^eKuan~yq@AX;4C(Qz-1 z$Liz9kAy@-2ikolE4TfTv%+hAby9Zu8dZhl<(uB#QbO@*L$`I6HHZ3!#`=cRtB(8? zQKiMqhx>_i3>k`^n{W|^Xblb6M75`)o{h}o)GY?~_5Q)nbAizM&Y%dFtc!y^gPX&A zyaoy`(X4dmlhmi$HdYSnW};a>Q!ZTetW;cemZ!OFl<}v|2)-|fXI%JN@DO!7gW-&- zJSJgYXZ0m{N{?+dJs5kv-_cF=H4mqvM|Blz<;NCf+!08q=uSaCpVn&x>4kei7L|(M z^L`w|q^G>ttS_{*a^R6^Qo)ZxaeItKhEIe&k^4Zb!2g-4+X%un{v5(SVa0r>y6GTz z!6ZWoq{%Cv)5+qsMVHsr6gjzJ+Bq+5+9QrWt~z)B-{FIP1h_L6HcAt0ISe{kTS+hMV&h<&USofnzivD&>8F^iEf%)Eaeom7o25v^}LI~Tb7y2p% zlODAclt+jWc}Md^;g=;Tbip~F`F;~QYhyz{Ut(C9pWjL~gM7>FZdKX`YA87xU zVEqBcX|I}_kfnrBG;{vj(%=8W0o-u@u~GktKa$KpC*U971N1N-B{W&~ z%O9mv9mwwWuU2qzJOpn@S8@c`|LRjKUWhjIn4lyzf)IGWb5qME|6OhP)yJR6e*lPF zPnqid-(LLPseN%^f?Bbp|EK4lDu5Ajh>p1QpW|hWKpGWCXz`yB{6r3vFH;N|kNFUXg=tiMD*N<<0liV}E@B@Nwk8%``{fO`x0f_4-(i>9F1`6884X zmzj5$@Zc3gzcqpY1}(x8#v4;E`c-NPi)ehscb@NYv~_W^*Ia+;vbC+Pex+55-)dO9c2}5FS?%4x!5#_`!{}93px8vvXck)k_LjX*Rc*TJRrKJH z>WdLEc-hlol>OXW0D5W=E!mA#Fg6^D$RZOSo^Qn7Rb=_j@nKf(A1ll-J@6D8NJq5v z2$V17hj}QE#MDn<%WI;#tK58Txtd`Hf}XO`<6IT+Bd_SfX%1Z>8x*9R+O;5mB~fFL z4PM9%(|H~GTEgu4>Apr_jWdza=YJ>64}P4+y5B-{k}KJIHAZ_-gUO;DZy`uxYi8W& z_4U~#k-QqAZc08>vH#DNWUKL?{1+2$h$f$lBRB(jYt?|(KZd7o?=-#&d+WfEjR&UV z6&xQwc^zJ_0~Wje_1y8BdQwTlCK0T2-fFZ}JK?Nq-!XHn|LLr+=)|VgIprFdRpJkE zDlXP4GIys5)&}_CsuyI&eRC}L9$)&`)OZ|SRjXJKu*1OxfLU&CE?6r;jf(2)l5c@b zIh)*@0y*udV<+W%eVWCPUPqWy(i7k8?6*mHB%ik4RBN$47Mn)iM1Z{Z@#AmHM{%6l zD`A+D%^uhC+3hsTGQ9eu{PLT!))#=iY-bdkRo*@UIU@xnWqCPY`sy&a`eJR_gPqlv zimCNWgi;4cKNLy^RSN1M36gIVXqkN~fuPTi+Ex>ULx9r04Zk>rf3no!r*;HdjcKpb z%^hd>4?t6_-6%mf@kOcqr>%nnpC^2mIdq3Mf3_XJP{sK8Vg<6C=qkG1hz*o)X!XlV ze;W3SoHEthy{`-7#!(T)75-wph$EC-uGfXSns!h0@Do1UDSnfVh>F~lF}Jhh9h*&J z3uG|g%rOc96>Hel&~qL>7&W++Om|2zVO40T3bPxVO&$^p1an;L+w#!Wt7GL>quqS7 zXMh7Pl`hh6+4>ldmK$8tH6BfXJz`w9_3>kyoQhA>Ix5cYQBB)xYdG8OL=%oHtlI-> z!J;Tb6D%ucpgsH&>&uaDY+9L;g5E5KT%dA#T+ z;0DSWAeH&olYF)LHb=Ur0j3Ge?W^wskH+{y!61b=fIH+-Aj2@qFm&*#)jLwAO?2U{vHxFU@`V+HM z2uL5Xe6Yu`jtBNT=vQt5lpOCu11t zU&)KUrjm0no`JS6XSvFF9sO5ANoXhsB#;$O7b85pcCZln_lFrd#p4^$BdNvFELYa= zy<8B9i87Zw6euP?PkVc?ZgjXYo_xNa#Q{Itbs;6N^8;8nS^*1K7B7JUFhqFw`=Ejl zTb+$e=298oO@>|iFcmOG1)Qrb9$*PDrNly}ZIS@K59o|4BKYlz-)kuE(=AV)wTfOd z$!n9<#zjW8wu)&CwSJjpMiH4^W|>i%-XHVlxh}ssQP1DIA#a6SqFLohH{i(DnJFL7 zT`h5K{qu;*1sEIWXy{r_mMF;dGBRA&?2dZw=MT=98_eFXleUm(#WjL%YJy0}XNBN- z0>2WUjn_e*LH%aM9!cG=LPZW+IK28ZOXsiNkL7OgIHykfrpl-GEZ09wY#nQ{yf^^% zktpGvP7Ty`gd64PzR@ooEZ?QZ${RsH^V^FuXYcHCF~XwPLc zXbuAkY9C#6RcA3zaq{YSHcX+^tC;q0iDoy++Ze~<7uXF5Wp;Iq_v9M8JTp>zYECQb zGFSUehFP@uu;X{ru4A;-0@d;OH=0=HMuqv9yO<9*%MteKSH7EoWm&fYSw`zKMW$J7 zL>A@J8Ow&Rrv} zRvvC1c(s7#PezKZ+)j<7uW_t{Lqkr0vKQ}Mp9DY$7m+5`!0R09>m&jTVO!Il5FG8W zYID0V7>Tn4{VVWN+)P^nxj>iz=7H(iinQ3k8;$Pn;z_DFMAUaBd-~c!}2)KWp*Z_To1k4}~j*>yK zrAD>P01Jvf!-j5hsQ^lgw7UyQD;eq{8KyuGNckr!o-iger?{8?>(MX`c zxo6n$WAFoA!FqXdaep1CsgLp8%+=>%Wx;%ixP~LhkaA`H8@Zx&IlkwM1|aIunU*xV zgrg4A_XK3ys^K+HB}}FKo-JgMgO* zVT}LT94WL4L=`#(@kum}->!;2zcFGtxApD$rA9%-ZSc%-96hW>ULVaF|qf@G@UOd_OlCDD(Ax)bjFQh1H3+ez4O;k$yky8&R&EuX&4t; zLiq_SRXlH2JS(@3yJ;awo-G#U_o+m_Ov<#m#PUcG7wT`fCYkBy7*w}1KNwPho_g(t zKlw8Iz2U0)jo%4gKM0RPddkxoz@qJqz#_EjJU?QQz97++%~IId8}6lzY5i;CK!3#v zBu@I5e~g*@o1`UG}p*si`Gas87ly20&4 z+kv9)tbaF25l_dVK#6j}>T;MzvLn~66a{mybK0k~KMY3q%{(2RT>4i@=8K7~H_a5wTb zTo~H1+x^hl@D(ZW)qbeYvljG!&yB-Mz^)p5 zK7IFd8_1)nGDrUleKX;UioHi~^c`z_JUco(NV__61emcJh|YebTX}MP#=5$#0hIFw zihsNl8Wnvkumdv{H$8TIQK$#pjV5|;U}@e(Rx(FUqs3#ZJQdqhY{4))JALaO9*@2l zFL(b4N}7f*bE$$^!qb!w7wy<)6Jp6<|K>2>7039)%S(~>sqxd^WMzL~!G$Mu<=H#& zBlhWm7i*Z3QM^a%17lG0x#DtM*2LDaLO_J^4tgIMEP#z3Av&MkTv|R6cN-OzeCdrHzVi~JL5Ny;Y{P9X zVeKLADXiMvSv71AK6%dWhpdd8Q^aY9zlO{PNM|c;n(5H3I>K>6EY1%TogdE!69*G3 z)}y>J)NS-EZ@6e6JPw+dE7h&LL@2dsd^4|zQvZd6p#4qu zq<+BwV;m_MRb^G{dk9e^R8dz&i@a83=U|az^xg8?`d!$xNEu>< zD5SdUp8ANY^<+KssIj@-K90*G{HJY%*Q_(kuGdt;kY0_sAwz-TFw#r427YcMhYbH$tMo+0Dl&%L44b!Jwyw<|r%?7?i=iWp)_r@?hOw5NJH3cQe^ zWZ5#4TCo+kVa4zX*n1ZdOZ;v>3>S9oDvdz8O#r?R?gIyeLWP)Xo~jE=7VPES>VeCg5tLtXts@ z_YJHo-WWLH4C44sCExCoV&aQ8is6(sBF6o7T_!c?VqfncIg~cD!T`I`nq!v2CiN*| zaBnLbIh@Me9Ll^UdcWqfz?JPgJ|Y%*YfmVVDouf#S>5PM(uYxC;@B%!-M@^!ra#Uj zu1BHC_B}9mFZ*W*rS4b_wOWm4@9zHB*jeTUKlsGn=%@Dg7_Kh$Fx8O2VTAuHoRtmA$G=t!5QV6B|(9PuFH3CfEw`~ngw?x~%j5wB2t8v}h687%xC z_k5ISK-;`S@rQ;-Q3r5%!ro4&%((FoOyQ(eMr?JuoIl5r8QlfwN5H@l48y>x!k%-q zKp&ueZrHZf{SFekHtM-OqDMWtsab3~*%rWO{`{i-Cmvn6o4f3j-{*mmCt>o&*xF`K z!TfcdAn{pSYz2XBxx-x(N(K%#w~Y+k(22Z!S zLUcEtUhTWwtJqU-H`Bofo;lAhTW-5)mzqg4-00W1p`Wdh0rU|2Dt_FY4Hf`%tVRpj zn3-?#SgHb+Or1?on~s5ohK80_EcgQq*|rFrz?Ph^ivANXwyQ z{57>b+3i6yYcI3-HY*_^xn0LPC_T%DpLD|7m%_vBWXu0{;x*e#SV&I zcl^?Q2YRSn50Ag>^qAh_JpMl0jV$15BVSD!Uk^}7{geb&_`|1s{m-Jw#%i3`j{wOF z5SWfu2bafQg+^MLFLlLE)Gx(Z0oXvS-F<$*<+NDK#KZ(x7qk&^)Loga0{sd529v&C z=zf}9ikQ-@5e)Hhw+NPA7`~+wp)w>Q<~gV+5x@U1_Erb&HL5p~*?VF9OddOZXqHRu zR*jstYexigvp=Z;DA)n-CXntZ)gRUXwMm`1QlO1~rETH@3hV>NC?9t(&t!M1+7XR< z&45rJxX9Sxkz9UKb}z z$@;q6=DJ+e9Gmg4-8@VRpLj!DNKDfGb8{sic@nq>scVv{p@jq)l)kM~6djx7m1G>W z4_H{%X6jE#0Y&m|2!B)AY0Gtkp~OtPqhSq$&CahR7iY_28D3cmiMfqYK|z;;DZ<(y zr-z$p!>UpExig2eez?u_i;9F<^JkC79sjUjJiAHf@ zhQ%?O_%QOyW3)yWs@EUaX9&-*sy6D~8{4#W2m4>FO(!7-ul zBgFE+Qg7V6K+Rr^&t-zehbhVJOD+;PN@*+I&MFJ>7X0$%ixHsKJB-u18<%nhgNd(4 zE$pc-IlLtyO?MQ3GxTgN+~3P#CQ*aVb$W-qhmD=aIJiUJ>TK}+*-t;`PTDc1p*)Bt z%5t)t>8#R1C%c%Ff`nj(O#6+A`MTBN0~iy4E15}UbI{SW&bN}Wj ze~5;Gr`k!hzDLdwjEw9I_>?KlpDZVjKF90NI%YI$E+ppC=yp0E=JS}GiDY0M66&mA zuXXI9CE{f_FU%r)Ffz5e)vnkV_S&z@m$;ic>i(*)N74FWhquPwL?C-c}C@4QcobK$-{ zk`St&RKA~+Oq{nBH%sx1Suf&xl+??9E8E?(I>rNP!pI2UlF}zuP-;tS`~#7hk)_qA z-zsZWvJ`5%d2Of0`ucuY)Ee=g0}C}(r^+QH6_;rc;dj(&BR1*%Ye=AS8o3N5wC;EN z5yOO}=VD?tGY@?!5Sz%o@47sbC|YEG-g>g`QplR~Aiz%UZOgpIyk8mt)xceNmNx67 z-R))5#ly9!+I$By_wplCA)>f?+p0G6cgY~ALk-1`%@bk)wOYW$+)pM+jTa%7PXpIR z6FwOhXRl$$rK-K{P4s9P$E6-T>LIcjqw44(abBlci=b`29c?$8lo~e?7E7rWFdaE} zhV<*JnUv6L^L04&b48Ui_Zb4)wvzVfyJvE8&5z+8zGkL1UT8kUX|$A(6;%->tNJtS zQy}zK-noa0tyLRX>BkK-GNpL;s)WaqzZuW0Uj6AuiNgD9%GNXX ztC4Mcok==sH618kL7f|aUT!ZkdK?ciOb0<6QEBn!hpg$XURPEpt}nJY9AgDjg} z29|I=H=mGovR*B9BD3nniB~X@D*J2~qkmCI+|%W{7gE~htD&9&Yjgg$q7WK+i3jVW zz0gZrV^@<7+k#~YOZM@eP{BiBQ6vi4vB2x!;!^&)5eibFPx6OJ1pLywD^y#Zm zl)vmXUytY86-v0<0JlGA62 zxk|#asHO&4x7RaRG=|f&t^cV|ulZ-Am8B*3Y~&Q2iVGGQ7E$Vo(D$V3Ejr4#Y-q`o zRY$oj**VoUq&DQT*M-%Zjzrq^CG^AdNE6SxoAJMW+I~M49PDsb$gXv8Jbw|BBY4eL zdkkiE@owK>^jtLU3OB@a#53lzGS++XEDDZDybs<{Gbg}_D+X*p+R0YAk3dvHrLl)T z0n3h**NS7atfPe$+p4|tI!A@KtKZZv_&cV*uZ(!lYfE&!G@g)R8-o*CasZgJR|N`1GYdr%EE$yaycs?Wr2jp(C#m9+J3rPV5F`C}X*>gGWnmZ_

    Y-?J@Z3k4x)%*7{pyUjL%U% z^bcLPrl@VJ9{apSd#5leB;3G{=X^wuQ+6XL({-jyMDTvby+JHb^UWWJj zS~FC+X=2}EsD!bN1;U_eK-;sw%Ls!FA-+fkwuaE2f>J`)H*Cmwis|G`shQi`DG)4l zqVf*P0LSLVrlrk2F5WRHmgdO8_w|>rjLIiPj!I(}SgmL|oScHXuj9CEvVEqV-XlyL zF=sd592p)}8+AWF>ad6&cwEZY^afZL(}0lQ(fGT+_!pcRuTvZOh}@jlwg6xPC^~Qd z;C(zy#VgtU4$|_(-K!Mm$?ppziX3G$7g+_hzxHaA%EV`OhHKB1+|^YseOcgnKVfye zJKLasZlLiTtmCfQIAf)mWmEP6GjHF%Eh#A>rsmZ=LewG}Sn=GFMLjPOQw=##Cb4i< z^p3%99R_!jR)4X3JR4LNgzXWUG=vbh`kEUa$t&N)hvULa@I$Gdw1~^Fz3l5W-nyg8 zN#-TCxg{7I7=^r}B1T^@3e4~2h9A?@A3y*Dk#T!#ye)aPJ$e zfGE?{)PFaVnV?)yNaz$Gld4Jr5K^IyhYZ}vBcPX6nJ*hua3u75pX=zF`l((jk?VXa zXg?_FL@0o zkv{B(4>#(Hxv3nnZdX^=keSqA<`oXqERIL4qURu4UYPO~{41Cux6o#T+8j^4Qg2K6 zJ=H8MYG3L4@Mxogz-MAYTWBYymEDr{!2=pjqxUSgMBf~PILM|j+_0F}A{@%%Zh-w* zPL#{cN``+)%Sba2!-{p5JPB7&?5K3^D_JX9yE2?nyhfPGsVglxaVuv?|6@!6+oaB6+c@;0KiPw8rF0r5jJDs~(-e{(ZHIkmwVNuR`+9UMzrZeTgML1S z)pgn$p{P*xHL}ZGl^#E44)w7kq8vGt5VE8~AcC_N02pj|B*xcP#HWaF#NNR|3a-|s zcPuh#Rak3%Q>$^Fto$PNRJ7JcK^*StFes2WUG13F&PhHNCqZz{aYRB;)y>+xvpHEF zk)H}5<~c1|xfPRG8=v{EkmLZVN^5a5y(u#%RqYuAtFUmU!o_ncIgR`ePM1L8VIcZQ zN=d07_?7JpCZ9V2Jvf+)l!c&ks?w&1>}r+R3LexSO+yq!e?(w=(1v{;F00aT!+`ue z{J3cci!C+Z`{H6H^O0RuOXaqskq1YDiU@<|1Ib|PW=nFug>p*RqVXaSWb`!lR~dBU zoo|$w38#CtfLEzvuHcEF?WtM%yKFGMf{L&|SaYl?hUn9Awo*`Xyaz?@k z4p!+px(S$MU45HV<%`O4N^01hj`zApdp<=Glx5+KRhR-h+K(jE!fy7bcwUvEcw^eR z?=zd$RQj@VD=c4~v>qm3VwR*c%<@T3BQW1B-A*vGNM)<7T#+@HP2O);PX;9JS4|{i?X+l_@ zP(qralJ!p#MQM#=WD-7g3UU`OIc-h3cg1i9L>|1!3GX07xzAj_{LIgL#>sl7ULcmM zPLJ38(4)4}+;fe9TV6RF*=dja3j)|?@lg`HJe=WcR;k>z$6oMde1Xk^%8=Xzf7w$I zqw4Yb+>84{R>?@U_C#&5C?5>{O_vGc-{`?M1LjQkX^9T!uPh#v<$6Nd1iwHC0H5G*aDZ_+ zl?&TLqXfsvjVVMD0wT>^Vn^n?E3!zIT`enGHK?FPt`?K;Gtz&WQss?-HHMogPL952RkHYbpE*8z=Rf}EDF8$g55nf@A?h+*&@Asxf_))2?#0v> zhhOKs&1&08r|PY;&0g|Lq(Sj?DdG;JUu4l8f}rUSs6+pQ466%kGdv{aCVEy0KcUZp ztn?Z9xv&!ypscb@CCq!1Uf2!}^#zZoPR*r$((ogca{ERWhsL6QAzMGd&9O%pZ=i9a z6k>de5F*N%nM(&RaQ}Am6?172%T(UAX2<<=4tUJpBR~r@*i)}JsvrG5@E61>y9*X$ zp{6C}zX$%dw(m7q-e}vBSS9{qOMlxTL?({$|8f)HC@mBsqVWra1V5+$C@!hil2lKr zG;aLox)a_2+9IY0V(C9Evyl{EbM2i`vr&sbU2pGgcZJ@y_%(560l0gb$;yfIdkXw; z$C?$|!b_^W?T+1gEmZ#eUJrUmF?>eA?<)PMmj5<9j1cSDSJf983ctFo2qy-o^9w!Z ze>=nrpgxC=2qWsh?fCornqZ0qzQy_P;p12^{d(&IZT|l5@9hi$YQ{!L_TR%l%)oWT zj@T=Hp_2cM1DG}p-WV14pZz`k_wazj$>2JusoJqi|7~d$xbgqT1NiXcXju7(|Mdcp zb+ywebFr0pC6Vxm=IuZC_PP_c%JixTJ$Es1$e6a*!kLnjj>MHl=jBL6Dm4)AAkMdv zj;kZy|8s`^J&+X#)=R~sU;i>ppYdA_+1jm#EL19E<;ydn>ISb*cH!p+i#;|rFL>1c zeIyOCM(m9a+f*g=os^#&O0?$=h`cO-tax)G^WT`M{e-M@&Fa7plK9q0X(QYx`rj$U zBt`Ye4N$mZ5C5HSeGOm&Zg$4!p?lCC&xe$+OGCopQK|I)2JC5z5UYGGRi5RbS)qAM zwUPX+v{eY*@LnCq!41xSTKP=Mf5yWQ3Oe>EntmC5T>wD`XXFiUVCq3kxU1dsuGs_? zeR(bCJ_29b^e~<&+>tzBq7mK>g)`xsH&?I7E}??t8og5Mf~h4sBQXKnJq`^$*Vd# zVlz$&j9g!s_b&xW_FC7RF4i<(T264@_2x*Rt4)VG%E`*A7ibRy-VTVFRz`MVsmWRu zHj})lOc2ZFONVaL&LeE>2Gvzm20?{=YHBJWfo9`?91}UJSRcT4#he_==h(ka{+SX1 zjuF`yK+w=@dJ_@H(CBD9R!d6@V9QQ}B=%Jl6ITY#Qg1SJ(x$RNul6wJRY)~(#{2?v z237U3BnLXr#Ym0)AM0H^o;4*Nq1Ii!Bs$YYR(9{}iGzbzEi8JvT1AF$!rthfcC_W; z%@{zsVrB+8R(TEImwIyy+QCtWz%sX1KA!jHdP!+%sL)o`t+O9^vd07ZHyLuEAYMj$ zGT-2L0^dMjDvQ+G%c~DZ0@zVuJ^jS&6e*E6y7Xb5;{iZsRXT_uPhPIQ^D=rKwZawUmF zMxK;A^B}u=@Gn^VUg$03JhF0ax2wo^v;3CY^E}>GS0l>(B4NOPhB0(34IBQ)6Qq=-JMIPiHhoG~LJ~yXPXHB`6SmQw(rkAMsq*nVkRxKj1P6 z+}lgc`@d8>E_?08&(;Ac3*=~PlSwX^4VYF-q1J+>yu2FTN`ncrxIG%`K3XVx`M6CV zISQRMdhk}96@27%FoRX-i$cAVg5ao4I}ZK2V^H+x^Z8_OfgDBmYa1P8qn#=FDMsaF z@FU<89BN1Z_v=I~7g+L%;aN)EdH!re&wOUId&tR(jAg}(C zfMbNdu9ny7h}zNR*fI`y!)==xWMAFEI>N2+@Wac*UXirpAFu8z>>#%Q#j?n_J-oC> z&}t}KsWl2%Ya0NScepQE^IhqX{TiSC&5LER3+98oQFlNz%WGh!dSMTEhjz0Kr~sy< zzEB+}cg1CHUS913A)uSu-*;;1C^UfUOEJ$vv2H+C4HTg-Erf91=-m#qMk@JsC+k6xa6JoTyf`;D|v~e$|;Ge5~F^;#E1!PAny*w^!8OLN}%D=2^x99>_%sXYJm)M32O ztI`2c^7j4~Dvgcrx;w3haCo(Ihxfip2#wR4SODt)EWMsVd7d{f&l-H1FNg>XX@*Rq$smDGwpGvq+6zni>4 zPT$9fGcVFz){dKvgXME4#q1kot6G{7*bI*QJgRFBme)nGki2j-nT8OV%Lg|>;9wdX#d>a$`YSeU>6SyqIpK|3=Q*F%__n5Y7z$-MfD?~ke;`$Wnv|R`G z7x~T&C&R0gX?NXnQ&Z>cpr`Rt?Ia-i+ZZdRt7Meg3x+<|xBW@-OZ;>i-RcV1sfrmH zKCcpfE?7=`Y8mPv1KCHZl}{H-An!d@m3C)u2N$x>m#H`% z^a_s-;-%nXrW097hG;k~oQcWTuh%9~ZW%*++8!-F=o4~zFM-h&Zb&8voDeVG6OePK z*1ry$*jBPT&`TJJOR-Hm3_jJ6)m)MzSPv7RT6XHCeRS<57@KOH=t~kQ-2^`{Z5`89 zZ#=~sPz8`Ucf(VWJ8)2?>Mr1;eky2NqLp+47)l869#%^cS&f)?JLaWS>DX4cTy}C& zkyDw8+De2FuTko^nIFb&+m9r?MwMKG+O~}Pj91Shgr>$r#v!uoI;hxZ`g(d=N*C3u zY3o$fvhj@3E=iO~^U1LJv%W(OvwG;!u6&@V>-W1t;SC;@8L*0pT8G6BKy>Q8ORQKJ zt34#o6oXp^cKtA^hVuN3b_gg$Be>&r!)~0}K%Yf^~7`~+A(Db!?qkqZz`BkBU~0VpTd`>}8wE5bt9Z6T7}IIemId ztuL*ixPNTlzaWo}ZMsstu>E9h&6jiwU-8TOl;ABf!92z2gszR~s0OD&IIQ7xF+NdR zk?d_Ps= zNfvr7a6b|fK|wtI&A5ZZRL{SjOef6ssSzKDl$LN@)R`(E!UunTv|tg24l=EGR4{Xz z9QPyl=hcDc5?Gd6KF_Cu?G)>C-}Z{{7WWXSSv()sZfiy5sXP}B9Jvt)tyvCw;*ck#$0v&Cky|LFyzmCaWwAOKg-I_4K!(flC#w2Ki*@=DW{WAFg0QBm~ z$}4d(h31RdSoSn&-4g|qADv+FAP>~`G@@pKqD4+eBmWN+eR^+O$4#eb%h( zq_jb)iR6=Xy*~V;Q3nLUGiw~$>+D#B3AM0x*w)ym9X|wqjY+rOPryEOGwXs$~in!n{+kFy;qiCIf!`+vL~>5@`*KN zrp8^ex9j`!X3N(mCt`BEPwiO~Ub|JU&IM%7>ejCeN8Y;sde_H*N{O5OOH90LGALbx zbwD$QhI) zbX)-?oFu?}4!X$&D$3~(o!M3dfn-5s#K3jh+0=o;PHS4dfOZ+bJ)P+WPj+f!qb zSn6m-fbG@wd$d6 zvgcP{-de3AzJ+q0DU}U+5NCfrH~Ub{eNn~rlltKZD$TAEOYmOs%$q%V-|w=hq}Q!C z^^5Q1xaiw|#RZ!$46zEcQ8GruZuh~V7iiH!jz^ucrn12u1BBjuB)zv`J$Acq#{OW#=M$*Lox}j)Gi}1Ho#IT)VElu-el=n#MS+|{6 z+Wp5|O1bF>OgtLSR{w$BN;cu%hI!QIyW(E;h#df2p}qlIP;9jEzJq>2KVEav$+tPo@U@w7dvJh93V zrRNVEutK}q5fMqBp!?Z$8Va2odMQRQSOW9{kHa#Yt)S8gRSg{q`w2?Y0xBxCow$~1 zmym~>LN*iJGV_NCw`H{VL8+7bvt2nh>&-m1?t?puuD(e)+}HPky2jiOud;b~@`4Px zI+8+3-I6xD{pHIY`Nh6-Hj89dBM#K0s4JC*kGTJpKDcg8VbW1oR;GZecJjo&5R75# zAjer8fEWWpx4V5_5o*MeZv5NeEZeGM0cGZhu} zmgm8UR%en5xxWiXgXNl*Cs7TnUM1K>BnP+FKwI0Ogh^qY$GU1yDXk`fvv~0Pc9|xR zb2(`ja0bf7l|5P)T>yNeJ_gvso>g|Sbl4%t71O=Ch1)l$CS0yHwhH0~t}Pye9i6au zX*m@omEi+tn^SOF2Z4>vD;t|G{V4}_wLXhawg8XE1$wLh2^z)Ny0g7d3x%E~LZ~Hz z$8W}(1&N*wDXElll@2{c1uCQZmvVinb2V&WlO=gC6`fysN_g11NQL$!O6~XD)Jfnf zOm+$8B+NJ;ZPJj78@VQGzETEDK89D12lPDHWT)9Om-ojvbd6D9{e#KKTzU2agXUFX-AToJgYCOW4TnBCX@Gg`}u=rsmy_r+-`FsqlJY=a* z=Wy*KsXAgR-$?7`__9L%{Qf)k5Jy7)e^4}WAwb2bCLRX0ggGYxqQg@ShxEA1ix8H8$q* z{lhB`;rX>BJy9Kov0>gp{$HpMfE}>F4p}rYg{?2p2Li5sZ|@Ph*oGx@ubydKJ3kH1 z=-tSDfHFT7M8%e7csV{Q4B+S6&7{$Ew#^+mgjhB}c`&7ZF9+}UTheEWGh;cS znwUg?Kx08>qb@{=2KAp|V&)jg7@$0O2^Z8 z6tC+^TtCBkL)Gqi%gZ*HSOp_PomJ~`9e?4Ewi)Pt@lkV#0n}QGKvB%xgGOV!E0W8s zAa0cuV&ePBchHZHa+Y#~DUWlU&y~aP7`^_(L<04G4cQM7EYpgZm_o94NwFM*Nxnyx z%82|JFT-}pEF`Tqx2oCV^1`HJeRy_uZr|)kqK^DKlcAU|#hj={dHoj`*#?u3!}s6S zl73vht$NcyJGedbs2g^3&u|=pMZkB(kn929K9EsjbYOj-;VLB?MZkarvY{p0bV_yf&aaw}Vv2Tlvt`h;G* zKTXyAcTkM#v3YV@63-tvJ=LMSWZl}$0x#yAPtkBKxcx{zX~vxE%#4Ro`sk6s6?;k$ z8sycyYyGn(zkQY>z*MBU&%E)I>5e)!-!jhO=i;jsIDH}?+OHmA)Lya21}9K^`0)NY zVg730`}Q4@)14T`=e(xw+#A)Do?i$)>%5K{RU}9m#^fGfcV*r@02RyqBuz4d5!DT@K|a5>GIrBUxcp(T^No2@bWJl-0{S-AMCS3kktnP3amOQduOvE#^F?>8qa!HyA*$G{3iD(HLfg>}eCTpU%w`DkBX z$|0@ukkZYYHw&z@trTK(A1PUfT(z|37B#Pf>Iv8l z7F6k`$);;vLse2OAz)WIu&0%P*V!?u+I?R4t;Fx|26(elYzWHE-79S-AA>S4c;}A6 z?y_ixnjV^i1*GF>ZNqxpZOZywi_e$Op%;jYaoLWJ4!~5`D}^k4l5SOUrKF?;Qi#KE zIOMqxwG|oNY*|0r%+vuwra{#9XAfZ@e$SGRk1}LHZ;9qB?F(rC_P}~o(*79iX8>%i zFZnelEk1K)wEVY_dD6F|@knPA!xS>2ZhJXo6V&AQ{9yExm%$km305OuFyhKay- zxF!Ax0eZFn7kh6NP{+1y?E(P;B)DsEcXtTx?wa7P!6lFY!QI{6-5r9vyE_DT`D?DV z_B#LF_vJp`hw~ItOsZy8jT+TQ@2!8$8E_oV=A|>&e zwHHe+o<2=M(7pJH$UmVe_3d;?HEgljX2A5AipQwdklz(Lushc^8hGe@?h z0CrsFt58 z*VR-y({E`3jIBXVxo6y{mucKCaq$t8a2tWbHp^wDrO)HvUzPRrY;eghxPat>KiqOE zXAn@K@w_>jO<}iI!hVoIcU>t_XHfPAJh5-td{;Vxe}tEt0+r1p{5BuPUufcG(UW~! z`x4O7WJAU@+SlfA(!FzY0jBCWeD1X(PLh&Av6Wo8+Qf4HM`qiUYS@~X1x7~*lI51m z)3t*?)0#g(73}f59qMC6@@2WG6@=DK6$kV!Tdr$pD{Jm2K)La+N9~8dZSAx0H#D>~ zKgaD>c+wB2@L-!`zenV3%v4S&^Qfm?YvpW5mP8NNSA9<)yAnKdJ@WJM(@e**&q-)$ zY3Rw+&=i?=d?}Ugs&AO>P2Lm6HU7LyYJ16_Qm>Y5lXU@%$-0NZ9kXNL_4NZ7MzB8s z_|mhOV7kolYg}m>x6@J0IRA2X)D$kVlN8Y~lqp=72Q~(rp62#U1`1hMx^{{z0x_OSpcuMx1%{2JE**5SkvOi$Sz#{*PbI;5}#d>xZFQ8MKML150eOPho;L< z?A5yA24SDrO}PXRIXA!__GQD*YLAXA_7$h-Nf~o3`)$LIwGyv1B_0Na9mlZro_@{` z-_6`Byo)&HVV7fwRe$&lhDeX+d2}%CzVqvy_4P2SJ{W~6Cqi>LUu+|K+ogvt>%$g3 zm#)>%_vi@RmFvFp^xg&+@fd4p3(6~z17idH?h8!H&~lRUAHKU;9$BvGpz|;faFDlv z^H`s=0<&~MOGLkL<_#okQp`>~mwxilVNe`F{pARn!cG}{yOUVIl!9bR67!TwL56qp zbCOyymA>iXfthaK5xryZRW`YJ22OwlUvTm0eHgl?7mycRWLDA#TDt>QWzB^Qjc+Uj z^^_gaqXq{DnH1Zr&j0{Qlt!3c5YIBnSg@Y+9wnW6R)AOmx=jw+0ltDD4jS(mI$ZTOEOFDar>hw1$dT$LjnL&16kDzV0r)#d$eBG z*k0jGqfQrqCCMU=6zxyX#;|awt0@VpD@|C&Al#&e@aHn`;Ayv(e*2pKh+I#t3SUx}QG zhTG-H^I}ID&H-*_Ce6dehH{yZY^B5U?Wp-^mnYsu;z`?O643tgo6ktfVXy1(wKFww zy&L=`xCIQ^pqgw_$Y)_e@1=SD@2g8Wky?E%2)OK4yF+n9Dn8}pz18Az6P0?X3#E%G zsl}>48QH)VVq;q>0K9^djgynpt&Z+MaB~j=&&UXiRk#!_m&+;DnWV&ZaI~tsMPlz8 z;LS?GaR{_{*klLh-I6J=g#akb4BEK0y|8(_VTTOr0d-q4RtF7;8=(0h;Z>!FHT*7= zWz1}u#v;5NgC_QqB>Ay28CfR~vAY8FqonY7f!4lhstq{^3iv^$qq|jEHi2Z#wFxZh z0XdJy*2;Z5nekG4K9+=&`xB2-In>(%)o4CkXRot8{VK!tR_G^XYw5M?CE#dgMNg_ovjPVs@}(blL6Wxtkk^E={O?uTe9RCaNE zX*=9|RcFeS=Bt;&-_0Mxwv9#REF(?eXRDK%yt1_2ROWo&-J6B6L%l1HX8)meGfZJ+ z+EeYl7kErwS7P_>p@+?|-Ue~8{7m7e;anzLUMPP;hgQW$qiSx6?S@(2K_{n}X=VJH z05FquW?H-#dYg{)7V#tXb;KjV9Pjy4 zox5)Fad{w;|CYftnij3Hxkr6?{+wHilq%U5Gh+hPEKq;Wl;d4jvhnrBsgW*-gOGv- z?^_Q0xE7bC(_wLi+>4gu{$u~n-bd0;5dY=_&w0U%V1n@Iu5qk`cst3J)8)fX>S~xI z8Knghg@M*gHduJ1IEc`4(hIx-2$%G{dQlE}|Fjrsp5X>-t#-B14`e|HvF*)%wIwTM zLZ&-yMsclaQ{E=XROpddG0ZbfF% zrF5TB1>U{Xfn24dnx#7G&qb+P_ZY`DTE}d@<~LZVZ0VO|chDoB%eEia${_DYwYk{i(kfje)_WIs`

    diff --git a/chapter_graph/graph_operations/index.html b/chapter_graph/graph_operations/index.html index b10135ee5..b275c639c 100644 --- a/chapter_graph/graph_operations/index.html +++ b/chapter_graph/graph_operations/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -3326,8 +3342,8 @@ panic("error") } // 删除边 vet1 - vet2 - DeleteSliceElms(g.adjList[vet1], vet2) - DeleteSliceElms(g.adjList[vet2], vet1) + g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2) + g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1) } /* 添加顶点 */ @@ -3349,8 +3365,8 @@ // 在邻接表中删除顶点 vet 对应的链表 delete(g.adjList, vet) // 遍历其他顶点的链表，删除所有包含 vet 的边 - for _, list := range g.adjList { - DeleteSliceElms(list, vet) + for v, list := range g.adjList { + g.adjList[v] = DeleteSliceElms(list, vet) } } diff --git a/chapter_graph/graph_traversal/index.html b/chapter_graph/graph_traversal/index.html index 92d82463b..4da56686b 100644 --- a/chapter_graph/graph_traversal/index.html +++ b/chapter_graph/graph_traversal/index.html @@ -1990,6 +1990,8 @@ + + @@ -2040,6 +2042,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_graph/index.html b/chapter_graph/index.html index d31ebd8a3..5a66449a7 100644 --- a/chapter_graph/index.html +++ b/chapter_graph/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_graph/summary/index.html b/chapter_graph/summary/index.html index 185363749..4b4722411 100644 --- a/chapter_graph/summary/index.html +++ b/chapter_graph/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_hashing/hash_algorithm/index.html b/chapter_hashing/hash_algorithm/index.html index e720c6b82..246ae630f 100644 --- a/chapter_hashing/hash_algorithm/index.html +++ b/chapter_hashing/hash_algorithm/index.html @@ -1964,6 +1964,8 @@ + + @@ -2014,6 +2016,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_hashing/hash_collision/index.html b/chapter_hashing/hash_collision/index.html index 9123016d6..cdd76343c 100644 --- a/chapter_hashing/hash_collision/index.html +++ b/chapter_hashing/hash_collision/index.html @@ -1970,6 +1970,8 @@ + + @@ -2020,6 +2022,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_hashing/hash_map/index.html b/chapter_hashing/hash_map/index.html index ded058b01..8b8fcb69a 100644 --- a/chapter_hashing/hash_map/index.html +++ b/chapter_hashing/hash_map/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -2203,11 +2219,11 @@

    哈希表的抽象表示

    Fig. 哈希表的抽象表示

    -

    除哈希表外，我们还可以使用数组或链表实现查询功能，其中：

    +

    除哈希表外，我们还可以使用数组或链表实现查询功能。若将学生数据看作数组（链表）元素，则有：

      -
    • 查询元素需要遍历数组（链表）中的所有元素，使用 \(O(n)\) 时间；
      • -
    • 添加元素仅需添加至数组（链表）的尾部即可，使用 \(O(1)\) 时间；
      • -
    • 删除元素需要先查询再删除，使用 \(O(n)\) 时间；
      • +
    • 添加元素：仅需将元素添加至数组（链表）的尾部即可，使用 \(O(1)\) 时间；
      • +
    • 查询元素：由于数组（链表）是乱序的，因此需要遍历数组（链表）中的所有元素，使用 \(O(n)\) 时间；
      • +
    • 删除元素：需要先查询到元素，再从数组中删除，使用 \(O(n)\) 时间；
    diff --git a/chapter_hashing/index.html b/chapter_hashing/index.html index 28d59a235..ecb901b7a 100644 --- a/chapter_hashing/index.html +++ b/chapter_hashing/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_hashing/summary/index.html b/chapter_hashing/summary/index.html index 873a44a86..7cd3be51b 100644 --- a/chapter_hashing/summary/index.html +++ b/chapter_hashing/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_heap/build_heap/index.html b/chapter_heap/build_heap/index.html index ff55e9006..d7d463f8f 100644 --- a/chapter_heap/build_heap/index.html +++ b/chapter_heap/build_heap/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_heap/heap/index.html b/chapter_heap/heap/index.html index dd1a4a7b8..6d43e3495 100644 --- a/chapter_heap/heap/index.html +++ b/chapter_heap/heap/index.html @@ -1991,6 +1991,8 @@ + + @@ -2041,6 +2043,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_heap/index.html b/chapter_heap/index.html index 9cdc2dc51..f6720a83f 100644 --- a/chapter_heap/index.html +++ b/chapter_heap/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_heap/summary/index.html b/chapter_heap/summary/index.html index fd2a33ebc..7b694bd66 100644 --- a/chapter_heap/summary/index.html +++ b/chapter_heap/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_heap/top_k/index.html b/chapter_heap/top_k/index.html index 4fe45d15d..12955c9a3 100644 --- a/chapter_heap/top_k/index.html +++ b/chapter_heap/top_k/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_introduction/algorithms_are_everywhere/index.html b/chapter_introduction/algorithms_are_everywhere/index.html index b40bcced9..04f114f39 100644 --- a/chapter_introduction/algorithms_are_everywhere/index.html +++ b/chapter_introduction/algorithms_are_everywhere/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -2125,7 +2141,7 @@

    1.1.   算法无处不在

    当我们听到“算法”这个词时，很自然地会想到数学。然而实际上，许多算法并不涉及复杂数学，而是更多地依赖于基本逻辑，这些逻辑在我们的日常生活中处处可见。

    -

    在正式探讨算法之前，有一个有趣的事实值得分享：你已经在不知不觉中学会了许多算法，并习惯将它们应用到日常生活中了。下面，我将举即个具体例子来证实这一点。

    +

    在正式探讨算法之前，有一个有趣的事实值得分享：你已经在不知不觉中学会了许多算法，并习惯将它们应用到日常生活中了。下面，我将举几个具体例子来证实这一点。

    例一：拼装积木。一套积木，除了包含许多零件之外，还附有详细的组装说明书。我们按照说明书一步步操作，就能组装出精美的积木模型。

    从数据结构与算法的角度来看，积木的各种形状和连接方式代表数据结构，而组装说明书上的一系列步骤则是算法。

    拼装积木

    diff --git a/chapter_introduction/index.html b/chapter_introduction/index.html index f5ec28849..c7ff17e70 100644 --- a/chapter_introduction/index.html +++ b/chapter_introduction/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_introduction/summary/index.html b/chapter_introduction/summary/index.html index 0af4d64f2..6ccbecf33 100644 --- a/chapter_introduction/summary/index.html +++ b/chapter_introduction/summary/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_introduction/what_is_dsa/index.html b/chapter_introduction/what_is_dsa/index.html index 52d4eba05..798ff630c 100644 --- a/chapter_introduction/what_is_dsa/index.html +++ b/chapter_introduction/what_is_dsa/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_preface/about_the_book/index.html b/chapter_preface/about_the_book/index.html index 01422d7fb..0d15d5202 100644 --- a/chapter_preface/about_the_book/index.html +++ b/chapter_preface/about_the_book/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_preface/index.html b/chapter_preface/index.html index f9d7721d4..4c2ac4931 100644 --- a/chapter_preface/index.html +++ b/chapter_preface/index.html @@ -1900,6 +1900,8 @@ + + @@ -1950,6 +1952,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_preface/suggestions/index.html b/chapter_preface/suggestions/index.html index 723d0b1dd..6b21c1c6e 100644 --- a/chapter_preface/suggestions/index.html +++ b/chapter_preface/suggestions/index.html @@ -1971,6 +1971,8 @@ + + @@ -2021,6 +2023,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_preface/summary/index.html b/chapter_preface/summary/index.html index 350c47e3f..a3eb4efad 100644 --- a/chapter_preface/summary/index.html +++ b/chapter_preface/summary/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_reference/index.html b/chapter_reference/index.html index 38d1f54b1..9be5ebccc 100644 --- a/chapter_reference/index.html +++ b/chapter_reference/index.html @@ -1898,6 +1898,8 @@ + + @@ -1948,6 +1950,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/binary_search/index.html b/chapter_searching/binary_search/index.html index d4e50ac4e..389c7e9c0 100644 --- a/chapter_searching/binary_search/index.html +++ b/chapter_searching/binary_search/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/binary_search_edge/index.html b/chapter_searching/binary_search_edge/index.html index 7ba25bba3..5aed92a9c 100644 --- a/chapter_searching/binary_search_edge/index.html +++ b/chapter_searching/binary_search_edge/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/index.html b/chapter_searching/index.html index f50453fa9..e0aa99a41 100644 --- a/chapter_searching/index.html +++ b/chapter_searching/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/replace_linear_by_hashing/index.html b/chapter_searching/replace_linear_by_hashing/index.html index a607b5a94..420c67561 100644 --- a/chapter_searching/replace_linear_by_hashing/index.html +++ b/chapter_searching/replace_linear_by_hashing/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/searching_algorithm_revisited/index.html b/chapter_searching/searching_algorithm_revisited/index.html index 4df1cdc24..6902ee891 100644 --- a/chapter_searching/searching_algorithm_revisited/index.html +++ b/chapter_searching/searching_algorithm_revisited/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_searching/summary/index.html b/chapter_searching/summary/index.html index c703a6611..41968a619 100644 --- a/chapter_searching/summary/index.html +++ b/chapter_searching/summary/index.html @@ -1912,6 +1912,8 @@ + + @@ -1962,6 +1964,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/bubble_sort/index.html b/chapter_sorting/bubble_sort/index.html index 36856b4c5..cbc0d4ac0 100644 --- a/chapter_sorting/bubble_sort/index.html +++ b/chapter_sorting/bubble_sort/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/bucket_sort/index.html b/chapter_sorting/bucket_sort/index.html index 46a9174d4..2ecc325e3 100644 --- a/chapter_sorting/bucket_sort/index.html +++ b/chapter_sorting/bucket_sort/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + @@ -2283,7 +2299,7 @@ i=int(num*k)# 将 num 添加进桶 ibuckets[i].append(num) -# 2. 对各个桶执行排序5 +# 2. 对各个桶执行排序forbucketinbuckets:# 使用内置排序函数，也可以替换成其他排序算法bucket.sort() diff --git a/chapter_sorting/counting_sort/index.html b/chapter_sorting/counting_sort/index.html index a44f7ca32..8d8f1f2bb 100644 --- a/chapter_sorting/counting_sort/index.html +++ b/chapter_sorting/counting_sort/index.html @@ -1964,6 +1964,8 @@ + + @@ -2014,6 +2016,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/heap_sort/index.html b/chapter_sorting/heap_sort/index.html index b1e7a731f..5d687eef1 100644 --- a/chapter_sorting/heap_sort/index.html +++ b/chapter_sorting/heap_sort/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/index.html b/chapter_sorting/index.html index 6cccb2f4e..20bfd6f5e 100644 --- a/chapter_sorting/index.html +++ b/chapter_sorting/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/insertion_sort/index.html b/chapter_sorting/insertion_sort/index.html index 4cfffd5e0..d6346aef2 100644 --- a/chapter_sorting/insertion_sort/index.html +++ b/chapter_sorting/insertion_sort/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/merge_sort/index.html b/chapter_sorting/merge_sort/index.html index 6e6de2afd..7c6f4b3f7 100644 --- a/chapter_sorting/merge_sort/index.html +++ b/chapter_sorting/merge_sort/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/quick_sort/index.html b/chapter_sorting/quick_sort/index.html index 22001ae59..4155602ae 100644 --- a/chapter_sorting/quick_sort/index.html +++ b/chapter_sorting/quick_sort/index.html @@ -1971,6 +1971,8 @@ + + @@ -2021,6 +2023,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/radix_sort/index.html b/chapter_sorting/radix_sort/index.html index 8cabf4ce4..415cdf275 100644 --- a/chapter_sorting/radix_sort/index.html +++ b/chapter_sorting/radix_sort/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/selection_sort/index.html b/chapter_sorting/selection_sort/index.html index 490207c61..5dc1d408d 100644 --- a/chapter_sorting/selection_sort/index.html +++ b/chapter_sorting/selection_sort/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/sorting_algorithm/index.html b/chapter_sorting/sorting_algorithm/index.html index 59aea6fbb..09f0a75ae 100644 --- a/chapter_sorting/sorting_algorithm/index.html +++ b/chapter_sorting/sorting_algorithm/index.html @@ -1950,6 +1950,8 @@ + + @@ -2000,6 +2002,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_sorting/summary/index.html b/chapter_sorting/summary/index.html index ca33e48e1..954b6324a 100644 --- a/chapter_sorting/summary/index.html +++ b/chapter_sorting/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_stack_and_queue/deque/index.html b/chapter_stack_and_queue/deque/index.html index ae892fc3d..5131ed8a5 100644 --- a/chapter_stack_and_queue/deque/index.html +++ b/chapter_stack_and_queue/deque/index.html @@ -1977,6 +1977,8 @@ + + @@ -2027,6 +2029,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_stack_and_queue/index.html b/chapter_stack_and_queue/index.html index 6c2cf4d41..c4c6ff779 100644 --- a/chapter_stack_and_queue/index.html +++ b/chapter_stack_and_queue/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_stack_and_queue/queue/index.html b/chapter_stack_and_queue/queue/index.html index dd96ec60a..49b06160b 100644 --- a/chapter_stack_and_queue/queue/index.html +++ b/chapter_stack_and_queue/queue/index.html @@ -1977,6 +1977,8 @@ + + @@ -2027,6 +2029,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_stack_and_queue/stack/index.html b/chapter_stack_and_queue/stack/index.html index eaf81b61e..48d3450d8 100644 --- a/chapter_stack_and_queue/stack/index.html +++ b/chapter_stack_and_queue/stack/index.html @@ -2011,6 +2011,8 @@ + + @@ -2061,6 +2063,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_stack_and_queue/summary/index.html b/chapter_stack_and_queue/summary/index.html index 298a67f40..e1a645ca9 100644 --- a/chapter_stack_and_queue/summary/index.html +++ b/chapter_stack_and_queue/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/array_representation_of_tree/index.html b/chapter_tree/array_representation_of_tree/index.html index 5b436e57e..43bfe0dcc 100644 --- a/chapter_tree/array_representation_of_tree/index.html +++ b/chapter_tree/array_representation_of_tree/index.html @@ -1957,6 +1957,8 @@ + + @@ -2007,6 +2009,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/avl_tree/index.html b/chapter_tree/avl_tree/index.html index edff9c21e..351edaf62 100644 --- a/chapter_tree/avl_tree/index.html +++ b/chapter_tree/avl_tree/index.html @@ -2052,6 +2052,8 @@ + + @@ -2102,6 +2104,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/binary_search_tree/index.html b/chapter_tree/binary_search_tree/index.html index d2f27cfca..ae9c0ccd5 100644 --- a/chapter_tree/binary_search_tree/index.html +++ b/chapter_tree/binary_search_tree/index.html @@ -1991,6 +1991,8 @@ + + @@ -2041,6 +2043,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/binary_tree/index.html b/chapter_tree/binary_tree/index.html index 9443ac2fc..006ec34f4 100644 --- a/chapter_tree/binary_tree/index.html +++ b/chapter_tree/binary_tree/index.html @@ -1998,6 +1998,8 @@ + + @@ -2048,6 +2050,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/binary_tree_traversal/index.html b/chapter_tree/binary_tree_traversal/index.html index 7fd8b0938..700584d01 100644 --- a/chapter_tree/binary_tree_traversal/index.html +++ b/chapter_tree/binary_tree_traversal/index.html @@ -1990,6 +1990,8 @@ + + @@ -2040,6 +2042,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/index.html b/chapter_tree/index.html index d7ccbd959..3a4cccd7d 100644 --- a/chapter_tree/index.html +++ b/chapter_tree/index.html @@ -1902,6 +1902,8 @@ + + @@ -1952,6 +1954,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/chapter_tree/summary/index.html b/chapter_tree/summary/index.html index e3d37bb59..b65e68677 100644 --- a/chapter_tree/summary/index.html +++ b/chapter_tree/summary/index.html @@ -1943,6 +1943,8 @@ + + @@ -1993,6 +1995,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/index.html b/index.html index 26e07ff3a..b8bb9e41f 100644 --- a/index.html +++ b/index.html @@ -1893,6 +1893,8 @@ + + @@ -1943,6 +1945,20 @@ + + + + + +
  • + + 13.3.   0-1 背包问题（New） + +
    • + + + + diff --git a/search/search_index.json b/search/search_index.json index 5dbbdbfb9..0bd24cc4a 100644 --- a/search/search_index.json +++ b/search/search_index.json @@ -1 +1 @@ -{"config":{"lang":["ja"],"separator":"[\\s\\-\uff0c\u3002]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"\u300a Hello \u7b97\u6cd5 \u300b

    \u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b

    \u63a8\u8350\u8bed

    Quote

    \u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d

    \u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388

    Quote

    \u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d

    \u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6

    \u5168\u4e66\u52a8\u753b\u56fe\u89e3

    \u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u7535\u8111\u3001\u5e73\u677f\u3001\u624b\u673a\u5168\u7ec8\u7aef\u9605\u8bfb

    \"A picture is worth a thousand words.\"

    \u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d

    \u4ee3\u7801\u4e00\u952e\u8fd0\u884c

    \u63d0\u4f9b\u5404\u4e2a\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u7b80\u6d01\u5b9e\u73b0\u4e0e\u6d4b\u8bd5\u6837\u4f8b\uff0c\u7686\u53ef\u76f4\u63a5\u8fd0\u884c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00

    \"Talk is cheap. Show me the code.\"

    \u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d

    \u53ef\u8ba8\u8bba\u4e0e\u63d0\u95ee

    \u9f13\u52b1\u5c0f\u4f19\u4f34\u4eec\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u6210\u957f\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d

    \u201c\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d

    \u4e00\u8d77\u52a0\u6cb9\uff01

    \u5e8f

    \u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u300a\u5251\u6307 Offer\u300b\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u670b\u53cb\u7684\u559c\u7231\u4e0e\u652f\u6301\u3002\u5728\u6b64\u671f\u95f4\uff0c\u6211\u56de\u7b54\u4e86\u4f17\u591a\u8bfb\u8005\u7684\u8bc4\u8bba\u95ee\u9898\uff0c\u5176\u4e2d\u6700\u5e38\u89c1\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u5b66\u4e60\u7b97\u6cd5\u201d\u3002\u6211\u9010\u6e10\u4e5f\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002

    \u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\uff0c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5730\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e66\u7c4d\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u9762\u8bd5\u5df2\u7ecf\u5360\u636e\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u539a\u91cd\u7684\u4e66\u7c4d\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002

    \u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u6b64\u95ee\u9898\u7684\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u867d\u7136\u4e0d\u4e00\u5b9a\u6b63\u786e\uff0c\u4f46\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u8fd9\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5e94\u5bf9\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002

    \u4f5c\u8005\u7b80\u4ecb

    \u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u5176 LeetBook\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 24 \u4e07\u672c\u3002

    \u81f4\u8c22

    \u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u6210\u957f\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u662f\u4ed6\u4eec\u65e0\u79c1\u5949\u732e\u4f7f\u8fd9\u672c\u4e66\u8d8a\u53d8\u8d8a\u597d\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a

    \u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 Gonglja, gvenusleo, justin\u2010tse, krahets, nuomi1, Reanon, sjinzh \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002

    GongljaC / C++ gvenusleoDart hpstoryC# justin-tseJS / TS krahetsJava / Python nuomi1Swift ReanonGo / C sjinzhRust / Zig"},{"location":"chapter_appendix/contribution/","title":"14.2. \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"

    \u5f00\u6e90\u7684\u9b45\u529b

    \u7eb8\u8d28\u4e66\u7c4d\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u9700\u8981\u6570\u5e74\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002\u4f46\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002

    \u7531\u4e8e\u4f5c\u8005\u80fd\u529b\u6709\u9650\uff0c\u4e66\u4e2d\u96be\u514d\u5b58\u5728\u4e00\u4e9b\u9057\u6f0f\u548c\u9519\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u60a8\u53d1\u73b0\u4e86\u7b14\u8bef\u3001\u5931\u6548\u94fe\u63a5\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u534f\u52a9\u6211\u4eec\u8fdb\u884c\u4fee\u6b63\uff0c\u4ee5\u5e2e\u52a9\u5176\u4ed6\u8bfb\u8005\u83b7\u5f97\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u8d44\u6e90\u3002\u6240\u6709\u64b0\u7a3f\u4eba\u5c06\u5728\u4ed3\u5e93\u548c\u7f51\u7ad9\u4e3b\u9875\u4e0a\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\uff01

    "},{"location":"chapter_appendix/contribution/#1421","title":"14.2.1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"

    \u5728\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u6709\u4e00\u4e2a\u300c\u7f16\u8f91\u300d\u56fe\u6807\uff0c\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\uff1a

    1. \u70b9\u51fb\u7f16\u8f91\u6309\u94ae\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\uff1b
    2. \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u5e76\u786e\u4fdd\u5185\u5bb9\u6b63\u786e\uff0c\u540c\u65f6\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\uff1b
    3. \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\uff1b\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002

    Fig. \u9875\u9762\u7f16\u8f91\u6309\u952e

    \u7531\u4e8e\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u56e0\u6b64\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u56fe\u7247\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002

    "},{"location":"chapter_appendix/contribution/#1422","title":"14.2.2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"

    \u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd Pull Request \u5de5\u4f5c\u6d41\u7a0b\uff1a

    1. \u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\uff1b
    2. \u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 git clone \u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\uff1b
    3. \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u901a\u8fc7\u8fd0\u884c\u6d4b\u8bd5\u4ee5\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\uff1b
    4. \u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\uff1b
    5. \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\uff1b
    "},{"location":"chapter_appendix/contribution/#1423-docker","title":"14.2.3. \u00a0 Docker \u90e8\u7f72","text":"

    \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7 Docker \u6765\u90e8\u7f72\u672c\u9879\u76ee\u3002\u6267\u884c\u4ee5\u4e0b\u811a\u672c\uff0c\u7a0d\u7b49\u7247\u523b\u540e\uff0c\u5373\u53ef\u4f7f\u7528\u6d4f\u89c8\u5668\u6253\u5f00 http://localhost:8000 \u6765\u8bbf\u95ee\u672c\u9879\u76ee\u3002

    git clone https://github.com/krahets/hello-algo.git\ncd hello-algo\ndocker-compose up -d\n

    \u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\u3002

    docker-compose down\n
    "},{"location":"chapter_appendix/installation/","title":"14.1. \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1411-vscode","title":"14.1.1. \u00a0 VSCode","text":"

    \u672c\u4e66\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730 IDE \uff0c\u4e0b\u8f7d\u5e76\u5b89\u88c5 VSCode \u3002

    "},{"location":"chapter_appendix/installation/#1412-java","title":"14.1.2. \u00a0 Java \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 java \uff0c\u5b89\u88c5 Extension Pack for Java \u3002
    "},{"location":"chapter_appendix/installation/#1413-cc","title":"14.1.3. \u00a0 C/C++ \u73af\u5883","text":"
    1. Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff0cMacOS \u81ea\u5e26 Clang \u65e0\u9700\u5b89\u88c5\u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c++ \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002
    3. \uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 Clang_format_fallback Style \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a { BasedOnStyle: Microsoft, BreakBeforeBraces: Attach } \u3002
    "},{"location":"chapter_appendix/installation/#1414-python","title":"14.1.4. \u00a0 Python \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 python \uff0c\u5b89\u88c5 Python Extension Pack \u3002
    3. \uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 pip install black \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
    "},{"location":"chapter_appendix/installation/#1415-go","title":"14.1.5. \u00a0 Go \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 go \uff0c\u5b89\u88c5 Go \u3002
    3. \u5feb\u6377\u952e Ctrl + Shift + P \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 Go: Install/Update Tools \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
    "},{"location":"chapter_appendix/installation/#1416-javascript","title":"14.1.6. \u00a0 JavaScript \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 javascript \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002
    3. \uff08\u53ef\u9009\uff09\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 Prettier \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
    "},{"location":"chapter_appendix/installation/#1417-c","title":"14.1.7. \u00a0 C# \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 6.0 \uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c# \uff0c\u5b89\u88c5 c# \u3002
    "},{"location":"chapter_appendix/installation/#1418-swift","title":"14.1.8. \u00a0 Swift \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift\uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 swift \uff0c\u5b89\u88c5 Swift for Visual Studio Code\u3002
    "},{"location":"chapter_appendix/installation/#1419-rust","title":"14.1.9. \u00a0 Rust \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust\uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 rust \uff0c\u5b89\u88c5 rust-analyzer\u3002
    "},{"location":"chapter_array_and_linkedlist/","title":"4. \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":""},{"location":"chapter_array_and_linkedlist/array/","title":"4.1. \u00a0 \u6570\u7ec4","text":"

    \u300c\u6570\u7ec4 Array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u5143\u7d20\u7684\u300c\u7d22\u5f15 Index\u300d\u3002

    Fig. \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f

    \u6570\u7ec4\u521d\u59cb\u5316\u3002\u901a\u5e38\u6709\u65e0\u521d\u59cb\u503c\u548c\u7ed9\u5b9a\u521d\u59cb\u503c\u4e24\u79cd\u65b9\u5f0f\uff0c\u6211\u4eec\u53ef\u6839\u636e\u9700\u6c42\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u3002\u5728\u672a\u7ed9\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u901a\u5e38\u4f1a\u88ab\u521d\u59cb\u5316\u4e3a\u9ed8\u8ba4\u503c \\(0\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
    array.cpp
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
    array.py
    # \u521d\u59cb\u5316\u6570\u7ec4\narr: List[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: List[int] = [1, 3, 2, 5, 4]  \n
    array.go
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
    array.js
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
    array.ts
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
    array.c
    int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
    array.cs
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
    array.swift
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
    array.zig
    // \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n
    array.dart
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nList<int> arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList<int> nums = [1, 3, 2, 5, 4];\n
    "},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1. \u00a0 \u6570\u7ec4\u4f18\u70b9","text":"

    \u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\u3002\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u6b64\u5143\u7d20\u3002

    Fig. \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 

    # \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n

    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u5143\u7d20\u7684\u7d22\u5f15\u8981\u4ece \\(0\\) \u5f00\u59cb\u7f16\u53f7\u5462\uff1f

    \u89c2\u5bdf\u4e0a\u56fe\uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002

    \u7136\u800c\uff0c\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u8868\u793a\u7684\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u7d22\u5f15\u4e3a \\(0\\) \u4e5f\u662f\u5408\u7406\u7684\u3002

    \u8bbf\u95ee\u5143\u7d20\u7684\u9ad8\u6548\u6027\u5e26\u6765\u4e86\u8bf8\u591a\u4fbf\u5229\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.cpp
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.py
    def random_access(nums: list[int]) -> int:\n\"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandom_index = random.randint(0, len(nums) - 1)\n# \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandom_num = nums[random_index]\nreturn random_num\n
    array.go
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandomIndex := rand.Intn(len(nums))\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandomNum = nums[randomIndex]\nreturn\n}\n
    array.js
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
    array.ts
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
    array.c
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.cs
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\nRandom random = new();\n// \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = random.Next(nums.Length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.swift
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n// \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet randomIndex = nums.indices.randomElement()!\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet randomNum = nums[randomIndex]\nreturn randomNum\n}\n
    array.zig
    // \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n// \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\nvar randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nvar randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.dart
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(List nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = Random().nextInt(nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2. \u00a0 \u6570\u7ec4\u7f3a\u70b9","text":"

    \u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7531\u4e8e\u7cfb\u7edf\u65e0\u6cd5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u56e0\u6b64\u6570\u7ec4\u957f\u5ea6\u65e0\u6cd5\u6269\u5c55\u3002\u800c\u82e5\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u65b0\u5efa\u4e00\u4e2a\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u662f\u975e\u5e38\u8017\u65f6\u7684\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.cpp
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = new int[size + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] nums;\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.py
    def extend(nums: list[int], enlarge: int) -> list[int]:\n\"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n# \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres = [0] * (len(nums) + enlarge)\n# \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in range(len(nums)):\nres[i] = nums[i]\n# \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n
    array.go
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres := make([]int, len(nums)+enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i, num := range nums {\nres[i] = num\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
    array.js
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.ts
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.c
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = (int *)malloc(sizeof(int) * (size + enlarge));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\nfor (int i = size; i < size + enlarge; i++) {\nres[i] = 0;\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.cs
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.Length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.swift
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = Array(repeating: 0, count: nums.count + enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in nums.indices {\nres[i] = nums[i]\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
    array.zig
    // \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = try mem_allocator.alloc(i32, nums.len + enlarge);\n@memset(res, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(i32, res, nums);\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.dart
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList extend(List nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nList<int> res = List.filled(nums.length + enlarge, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (var i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n

    \u6570\u7ec4\u4e2d\u63d2\u5165\u6216\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\u4e0b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u653e\u4efb\u4f55\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4e0d\u5f97\u4e0d\u5c06\u6b64\u7d22\u5f15\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002

    Fig. \u6570\u7ec4\u63d2\u5165\u5143\u7d20

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.cpp
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.py
    def insert(nums: list[int], num: int, index: int) -> None:\n\"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n# \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in range(len(nums) - 1, index, -1):\nnums[i] = nums[i - 1]\n# \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n
    array.go
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i := len(nums) - 1; i > index; i-- {\nnums[i] = nums[i-1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
    array.js
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.ts
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.c
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.cs
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.Length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.swift
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: nums.count - 1, next: { $0 > index + 1 ? $0 - 1 : nil }) {\nnums[i] = nums[i - 1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
    array.zig
    // \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar i = nums.len - 1;\nwhile (i > index) : (i -= 1) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.dart
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(List nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n

    \u5220\u9664\u5143\u7d20\u4e5f\u7c7b\u4f3c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5220\u9664\u5143\u7d20\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6211\u4eec\u65e0\u9700\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002

    Fig. \u6570\u7ec4\u5220\u9664\u5143\u7d20

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.cpp
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.py
    def remove(nums: list[int], index: int) -> None:\n\"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\"\"\"\n# \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in range(index, len(nums) - 1):\nnums[i] = nums[i + 1]\n
    array.go
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i := index; i < len(nums)-1; i++ {\nnums[i] = nums[i+1]\n}\n}\n
    array.js
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums, index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.ts
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.c
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.cs
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.Length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.swift
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\nlet count = nums.count\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: index, next: { $0 < count - 1 - 1 ? $0 + 1 : nil }) {\nnums[i] = nums[i + 1]\n}\n}\n
    array.zig
    // \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar i = index;\nwhile (i < nums.len - 1) : (i += 1) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.dart
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(List nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n

    \u603b\u7ed3\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\uff1a

    • \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
    • \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002
    • \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u6211\u4eec\u4e0d\u5173\u5fc3\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u540c\u65f6\u4e5f\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
    "},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3. \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":"

    \u6570\u7ec4\u904d\u5386\u3002\u4ee5\u4e0b\u4ecb\u7ecd\u4e24\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int num : nums) {\ncount++;\n}\n}\n
    array.cpp
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
    array.py
    def traverse(nums: list[int]) -> None:\n\"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\ncount = 0\n# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor num in nums:\ncount += 1\n# \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\nfor i, num in enumerate(nums):\ncount += 1\n
    array.go
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\ncount := 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\ncount++\n}\ncount = 0\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor range nums {\ncount++\n}\n}\n
    array.js
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
    array.ts
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
    array.c
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
    array.cs
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nforeach (int num in nums) {\ncount++;\n}\n}\n
    array.swift
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\nvar count = 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor _ in nums.indices {\ncount += 1\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor _ in nums {\ncount += 1\n}\n}\n
    array.zig
    // \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\nvar count: i32 = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nvar i: i32 = 0;\nwhile (i < nums.len) : (i += 1) {\ncount += 1;\n}\ncount = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (nums) |_| {\ncount += 1;\n}\n}\n
    array.dart
    /* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List nums) {\nvar count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (var i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (var num in nums) {\ncount++;\n}\n// \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\nnums.forEach((element) {\ncount++;\n});\n}\n

    \u6570\u7ec4\u67e5\u627e\u3002\u901a\u8fc7\u904d\u5386\u6570\u7ec4\uff0c\u67e5\u627e\u6570\u7ec4\u5185\u7684\u6307\u5b9a\u5143\u7d20\uff0c\u5e76\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.cpp
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.py
    def find(nums: list[int], target: int) -> int:\n\"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\nfor i in range(len(nums)):\nif nums[i] == target:\nreturn i\nreturn -1\n
    array.go
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\nindex = -1\nfor i := 0; i < len(nums); i++ {\nif nums[i] == target {\nindex = i\nbreak\n}\n}\nreturn\n}\n
    array.js
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
    array.ts
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] === target) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    array.c
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.cs
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.Length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.swift
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\nfor i in nums.indices {\nif nums[i] == target {\nreturn i\n}\n}\nreturn -1\n}\n
    array.zig
    // \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\nfor (nums, 0..) |num, i| {\nif (num == target) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    array.dart
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List nums, int target) {\nfor (var i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#414","title":"4.1.4. \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"

    \u968f\u673a\u8bbf\u95ee\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u6837\u672c\u7684\u968f\u673a\u62bd\u53d6\u3002

    \u4e8c\u5206\u67e5\u627e\u3002\u4f8b\u5982\u524d\u6587\u67e5\u5b57\u5178\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u4e2d\u7684\u6240\u6709\u5b57\u6309\u7167\u62fc\u97f3\u987a\u5e8f\u5b58\u50a8\u5728\u6570\u7ec4\u4e2d\uff0c\u7136\u540e\u4f7f\u7528\u4e0e\u65e5\u5e38\u67e5\u7eb8\u8d28\u5b57\u5178\u76f8\u540c\u7684\u201c\u7ffb\u5f00\u4e2d\u95f4\uff0c\u6392\u9664\u4e00\u534a\u201d\u7684\u65b9\u5f0f\uff0c\u6765\u5b9e\u73b0\u4e00\u4e2a\u67e5\u7535\u5b50\u5b57\u5178\u7684\u7b97\u6cd5\u3002

    \u6df1\u5ea6\u5b66\u4e60\u3002\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2. \u00a0 \u94fe\u8868","text":"

    \u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u6392\u9664\u5df2\u88ab\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u7a7a\u95f2\u5185\u5b58\u7a7a\u95f4\u901a\u5e38\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6211\u4eec\u9700\u8981\u7533\u8bf7\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6570\u7ec4\u65f6\uff0c\u7a7a\u95f2\u5185\u5b58\u4e2d\u53ef\u80fd\u6ca1\u6709\u8fd9\u4e48\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u4e0e\u6570\u7ec4\u76f8\u6bd4\uff0c\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u5b83\u53ef\u4ee5\u88ab\u5b58\u50a8\u5728\u975e\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002

    \u300c\u94fe\u8868 Linked List\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u8fde\u63a5\uff0c\u4ece\u5f53\u524d\u8282\u70b9\u901a\u8fc7\u6307\u9488\u53ef\u4ee5\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002\u7531\u4e8e\u6307\u9488\u8bb0\u5f55\u4e86\u4e0b\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u56e0\u6b64\u65e0\u9700\u4fdd\u8bc1\u5185\u5b58\u5730\u5740\u7684\u8fde\u7eed\u6027\uff0c\u4ece\u800c\u53ef\u4ee5\u5c06\u5404\u4e2a\u8282\u70b9\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002

    \u94fe\u8868\u300c\u8282\u70b9 Node\u300d\u5305\u542b\u4e24\u9879\u6570\u636e\uff0c\u4e00\u662f\u8282\u70b9\u300c\u503c Value\u300d\uff0c\u4e8c\u662f\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u300c\u6307\u9488 Pointer\u300d\uff0c\u6216\u79f0\u300c\u5f15\u7528 Reference\u300d\u3002

    Fig. \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
    class ListNode:\n\"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                  # \u8282\u70b9\u503c\nself.next: Optional[ListNode] = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\nVal  int       // \u8282\u70b9\u503c\nNext *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\nreturn &ListNode{\nVal:  val,\nNext: nil,\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nconstructor(val, next) {\nthis.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\nthis.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nconstructor(val?: number, next?: ListNode | null) {\nthis.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nreturn node;\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode\n{\nint val;         // \u8282\u70b9\u503c\nListNode next;   // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\nListNode(int x) => val = x;  //\u6784\u9020\u51fd\u6570\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
    // \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\n}\n};\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n

    \u5c3e\u8282\u70b9\u6307\u5411\u4ec0\u4e48\uff1f

    \u6211\u4eec\u5c06\u94fe\u8868\u7684\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u300c\u5c3e\u8282\u70b9\u300d\uff0c\u5176\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5728 Java, C++, Python \u4e2d\u5206\u522b\u8bb0\u4e3a \\(\\text{null}\\) , \\(\\text{nullptr}\\) , \\(\\text{None}\\) \u3002\u5728\u4e0d\u5f15\u8d77\u6b67\u4e49\u7684\u524d\u63d0\u4e0b\uff0c\u672c\u4e66\u90fd\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u7a7a\u3002

    \u5982\u4f55\u79f0\u547c\u94fe\u8868\uff1f

    \u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u6574\u4f53\u5c31\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u4f8b\u5982\u6570\u7ec4 nums \uff0c\u5305\u542b\u5404\u4e2a\u5143\u7d20 nums[0] , nums[1] \u7b49\u7b49\u3002\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u4f8b\u5982\u5934\u8282\u70b9 head \u548c\u94fe\u8868 head \u5b9e\u9645\u4e0a\u662f\u540c\u4e49\u7684\u3002

    \u94fe\u8868\u521d\u59cb\u5316\u65b9\u6cd5\u3002\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u5f15\u7528\u6307\u5411\u5173\u7cfb\u3002\u5b8c\u6210\u540e\uff0c\u5373\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\uff08\u5373\u9996\u4e2a\u8282\u70b9\uff09\u51fa\u53d1\uff0c\u901a\u8fc7\u6307\u9488 next \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.cpp
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
    linked_list.py
    # \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
    linked_list.go
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
    linked_list.js
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.ts
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.c
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
    linked_list.cs
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.swift
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
    linked_list.zig
    // \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n
    linked_list.dart
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1. \u00a0 \u94fe\u8868\u4f18\u70b9","text":"

    \u94fe\u8868\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u7684\u64cd\u4f5c\u6548\u7387\u9ad8\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u5728\u94fe\u8868\u4e2d\u95f4\u7684\u4e24\u4e2a\u8282\u70b9 A , B \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 P \uff0c\u6211\u4eec\u53ea\u9700\u8981\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u6307\u9488\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u64cd\u4f5c\u6548\u7387\u8981\u4f4e\u5f97\u591a\u3002

    Fig. \u94fe\u8868\u63d2\u5165\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.cpp
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
    linked_list.py
    def insert(n0: ListNode, P: ListNode) -> None:\n\"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\nn1 = n0.next\nP.next = n1\nn0.next = P\n
    linked_list.go
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\nn1 := n0.Next\nP.Next = n1\nn0.Next = P\n}\n
    linked_list.js
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.ts
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.c
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
    linked_list.cs
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.swift
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\nlet n1 = n0.next\nP.next = n1\nn0.next = P\n}\n
    linked_list.zig
    // \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\nvar n1 = n0.?.next;\nP.?.next = n1;\nn0.?.next = P;\n}\n
    linked_list.dart
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n

    \u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u6307\u9488\u5373\u53ef\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u8282\u70b9 P \u4ecd\u7136\u6307\u5411 n1 \uff0c\u4f46\u5b9e\u9645\u4e0a P \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u6b64\u94fe\u8868\uff0c\u56e0\u4e3a\u904d\u5386\u6b64\u94fe\u8868\u65f6\u65e0\u6cd5\u8bbf\u95ee\u5230 P \u3002

    Fig. \u94fe\u8868\u5220\u9664\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.cpp
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\nif (n0->next == nullptr)\nreturn;\n// n0 -> P -> n1\nListNode *P = n0->next;\nListNode *n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\ndelete P;\n}\n
    linked_list.py
    def remove(n0: ListNode) -> None:\n\"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nif not n0.next:\nreturn\n# n0 -> P -> n1\nP = n0.next\nn1 = P.next\nn0.next = n1\n
    linked_list.go
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeNode(n0 *ListNode) {\nif n0.Next == nil {\nreturn\n}\n// n0 -> P -> n1\nP := n0.Next\nn1 := P.Next\nn0.Next = n1\n}\n
    linked_list.js
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\nif (!n0.next) return;\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.ts
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\nif (!n0.next) {\nreturn;\n}\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.c
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeNode(ListNode *n0) {\nif (!n0->next)\nreturn;\n// n0 -> P -> n1\nListNode *P = n0->next;\nListNode *n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\nfree(P);\n}\n
    linked_list.cs
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.swift
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\nif n0.next == nil {\nreturn\n}\n// n0 -> P -> n1\nlet P = n0.next\nlet n1 = P?.next\nn0.next = n1\nP?.next = nil\n}\n
    linked_list.zig
    // \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\nif (n0.?.next == null) return;\n// n0 -> P -> n1\nvar P = n0.?.next;\nvar n1 = P.?.next;\nn0.?.next = n1;\n}\n
    linked_list.dart
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null) return;\n// n0 -> P -> n1\nListNode P = n0.next!;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#422","title":"4.2.2. \u00a0 \u94fe\u8868\u7f3a\u70b9","text":"

    \u94fe\u8868\u8bbf\u95ee\u8282\u70b9\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u8282\u6240\u8ff0\uff0c\u6570\u7ec4\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u4efb\u610f\u5143\u7d20\u3002\u7136\u800c\uff0c\u94fe\u8868\u65e0\u6cd5\u76f4\u63a5\u8bbf\u95ee\u4efb\u610f\u8282\u70b9\uff0c\u8fd9\u662f\u56e0\u4e3a\u7cfb\u7edf\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4f8b\u5982\uff0c\u82e5\u8981\u8bbf\u95ee\u94fe\u8868\u7d22\u5f15\u4e3a index\uff08\u5373\u7b2c index + 1 \u4e2a\uff09\u7684\u8282\u70b9\uff0c\u5219\u9700\u8981\u5411\u540e\u904d\u5386 index \u8f6e\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.cpp
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == nullptr)\nreturn nullptr;\nhead = head->next;\n}\nreturn head;\n}\n
    linked_list.py
    def access(head: ListNode, index: int) -> ListNode | None:\n\"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\nfor _ in range(index):\nif not head:\nreturn None\nhead = head.next\nreturn head\n
    linked_list.go
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\nfor i := 0; i < index; i++ {\nif head == nil {\nreturn nil\n}\nhead = head.Next\n}\nreturn head\n}\n
    linked_list.js
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.ts
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.c
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nwhile (head && head->next && index) {\nhead = head->next;\nindex--;\n}\nreturn head;\n}\n
    linked_list.cs
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.swift
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\nvar head: ListNode? = head\nfor _ in 0 ..< index {\nif head == nil {\nreturn nil\n}\nhead = head?.next\n}\nreturn head\n}\n
    linked_list.zig
    // \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\nvar head = node;\nvar i: i32 = 0;\nwhile (i < index) : (i += 1) {\nhead = head.?.next;\nif (head == null) return null;\n}\nreturn head;\n}\n
    linked_list.dart
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\nfor (var i = 0; i < index; i++) {\nif (head == null) return null;\nhead = head.next;\n}\nreturn head;\n}\n

    \u94fe\u8868\u7684\u5185\u5b58\u5360\u7528\u8f83\u5927\u3002\u94fe\u8868\u4ee5\u8282\u70b9\u4e3a\u5355\u4f4d\uff0c\u6bcf\u4e2a\u8282\u70b9\u9664\u4e86\u4fdd\u5b58\u503c\u4e4b\u5916\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u8fd9\u610f\u5473\u7740\u5728\u76f8\u540c\u6570\u636e\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    "},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3. \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u904d\u5386\u94fe\u8868\u67e5\u627e\u3002\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u94fe\u8868\u5185\u503c\u4e3a target \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.cpp
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head != nullptr) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.py
    def find(head: ListNode, target: int) -> int:\n\"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nindex = 0\nwhile head:\nif head.val == target:\nreturn index\nhead = head.next\nindex += 1\nreturn -1\n
    linked_list.go
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\nindex := 0\nfor head != nil {\nif head.Val == target {\nreturn index\n}\nhead = head.Next\nindex++\n}\nreturn -1\n}\n
    linked_list.js
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.ts
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.c
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.cs
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.swift
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\nvar head: ListNode? = head\nvar index = 0\nwhile head != nil {\nif head?.val == target {\nreturn index\n}\nhead = head?.next\nindex += 1\n}\nreturn -1\n}\n
    linked_list.zig
    // \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\nvar head = node;\nvar index: i32 = 0;\nwhile (head != null) {\nif (head.?.val == target) return index;\nhead = head.?.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.dart
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target) {\nreturn index;\n}\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4. \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"

    \u5355\u5411\u94fe\u8868\u3002\u5373\u4e0a\u8ff0\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u6210\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002

    \u73af\u5f62\u94fe\u8868\u3002\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u5373\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002

    \u53cc\u5411\u94fe\u8868\u3002\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u8282\u70b9\uff09\u7684\u6307\u9488\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
    class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                   # \u8282\u70b9\u503c\nself.next: Optional[ListNode] = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nself.prev: Optional[ListNode] = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\nVal  int             // \u8282\u70b9\u503c\nNext *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nPrev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\nreturn &DoublyListNode{\nVal:  val,\nNext: nil,\nPrev: nil,\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nprev;\nconstructor(val, next, prev) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nprev: ListNode | null;\nconstructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nstruct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nnode->prev = NULL;\nreturn node;\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) => val = x;  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nvar prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
    // \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nprev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n

    Fig. \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b

    "},{"location":"chapter_array_and_linkedlist/list/","title":"4.3. \u00a0 \u5217\u8868","text":"

    \u6570\u7ec4\u957f\u5ea6\u4e0d\u53ef\u53d8\u5bfc\u81f4\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u5728\u8bb8\u591a\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e8b\u5148\u65e0\u6cd5\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u8fd9\u4f7f\u6570\u7ec4\u957f\u5ea6\u7684\u9009\u62e9\u53d8\u5f97\u56f0\u96be\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u9700\u8981\u5728\u6301\u7eed\u6dfb\u52a0\u6570\u636e\u65f6\u9891\u7e41\u6269\u5bb9\u6570\u7ec4\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002

    \u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u51fa\u73b0\u4e86\u4e00\u79cd\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u7ec4 Dynamic Array\u300d\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5373\u957f\u5ea6\u53ef\u53d8\u7684\u6570\u7ec4\uff0c\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5217\u8868 List\u300d\u3002\u5217\u8868\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\uff0c\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u52a8\u6001\u6269\u5bb9\u3002\u5728\u5217\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u65e0\u9700\u62c5\u5fc3\u8d85\u8fc7\u5bb9\u91cf\u9650\u5236\u3002

    "},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1. \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u521d\u59cb\u5316\u5217\u8868\u3002\u901a\u5e38\u6211\u4eec\u4f1a\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> list1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> list = new ArrayList<>(Arrays.asList(numbers));\n
    list.cpp
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 list\n// \u65e0\u521d\u59cb\u503c\nvector<int> list1;\n// \u6709\u521d\u59cb\u503c\nvector<int> list = { 1, 3, 2, 5, 4 };\n
    list.py
    # \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nlist1: List[int] = []\n# \u6709\u521d\u59cb\u503c\nlist: List[int] = [1, 3, 2, 5, 4]\n
    list_test.go
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlist1 := []int\n// \u6709\u521d\u59cb\u503c\nlist := []int{1, 3, 2, 5, 4}\n
    list.js
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1 = [];\n// \u6709\u521d\u59cb\u503c\nconst list = [1, 3, 2, 5, 4];\n
    list.ts
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst list: number[] = [1, 3, 2, 5, 4];\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = new ();\n// \u6709\u521d\u59cb\u503c\nint[] numbers = new int[] { 1, 3, 2, 5, 4 };\nList<int> list = numbers.ToList();\n
    list.swift
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet list1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar list = [1, 3, 2, 5, 4]\n
    list.zig
    // \u521d\u59cb\u5316\u5217\u8868\nvar list = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list.deinit();\ntry list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n
    list.dart
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = [];\n// \u6709\u521d\u59cb\u503c\nList<int> list = [1, 3, 2, 5, 4];\n

    \u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\u3002\u7531\u4e8e\u5217\u8868\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.cpp
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.py
    # \u8bbf\u95ee\u5143\u7d20\nnum: int = list[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nlist[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list_test.go
    /* \u8bbf\u95ee\u5143\u7d20 */\nnum := list[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.js
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.ts
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.swift
    /* \u8bbf\u95ee\u5143\u7d20 */\nlet num = list[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.zig
    // \u8bbf\u95ee\u5143\u7d20\nvar num = list.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n// \u66f4\u65b0\u5143\u7d20\nlist.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n
    list.dart
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n

    \u5728\u5217\u8868\u4e2d\u6dfb\u52a0\u3001\u63d2\u5165\u3001\u5220\u9664\u5143\u7d20\u3002\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.cpp
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push_back(1);\nlist.push_back(3);\nlist.push_back(2);\nlist.push_back(5);\nlist.push_back(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(list.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.erase(list.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.py
    # \u6e05\u7a7a\u5217\u8868\nlist.clear()\n# \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n# \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nlist.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n# \u5220\u9664\u5143\u7d20\nlist.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list_test.go
    /* \u6e05\u7a7a\u5217\u8868 */\nlist = nil\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist = append(list, 1)\nlist = append(list, 3)\nlist = append(list, 2)\nlist = append(list, 5)\nlist = append(list, 4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist = append(list[:3], append([]int{6}, list[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist = append(list[:3], list[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.js
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
    list.ts
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.Clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.Add(1);\nlist.Add(3);\nlist.Add(2);\nlist.Add(5);\nlist.Add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.Insert(3, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.RemoveAt(3);\n
    list.swift
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.removeAll()\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.zig
    // \u6e05\u7a7a\u5217\u8868\nlist.clearRetainingCapacity();\n// \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry list.append(1);\ntry list.append(3);\ntry list.append(2);\ntry list.append(5);\ntry list.append(4);\n// \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry list.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n// \u5220\u9664\u5143\u7d20\n_ = list.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.dart
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n

    \u904d\u5386\u5217\u8868\u3002\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
    list.cpp
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
    list.py
    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(list)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0\nfor n in list:\ncount += 1\n
    list_test.go
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(list); i++ {\ncount++\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor range list {\ncount++\n}\n
    list.js
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
    list.ts
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.Count(); i++)\n{\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int n in list)\n{\ncount++;\n}\n
    list.swift
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor _ in list.indices {\ncount += 1\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _ in list {\ncount += 1\n}\n
    list.zig
    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < list.items.len) : (i += 1) {\ncount += 1;\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (list.items) |_| {\ncount += 1;\n}\n
    list.dart
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n in list) {\ncount++;\n}\n

    \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\u3002\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 list1 \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u5217\u8868\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.cpp
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> list1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\nlist.insert(list.end(), list1.begin(), list1.end());\n
    list.py
    # \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nlist1: List[int] = [6, 8, 7, 10, 9]\nlist += list1  # \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list_test.go
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlist1 := []int{6, 8, 7, 10, 9}\nlist = append(list, list1...)  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.js
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1 = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.ts
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1: number[] = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = new() { 6, 8, 7, 10, 9 };\nlist.AddRange(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.swift
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet list1 = [6, 8, 7, 10, 9]\nlist.append(contentsOf: list1) // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.zig
    // \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar list1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list1.deinit();\ntry list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry list.insertSlice(list.items.len, list1.items); // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.dart
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = [6, 8, 7, 10, 9];\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n

    \u6392\u5e8f\u5217\u8868\u3002\u6392\u5e8f\u4e5f\u662f\u5e38\u7528\u7684\u65b9\u6cd5\u4e4b\u4e00\u3002\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u5bdf\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u53cc\u6307\u9488\u300d\u7b97\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(list);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.cpp
    /* \u6392\u5e8f\u5217\u8868 */\nsort(list.begin(), list.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.py
    # \u6392\u5e8f\u5217\u8868\nlist.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list_test.go
    /* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(list)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.js
    /* \u6392\u5e8f\u5217\u8868 */  list.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.ts
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u6392\u5e8f\u5217\u8868 */\nlist.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.swift
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.zig
    // \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32));\n
    list.dart
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    "},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2. \u00a0 \u5217\u8868\u5b9e\u73b0 *","text":"

    \u4e3a\u4e86\u5e2e\u52a9\u52a0\u6df1\u5bf9\u5217\u8868\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5728\u6b64\u63d0\u4f9b\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\u5b9e\u73b0\u3002\u9700\u8981\u5173\u6ce8\u4e09\u4e2a\u6838\u5fc3\u70b9\uff1a

    • \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
    • \u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf size\uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002
    • \u6269\u5bb9\u673a\u5236\uff1a\u63d2\u5165\u5143\u7d20\u65f6\u53ef\u80fd\u8d85\u51fa\u5217\u8868\u5bb9\u91cf\uff0c\u6b64\u65f6\u9700\u8981\u6269\u5bb9\u5217\u8868\u3002\u6269\u5bb9\u65b9\u6cd5\u662f\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002

    \u672c\u793a\u4f8b\u65e8\u5728\u5e2e\u52a9\u8bfb\u8005\u76f4\u89c2\u7406\u89e3\u5217\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002\u5b9e\u9645\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u5217\u8868\u5b9e\u73b0\u66f4\u52a0\u6807\u51c6\u548c\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u6709\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_list.java
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int capacity = 10; // \u5217\u8868\u5bb9\u91cf\nprivate int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[capacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\npublic int size() {\nreturn size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\nnums[size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = Arrays.copyOf(nums, capacity() * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\ncapacity = nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\nint size = size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[size];\nfor (int i = 0; i < size; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
    my_list.cpp
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate:\nint *nums;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint numsCapacity = 10; // \u5217\u8868\u5bb9\u91cf\nint numsSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\nnums = new int[numsCapacity];\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~MyList() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\nnums[size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size() - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size() - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nint newCapacity = capacity() * extendRatio;\nint *tmp = nums;\nnums = new int[newCapacity];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size(); i++) {\nnums[i] = tmp[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] tmp;\nnumsCapacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> vec(size());\nfor (int i = 0; i < size(); i++) {\nvec[i] = nums[i];\n}\nreturn vec;\n}\n};\n
    my_list.py
    class MyList:\n\"\"\"\u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\nself.__nums: list[int] = [0] * self.__capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nself.__size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nself.__extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\nreturn self.__size\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\nreturn self.__capacity\ndef get(self, index: int) -> int:\n\"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nreturn self.__nums[index]\ndef set(self, num: int, index: int) -> None:\n\"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nself.__nums[index] = num\ndef add(self, num: int) -> None:\n\"\"\"\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size() == self.capacity():\nself.extend_capacity()\nself.__nums[self.__size] = num\nself.__size += 1\ndef insert(self, num: int, index: int) -> None:\n\"\"\"\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.__size == self.capacity():\nself.extend_capacity()\n# \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in range(self.__size - 1, index - 1, -1):\nself.__nums[j + 1] = self.__nums[j]\nself.__nums[index] = num\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size += 1\ndef remove(self, index: int) -> int:\n\"\"\"\u5220\u9664\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nnum = self.__nums[index]\n# \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in range(index, self.__size - 1):\nself.__nums[j] = self.__nums[j + 1]\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size -= 1\n# \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\ndef extend_capacity(self) -> None:\n\"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n# \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nself.__nums = self.__nums + [0] * self.capacity() * (self.__extend_ratio - 1)\n# \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.__capacity = len(self.__nums)\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\nreturn self.__nums[: self.__size]\n
    my_list.go
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\ntype myList struct {\nnumsCapacity int\nnums         []int\nnumsSize     int\nextendRatio  int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\nreturn &myList{\nnumsCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\nnums:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio:  2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\nreturn l.numsSize\n}\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\nreturn l.numsCapacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn l.nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nl.nums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\nl.nums[l.numsSize] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j := l.numsSize - 1; j >= index; j-- {\nl.nums[j+1] = l.nums[j]\n}\nl.nums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnum := l.nums[index]\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j := index; j < l.numsSize-1; j++ {\nl.nums[j] = l.nums[j+1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize--\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nl.nums = append(l.nums, make([]int, l.numsCapacity*(l.extendRatio-1))...)\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nl.numsCapacity = len(l.nums)\n}\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nreturn l.nums[:l.numsSize]\n}\n
    my_list.js
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\n#nums = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n#capacity = 10; // \u5217\u8868\u5bb9\u91cf\n#size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n#extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#nums = new Array(this.#capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nsize() {\nreturn this.#size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\ncapacity() {\nreturn this.#capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nget(index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.#nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nset(index, num) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.#nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nadd(num) {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.#nums[this.#size] = num;\nthis.#size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\ninsert(index, num) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this.#size - 1; j >= index; j--) {\nthis.#nums[j + 1] = this.#nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#nums[index] = num;\nthis.#size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nremove(index) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.#nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this.#size - 1; j++) {\nthis.#nums[j] = this.#nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nextendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.#nums = this.#nums.concat(\nnew Array(this.capacity() * (this.#extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis.#capacity = this.#nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\ntoArray() {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
    my_list.ts
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate nums: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\nprivate _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.nums = new Array(this._capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic size(): number {\nreturn this._size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic capacity(): number {\nreturn this._capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic get(index: number): number {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic set(index: number, num: number): void {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic add(num: number): void {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this._size === this._capacity) this.extendCapacity();\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.nums[this._size] = num;\nthis._size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic insert(index: number, num: number): void {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this._size === this._capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this._size - 1; j >= index; j--) {\nthis.nums[j + 1] = this.nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.nums[index] = num;\nthis._size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic remove(index: number): number {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this._size - 1; j++) {\nthis.nums[j] = this.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis._size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic extendCapacity(): void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.nums = this.nums.concat(\nnew Array(this.capacity() * (this.extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis._capacity = this.nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic toArray(): number[] {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
    my_list.c
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nstruct myList {\nint *nums;       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint capacity;    // \u5217\u8868\u5bb9\u91cf\nint size;        // \u5217\u8868\u5927\u5c0f\nint extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n};\ntypedef struct myList myList;\n/* \u6784\u9020\u51fd\u6570 */\nmyList *newMyList() {\nmyList *list = malloc(sizeof(myList));\nlist->capacity = 10;\nlist->nums = malloc(sizeof(int) * list->capacity);\nlist->size = 0;\nlist->extendRatio = 2;\nreturn list;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(myList *list) {\nfree(list->nums);\nfree(list);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(myList *list) {\nreturn list->size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(myList *list) {\nreturn list->capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(myList *list, int index) {\nassert(index >= 0 && index < list->size);\nreturn list->nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(myList *list, int index, int num) {\nassert(index >= 0 && index < list->size);\nlist->nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(myList *list, int num) {\nif (size(list) == capacity(list)) {\nextendCapacity(list); // \u6269\u5bb9\n}\nlist->nums[size(list)] = num;\nlist->size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(myList *list, int index, int num) {\nassert(index >= 0 && index < size(list));\nfor (int i = size(list); i > index; --i) {\nlist->nums[i] = list->nums[i - 1];\n}\nlist->nums[index] = num;\nlist->size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeNum(myList *list, int index) {\nassert(index >= 0 && index < size(list));\nint num = list->nums[index];\nfor (int i = index; i < size(list) - 1; i++) {\nlist->nums[i] = list->nums[i + 1];\n}\nlist->size--;\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(myList *list) {\n// \u5148\u5206\u914d\u7a7a\u95f4\nint newCapacity = capacity(list) * list->extendRatio;\nint *extend = (int *)malloc(sizeof(int) * newCapacity);\nint *temp = list->nums;\n// \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\nfor (int i = 0; i < size(list); i++)\nextend[i] = list->nums[i];\n// \u91ca\u653e\u65e7\u6570\u636e\nfree(temp);\n// \u66f4\u65b0\u65b0\u6570\u636e\nlist->nums = extend;\nlist->capacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(myList *list) {\nreturn list->nums;\n}\n
    my_list.cs
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[numsCapacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\nnums[numsSize] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = numsSize - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < numsSize - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a numsCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nSystem.Array.Resize(ref nums, numsCapacity * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nnumsCapacity = nums.Length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[numsSize];\nfor (int i = 0; i < numsSize; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
    my_list.swift
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate var nums: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\nprivate var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nnums = Array(repeating: 0, count: _capacity)\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nfunc size() -> Int {\n_size\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc capacity() -> Int {\n_capacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc get(index: Int) -> Int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc set(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc add(num: Int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\nnums[_size] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc insert(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\nnums[j + 1] = nums[j]\n}\nnums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5220\u9664\u5143\u7d20 */\n@discardableResult\nfunc remove(index: Int) -> Int {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nlet num = nums[index]\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in index ..< (_size - 1) {\nnums[j] = nums[j + 1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size -= 1\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = nums + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = nums.count\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nfunc toArray() -> [Int] {\nvar nums = Array(repeating: 0, count: _size)\nfor i in 0 ..< _size {\nnums[i] = get(index: i)\n}\nreturn nums\n}\n}\n
    my_list.zig
    // \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\nfn MyList(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\nnumSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.nums = try self.mem_allocator.alloc(T, self.numsCapacity);\n@memset(self.nums, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\npub fn size(self: *Self) usize {\nreturn self.numSize;\n}\n// \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.numsCapacity;\n}\n// \u8bbf\u95ee\u5143\u7d20\npub fn get(self: *Self, index: usize) T {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn self.nums[index];\n}  // \u66f4\u65b0\u5143\u7d20\npub fn set(self: *Self, index: usize, num: T) void {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nself.nums[index] = num;\n}  // \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\npub fn add(self: *Self, num: T) !void {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\nself.nums[self.size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}  // \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\npub fn insert(self: *Self, index: usize, num: T) !void {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar j = self.size() - 1;\nwhile (j >= index) : (j -= 1) {\nself.nums[j + 1] = self.nums[j];\n}\nself.nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}\n// \u5220\u9664\u5143\u7d20\npub fn remove(self: *Self, index: usize) T {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nvar num = self.nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar j = index;\nwhile (j < self.size() - 1) : (j += 1) {\nself.nums[j] = self.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n// \u5217\u8868\u6269\u5bb9\npub fn extendCapacity(self: *Self) !void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nvar newCapacity = self.capacity() * self.extendRatio;\nvar extend = try self.mem_allocator.alloc(T, newCapacity);\n@memset(extend, @as(T, 0));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(T, extend, self.nums);\nself.nums = extend;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.numsCapacity = newCapacity;\n}\n// \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar nums = try self.mem_allocator.alloc(T, self.size());\n@memset(nums, @as(T, 0));\nfor (nums, 0..) |*num, i| {\nnum.* = self.get(i);\n}\nreturn nums;\n}\n};\n}\n
    my_list.dart
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nlate List<int> _nums; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint _capacity = 10; // \u5217\u8868\u5bb9\u91cf\nint _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\n_nums = List.filled(_capacity, 0);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() => _size;\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() => _capacity;\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nreturn _nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n_nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n_nums[_size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var j = _size - 1; j >= index; j--) {\n_nums[j + 1] = _nums[j];\n}\n_nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nint num = _nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var j = index; j < _size - 1; j++) {\n_nums[j] = _nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nfinal _newNums = List.filled(_capacity * _extendRatio, 0);\n// \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nList.copyRange(_newNums, 0, _nums);\n// \u66f4\u65b0 _nums \u7684\u5f15\u7528\n_nums = _newNums;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = _nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nList<int> toArray() {\nList<int> nums = [];\nfor (var i = 0; i < _size; i++) {\nnums.add(get(i));\n}\nreturn nums;\n}\n}\n
    "},{"location":"chapter_array_and_linkedlist/summary/","title":"4.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u65b9\u5f0f\u3002\u4e24\u8005\u7684\u4f18\u7f3a\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002
    • \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
    • \u94fe\u8868\u901a\u8fc7\u66f4\u6539\u6307\u9488\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u5faa\u73af\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002
    • \u52a8\u6001\u6570\u7ec4\uff0c\u53c8\u79f0\u5217\u8868\uff0c\u662f\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u4e00\u79cd\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u6613\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
    • \u4e0b\u8868\u603b\u7ed3\u5e76\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u9879\u7279\u6027\u4e0e\u64cd\u4f5c\u6548\u7387\u3002
    \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u79bb\u6563\u5185\u5b58\u7a7a\u95f4 \u6570\u636e\u7ed3\u6784\u957f\u5ea6 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u957f\u5ea6\u53ef\u53d8 \u5185\u5b58\u4f7f\u7528\u7387 \u5360\u7528\u5185\u5b58\u5c11\u3001\u7f13\u5b58\u5c40\u90e8\u6027\u597d \u5360\u7528\u5185\u5b58\u591a \u4f18\u52bf\u64cd\u4f5c \u968f\u673a\u8bbf\u95ee \u63d2\u5165\u3001\u5220\u9664 \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(N)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(N)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(N)\\) \\(O(1)\\)

    \u7f13\u5b58\u5c40\u90e8\u6027

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6570\u636e\u8bfb\u5199\u901f\u5ea6\u6392\u5e8f\u662f\u201c\u786c\u76d8 < \u5185\u5b58 < CPU \u7f13\u5b58\u201d\u3002\u5f53\u6211\u4eec\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u8ba1\u7b97\u673a\u53ea\u80fd\u6328\u4e2a\u5730\u7f13\u5b58\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u6837\u7684\u591a\u6b21\u201c\u642c\u8fd0\u201d\u964d\u4f4e\u4e86\u6574\u4f53\u6548\u7387\u3002

    "},{"location":"chapter_array_and_linkedlist/summary/#441-q-a","title":"4.4.1. \u00a0 Q & A","text":"

    \u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f

    \u6808\u5185\u5b58\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff0c\u800c\u5806\u5185\u5b58\u7531\u7a0b\u5e8f\u5458\u5728\u4ee3\u7801\u4e2d\u5206\u914d\uff08\u6ce8\u610f\uff0c\u8fd9\u91cc\u7684\u6808\u548c\u5806\u548c\u6570\u636e\u7ed3\u6784\u4e2d\u7684\u6808\u548c\u5806\u4e0d\u662f\u540c\u4e00\u6982\u5ff5\uff09\u3002

    1. \u6808\u4e0d\u7075\u6d3b\uff0c\u5206\u914d\u7684\u5185\u5b58\u5927\u5c0f\u4e0d\u53ef\u66f4\u6539\uff1b\u5806\u76f8\u5bf9\u7075\u6d3b\uff0c\u53ef\u4ee5\u52a8\u6001\u5206\u914d\u5185\u5b58\uff1b
    2. \u6808\u662f\u4e00\u5757\u6bd4\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5bb9\u6613\u51fa\u73b0\u5185\u5b58\u4e0d\u8db3\uff1b\u5806\u5185\u5b58\u5f88\u5927\uff0c\u4f46\u662f\u7531\u4e8e\u662f\u52a8\u6001\u5206\u914d\uff0c\u5bb9\u6613\u788e\u7247\u5316\uff0c\u7ba1\u7406\u5806\u5185\u5b58\u7684\u96be\u5ea6\u66f4\u5927\u3001\u6210\u672c\u66f4\u9ad8\uff1b
    3. \u8bbf\u95ee\u6808\u6bd4\u8bbf\u95ee\u5806\u66f4\u5feb\uff0c\u56e0\u4e3a\u6808\u5185\u5b58\u8f83\u5c0f\u3001\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u5806\u5e27\u5206\u6563\u5728\u5f88\u5927\u7684\u7a7a\u95f4\u5185\uff0c\u4f1a\u51fa\u73b0\u66f4\u591a\u7684\u7f13\u5b58\u672a\u547d\u4e2d\uff1b

    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u4f1a\u5f3a\u8c03\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f

    \u94fe\u8868\u7531\u7ed3\u70b9\u7ec4\u6210\uff0c\u7ed3\u70b9\u4e4b\u95f4\u7531\u6307\u9488\u8fde\u63a5\uff0c\u5404\u4e2a\u7ed3\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int, double, string, object \u7b49\u3002

    \u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u90a3\u4e48\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd elementLength \u3002

    // \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n

    \u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a P.next \u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f

    \u4e0d\u4fee\u6539 P.next \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u7ed3\u70b9\u904d\u5386\u5230\u5c3e\u7ed3\u70b9\u5df2\u7ecf\u9047\u4e0d\u5230 P \u4e86\u3002\u8fd9\u610f\u5473\u7740\u7ed3\u70b9 P \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u7ed3\u70b9 P \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8fd9\u6761\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u4e86\u3002

    \u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java, Python, Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9 P \u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u6709\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f P.next \u7684\u503c\u3002\u5728 C, C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002

    \u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f

    \u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u7ed3\u70b9\u3001\u5c3e\u7ed3\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002

    \u56fe\u7247\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u7ed3\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u7ed3\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f

    \u6587\u4e2d\u53ea\u662f\u4e00\u4e2a\u793a\u610f\u56fe\uff0c\u53ea\u662f\u5b9a\u6027\u8868\u793a\u3002\u5b9a\u91cf\u7684\u8bdd\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u5206\u6790\uff1a

    • \u4e0d\u540c\u7c7b\u578b\u7684\u7ed3\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int, long, double, \u6216\u8005\u662f\u7c7b\u7684\u5b9e\u4f8b\u7b49\u7b49\u3002
    • \u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002

    \u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f

    \u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002

    \u201c\u5217\u8868\u7684\u51fa\u73b0\u5927\u5927\u63d0\u5347\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f

    \u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u90fd\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002

    \u5728 Python \u4e2d\u521d\u59cb\u5316 n = [1, 2, 3] \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 m = [2, 1, 3] \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf n \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 m \u8fd8\u662f\u6570\u7ec4\u5417\uff1f

    \u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 n = [n1, n2, n3, n4, n5] \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9\u4e94\u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u662f\u88ab\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u7684\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5230\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002

    \u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\u7684\u662f\uff0c\u5728 Python \u4e2d\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u662f\u65e0\u9700\u8fde\u7eed\u7684\u3002

    "},{"location":"chapter_backtracking/","title":"12. \u00a0 \u56de\u6eaf","text":""},{"location":"chapter_backtracking/backtracking_algorithm/","title":"12.1. \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"

    \u300c\u56de\u6eaf\u7b97\u6cd5 Backtracking Algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002

    \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u4e0b\u9762\uff0c\u6211\u4eec\u5c06\u4ece\u524d\u5e8f\u904d\u5386\u5165\u624b\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002

    \u4f8b\u9898\u4e00

    \u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u524d\u5e8f\u904d\u5386\u8fd9\u9897\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u5230\u7ed3\u679c\u5217\u8868 res \u4e4b\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_i_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(root);\n}\npreOrder(root.left);\npreOrder(root.right);\n}\n
    preorder_traversal_i_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(root);\n}\npreOrder(root->left);\npreOrder(root->right);\n}\n
    preorder_traversal_i_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\nif root is None:\nreturn\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(root)\npre_order(root.left)\npre_order(root.right)\n
    preorder_traversal_i_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\nif root == nil {\nreturn\n}\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, root)\n}\npreOrderI(root.Left, res)\npreOrderI(root.Right, res)\n}\n
    preorder_traversal_i_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n
    preorder_traversal_i_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n
    preorder_traversal_i_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_i_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(root);\n}\npreOrder(root.left);\npreOrder(root.right);\n}\n
    preorder_traversal_i_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(root)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n
    preorder_traversal_i_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_i_compact.dart
    [class]{}-[func]{preOrder}\n

    Fig. \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1211","title":"12.1.1. \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"

    \u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002

    \u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 return \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002

    \u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u7b49\u4ef7\u4e8e\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002

    \u4f8b\u9898\u4e8c

    \u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 path \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 path \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 res \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0cres \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_ii_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList<>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n
    preorder_traversal_ii_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root->left);\npreOrder(root->right);\n// \u56de\u9000\npath.pop_back();\n}\n
    preorder_traversal_ii_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\nif root is None:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n
    preorder_traversal_ii_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\nif root == nil {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderII(root.Left, res, path)\npreOrderII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n
    preorder_traversal_ii_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_ii_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_ii_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_ii_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List<TreeNode>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n
    preorder_traversal_ii_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n
    preorder_traversal_ii_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_ii_compact.dart
    [class]{}-[func]{preOrder}\n

    \u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb path \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece path \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u662f\u4e92\u4e3a\u76f8\u53cd\u7684\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1212","title":"12.1.2. \u00a0 \u526a\u679d","text":"

    \u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002

    \u4f8b\u9898\u4e09

    \u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u8def\u5f84\u4e2d\u4e0d\u80fd\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u5728\u4f8b\u9898\u4e8c\u7684\u57fa\u7840\u4e0a\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff0c\u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u7ec8\u6b62\u7ee7\u7eed\u641c\u7d22\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_iii_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList<>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n
    preorder_traversal_iii_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n// \u526a\u679d\nif (root == nullptr || root->val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root->left);\npreOrder(root->right);\n// \u56de\u9000\npath.pop_back();\n}\n
    preorder_traversal_iii_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u526a\u679d\nif root is None or root.val == 3:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n
    preorder_traversal_iii_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n// \u526a\u679d\nif root == nil || root.Val == 3 {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderIII(root.Left, res, path)\npreOrderIII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n
    preorder_traversal_iii_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_iii_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_iii_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_iii_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List<TreeNode>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n
    preorder_traversal_iii_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n// \u526a\u679d\nguard let root = root, root.val != 3 else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n
    preorder_traversal_iii_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_iii_compact.dart
    [class]{}-[func]{preOrder}\n

    \u526a\u679d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5229\u7528\u7ea6\u675f\u6761\u4ef6\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002

    Fig. \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1213","title":"12.1.3. \u00a0 \u5e38\u7528\u672f\u8bed","text":"

    \u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\u3002

    \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3 Solution \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\u3002\u56de\u6eaf\u7b97\u6cd5\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a\u6216\u591a\u4e2a\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3 \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6240\u6709\u8def\u5f84\uff0c\u4e14\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9 \u72b6\u6001 State \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 path \u8282\u70b9\u5217\u8868 \u7ea6\u675f\u6761\u4ef6 Constraint \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u80fd\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9 \u5c1d\u8bd5 Attempt \u5c1d\u8bd5\u662f\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6839\u636e\u5f53\u524d\u72b6\u6001\u548c\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\u3002\u5c1d\u8bd5\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb path \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000 Backtracking \u56de\u9000\u6307\u5728\u641c\u7d22\u4e2d\u9047\u5230\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u6216\u65e0\u6cd5\u7ee7\u7eed\u641c\u7d22\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u3001\u7ed3\u675f\u7ed3\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d Pruning \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u7ec8\u6b62\u7ee7\u7eed\u641c\u7d22

    Tip

    \u89e3\u3001\u72b6\u6001\u3001\u7ea6\u675f\u6761\u4ef6\u7b49\u672f\u8bed\u662f\u901a\u7528\u7684\uff0c\u9002\u7528\u4e8e\u56de\u6eaf\u7b97\u6cd5\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1214","title":"12.1.4. \u00a0 \u6846\u67b6\u4ee3\u7801","text":"

    \u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002\u4e3a\u63d0\u5347\u4ee3\u7801\u901a\u7528\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5c06\u56de\u6eaf\u7b97\u6cd5\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\u3002

    \u8bbe state \u4e3a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0cchoices \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff0c\u5219\u53ef\u5f97\u5230\u4ee5\u4e0b\u56de\u6eaf\u7b97\u6cd5\u7684\u6846\u67b6\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector<Choice *> &choices, vector<State *> &res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    def backtrack(state: State, choices: list[choice], res: list[state]) -> None:\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n# \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\nbacktrack(state, choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\nbacktrack(state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res, numRes);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < numChoices; i++) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, &choices[i])) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, &choices[i]);\nbacktrack(state, choices, numChoices, res, numRes);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, &choices[i]);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state: state, res: &res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &state, choice: choice)\nbacktrack(state: &state, choices: choices, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &state, choice: choice)\n}\n}\n}\n
    \n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice>, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n

    \u4e0b\u9762\uff0c\u6211\u4eec\u5c1d\u8bd5\u57fa\u4e8e\u6b64\u6846\u67b6\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u5728\u4f8b\u9898\u4e09\u4e2d\uff0c\u72b6\u6001 state \u662f\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 choices \u662f\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c res \u662f\u8def\u5f84\u5217\u8868\uff0c\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_iii_template.java
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List<TreeNode> state) {\nreturn !state.isEmpty() && state.get(state.size() - 1).val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\nres.add(new ArrayList<>(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List<TreeNode> state, TreeNode choice) {\nreturn choice != null && choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\nstate.add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\nstate.remove(state.size() - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, Arrays.asList(choice.left, choice.right), res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.cpp
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector<TreeNode *> &state) {\nreturn !state.empty() && state.back()->val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {\nres.push_back(state);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector<TreeNode *> &state, TreeNode *choice) {\nreturn choice != nullptr && choice->val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector<TreeNode *> &state, TreeNode *choice) {\nstate.push_back(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector<TreeNode *> &state, TreeNode *choice) {\nstate.pop_back();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode *choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nvector<TreeNode *> nextChoices{choice->left, choice->right};\nbacktrack(state, nextChoices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.py
    def is_solution(state: list[TreeNode]) -> bool:\n\"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\nreturn state and state[-1].val == 7\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n\"\"\"\u8bb0\u5f55\u89e3\"\"\"\nres.append(list(state))\ndef is_valid(state: list[TreeNode], choice: TreeNode) -> bool:\n\"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\nreturn choice is not None and choice.val != 3\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\nstate.append(choice)\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u6062\u590d\u72b6\u6001\"\"\"\nstate.pop()\ndef backtrack(\nstate: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n
    preorder_traversal_iii_template.go
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\nreturn len(*state) != 0 && (*state)[len(*state)-1].Val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n*res = append(*res, *state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\nreturn choice != nil && choice.Val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = append(*state, choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = (*state)[:len(*state)-1]\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range *choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\ntemp := make([]*TreeNode, 0)\ntemp = append(temp, choice.Left, choice.Right)\nbacktrackIII(state, &temp, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n
    preorder_traversal_iii_template.js
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\nreturn state && state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\nreturn choice !== null && choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n
    preorder_traversal_iii_template.ts
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\nreturn state && state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\nreturn choice !== null && choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\nstate: TreeNode[],\nchoices: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n
    preorder_traversal_iii_template.c
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n
    preorder_traversal_iii_template.cs
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List<TreeNode> state) {\nreturn state.Count != 0 && state[^1].val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\nres.Add(new List<TreeNode>(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List<TreeNode> state, TreeNode choice) {\nreturn choice != null && choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\nstate.Add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\nstate.RemoveAt(state.Count - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (TreeNode choice in choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, new List<TreeNode> { choice.left, choice.right }, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.swift
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -> Bool {\n!state.isEmpty && state.last!.val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\nres.append(state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -> Bool {\nchoice != nil && choice!.val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.append(choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.removeLast()\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\nrecordSolution(state: state, res: &res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &state, choice: choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: [choice.left, choice.right].compactMap { $0 }, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &state, choice: choice)\n}\n}\n}\n
    preorder_traversal_iii_template.zig
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n
    preorder_traversal_iii_template.dart
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n

    \u76f8\u8f83\u4e8e\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u5b9e\u73b0\u4ee3\u7801\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u5b9e\u73b0\u4ee3\u7801\u867d\u7136\u663e\u5f97\u5570\u55e6\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u6240\u6709\u56de\u6eaf\u95ee\u9898\u90fd\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u9700\u8981\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 state \u548c choices \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1215","title":"12.1.5. \u00a0 \u5178\u578b\u4f8b\u9898","text":"

    \u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002

    • \u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002
    • \u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002
    • \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u4e2a\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u4e2a\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u4e2a\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002

    \u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002

    • \\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002
    • \u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002
    • \u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002

    \u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002

    • 0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002
    • \u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002
    • \u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002

    \u8bf7\u6ce8\u610f\uff0c\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u4e0d\u662f\u89e3\u51b3\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u7684\u6700\u4f18\u65b9\u6cd5\u30020-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff1b\u65c5\u884c\u5546\u662f\u4e00\u4e2a NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u51b3\u65b9\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\uff1b\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8f6e\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178 NP-Hard \u95ee\u9898\uff0c\u901a\u5e38\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1216","title":"12.1.6. \u00a0 \u4f18\u52bf\u4e0e\u5c40\u9650\u6027","text":"

    \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u5b83\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002

    \u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u6700\u574f\u7684\u60c5\u51b5\u4e0b\uff0c\u56de\u6eaf\u7b97\u6cd5\u9700\u8981\u904d\u5386\u89e3\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\u89e3\u3002\u4f8b\u5982\uff0c\u6c42\u89e3 \\(n\\) \u7687\u540e\u95ee\u9898\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n!)\\) \u3002\u56de\u6eaf\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u53ef\u80fd\u8f83\u9ad8\u3002\u56e0\u4e3a\u5728\u6bcf\u4e00\u6b21\u9012\u5f52\u8c03\u7528\u65f6\uff0c\u90fd\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u9009\u62e9\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5bf9\u4e8e\u6df1\u5ea6\u5f88\u5927\u7684\u9012\u5f52\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u975e\u5e38\u5927\u3002

    \u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u6211\u4eec\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u8fdb\u884c\u6548\u7387\u4f18\u5316\uff1a

    • \u4e0a\u6587\u4ecb\u7ecd\u8fc7\u7684\u526a\u679d\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u3002\u5b83\u53ef\u4ee5\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002
    • \u53e6\u4e00\u4e2a\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u662f\u52a0\u5165\u300c\u542f\u53d1\u5f0f\u641c\u7d22 Heuristic Search\u300d\u7b56\u7565\uff0c\u5b83\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002
    "},{"location":"chapter_backtracking/n_queens_problem/","title":"12.4. \u00a0 N \u7687\u540e\u95ee\u9898","text":"

    Question

    \u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u4e4b\u5904\u5728\u540c\u4e00\u884c\u6216\u540c\u4e00\u5217\u6216\u540c\u4e00\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 choices \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 state \u3002

    Fig. 4 \u7687\u540e\u95ee\u9898\u7684\u89e3

    \u672c\u9898\u5171\u6709\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u548c\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf \\ \u548c\u526f\u5bf9\u89d2\u7ebf / \u4e24\u79cd\u3002

    Fig. n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6

    \u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u7b2c\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002\u8fd9\u610f\u5473\u7740\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002\u6b64\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86 \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u7bc7\u5e45\u9650\u5236\uff0c\u4e0b\u56fe\u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u4e00\u4e2a\u641c\u7d22\u5206\u652f\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u526a\u679d\u4e86\u3002

    Fig. \u9010\u884c\u653e\u7f6e\u7b56\u7565

    \u4e3a\u4e86\u5b9e\u73b0\u6839\u636e\u5217\u7ea6\u675f\u526a\u679d\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 cols \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 cols \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 cols \u7684\u72b6\u6001\u3002

    \u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a (row, col) \uff0c\u89c2\u5bdf\u77e9\u9635\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u76f8\u7b49\uff0c\u5373 row - col \u4e3a\u6052\u5b9a\u503c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u82e5\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 row1 - col1 == row2 - col2 \uff0c\u5219\u8fd9\u4e24\u4e2a\u683c\u5b50\u4e00\u5b9a\u5904\u5728\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002

    \u5229\u7528\u8be5\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4 diag1 \u6765\u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635 row - col \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\u56e0\u6b64\u5171\u6709 \\(2n - 1\\) \u6761\u4e3b\u5bf9\u89d2\u7ebf\u3002

    Fig. \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f

    \u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 row + col \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u540c\u6837\u7684\u65b9\u6cd5\uff0c\u501f\u52a9\u6570\u7ec4 diag2 \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002

    \u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u5199\u51fa \\(n\\) \u7687\u540e\u7684\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart n_queens.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<String>> state, List<List<List<String>>> res,\nboolean[] cols, boolean[] diags1, boolean[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList<List<String>> copyState = new ArrayList<>();\nfor (List<String> sRow : state) {\ncopyState.add(new ArrayList<>(sRow));\n}\nres.add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate.get(row).set(col, \"Q\");\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate.get(row).set(col, \"#\");\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList<List<String>> state = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<String> row = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\nrow.add(\"#\");\n}\nstate.add(row);\n}\nboolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nboolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nboolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nList<List<List<String>>> res = new ArrayList<>();\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,\nvector<bool> &diags1, vector<bool> &diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nvector<vector<vector<string>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvector<vector<string>> state(n, vector<string>(n, \"#\"));\nvector<bool> cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvector<bool> diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvector<bool> diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvector<vector<vector<string>>> res;\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.py
    def backtrack(\nrow: int,\nn: int,\nstate: list[list[str]],\nres: list[list[list[str]]],\ncols: list[bool],\ndiags1: list[bool],\ndiags2: list[bool],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n# \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n:\nres.append([list(row) for row in state])\nreturn\n# \u904d\u5386\u6240\u6709\u5217\nfor col in range(n):\n# \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 = row - col + n - 1\ndiag2 = row + col\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif not cols[col] and not diags1[diag1] and not diags2[diag2]:\n# \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = diags1[diag1] = diags2[diag2] = True\n# \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2)\n# \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = diags1[diag1] = diags2[diag2] = False\ndef n_queens(n: int) -> list[list[list[str]]]:\n\"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n# \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate = [[\"#\" for _ in range(n)] for _ in range(n)]\ncols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ndiags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\ndiags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nres = []\nbacktrack(0, n, state, res, cols, diags1, diags2)\nreturn res\n
    n_queens.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col < n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col < n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\nfunc nQueens(n int) [][][]string {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate := make([][]string, n)\nfor i := 0; i < n; i++ {\nrow := make([]string, n)\nfor i := 0; i < n; i++ {\nrow[i] = \"#\"\n}\nstate[i] = row\n}\n// \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ncols := make([]bool, n)\ndiags1 := make([]bool, 2*n-1)\ndiags2 := make([]bool, 2*n-1)\nres := make([][][]string, 0)\nbacktrack(0, n, &state, &res, &cols, &diags1, &diags2)\nreturn res\n}\n
    n_queens.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) => row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () => Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst res = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\nrow: number,\nn: number,\nstate: string[][],\nres: string[][][],\ncols: boolean[],\ndiags1: boolean[],\ndiags2: boolean[]\n): void {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) => row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () => Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst res: string[][][] = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    n_queens.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<string>> state, List<List<List<string>>> res,\nbool[] cols, bool[] diags1, bool[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList<List<string>> copyState = new List<List<string>>();\nforeach (List<string> sRow in state) {\ncopyState.Add(new List<string>(sRow));\n}\nres.Add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<string>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList<List<string>> state = new List<List<string>>();\nfor (int i = 0; i < n; i++) {\nList<string> row = new List<string>();\nfor (int j = 0; j < n; j++) {\nrow.Add(\"#\");\n}\nstate.Add(row);\n}\nbool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nbool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nbool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nList<List<List<string>>> res = new List<List<List<string>>>();\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col in 0 ..< n {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nlet diag1 = row - col + n - 1\nlet diag2 = row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !cols[col] && !diags1[diag1] && !diags2[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = true\ndiags1[diag1] = true\ndiags2[diag2] = true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row: row + 1, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = false\ndiags1[diag1] = false\ndiags2[diag2] = false\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -> [[[String]]] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvar state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\nvar cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvar diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvar diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvar res: [[[String]]] = []\nbacktrack(row: 0, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\nreturn res\n}\n
    n_queens.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    n_queens.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    "},{"location":"chapter_backtracking/n_queens_problem/#1241","title":"12.4.1. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n, n-1, \\cdots, 2, 1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u5730\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002

    state \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0ccols , diags1 , diags2 \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_backtracking/permutations_problem/","title":"12.2. \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"

    \u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u8fd9\u4e2a\u96c6\u5408\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002

    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u7ec4\u548c\u5176\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002

    \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1221","title":"12.2.1. \u00a0 \u65e0\u91cd\u590d\u7684\u60c5\u51b5","text":"

    Question

    \u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002

    \u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \u3001\u518d\u9009\u62e9 \\(3\\) \u3001\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002

    \u4ece\u56de\u6eaf\u7b97\u6cd5\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 choices \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 state \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64\u5728\u904d\u5386\u9009\u62e9\u65f6\uff0c\u5e94\u5f53\u6392\u9664\u5df2\u7ecf\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u4e2a\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 state \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002

    Fig. \u5168\u6392\u5217\u7684\u9012\u5f52\u6811

    \u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u4ee3\u7801\u884c\u6570\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u4ed6\u4eec\u5c55\u5f00\u5728 backtrack() \u51fd\u6570\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart permutations_i.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList<Integer>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList<List<Integer>> permutationsI(int[] nums) {\nList<List<Integer>> res = new ArrayList<List<Integer>>();\nbacktrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\nreturn res;\n}\n
    permutations_i.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nvector<vector<int>> permutationsI(vector<int> nums) {\nvector<int> state;\nvector<bool> selected(nums.size(), false);\nvector<vector<int>> res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n
    permutations_i.py
    def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif not selected[i]:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_i(nums: list[int]) -> list[list[int]]:\n\"\"\"\u5168\u6392\u5217 I\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n
    permutations_i.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i < len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackI(&state, &nums, &selected, &res)\nreturn res\n}\n
    permutations_i.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_i.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_i.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n
    permutations_i.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList<List<int>> permutationsI(int[] nums) {\nList<List<int>> res = new List<List<int>>();\nbacktrack(new List<int>(), nums, new bool[nums.Length], res);\nreturn res;\n}\n
    permutations_i.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: choices, selected: &selected, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -> [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &state, choices: nums, selected: &selected, res: &res)\nreturn res\n}\n
    permutations_i.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n
    permutations_i.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n

    \u9700\u8981\u91cd\u70b9\u5173\u6ce8\u7684\u662f\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 selected \uff0c\u5b83\u7684\u957f\u5ea6\u4e0e\u8f93\u5165\u6570\u7ec4\u957f\u5ea6\u76f8\u7b49\uff0c\u5176\u4e2d selected[i] \u8868\u793a choices[i] \u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u5229\u7528 selected \u907f\u514d\u67d0\u4e2a\u5143\u7d20\u88ab\u91cd\u590d\u9009\u62e9\uff0c\u4ece\u800c\u5b9e\u73b0\u526a\u679d\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1, 3 \u7684\u5206\u652f\u3002\u4ece\u672c\u8d28\u4e0a\u7406\u89e3\uff0c\u6b64\u526a\u679d\u64cd\u4f5c\u53ef\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u964d\u4f4e\u81f3 \\(O(n!)\\) \u3002

    Fig. \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b

    "},{"location":"chapter_backtracking/permutations_problem/#1222","title":"12.2.2. \u00a0 \u8003\u8651\u91cd\u590d\u7684\u60c5\u51b5","text":"

    Question

    \u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002

    \u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u7684\u5143\u7d20 \\(1\\) \uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a\u5143\u7d20\u8bb0\u4e3a \\(\\hat{1}\\) \u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u90fd\u662f\u91cd\u590d\u7684\u3002

    Fig. \u91cd\u590d\u6392\u5217

    \u90a3\u4e48\uff0c\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\uff0c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u662f\u6ca1\u6709\u5fc5\u8981\u7684\uff0c\u5e94\u5f53\u88ab\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u56e0\u4e3a\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u6389\u3002\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u9700\u8981\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002

    Fig. \u91cd\u590d\u6392\u5217\u526a\u679d

    \u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5b9e\u73b0\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u56e0\u6b64\uff0c\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 duplicated \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart permutations_ii.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList<Integer>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nSet<Integer> duplicated = new HashSet<Integer>();\nfor (int i = 0; i < choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList<List<Integer>> permutationsII(int[] nums) {\nList<List<Integer>> res = new ArrayList<List<Integer>>();\nbacktrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\nreturn res;\n}\n
    permutations_ii.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nunordered_set<int> duplicated;\nfor (int i = 0; i < choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && duplicated.find(choice) == duplicated.end()) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nvector<vector<int>> permutationsII(vector<int> nums) {\nvector<int> state;\nvector<bool> selected(nums.size(), false);\nvector<vector<int>> res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n
    permutations_ii.py
    def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated = set[int]()\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif not selected[i] and choice not in duplicated:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_ii(nums: list[int]) -> list[list[int]]:\n\"\"\"\u5168\u6392\u5217 II\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n
    permutations_ii.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated := make(map[int]struct{}, 0)\nfor i := 0; i < len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif _, ok := duplicated[choice]; !ok && !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n// \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nduplicated[choice] = struct{}{}\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackII(&state, &nums, &selected, &res)\nreturn res\n}\n
    permutations_ii.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_ii.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_ii.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n
    permutations_ii.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nISet<int> duplicated = new HashSet<int>();\nfor (int i = 0; i < choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.Contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList<List<int>> permutationsII(int[] nums) {\nList<List<int>> res = new List<List<int>>();\nbacktrack(new List<int>(), nums, new bool[nums.Length], res);\nreturn res;\n}\n
    permutations_ii.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nvar duplicated: Set<Int> = []\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i], !duplicated.contains(choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: choices, selected: &selected, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -> [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &state, choices: nums, selected: &selected, res: &res)\nreturn res\n}\n
    permutations_ii.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n
    permutations_ii.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n

    \u6ce8\u610f\uff0c\u867d\u7136 selected \u548c duplicated \u90fd\u8d77\u5230\u526a\u679d\u7684\u4f5c\u7528\uff0c\u4f46\u4ed6\u4eec\u526a\u6389\u7684\u662f\u4e0d\u540c\u7684\u5206\u652f\uff1a

    • \u526a\u679d\u6761\u4ef6\u4e00\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a selected \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u4f5c\u7528\u662f\u907f\u514d\u67d0\u4e2a\u5143\u7d20\u5728 state \u4e2d\u91cd\u590d\u51fa\u73b0\u3002
    • \u526a\u679d\u6761\u4ef6\u4e8c\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u5373\u6bcf\u4e2a\u5f00\u542f\u7684 backtrack \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a duplicated \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u904d\u5386\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u4ee5\u907f\u514d\u4ea7\u751f\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002

    Fig. \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4

    "},{"location":"chapter_backtracking/permutations_problem/#1223","title":"12.2.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\) \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002

    \u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002selected \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a duplicated \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u5168\u6392\u5217 I \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5168\u6392\u5217 II \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_backtracking/subset_sum_problem/","title":"12.3. \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 target \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e target \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002

    \u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u7531\u4e8e\u96c6\u5408\u4e2d\u7684\u6570\u5b57\u53ef\u4ee5\u88ab\u91cd\u590d\u9009\u53d6\uff0c\u56e0\u6b64\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u5b50\u96c6\u662f\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\u7684\uff0c\u4f8b\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002

    "},{"location":"chapter_backtracking/subset_sum_problem/#1231","title":"12.3.1. \u00a0 \u4ece\u5168\u6392\u5217\u5f15\u51fa\u89e3\u6cd5","text":"

    \u7c7b\u4f3c\u4e8e\u4e0a\u8282\u5168\u6392\u5217\u95ee\u9898\u7684\u89e3\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e target \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002

    \u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u5141\u8bb8\u91cd\u590d\u9009\u53d6\u540c\u4e00\u5143\u7d20\uff0c\u56e0\u6b64\u65e0\u9700\u501f\u52a9 selected \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_i_naive.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int total, int[] choices, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<Integer>> subsetSumINaive(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, int total, vector<int> &choices, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (size_t i = 0; i < choices.size(); i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector<vector<int>> subsetSumINaive(vector<int> &nums, int target) {\nvector<int> state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0;           // \u5b50\u96c6\u548c\nvector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.py
    def backtrack(\nstate: list[int],\ntarget: int,\ntotal: int,\nchoices: list[int],\nres: list[list[int]],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif total == target:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in range(len(choices)):\n# \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total + choices[i] > target:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i_naive(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal = 0  # \u5b50\u96c6\u548c\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res)\nreturn res\n
    subset_sum_i_naive.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == total {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i < len(*choices); i++ {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total+(*choices)[i] > target {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal := 0              // \u5b50\u96c6\u548c\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumINaive(total, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_i_naive.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<int> state, int target, int total, int[] choices, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.Length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> subsetSumINaive(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n

    \u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e24\u79cd\u4e0d\u540c\u7684\u60c5\u51b5\u3002

    Fig. \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d

    "},{"location":"chapter_backtracking/subset_sum_problem/#1232","title":"12.3.2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"

    \u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u56e0\u4e3a\uff1a

    • \u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 target \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002
    • \u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u662f\u5f88\u8017\u65f6\u7684\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002

    \u4e3a\u4e86\u8fbe\u5230\u6700\u4f73\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u4e0b\u56fe\uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u5177\u4f53\u6765\u770b\uff1a

    1. \u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) , \\(4\\) \uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\cdots]\\) \u3002
    2. \u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\cdots]\\) \u548c 1. \u4e2d\u63d0\u5230\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
    3. \u540c\u7406\uff0c\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\cdots]\\) \u548c\u5b50\u96c6 \\([5, 4, \\cdots]\\) \u548c\u4e4b\u524d\u7684\u5b50\u96c6\u91cd\u590d\u3002

    Fig. \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6

    \u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\cdots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\cdots , x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\cdots \\leq i_m\\) \u3002\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u662f\u91cd\u590d\u5b50\u96c6\u3002

    \u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf start \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\uff0c\u4ece\u800c\u5b8c\u6210\u5b50\u96c6\u53bb\u91cd\u3002

    \u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4e24\u9879\u4f18\u5316\u3002\u9996\u5148\uff0c\u6211\u4eec\u5728\u5f00\u542f\u641c\u7d22\u524d\u5c06\u6570\u7ec4 nums \u6392\u5e8f\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 target \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u90fd\u4e00\u5b9a\u4f1a\u8d85\u8fc7 target \u3002\u5176\u6b21\uff0c\u6211\u4eec\u901a\u8fc7\u5728 target \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 target \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\uff0c\u7701\u53bb\u4e86\u5143\u7d20\u548c\u53d8\u91cf total \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_i.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<Integer>> subsetSumI(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector<vector<int>> subsetSumI(vector<int> &nums, int target) {\nvector<int> state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector<vector<int>> res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.py
    def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] < 0:\nbreak\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n
    subset_sum_i.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i := start; i < len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] < 0 {\nbreak\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumI(start, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_i.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> subsetSumI(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u5230\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002

    Fig. \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b

    "},{"location":"chapter_backtracking/subset_sum_problem/#1233","title":"12.3.3. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 target \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e target \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002

    \u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u4e5f\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002

    Fig. \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6

    \u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u5229\u7528\u8be5\u7279\u6027\uff0c\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002

    \u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf start \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u5373\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_ii.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<Integer>> subsetSumII(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector<vector<int>> subsetSumII(vector<int> &nums, int target) {\nvector<int> state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector<vector<int>> res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.py
    def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n# \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] < 0:\nbreak\n# \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i > start and choices[i] == choices[i - 1]:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_ii(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n
    subset_sum_ii.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i := start; i < len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] < 0 {\nbreak\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i > start && (*choices)[i] == (*choices)[i-1] {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumII(start, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_ii.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> subsetSumII(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n

    \u4e0b\u56fe\u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u5efa\u8bae\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002

    Fig. \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b

    "},{"location":"chapter_backtracking/summary/","title":"12.5. \u00a0 \u5c0f\u7ed3","text":"
    • \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002
    • \u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u66f4\u9ad8\u6548\u7387\u6216\u66f4\u597d\u6548\u679c\u7684\u89e3\u6cd5\u3002
    • \u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u679d\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
    • \u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
    • \u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002
    • \u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002
    • \\(n\\) \u7687\u540e\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
    • \u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002
    "},{"location":"chapter_computational_complexity/","title":"2. \u00a0 \u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1. \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1. \u00a0 \u7b97\u6cd5\u8bc4\u4ef7\u7ef4\u5ea6","text":"

    \u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u7b97\u6cd5\u8bbe\u8ba1\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\uff1a

    1. \u627e\u5230\u95ee\u9898\u89e3\u6cd5\u3002\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\uff0c\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
    2. \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\u3002\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002

    \u56e0\u6b64\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u6210\u4e3a\u4e3b\u8981\u7684\u8bc4\u4ef7\u7ef4\u5ea6\uff0c\u4e3b\u8981\u5305\u62ec\uff1a

    • \u65f6\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
    • \u7a7a\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002

    \u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u638c\u63e1\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u65b9\u6cd5\u5219\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u4e86\u89e3\u8bc4\u4ef7\u6807\u51c6\uff0c\u6211\u4eec\u624d\u80fd\u8fdb\u884c\u7b97\u6cd5\u4e4b\u95f4\u7684\u5bf9\u6bd4\u5206\u6790\uff0c\u4ece\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2. \u00a0 \u6548\u7387\u8bc4\u4f30\u65b9\u6cd5","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#_1","title":"\u5b9e\u9645\u6d4b\u8bd5","text":"

    \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 A \u548c\u7b97\u6cd5 B\uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6211\u4eec\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u5c31\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u5c40\u9650\u6027\u3002

    \u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u8868\u73b0\u3002\u4f8b\u5982\uff0c\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 A \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 B \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u6211\u4eec\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002

    \u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 A \u7684\u8fd0\u884c\u65f6\u95f4\u53ef\u80fd\u77ed\u4e8e\u7b97\u6cd5 B\uff1b\u800c\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u8fd9\u6837\u9700\u8981\u5360\u7528\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#_2","title":"\u7406\u8bba\u4f30\u7b97","text":"

    \u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u590d\u6742\u5ea6\u5206\u6790 Complexity Analysis\u300d\u6216\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 Asymptotic Complexity Analysis\u300d\u3002

    \u590d\u6742\u5ea6\u5206\u6790\u8bc4\u4f30\u7684\u662f\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\uff1a

    • \u201c\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u201d\u53ef\u5206\u4e3a\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u548c\u201c\u5360\u7528\u7a7a\u95f4\u201d\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5c06\u590d\u6742\u5ea6\u5206\u4e3a\u300c\u65f6\u95f4\u590d\u6742\u5ea6 Time Complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\uff1b
    • \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u201d\u8868\u793a\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u6709\u5173\uff0c\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff1b
    • \u201c\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u662f\u7b97\u6cd5\u65f6\u95f4\u4e0e\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u975e\u5177\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\uff1b

    \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\u3002\u9996\u5148\uff0c\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u56e0\u6b64\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002\u5176\u6b21\uff0c\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002

    \u5982\u679c\u4f60\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u7684\u6982\u5ff5\u4ecd\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u8be6\u7ec6\u4ecb\u7ecd\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#213","title":"2.1.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790\u91cd\u8981\u6027","text":"

    \u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u544a\u8bc9\u6211\u4eec\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5e76\u4f7f\u6211\u4eec\u80fd\u591f\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002

    \u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u7ae0\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u6211\u4eec\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002\u56e0\u6b64\uff0c\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5efa\u8bae\u8bfb\u8005\u5148\u5bf9\u590d\u6742\u5ea6\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u5e76\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u6848\u4f8b\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/","title":"2.3. \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"

    \u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u4f7f\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#231","title":"2.3.1. \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"

    \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\uff1a

    • \u300c\u8f93\u5165\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\uff1b
    • \u300c\u6682\u5b58\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\uff1b
    • \u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\uff1b

    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u8303\u56f4\u662f\u300c\u6682\u5b58\u7a7a\u95f4\u300d+\u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u3002

    \u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a

    • \u300c\u6682\u5b58\u6570\u636e\u300d\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002
    • \u300c\u6808\u5e27\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
    • \u300c\u6307\u4ee4\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002

    \u56e0\u6b64\uff0c\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u4e00\u822c\u7edf\u8ba1 \u6682\u5b58\u6570\u636e\u3001\u8f93\u51fa\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4 \u4e09\u90e8\u5206\u3002

    Fig. \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nfinal int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7ed3\u6784\u4f53 */\nstruct Node {\nint val;\nNode *next;\nNode(int x) : val(x), next(nullptr) {}\n};\n/* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    class Node:\n\"\"\"\u7c7b\"\"\"\ndef __init__(self, x: int):\nself.val: int = x                 # \u8282\u70b9\u503c\nself.next: Optional[Node] = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ndef function() -> int:\n\"\"\"\u51fd\u6570\"\"\"\n# do something...\nreturn 0\ndef algorithm(n) -> int:  # \u8f93\u5165\u6570\u636e\nA = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\nb = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnode = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn A + b + c      # \u8f93\u51fa\u6570\u636e\n
    /* \u7ed3\u6784\u4f53 */\ntype node struct {\nval  int\nnext *node\n}\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\nreturn &node{val: val}\n}\n/* \u51fd\u6570 */\nfunc function() int {\n// do something...\nreturn 0\n}\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\nconst a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nb := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnewNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nval;\nnext;\nconstructor(val) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc() {\n// do something\nreturn 0;\n}\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\nconst a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nval: number;\nnext: Node | null;\nconstructor(val?: number) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n// do something\nreturn 0;\n}\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\nconst a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\nconst int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nint c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node\n{\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function()\n{\n// do something...\nreturn 0;\n}\nint algorithm(int n)          // \u8f93\u5165\u6570\u636e\n{\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nvar val: Int\nvar next: Node?\ninit(x: Int) {\nval = x\n}\n}\n/* \u51fd\u6570 */\nfunc function() -> Int {\n// do something...\nreturn 0\n}\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\nlet a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nvar b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n
    \n
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(this.val, [this.next]);\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\nconst int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#232","title":"2.3.2. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"

    \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u662f\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u300c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002

    \u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\uff0c\u5206\u522b\u662f\u8f93\u5165\u6570\u636e\u7684\u6700\u5dee\u5206\u5e03\u548c\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u6700\u5dee\u65f6\u95f4\u70b9\u3002

    • \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\u3002\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 nums \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    • \u4ee5\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\u3002\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 nums \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\nint[] nums = new int[n]; // O(n)\n}\n
    void algorithm(int n) {\nint a = 0;               // O(1)\nvector<int> b(10000);    // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
    def algorithm(n: int) -> None:\na = 0               # O(1)\nb = [0] * 10000     # O(1)\nif n > 10:\nnums = [0] * n  # O(n)\n
    func algorithm(n int) {\na := 0                      // O(1)\nb := make([]int, 10000)     // O(1)\nvar nums []int\nif n > 10 {\nnums := make([]int, n)  // O(n)\n}\nfmt.Println(a, b, nums)\n}\n
    function algorithm(n) {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
    function algorithm(n: number): void {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
    void algorithm(int n) {\nint a = 0;               // O(1)\nint b[10000];            // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
    void algorithm(int n)\n{\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\n{\nint[] nums = new int[n]; // O(n)\n}\n}\n
    func algorithm(n: Int) {\nlet a = 0 // O(1)\nlet b = Array(repeating: 0, count: 10000) // O(1)\nif n > 10 {\nlet nums = Array(repeating: 0, count: n) // O(n)\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 0;                            // O(1)\nList<int> b = List.filled(10000, 0);  // O(1)\nif (n > 10) {\nList<int> nums = List.filled(n, 0); // O(n)\n}\n}\n

    \u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u4f8b\u5982\uff0c\u51fd\u6570 loop() \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 function() \uff0c\u6bcf\u8f6e\u4e2d\u7684 function() \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002\u800c\u9012\u5f52\u51fd\u6570 recur() \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 recur() \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    def function() -> int:\n# do something\nreturn 0\ndef loop(n: int) -> None:\n\"\"\"\u5faa\u73af O(1)\"\"\"\nfor _ in range(n):\nfunction()\ndef recur(n: int) -> int:\n\"\"\"\u9012\u5f52 O(n)\"\"\"\nif n == 1: return\nreturn recur(n - 1)\n
    func function() int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\nfor i := 0; i < n; i++ {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\nif n == 1 {\nreturn\n}\nrecur(n - 1)\n}\n
    function constFunc() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
    function constFunc(): number {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    int function()\n{\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nint recur(int n)\n{\nif (n == 1) return 1;\nreturn recur(n - 1);\n}\n
    @discardableResult\nfunc function() -> Int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\nfor _ in 0 ..< n {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\nif n == 1 {\nreturn\n}\nrecur(n: n - 1)\n}\n
    \n
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#233","title":"2.3.3. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09

    \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\]

    Fig. \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b

    Tip

    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u6211\u4eec\u5148\u4e13\u6ce8\u4e8e\u7406\u89e3\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"

    \u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u5373\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
    space_complexity.cpp
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nvector<int> nums(10000);\nListNode node(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
    space_complexity.py
    def constant(n: int) -> None:\n\"\"\"\u5e38\u6570\u9636\"\"\"\n# \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\na = 0\nnums = [0] * 10000\nnode = ListNode(0)\n# \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nc = 0\n# \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nfunction()\n
    space_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0\nb := 0\nnums := make([]int, 10000)\nListNode := newNode(0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nvar c int\nfor i := 0; i < n; i++ {\nc = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i := 0; i < n; i++ {\nfunction()\n}\nfmt.Println(a, b, nums, c, ListNode)\n}\n
    space_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
    space_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
    space_complexity.c
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nint nums[1000];\nListNode *node = newListNode(0);\nfree(node);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
    space_complexity.cs
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nint a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
    space_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nlet a = 0\nvar b = 0\nlet nums = Array(repeating: 0, count: 10000)\nlet node = ListNode(x: 0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nlet c = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nfunction()\n}\n}\n
    space_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a: i32 = 0;\nvar b: i32 = 0;\nvar nums = [_]i32{0}**10000;\nvar node = inc.ListNode(i32){.val = 0};\nvar i: i32 = 0;\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nwhile (i < n) : (i += 1) {\nvar c: i32 = 0;\n_ = c;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\ni = 0;\nwhile (i < n) : (i += 1) {\n_ = function();\n}\n_ = a;\n_ = b;\n_ = nums;\n_ = node;\n}\n
    space_complexity.dart
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nList<int> nums = List.filled(10000, 0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i < n; i++) {\nfunction();\n}\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"

    \u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nnodes.add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<Integer, String> map = new HashMap<>();\nfor (int i = 0; i < n; i++) {\nmap.put(i, String.valueOf(i));\n}\n}\n
    space_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvector<int> nums(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvector<ListNode> nodes;\nfor (int i = 0; i < n; i++) {\nnodes.push_back(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nunordered_map<int, string> map;\nfor (int i = 0; i < n; i++) {\nmap[i] = to_string(i);\n}\n}\n
    space_complexity.py
    def linear(n: int) -> None:\n\"\"\"\u7ebf\u6027\u9636\"\"\"\n# \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nnums = [0] * n\n# \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nmapp = dict[int, str]()\nfor i in range(n):\nmapp[i] = str(i)\n
    space_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n_ = make([]int, n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes []*node\nfor i := 0; i < n; i++ {\nnodes = append(nodes, newNode(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nm := make(map[int]string, n)\nfor i := 0; i < n; i++ {\nm[i] = strconv.Itoa(i)\n}\n}\n
    space_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
    space_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes: ListNode[] = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
    space_complexity.c
    /* \u54c8\u5e0c\u8868 */\nstruct hashTable {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n};\ntypedef struct hashTable hashTable;\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint *nums = malloc(sizeof(int) * n);\nfree(nums);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nListNode **nodes = malloc(sizeof(ListNode *) * n);\nfor (int i = 0; i < n; i++) {\nnodes[i] = newListNode(i);\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i < n; i++) {\nfree(nodes[i]);\n}\nfree(nodes);\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nhashTable *h = NULL;\nfor (int i = 0; i < n; i++) {\nhashTable *tmp = malloc(sizeof(hashTable));\ntmp->key = i;\ntmp->val = i;\nHASH_ADD_INT(h, key, tmp);\n}\n// \u5185\u5b58\u91ca\u653e\nhashTable *curr, *tmp;\nHASH_ITER(hh, h, curr, tmp) {\nHASH_DEL(h, curr);\nfree(curr);\n}\n}\n
    space_complexity.cs
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new();\nfor (int i = 0; i < n; i++) {\nnodes.Add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nDictionary<int, string> map = new();\nfor (int i = 0; i < n; i++) {\nmap.Add(i, i.ToString());\n}\n}\n
    space_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet nums = Array(repeating: 0, count: n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet nodes = (0 ..< n).map { ListNode(x: $0) }\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
    space_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvar nums = [_]i32{0}**n;\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ntry nodes.append(i);\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\ndefer map.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\nconst string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\ndefer std.heap.page_allocator.free(string);\ntry map.put(i, string);\n}\n_ = nums;\n}\n
    space_complexity.dart
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nList<int> nums = List.filled(n, 0);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = [];\nfor (var i = 0; i < n; i++) {\nnodes.add(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<int, String> map = HashMap();\nfor (var i = 0; i < n; i++) {\nmap.putIfAbsent(i, () => i.toString());\n}\n}\n

    \u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nSystem.out.println(\"\u9012\u5f52 n = \" + n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\ncout << \"\u9012\u5f52 n = \" << n << endl;\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.py
    def linear_recur(n: int) -> None:\n\"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nprint(\"\u9012\u5f52 n =\", n)\nif n == 1:\nreturn\nlinear_recur(n - 1)\n
    space_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\nfmt.Println(\"\u9012\u5f52 n =\", n)\nif n == 1 {\nreturn\n}\nspaceLinearRecur(n - 1)\n}\n
    space_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.c
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprintf(\"\u9012\u5f52 n = %d\\r\\n\", n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\nprint(\"\u9012\u5f52 n = \\(n)\")\nif n == 1 {\nreturn\n}\nlinearRecur(n: n - 1)\n}\n
    space_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\nstd.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.dart
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprint('\u9012\u5f52 n = $n');\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n

    Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"

    \u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[][] numMatrix = new int[n][n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<Integer>> numList = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<Integer> tmp = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n
    space_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvector<vector<int>> numMatrix;\nfor (int i = 0; i < n; i++) {\nvector<int> tmp;\nfor (int j = 0; j < n; j++) {\ntmp.push_back(0);\n}\nnumMatrix.push_back(tmp);\n}\n}\n
    space_complexity.py
    def quadratic(n: int) -> None:\n\"\"\"\u5e73\u65b9\u9636\"\"\"\n# \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nnum_matrix = [[0] * n for _ in range(n)]\n
    space_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nnumMatrix := make([][]int, n)\nfor i := 0; i < n; i++ {\nnumMatrix[i] = make([]int, n)\n}\n}\n
    space_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
    space_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
    space_complexity.c
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nint **numMatrix = malloc(sizeof(int *) * n);\nfor (int i = 0; i < n; i++) {\nint *tmp = malloc(sizeof(int) * n);\nfor (int j = 0; j < n; j++) {\ntmp[j] = 0;\n}\nnumMatrix[i] = tmp;\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i < n; i++) {\nfree(numMatrix[i]);\n}\nfree(numMatrix);\n}\n
    space_complexity.cs
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[,] numMatrix = new int[n, n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = new();\nfor (int i = 0; i < n; i++) {\nList<int> tmp = new();\nfor (int j = 0; j < n; j++) {\ntmp.Add(0);\n}\nnumList.Add(tmp);\n}\n}\n
    space_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
    space_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvar nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\nvar tmp = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer tmp.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ntry tmp.append(0);\n}\ntry nodes.append(tmp);\n}\n}\n
    space_complexity.dart
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = [];\nfor (var i = 0; i < n; i++) {\nList<int> tmp = [];\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n

    \u5728\u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \uff0c\u5e76\u4e14\u6bcf\u4e2a\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n, n-1, n-2, ..., 2, 1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nint[] nums = new int[n];\nSystem.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\nvector<int> nums(n);\ncout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.py
    def quadratic_recur(n: int) -> int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 0:\nreturn 0\n# \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nnums = [0] * n\nreturn quadratic_recur(n - 1)\n
    space_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\nif n <= 0 {\nreturn 0\n}\nnums := make([]int, n)\nfmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\nreturn spaceQuadraticRecur(n - 1)\n}\n
    space_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.c
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\nint *nums = malloc(sizeof(int) * n);\nprintf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\nfree(nums);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nint[] nums = new int[n];\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\nif n <= 0 {\nreturn 0\n}\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = Array(repeating: 0, count: n)\nprint(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\nreturn quadraticRecur(n: n - 1)\n}\n
    space_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\nif (n <= 0) return 0;\nvar nums = [_]i32{0}**n;\nstd.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.dart
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nList<int> nums = List.filled(n, 0);\nprint('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\nreturn quadraticRecur(n - 1);\n}\n

    Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"

    \u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\nif (n == 0)\nreturn null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn nullptr;\nTreeNode *root = new TreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.py
    def build_tree(n: int) -> TreeNode | None:\n\"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\nif n == 0:\nreturn None\nroot = TreeNode(0)\nroot.left = build_tree(n - 1)\nroot.right = build_tree(n - 1)\nreturn root\n
    space_complexity.go
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\nif n == 0 {\nreturn nil\n}\nroot := newTreeNode(0)\nroot.left = buildTree(n - 1)\nroot.right = buildTree(n - 1)\nreturn root\n}\n
    space_complexity.js
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.c
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn NULL;\nTreeNode *root = newTreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\nif n == 0 {\nreturn nil\n}\nlet root = TreeNode(x: 0)\nroot.left = buildTree(n: n - 1)\nroot.right = buildTree(n: n - 1)\nreturn root\n}\n
    space_complexity.zig
    // \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\nif (n == 0) return null;\nconst root = try mem_allocator.create(inc.TreeNode(i32));\nroot.init(0);\nroot.left = try buildTree(mem_allocator, n - 1);\nroot.right = try buildTree(mem_allocator, n - 1);\nreturn root;\n}\n
    space_complexity.dart
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n

    Fig. \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"

    \u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u548c\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u7b49\u3002

    \u4f8b\u5982\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002

    \u518d\u4f8b\u5982\u201c\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\u201d\uff0c\u8f93\u5165\u4efb\u610f\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n) = O(\\log n)\\) \u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#234","title":"2.3.4. \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"

    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u662f\u975e\u5e38\u56f0\u96be\u7684\u3002

    \u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002

    \u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002

    "},{"location":"chapter_computational_complexity/summary/","title":"2.4. \u00a0 \u5c0f\u7ed3","text":"

    \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30

    • \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8bc4\u4ef7\u7b97\u6cd5\u6027\u80fd\u7684\u4e24\u4e2a\u5173\u952e\u7ef4\u5ea6\u3002
    • \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
    • \u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u514b\u670d\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6

    • \u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002
    • \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5373\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002
    • \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
    • \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n \\log n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) , \\(O(n!)\\) \u7b49\u3002
    • \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
    • \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6

    • \u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002
    • \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u8ba1\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6307\u4ee4\u7a7a\u95f4\u3001\u6570\u636e\u7a7a\u95f4\u3001\u6808\u5e27\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
    • \u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u95f4\u70b9\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
    • \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) \u7b49\u3002
    "},{"location":"chapter_computational_complexity/summary/#241-q-a","title":"2.4.1. \u00a0 Q & A","text":"

    \u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f

    \u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java, Python, C++, Go, C# \u7b49\uff09 \u90fd\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u4e00\u822c\u6765\u8bf4\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002

    \u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f

    \u51fd\u6570\uff08function\uff09\u53ef\u4ee5\u72ec\u7acb\u88ab\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002 \u65b9\u6cd5\uff08method\uff09\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u65b9\u6cd5\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u65b9\u6cd5\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002

    \u56e0\u6b64\uff0cC \u548c Go \u53ea\u6709\u51fd\u6570\uff0cJava \u548c C# \u53ea\u6709\u65b9\u6cd5\uff0c\u5728 C++, Python \u4e2d\u53d6\u51b3\u4e8e\u5b83\u662f\u5426\u5c5e\u4e8e\u4e00\u4e2a\u7c7b\u3002

    \u56fe\u7247\u201c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f

    \u4e0d\u662f\uff0c\u8be5\u56fe\u7247\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff08\u5373\u589e\u957f\u8d8b\u52bf\uff09\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002\u6bcf\u4e2a\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u6765\u628a\u6240\u6709\u66f2\u7ebf\u7684\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002 \u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\uff1b\u4f46\u76f8\u5bf9\u5730 \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u662f\u590d\u6742\u5ea6\u5360\u4e3b\u5bfc\u7684\u60c5\u51b5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/","title":"2.2. \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/time_complexity/#221","title":"2.2.1. \u00a0 \u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4","text":"

    \u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f

    1. \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
    2. \u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c + \u9700\u8981 1 ns\uff0c\u4e58\u6cd5\u64cd\u4f5c * \u9700\u8981 10 ns\uff0c\u6253\u5370\u64cd\u4f5c\u9700\u8981 5 ns \u7b49\u3002
    3. \u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002

    \u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e3a \\(6n + 12\\) ns \u3002

    \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nSystem.out.println(0);     // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\ncout << 0 << endl;         // 5 ns\n}\n}\n
    # \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int) -> None:\na = 2      # 1 ns\na = a + 1  # 1 ns\na = a * 2  # 10 ns\n# \u5faa\u73af n \u6b21\nfor _ in range(n):  # 1 ns\nprint(0)        # 5 ns\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\na := 2      // 1 ns\na = a + 1   // 1 ns\na = a * 2   // 10 ns\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {    // 1 ns\nfmt.Println(a)          // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\nvar a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\nvar a: number = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintf(\"%d\", 0);            // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n)\n{\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++)\n{  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nConsole.WriteLine(0);     // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\nvar a = 2 // 1 ns\na = a + 1 // 1 ns\na = a * 2 // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // 1 ns\nprint(0) // 5 ns\n}\n}\n
    \n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprint(0); // 5 ns\n}\n}\n

    \u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#222","title":"2.2.2. \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"

    \u300c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u300d\u91c7\u53d6\u4e86\u4e00\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u5176\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002

    \u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u8f83\u4e3a\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 A , B , C \u3002

    • \u7b97\u6cd5 A \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
    • \u7b97\u6cd5 B \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u300c\u7ebf\u6027\u9636\u300d\u3002
    • \u7b97\u6cd5 C \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u4ecd\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c A \u76f8\u540c\uff0c\u4ecd\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nSystem.out.println(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nSystem.out.println(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nSystem.out.println(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\ncout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\ncout << 0 << endl;\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\ncout << 0 << endl;\n}\n}\n
    # \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int) -> None:\nprint(0)\n# \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int) -> None:\nfor _ in range(n):\nprint(0)\n# \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int) -> None:\nfor _ in range(1000000):\nprint(0)\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\nfmt.Println(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\nfor i := 0; i < n; i++ {\nfmt.Println(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\nfor i := 0; i < 1000000; i++ {\nfmt.Println(0)\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nprintf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nprintf(\"%d\", 0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprintf(\"%d\", 0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n)\n{\nConsole.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nConsole.WriteLine(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n)\n{\nfor (int i = 0; i < 1000000; i++)\n{\nConsole.WriteLine(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\nprint(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\nfor _ in 0 ..< n {\nprint(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\nfor _ in 0 ..< 1000000 {\nprint(0)\n}\n}\n
    \n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\nprint(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\nfor (int i = 0; i < n; i++) {\nprint(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprint(0);\n}\n}\n

    Fig. \u7b97\u6cd5 A, B, C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf

    \u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u4f18\u52bf\u548c\u5c40\u9650\u6027\u5462\uff1f

    \u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 B \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n > 1\\) \u65f6\u6bd4\u7b97\u6cd5 A \u6162\uff0c\u5728 \\(n > 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 C \u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u300c\u7ebf\u6027\u9636\u300d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u6240\u8868\u8fbe\u7684\u542b\u4e49\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\u7684\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf\u7684\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u7684\u7b80\u5316\u65b9\u6cd5\u5927\u5927\u964d\u4f4e\u4e86\u4f30\u7b97\u96be\u5ea6\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 A \u548c C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 C \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 B \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 C \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#223","title":"2.2.3. \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"

    \u8bbe\u7b97\u6cd5\u7684\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0b\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a

    \\[ T(n) = 3 + 2n \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nSystem.out.println(0);    // +1\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\ncout << 0 << endl;    // +1\n}\n}\n
    def algorithm(n: int) -> None:\na = 1      # +1\na = a + 1  # +1\na = a * 2  # +1\n# \u5faa\u73af n \u6b21\nfor i in range(n):  # +1\nprint(0)        # +1\n
    func algorithm(n int) {\na := 1      // +1\na = a + 1   // +1\na = a * 2   // +1\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {   // +1\nfmt.Println(a)         // +1\n}\n}\n
    function algorithm(n) {\nvar a = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
    function algorithm(n: number): void{\nvar a: number = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintf(\"%d\", 0);            // +1\n}\n}  
    void algorithm(int n)\n{\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n{\nConsole.WriteLine(0);   // +1\n}\n}\n
    func algorithm(n: Int) {\nvar a = 1 // +1\na = a + 1 // +1\na = a * 2 // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // +1\nprint(0) // +1\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 1; // +1\na = a + 1; // +1\na = a * 2; // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprint(0); // +1\n}\n}\n

    \\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u53ef\u4ee5\u5f97\u51fa\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002

    \u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 Big-\\(O\\) Notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c Asymptotic Upper Bound\u300d\u3002

    \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf\u51fd\u6570 \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u6765\u770b\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5b9a\u4e49\u3002

    \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c

    \u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 $$ T(n) \\leq c \\cdot f(n) $$ \u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a $$ T(n) = O(f(n)) $$

    Fig. \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c

    \u4ece\u672c\u8d28\u4e0a\u8bb2\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#224","title":"2.2.4. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"

    \u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u9700\u62c5\u5fc3\u3002\u56e0\u4e3a\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\uff0c\u6211\u4eec\u53ea\u9700\u8981\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u6570\u5b66\u610f\u4e49\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u3002

    \u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#_1","title":"\u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"

    \u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\uff1a

    1. \u5ffd\u7565\u4e0e \\(n\\) \u65e0\u5173\u7684\u64cd\u4f5c\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u662f \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\uff0c\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
    2. \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002
    3. \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u4e0a\u8ff0 1. \u548c 2. \u6280\u5de7\u3002

    \u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u3001\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\u3002

    \\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]

    \u6700\u7ec8\uff0c\u4e24\u8005\u90fd\u80fd\u63a8\u51fa\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7ed3\u679c\uff0c\u5373 \\(O(n^2)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nSystem.out.println(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nSystem.out.println(0);\n}\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\ncout << 0 << endl;\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\ncout << 0 << endl;\n}\n}\n}\n
    def algorithm(n: int) -> None:\na = 1      # +0\uff08\u6280\u5de7 1\uff09\na = a + n  # +0\uff08\u6280\u5de7 1\uff09\n# +n\uff08\u6280\u5de7 2\uff09\nfor i in range(5 * n + 1):\nprint(0)\n# +n*n\uff08\u6280\u5de7 3\uff09\nfor i in range(2 * n):\nfor j in range(n + 1):\nprint(0)\n
    func algorithm(n int) {\na := 1     // +0\uff08\u6280\u5de7 1\uff09\na = a + n  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i := 0; i < 5 * n + 1; i++ {\nfmt.Println(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i := 0; i < 2 * n; i++ {\nfor j := 0; j < n + 1; j++ {\nfmt.Println(0)\n}\n}\n}\n
    function algorithm(n) {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
    function algorithm(n: number): void {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprintf(\"%d\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprintf(\"%d\", 0);\n}\n}\n}\n
    void algorithm(int n)\n{\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++)\n{\nConsole.WriteLine(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++)\n{\nfor (int j = 0; j < n + 1; j++)\n{\nConsole.WriteLine(0);\n}\n}\n}\n
    func algorithm(n: Int) {\nvar a = 1 // +0\uff08\u6280\u5de7 1\uff09\na = a + n // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor _ in 0 ..< (5 * n + 1) {\nprint(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor _ in 0 ..< (2 * n) {\nfor _ in 0 ..< (n + 1) {\nprint(0)\n}\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 1; // +0\uff08\u6280\u5de7 1\uff09\na = a + n; // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprint(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprint(0);\n}\n}\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#_2","title":"\u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\u7531\u591a\u9879\u5f0f \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002

    \u4ee5\u4e0b\u8868\u683c\u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002

    \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#225","title":"2.2.5. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5305\u62ec\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a

    \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\]

    Fig. \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b

    Tip

    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u9884\u5907\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u4e0d\u7406\u89e3\u7684\u90e8\u5206\uff0c\u8bf7\u4e0d\u8981\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u56de\u987e\u3002\u73b0\u9636\u6bb5\uff0c\u8bf7\u5148\u4e13\u6ce8\u4e8e\u7406\u89e3\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"

    \u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002

    \u5bf9\u4e8e\u4ee5\u4e0b\u7b97\u6cd5\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf size \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.py
    def constant(n: int) -> int:\n\"\"\"\u5e38\u6570\u9636\"\"\"\ncount = 0\nsize = 100000\nfor _ in range(size):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\ncount := 0\nsize := 100000\nfor i := 0; i < size; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
    time_complexity.c
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nint i = 0;\nfor (int i = 0; i < size; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\nvar count = 0\nlet size = 100_000\nfor _ in 0 ..< size {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n_ = n;\nvar count: i32 = 0;\nconst size: i32 = 100_000;\nvar i: i32 = 0;\nwhile(i<size) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (var i = 0; i < size; i++) {\ncount++;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"

    \u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.py
    def linear(n: int) -> int:\n\"\"\"\u7ebf\u6027\u9636\"\"\"\ncount = 0\nfor _ in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\ncount := 0\nfor i := 0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\nvar count = 0\nfor _ in 0 ..< n {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (var i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n

    \u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\u3002

    \u5982\u4f55\u786e\u5b9a\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \uff1f

    \u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u4f8b\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u76f4\u63a5\u5c06 \\(n\\) \u89c6\u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u4e0b\u9762\u904d\u5386\u6570\u7ec4\u7684\u793a\u4f8b\u4e2d\uff0c\u6570\u636e\u5927\u5c0f \\(n\\) \u4e3a\u6570\u7ec4\u7684\u957f\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def array_traversal(nums: list[int]) -> int:\n\"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor num in nums:\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor range nums {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nforeach (int num in nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\nvar count: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (nums) |_| {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (var num in nums) {\ncount++;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"

    \u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u4e3a \\(O(n^2)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.py
    def quadratic(n: int) -> int:\n\"\"\"\u5e73\u65b9\u9636\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i in range(n):\nfor j in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i := 0; i < n; i++ {\nfor j := 0; j < n; j++ {\ncount++\n}\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0 ..< n {\nfor _ in 0 ..< n {\ncount += 1\n}\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ncount += 1;\n}\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n

    Fig. \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u3001\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u4ee5\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1, n-2, \\cdots, 2, 1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(\\frac{n}{2}\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    \\[ O((n - 1) \\frac{n}{2}) = O(n^2) \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.py
    def bubble_sort(nums: list[int]) -> int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\ncount = 0  # \u8ba1\u6570\u5668\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(len(nums) - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp: int = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\nreturn count\n
    time_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\ncount := 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp := nums[j]\nnums[j] = nums[j+1]\nnums[j+1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = n - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\nvar count = 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0 ..< i {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\nvar count: i32 = 0;  // \u8ba1\u6570\u5668 \n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: i32 = @intCast(i32, nums.len ) - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (var i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (var j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"

    Note

    \u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002

    \u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u901a\u5e38\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002\u82e5\u4e00\u4e2a\u95ee\u9898\u4f7f\u7528\u300c\u66b4\u529b\u679a\u4e3e\u300d\u6c42\u89e3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u90a3\u4e48\u901a\u5e38\u9700\u8981\u4f7f\u7528\u300c\u52a8\u6001\u89c4\u5212\u300d\u6216\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u7b49\u65b9\u6cd5\u6765\u89e3\u51b3\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.py
    def exponential(n: int) -> int:\n\"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nbase = 1\n# cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in range(n):\nfor _ in range(base):\ncount += 1\nbase *= 2\n# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n
    time_complexity.go
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\ncount, base := 0, 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor i := 0; i < n; i++ {\nfor j := 0; j < base; j++ {\ncount++\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
    time_complexity.js
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.c
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0;\nint bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\nvar count = 0\nvar base = 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0 ..< n {\nfor _ in 0 ..< base {\ncount += 1\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
    time_complexity.zig
    // \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\nvar count: i32 = 0;\nvar bas: i32 = 1;\nvar i: i32 = 0;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < bas) : (j += 1) {\ncount += 1;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.dart
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (var i = 0; i < n; i++) {\nfor (var j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n

    Fig. \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u4e0d\u65ad\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.py
    def exp_recur(n: int) -> int:\n\"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 1:\nreturn 1\nreturn exp_recur(n - 1) + exp_recur(n - 1) + 1\n
    time_complexity.go
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n-1) + expRecur(n-1) + 1\n}\n
    time_complexity.js
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.c
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
    time_complexity.zig
    // \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.dart
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"

    \u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u7684\u60c5\u51b5\u201d\u3002\u5bf9\u6570\u9636\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\uff0c\u65f6\u95f4\u589e\u957f\u7f13\u6162\uff0c\u662f\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002

    \u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u5206\u6cbb\u7b97\u6cd5\u300d\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def logarithmic(n: float) -> int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nwhile n > 1:\nn = n / 2\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\ncount := 0\nfor n > 1 {\nn = n / 2\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\nvar count = 0\nvar n = n\nwhile n > 1 {\nn = n / 2\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\nvar count: i32 = 0;\nvar n_var = n;\nwhile (n_var > 1)\n{\nn_var = n_var / 2;\ncount +=1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n

    Fig. \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.py
    def log_recur(n: float) -> int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 1:\nreturn 0\nreturn log_recur(n / 2) + 1\n
    time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n/2) + 1\n}\n
    time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.c
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n: n / 2) + 1\n}\n
    time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.dart
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on-log-n","title":"\u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"

    \u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002

    \u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def linear_log_recur(n: float) -> int:\n\"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\nif n <= 1:\nreturn 1\ncount: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\nfor _ in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\nif n <= 1 {\nreturn 1\n}\ncount := linearLogRecur(n/2) +\nlinearLogRecur(n/2)\nfor i := 0.0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 1\n}\nvar count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\nfor _ in stride(from: 0, to: n, by: 1) {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\nif (n <= 1) return 1;\nvar count: i32 = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nvar i: f32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (var i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n

    Fig. \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/time_complexity/#on_1","title":"\u9636\u4e58\u9636 \\(O(n!)\\)","text":"

    \u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u300c\u5168\u6392\u5217\u300d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a

    \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\cdots \\times 2 \\times 1 \\]

    \u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u7ec8\u6b62\u5206\u88c2\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.py
    def factorial_recur(n: int) -> int:\n\"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 0:\nreturn 1\ncount = 0\n# \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in range(n):\ncount += factorial_recur(n - 1)\nreturn count\n
    time_complexity.go
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\nif n == 0 {\nreturn 1\n}\ncount := 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor i := 0; i < n; i++ {\ncount += factorialRecur(n - 1)\n}\nreturn count\n}\n
    time_complexity.js
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\nif n == 0 {\nreturn 1\n}\nvar count = 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0 ..< n {\ncount += factorialRecur(n: n - 1)\n}\nreturn count\n}\n
    time_complexity.zig
    // \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\nif (n == 0) return 1;\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nwhile (i < n) : (i += 1) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (var i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n

    Fig. \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/time_complexity/#226","title":"2.2.6. \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"

    \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5176\u4e2d nums \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff1b\u7b97\u6cd5\u7684\u4efb\u52a1\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a

    • \u5f53 nums = [?, ?, ..., 1] \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\uff1b
    • \u5f53 nums = [1, ?, ?, ...] \uff0c\u5373\u5f53\u9996\u4e2a\u6570\u5b57\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\)\uff1b

    \u201c\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u201d\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002\u76f8\u5e94\u5730\uff0c\u201c\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\u201d\u7528 \\(\\Omega\\) \u8bb0\u53f7\u6765\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart worst_best_time_complexity.java
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nInteger[] nums = new Integer[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nCollections.shuffle(Arrays.asList(nums));\n// Integer[] -> int[]\nint[] res = new int[n];\nfor (int i = 0; i < n; i++) {\nres[i] = nums[i];\n}\nreturn res;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.cpp
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\nvector<int> nums(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\nunsigned seed = chrono::system_clock::now().time_since_epoch().count();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nshuffle(nums.begin(), nums.end(), default_random_engine(seed));\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\nfor (int i = 0; i < nums.size(); i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.py
    def random_numbers(n: int) -> list[int]:\n\"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n# \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\nnums = [i for i in range(1, n + 1)]\n# \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrandom.shuffle(nums)\nreturn nums\ndef find_one(nums: list[int]) -> int:\n\"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\nfor i in range(len(nums)):\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1:\nreturn i\nreturn -1\n
    worst_best_time_complexity.go
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\nnums := make([]int, n)\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor i := 0; i < n; i++ {\nnums[i] = i + 1\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrand.Shuffle(len(nums), func(i, j int) {\nnums[i], nums[j] = nums[j], nums[i]\n})\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\nfor i := 0; i < len(nums); i++ {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
    worst_best_time_complexity.js
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    worst_best_time_complexity.ts
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    worst_best_time_complexity.c
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n// \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3an\uff0c\u5143\u7d20\u7c7b\u578b\u4e3aint\uff09\nint *nums = (int *)malloc(n * sizeof(int));\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = n - 1; i > 0; i--) {\nint j = rand() % (i + 1);\nint temp = nums[i];\nnums[i] = nums[j];\nnums[j] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\nfor (int i = 0; i < n; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.cs
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nint[] nums = new int[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = 0; i < nums.Length; i++) {\nvar index = new Random().Next(i, nums.Length);\nvar tmp = nums[i];\nvar ran = nums[index];\nnums[i] = ran;\nnums[index] = tmp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.Length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.swift
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nvar nums = Array(1 ... n)\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle()\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\nfor i in nums.indices {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
    worst_best_time_complexity.zig
    // \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\npub fn randomNumbers(comptime n: usize) [n]i32 {\nvar nums: [n]i32 = undefined;\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (nums) |*num, i| {\nnum.* = @intCast(i32, i) + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nconst rand = std.crypto.random;\nrand.shuffle(i32, &nums);\nreturn nums;\n}\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\npub fn findOne(nums: []i32) i32 {\nfor (nums) |num, i| {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (num == 1) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    worst_best_time_complexity.dart
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\nfinal nums = List.filled(n, 0);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (var i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle();\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\nfor (var i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1) return i;\n}\nreturn -1;\n}\n

    Tip

    \u5b9e\u9645\u5e94\u7528\u4e2d\u6211\u4eec\u5f88\u5c11\u4f7f\u7528\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u76f8\u53cd\uff0c\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u201c\u6548\u7387\u5b89\u5168\u503c\u201d\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002

    \u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6216\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u5728\u201c\u7279\u6b8a\u5206\u5e03\u7684\u6570\u636e\u201d\u4e2d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u56e0\u6b64\u5e76\u4e0d\u80fd\u6700\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u8f83\u4e4b\u4e0b\uff0c\u300c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002

    \u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5219\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(\\frac{n}{2}\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(\\frac{n}{2}) = \\Theta(n)\\) \u3002

    \u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5c24\u5176\u662f\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u7b80\u4fbf\u5730\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002

    \u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f

    \u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u300c\u5e73\u5747\u590d\u6742\u5ea6\u300d\uff0c\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002

    "},{"location":"chapter_data_structure/","title":"3. \u00a0 \u6570\u636e\u7ed3\u6784","text":""},{"location":"chapter_data_structure/basic_data_types/","title":"3.2. \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"

    \u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002

    \u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\u3002\u5b83\u5305\u62ec\uff1a

    • \u6574\u6570\u7c7b\u578b byte , short , int , long \uff1b
    • \u6d6e\u70b9\u6570\u7c7b\u578b float , double \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\uff1b
    • \u5b57\u7b26\u7c7b\u578b char \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u3001\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\uff1b
    • \u5e03\u5c14\u7c7b\u578b bool \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\uff1b

    \u6240\u6709\u57fa\u672c\u6570\u636e\u7c7b\u578b\u90fd\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6211\u4eec\u5c06 \\(1\\) \u4e2a\u4e8c\u8fdb\u5236\u4f4d\u79f0\u4e3a \\(1\\) \u6bd4\u7279\uff0c\u5e76\u89c4\u5b9a \\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bits\uff09\u7ec4\u6210\u3002\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\uff0c\u4f8b\u5982\uff1a

    • \u6574\u6570\u7c7b\u578b byte \u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u4e0d\u540c\u7684\u6570\u5b57\uff1b
    • \u6574\u6570\u7c7b\u578b int \u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\uff1b

    \u4e0b\u8868\u5217\u4e3e\u4e86\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u9700\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002

    \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 byte 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) short 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) int 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) long 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 float 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0 f\\) double 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 char 2 bytes / 1 byte \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 bool 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\)

    Tip

    \u5b57\u7b26\u7684\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u300c\u5b57\u7b26\u7f16\u7801\u300d\u7ae0\u8282\u3002

    \u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002\u56e0\u6b64\uff0c\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u88ab\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002

    \u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u4e0e\u533a\u522b\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u5b83\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\uff0c\u800c\u975e\u201c\u6570\u636e\u201d\u3002\u5982\u679c\u60f3\u8981\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 int \u3001\u5c0f\u6570 float \u3001\u8fd8\u662f\u5b57\u7b26 char \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002

    \u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e0d\u540c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] booleans = new boolean[5];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool booleans[5];\n
    # Python \u7684 list \u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nlist = [0, 0.0, 'a', False]\n
    // \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar booleans = [5]bool{}\n
    /* JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 */\nconst array = [0, 0.0, 'a', false];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst booleans: boolean[] = [];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool booleans[10];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] booleans = new bool[5];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet booleans = Array(repeating: Bool(), count: 5)\n
    \n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nList<int> numbers = List.filled(5, 0);\nList<double> decimals = List.filled(5, 0.0);\nList<String> characters = List.filled(5, 'a');\nList<bool> booleans = List.filled(5, false);\n
    "},{"location":"chapter_data_structure/character_encoding/","title":"3.4. \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 char \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u300c\u5b57\u7b26\u96c6\u300d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002

    "},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1. \u00a0 ASCII \u5b57\u7b26\u96c6","text":"

    \u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5168\u79f0\u4e3a\u201c\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\u201d\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u5373\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u8fd9\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0-9 \u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002

    Fig. ASCII \u7801

    \u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002

    "},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2. \u00a0 GBK \u5b57\u7b26\u96c6","text":"

    \u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u4f8b\u5982\uff0c\u6c49\u5b57\u5927\u7ea6\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002

    \u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7684\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u4e4b\u540e\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\uff0c\u6269\u5c55\u5f97\u5230\u4e86\u300cGBK\u300d\u5b57\u7b26\u96c6\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002

    "},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3. \u00a0 Unicode \u5b57\u7b26\u96c6","text":"

    \u968f\u7740\u8ba1\u7b97\u673a\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u4e5f\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u7535\u8111\u5b89\u88c5\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002

    \u90a3\u4e2a\u65f6\u4ee3\u7684\u4eba\u4eec\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002

    \u300cUnicode\u300d\u7684\u5168\u79f0\u4e3a\u201c\u7edf\u4e00\u5b57\u7b26\u7f16\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3\u4e00\u767e\u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u5230\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002

    \u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u6b62 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u3001\u751a\u81f3\u662f\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002

    Unicode \u662f\u4e00\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u540c\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f

    \u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \uff0c\u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u51fa\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002

    Fig. Unicode \u7f16\u7801\u793a\u4f8b

    \u7136\u800c\uff0cASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700\u8981 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u5927\u5c0f\u7684 2 \u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002

    "},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4. \u00a0 UTF-8 \u7f16\u7801","text":"

    \u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u4e2a\u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700\u8981 1 \u4e2a\u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u4e2a\u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u4e2a\u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u4e2a\u5b57\u8282\u3002

    UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\uff1a

    • \u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \u3001\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002
    • \u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n > 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \u3001\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u5c06\u6700\u9ad8 \\(n\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(1\\) \u6bd4\u8f83\u5bb9\u6613\u7406\u89e3\uff0c\u53ef\u4ee5\u5411\u7cfb\u7edf\u6307\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002\u90a3\u4e48\uff0c\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\uff0c\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u662f\u56e0\u4e3a\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\u7684\u6700\u9ad8\u4e00\u4f4d\u662f \\(0\\) \u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u7684\u5224\u65ad\u51fa\u5f02\u5e38\u3002

    Fig. UTF-8 \u7f16\u7801\u793a\u4f8b

    \u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec UTF-16 \u548c UTF-32 \u3002\u5b83\u4eec\u4e3a Unicode \u5b57\u7b26\u96c6\u63d0\u4f9b\u4e86\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6cd5\u3002

    • UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u4e2a\u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5f88\u591a\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u4e2a\u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u4e2a\u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002
    • UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u4e2a\u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u4f1a\u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u4e3b\u8981\u4f7f\u7528 ASCII \u5b57\u7b26\u7684\u6587\u672c\u3002

    \u4ece\u5b58\u50a8\u7a7a\u95f4\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u4e2a\u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u53ea\u9700\u8981 2 \u4e2a\u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u4e2a\u5b57\u8282\u3002\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u90fd\u4f18\u5148\u652f\u6301 UTF-8 \u3002

    "},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5. \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"

    \u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u5177\u4f53\u6765\u8bf4\uff1a

    • \u968f\u673a\u8bbf\u95ee: UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002
    • \u5b57\u7b26\u8ba1\u6570: \u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002
    • \u5b57\u7b26\u4e32\u64cd\u4f5c: \u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u90fd\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002

    \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u5230\u8bb8\u591a\u56e0\u7d20\uff1a

    • Java \u7684 String \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002
    • JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 JavaScript \u8bed\u8a00\u5728 1995 \u5e74\u88ab Netscape \u516c\u53f8\u9996\u6b21\u5f15\u5165\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u76f8\u5bf9\u65e9\u671f\u7684\u9636\u6bb5\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u591f\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002
    • C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff0c\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff0c\u90fd\u5e7f\u6cdb\u5730\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002

    \u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u56e0\u6b64\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u589e\u52a0\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c Debug \u96be\u5ea6\u3002

    \u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\uff1a

    • Python 3 \u4f7f\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u5bf9\u4e8e\u5168\u90e8\u662f ASCII \u5b57\u7b26\u7684\u5b57\u7b26\u4e32\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u5b57\u7b26\u4e32\u4e2d\u5305\u542b\u7684\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u5b57\u7b26\u4e32\u4e2d\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u90a3\u4e48\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u4e2a\u5b57\u8282\u3002
    • Go \u8bed\u8a00\u7684 string \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 rune \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002
    • Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 char \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002

    "},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1. \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"

    \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002

    "},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1. \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"

    \u300c\u903b\u8f91\u7ed3\u6784\u300d\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002

    \u903b\u8f91\u7ed3\u6784\u901a\u5e38\u5206\u4e3a\u300c\u7ebf\u6027\u300d\u548c\u300c\u975e\u7ebf\u6027\u300d\u4e24\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\uff0c\u4f8b\u5982\u7f51\u72b6\u6216\u6811\u72b6\u7ed3\u6784\u3002

    • \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff1b
    • \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u56fe\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff1b

    Fig. \u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784

    "},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2. \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u79bb\u6563","text":"

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u5185\u5b58\u548c\u786c\u76d8\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u5b58\u50a8\u786c\u4ef6\u8bbe\u5907\u3002\u300c\u786c\u76d8\u300d\u4e3b\u8981\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5bb9\u91cf\u8f83\u5927\uff08\u901a\u5e38\u53ef\u8fbe\u5230 TB \u7ea7\u522b\uff09\u3001\u901f\u5ea6\u8f83\u6162\u3002\u300c\u5185\u5b58\u300d\u7528\u4e8e\u8fd0\u884c\u7a0b\u5e8f\u65f6\u6682\u5b58\u6570\u636e\uff0c\u901f\u5ea6\u8f83\u5feb\uff0c\u4f46\u5bb9\u91cf\u8f83\u5c0f\uff08\u901a\u5e38\u4e3a GB \u7ea7\u522b\uff09\u3002

    \u5728\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u5173\u6570\u636e\u90fd\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8 1 byte \u7684\u6570\u636e\uff0c\u5728\u7b97\u6cd5\u8fd0\u884c\u65f6\uff0c\u6240\u6709\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u8fd9\u4e9b\u5355\u5143\u683c\u4e2d\u3002

    \u7cfb\u7edf\u901a\u8fc7\u300c\u5185\u5b58\u5730\u5740 Memory Location\u300d\u6765\u8bbf\u95ee\u76ee\u6807\u5185\u5b58\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002

    Fig. \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740

    \u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5176\u4ed6\u7a0b\u5e8f\u65e0\u6cd5\u540c\u65f6\u4f7f\u7528\u3002\u56e0\u6b64\uff0c\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u8fd0\u884c\u7684\u7a0b\u5e8f\u5f88\u591a\u5e76\u4e14\u7f3a\u5c11\u5927\u91cf\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u79bb\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002

    \u300c\u7269\u7406\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u6570\u7ec4\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u94fe\u8868\u7684\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u540c\u65f6\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002

    Fig. \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8

    \u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002

    • \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\uff1b
    • \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\uff1b

    \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u300c\u9759\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002

    Tip

    \u5982\u82e5\u611f\u89c9\u7406\u89e3\u7269\u7406\u7ed3\u6784\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\u201c\u6570\u7ec4\u4e0e\u94fe\u8868\u201d\uff0c\u7136\u540e\u518d\u56de\u5934\u7406\u89e3\u7269\u7406\u7ed3\u6784\u7684\u542b\u4e49\u3002\u6570\u7ec4\u4e0e\u94fe\u8868\u662f\u5176\u4ed6\u6240\u6709\u6570\u636e\u7ed3\u6784\u7684\u57fa\u77f3\uff0c\u5efa\u8bae\u4f60\u6295\u5165\u66f4\u591a\u65f6\u95f4\u6df1\u5165\u4e86\u89e3\u8fd9\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_data_structure/number_encoding/","title":"3.3. \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"

    Note

    \u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709\u7684 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002

    "},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1. \u00a0 \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801","text":"

    \u4ece\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\u3002\u4f8b\u5982\uff0cbyte \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u5230\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002\u5728\u5c55\u5f00\u5206\u6790\u4e4b\u524d\uff0c\u6211\u4eec\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\uff1a

    • \u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002
    • \u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002
    • \u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002

    Fig. \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362

    \u663e\u7136\uff0c\u300c\u539f\u7801\u300d\u6700\u4e3a\u76f4\u89c2\uff0c\u7136\u800c\u6570\u5b57\u5374\u662f\u4ee5\u300c\u8865\u7801\u300d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u539f\u7801\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002

    \u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002

    \\[ \\begin{aligned} & 1 + (-2) \\newline & = 0000 \\space 0001 + 1000 \\space 0010 \\newline & = 1000 \\space 0011 \\newline & = -3 \\end{aligned} \\]

    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801\u300d\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5e76\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u5316\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002

    \\[ \\begin{aligned} & 1 + (-2) \\newline & = 0000 \\space 0001 \\space \\text{(\u539f\u7801)} + 1000 \\space 0010 \\space \\text{(\u539f\u7801)} \\newline & = 0000 \\space 0001 \\space \\text{(\u53cd\u7801)} + 1111 \\space 1101 \\space \\text{(\u53cd\u7801)} \\newline & = 1111 \\space 1110 \\space \\text{(\u53cd\u7801)} \\newline & = 1000 \\space 0001 \\space \\text{(\u539f\u7801)} \\newline & = -1 \\end{aligned} \\]

    \u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u7740\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u800c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u53ef\u80fd\u4f1a\u5bfc\u81f4\u9519\u8bef\u7684\u5224\u65ad\u7ed3\u679c\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u5176\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002

    \\[ \\begin{aligned} +0 & = 0000 \\space 0000 \\newline -0 & = 1000 \\space 0000 \\end{aligned} \\]

    \u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\u3002\u4e3a\u6b64\uff0c\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801\u300d\u3002\u90a3\u4e48\uff0c\u8865\u7801\u6709\u4ec0\u4e48\u4f5c\u7528\u5462\uff1f\u6211\u4eec\u5148\u6765\u5206\u6790\u4e00\u4e0b\u8d1f\u96f6\u7684\u8865\u7801\u7684\u8ba1\u7b97\u8fc7\u7a0b\uff1a

    \\[ \\begin{aligned} -0 = \\space & 1000 \\space 0000 \\space \\text{(\u539f\u7801)} \\newline = \\space & 1111 \\space 1111 \\space \\text{(\u53cd\u7801)} \\newline = 1 \\space & 0000 \\space 0000 \\space \\text{(\u8865\u7801)} \\newline \\end{aligned} \\]

    \u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u800c\u7531\u4e8e byte \u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4ece\u800c\u5f97\u5230\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\space 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u4ece\u800c\u89e3\u51b3\u4e86\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\u3002

    \u8fd8\u5269\u4f59\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1abyte \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u662f\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u7684\u3002

    \u7136\u800c\uff0c\u8865\u7801 \\(1000 \\space 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\space 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\space 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002

    \\[ \\begin{aligned} & (-127) + (-1) \\newline & = 1111 \\space 1111 \\space \\text{(\u539f\u7801)} + 1000 \\space 0001 \\space \\text{(\u539f\u7801)} \\newline & = 1000 \\space 0000 \\space \\text{(\u53cd\u7801)} + 1111 \\space 1110 \\space \\text{(\u53cd\u7801)} \\newline & = 1000 \\space 0001 \\space \\text{(\u8865\u7801)} + 1111 \\space 1111 \\space \\text{(\u8865\u7801)} \\newline & = 1000 \\space 0000 \\space \\text{(\u8865\u7801)} \\newline & = -128 \\end{aligned} \\]

    \u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\uff0c\u4e0a\u8ff0\u7684\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u4ece\u800c\u63d0\u9ad8\u8fd0\u7b97\u901f\u5ea6\u3002

    \u7136\u800c\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002

    \u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u9700\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u5e76\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002

    \u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u7531\u4e8e\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\u3002\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5ea6\u4e86\u89e3\u3002

    "},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2. \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"

    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1aint \u548c float \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes\uff0c\u4f46\u4e3a\u4ec0\u4e48 float \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 float \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 float \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 float \u7531\u4ee5\u4e0b\u90e8\u5206\u6784\u6210\uff1a

    • \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff1b
    • \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff1b
    • \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 24 bits \uff0c\u5176\u4e2d 23 \u4f4d\u663e\u5f0f\u5b58\u50a8\uff1b

    \u8bbe 32-bit \u4e8c\u8fdb\u5236\u6570\u7684\u7b2c \\(i\\) \u4f4d\u4e3a \\(b_i\\) \uff0c\u5219 float \u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u5b9a\u4e49\u4e3a\uff1a

    \\[ \\text { val } = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\]

    \u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a

    \\[ \\text { val }=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\]

    \u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a

    \\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\} , \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\]

    Fig. IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u8868\u793a\u65b9\u5f0f

    \u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u6613\u5f97

    \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\]

    \u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1afloat \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0cfloat \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002

    \u5c3d\u7ba1\u6d6e\u70b9\u6570 float \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b int \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 float \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002

    \u8fdb\u4e00\u6b65\u5730\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002

    \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\)

    \u7279\u522b\u5730\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a

    • \u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\approx 1.18 \\times 10^{-38}\\) \uff1b
    • \u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23} \\approx 1.4 \\times 10^{-45}\\) \uff1b

    \u53cc\u7cbe\u5ea6 double \u4e5f\u91c7\u7528\u7c7b\u4f3c float \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u6b64\u5904\u4e0d\u518d\u8be6\u8ff0\u3002

    "},{"location":"chapter_data_structure/summary/","title":"3.5. \u00a0 \u5c0f\u7ed3","text":"

    \u6570\u636e\u7ed3\u6784\u5206\u7c7b

    • \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002
    • \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u548c\u975e\u7ebf\u6027\u7ed3\u6784\u3002
    • \u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002
    • \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002

    \u6570\u636e\u7c7b\u578b\u4e0e\u7f16\u7801

    • \u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 byte, short, int, long \u3001\u6d6e\u70b9\u6570 float, double \u3001\u5b57\u7b26 char \u548c\u5e03\u5c14 boolean \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002
    • \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u662f\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u7684\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002
    • \u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002
    • \u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002
    • ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002
    • UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java, C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
    "},{"location":"chapter_data_structure/summary/#351-q-a","title":"3.5.1. \u00a0 Q & A","text":"

    \u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f

    \u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u62c9\u94fe\u6cd5\u201d\uff08\u540e\u7eed\u6563\u5217\u8868\u7ae0\u8282\u4f1a\u8bb2\uff09\u3002\u5728\u62c9\u94fe\u6cd5\u4e2d\uff0c\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5730\u5740\uff08\u6876\uff09\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff1b\u5f53\u8fd9\u4e2a\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\uff09\u6570\u636e\u7ed3\u6784\u3002

    char \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 bytes \u5417\uff1f

    \u8fd9\u4e2a\u4e0e\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u6709\u5173\u3002\u4f8b\u5982\uff0cJava, JS, TS, C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes \u3002

    "},{"location":"chapter_dynamic_programming/","title":"13. \u00a0 \u52a8\u6001\u89c4\u5212","text":""},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"13.2. \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"

    \u5728\u4e0a\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u66b4\u529b\u89e3\u6cd5\uff0c\u4ece\u9012\u5f52\u6811\u4e2d\u89c2\u5bdf\u5230\u6d77\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u4e86\u89e3\u5230\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u8bb0\u5f55\u89e3\u6765\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u3002

    \u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u6700\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u65b9\u6848\u95ee\u9898\uff0c\u4f8b\u5982\u5bfb\u627e\u6700\u77ed\u8def\u5f84\u3001\u6700\u5927\u5229\u6da6\u3001\u6700\u5c11\u65f6\u95f4\u7b49\u3002\u8fd9\u7c7b\u95ee\u9898\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002

    "},{"location":"chapter_dynamic_programming/dp_problem_features/#1321","title":"13.2.1. \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"

    \u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002

    \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7

    \u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\u8d77\u59cb\u70b9\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u82e5\u7b2c \\(1\\) , \\(2\\) , \\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\) , \\(10\\) , \\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002

    Fig. \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7

    \u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff0c\u5373\uff1a

    \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\]

    \u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002\u5bf9\u4e8e\u672c\u9898\uff0c\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) , \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002

    \u76f8\u8f83\u4e8e\u5206\u6cbb\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u89e3\u4e5f\u662f\u7531\u5176\u5b50\u95ee\u9898\u7684\u89e3\u6784\u6210\u7684\u3002\u4e0d\u540c\u7684\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u5b50\u95ee\u9898\u7684\u89e3\u4e0d\u4ec5\u63ed\u793a\u4e86\u95ee\u9898\u7684\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u800c\u4e14\u8fd8\u901a\u8fc7\u7279\u5b9a\u7684\u9012\u63a8\u5173\u7cfb\u94fe\u63a5\u8d77\u6765\uff0c\u5171\u540c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u5168\u5c40\u6700\u4f18\u89e3\u3002

    \u90a3\u4e48\uff0c\u4e0a\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u8981\u6c42\u89e3\u7684\u662f\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u662f\u4e00\u4e2a\u6bd4\u8f83\u5bbd\u6cdb\u7684\u6982\u5ff5\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002

    \u6839\u636e\u4ee5\u4e0a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) , \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u52a8\u6001\u89c4\u5212\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart min_cost_climbing_stairs_dp.java
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector<int> &cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<int> dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.py
    def min_cost_climbing_stairs_dp(cost: list[int]) -> int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = cost[1], cost[2]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\nreturn dp[n]\n
    min_cost_climbing_stairs_dp.go
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.js
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.ts
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.c
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.swift
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.zig
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.dart
    [class]{}-[func]{minCostClimbingStairsDP}\n

    Fig. \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b

    \u8fd9\u9053\u9898\u540c\u6837\u4e5f\u53ef\u4ee5\u8fdb\u884c\u72b6\u6001\u538b\u7f29\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart min_cost_climbing_stairs_dp.java
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = Math.min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector<int> &cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.py
    def min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\na, b = cost[1], cost[2]\nfor i in range(3, n + 1):\na, b = b, min(a, b) + cost[i]\nreturn b\n
    min_cost_climbing_stairs_dp.go
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.js
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.ts
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.c
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = Math.Min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.swift
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.zig
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.dart
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    "},{"location":"chapter_dynamic_programming/dp_problem_features/#1322","title":"13.2.2. \u00a0 \u65e0\u540e\u6548\u6027","text":"

    \u300c\u65e0\u540e\u6548\u6027\u300d\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u5f53\u524d\u72b6\u6001\u8fc7\u53bb\u6240\u7ecf\u5386\u8fc7\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002

    \u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u9700\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5373\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002

    \u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5411\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002

    \u5e26\u7ea6\u675f\u722c\u697c\u68af

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u4f8b\u5982\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002

    Fig. \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf

    \u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002

    \u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002

    \u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u3001\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u6765\u51b3\u5b9a\u4e0b\u4e00\u6b65\u8be5\u600e\u4e48\u8df3\uff1a

    • \u5f53 \\(j\\) \u7b49\u4e8e \\(1\\) \uff0c\u5373\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u8fd9\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff1b
    • \u5f53 \\(j\\) \u7b49\u4e8e \\(2\\) \uff0c\u5373\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u8fd9\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff1b

    \u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u7531\u6b64\uff0c\u6211\u4eec\u4fbf\u80fd\u63a8\u5bfc\u51fa\u4ee5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a

    \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\]

    Fig. \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb

    \u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_constraint_dp.java
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[][] dp = new int[n + 1][3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n
    climbing_stairs_constraint_dp.cpp
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<vector<int>> dp(n + 1, vector<int>(3, 0));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n
    climbing_stairs_constraint_dp.py
    def climbing_stairs_constraint_dp(n: int) -> int:\n\"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [[0] * 3 for _ in range(n + 1)]\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1], dp[1][2] = 1, 0\ndp[2][1], dp[2][2] = 0, 1\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i][1] = dp[i - 1][2]\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2]\nreturn dp[n][1] + dp[n][2]\n
    climbing_stairs_constraint_dp.go
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.js
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.ts
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.c
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.cs
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[,] dp = new int[n + 1, 3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1, 1] = 1;\ndp[1, 2] = 0;\ndp[2, 1] = 0;\ndp[2, 2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i, 1] = dp[i - 1, 2];\ndp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n}\nreturn dp[n, 1] + dp[n, 2];\n}\n
    climbing_stairs_constraint_dp.swift
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.zig
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.dart
    [class]{}-[func]{climbingStairsConstraintDP}\n

    \u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u6062\u590d\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u8bb8\u591a\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\uff0c\u4f8b\u5982\uff1a

    \u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u7ed9\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2, 3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4, 6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u4e8e\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\uff0c\u6216\u662f\u56e0\u4e3a\u8ba1\u7b97\u590d\u6742\u5ea6\u8fc7\u9ad8\u800c\u96be\u4ee5\u5e94\u7528\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u8457\u540d\u7684\u65c5\u884c\u5546\u95ee\u9898\uff09\u90fd\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u80fd\u591f\u63a5\u53d7\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"13.1. \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"

    \u300c\u52a8\u6001\u89c4\u5212 Dynamic Programming\u300d\u662f\u4e00\u79cd\u7528\u4e8e\u89e3\u51b3\u590d\u6742\u95ee\u9898\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u5b83\u628a\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u628a\u5b50\u95ee\u9898\u7684\u89e3\u5b58\u50a8\u8d77\u6765\u4ee5\u4f9b\u540e\u7eed\u4f7f\u7528\uff0c\u4ece\u800c\u907f\u514d\u4e86\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u5347\u4e86\u89e3\u9898\u6548\u7387\u3002

    \u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u4ece\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u7684\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u4e00\u6b65\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002

    \u722c\u697c\u68af

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    Fig. \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf

    \u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_backtrack.java
    /* \u56de\u6eaf */\nvoid backtrack(List<Integer> choices, int state, int n, List<Integer> res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres.set(0, res.get(0) + 1);\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Integer choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList<Integer> choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList<Integer> res = new ArrayList<>();\nres.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res.get(0);\n}\n
    climbing_stairs_backtrack.cpp
    /* \u56de\u6eaf */\nvoid backtrack(vector<int> &choices, int state, int n, vector<int> &res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (auto &choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nvector<int> choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nvector<int> res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n
    climbing_stairs_backtrack.py
    def backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int:\n\"\"\"\u56de\u6eaf\"\"\"\n# \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n:\nres[0] += 1\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state + choice > n:\nbreak\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res)\n# \u56de\u9000\ndef climbing_stairs_backtrack(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\nchoices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nstate = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nres = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res)\nreturn res[0]\n
    climbing_stairs_backtrack.go
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.cs
    /* \u56de\u6eaf */\nvoid backtrack(List<int> choices, int state, int n, List<int> res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (int choice in choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList<int> choices = new List<int> { 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList<int> res = new List<int> { 0 }; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n
    climbing_stairs_backtrack.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1311","title":"13.1.1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"

    \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002

    \u5bf9\u4e8e\u672c\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u5c06\u95ee\u9898\u62c6\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec:

    \\[ dp[i-1] , dp[i-2] , \\cdots , dp[2] , dp[1] \\]

    \u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\uff0c\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u524d\u5f80\u7b2c \\(i\\) \u9636\u3002\u56e0\u6b64\uff0c\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\uff0c\u5373\uff1a

    \\[ dp[i] = dp[i-1] + dp[i-2] \\]

    Fig. \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb

    \u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u6210\u3002

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u6b64\u9012\u63a8\u516c\u5f0f\u5199\u51fa\u66b4\u529b\u641c\u7d22\u4ee3\u7801\uff1a\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373\u722c\u5230\u7b2c \\(1\\) , \\(2\\) \u9636\u5206\u522b\u6709 \\(1\\) , \\(2\\) \u79cd\u65b9\u6848\u3002

    \u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u4e0e\u56de\u6eaf\u89e3\u6cd5\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u6bd4\u56de\u6eaf\u7b97\u6cd5\u66f4\u52a0\u7b80\u6d01\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dfs.java
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.cpp
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.py
    def dfs(i: int) -> int:\n\"\"\"\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1) + dfs(i - 2)\nreturn count\ndef climbing_stairs_dfs(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\nreturn dfs(n)\n
    climbing_stairs_dfs.go
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.js
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.ts
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.cs
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.swift
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.dart
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n

    \u4e0b\u56fe\u5c55\u793a\u4e86\u8be5\u65b9\u6cd5\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u7684\u8fd0\u884c\u65f6\u95f4\u589e\u957f\u5730\u975e\u5e38\u5feb\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002

    Fig. \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811

    \u5b9e\u9645\u4e0a\uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7531\u4e8e\u300c\u91cd\u53e0\u5b50\u95ee\u9898\u300d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982\uff0c\u95ee\u9898 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\u95ee\u9898 \\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \uff0c\u800c\u5b50\u95ee\u9898\u4e2d\u53c8\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\uff0c\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1312","title":"13.1.2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"

    \u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4 mem \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u8fd9\u6837\u505a\uff1a

    • \u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 mem[i] \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\uff1b
    • \u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece mem[i] \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\uff1b
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dfs_mem.java
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArrays.fill(mem, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.cpp
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector<int> &mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nvector<int> mem(n + 1, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.py
    def dfs(i: int, mem: list[int]) -> int:\n\"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1:\nreturn mem[i]\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1, mem) + dfs(i - 2, mem)\n# \u8bb0\u5f55 dp[i]\nmem[i] = count\nreturn count\ndef climbing_stairs_dfs_mem(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nmem = [-1] * (n + 1)\nreturn dfs(n, mem)\n
    climbing_stairs_dfs_mem.go
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.js
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.ts
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.cs
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArray.Fill(mem, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.swift
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.dart
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n

    \u89c2\u5bdf\u4e0b\u56fe\uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002\u5b9e\u9645\u4e0a\uff0c\u5982\u679c\u4e0d\u8003\u8651\u9012\u5f52\u5e26\u6765\u7684\u989d\u5916\u5f00\u9500\uff0c\u8bb0\u5fc6\u5316\u641c\u7d22\u89e3\u6cd5\u5df2\u7ecf\u51e0\u4e4e\u7b49\u540c\u4e8e\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u3002

    Fig. \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1313","title":"13.1.3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"

    \u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\uff1b\u6700\u7ec8\u901a\u8fc7\u56de\u6eaf\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u9010\u5c42\u6536\u96c6\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002

    \u6211\u4eec\u4e5f\u53ef\u4ee5\u76f4\u63a5\u201c\u4ece\u5e95\u81f3\u9876\u201d\u8fdb\u884c\u6c42\u89e3\uff0c\u5f97\u5230\u6807\u51c6\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002

    \u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u65e0\u9700\u4f7f\u7528\u9012\u5f52\uff0c\u800c\u53ef\u4ee5\u76f4\u63a5\u57fa\u4e8e\u9012\u63a8\u5b9e\u73b0\u3002\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 dp \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6570\u7ec4 dp \u8d77\u5230\u4e86\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 mem \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dp.java
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<int> dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.py
    def climbing_stairs_dp(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = 1, 2\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = dp[i - 1] + dp[i - 2]\nreturn dp[n]\n
    climbing_stairs_dp.go
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.js
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.ts
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.c
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.swift
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.zig
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.dart
    [class]{}-[func]{climbingStairsDP}\n

    \u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u67d0\u4e2a\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\u5bf9\u4e8e\u722c\u697c\u68af\u95ee\u9898\uff0c\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\u3002\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u5305\u62ec\uff1a

    • \u5c06 \\(dp\\) \u6570\u7ec4\u79f0\u4e3a\u300c\u72b6\u6001\u5217\u8868\u300d\uff0c\\(dp[i]\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u72b6\u6001\u7684\u89e3\uff1b
    • \u5c06\u6700\u7b80\u5355\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u5373\u7b2c \\(1\\) , \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\uff1b
    • \u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\uff1b

    Fig. \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b

    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u9700\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 dp \u6765\u5b58\u50a8\u6240\u6709\u72b6\u6001\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 dp \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dp.java
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.py
    def climbing_stairs_dp_comp(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\na, b = 1, 2\nfor _ in range(3, n + 1):\na, b = b, a + b\nreturn b\n
    climbing_stairs_dp.go
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.js
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.ts
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.c
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.swift
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.zig
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.dart
    [class]{}-[func]{climbingStairsDPComp}\n

    \u6211\u4eec\u5c06\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u79f0\u4e3a\u300c\u72b6\u6001\u538b\u7f29\u300d\u3002\u5728\u8bb8\u591a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u4e0d\u5fc5\u4fdd\u5b58\u6240\u6709\u7684\u5386\u53f2\u72b6\u6001\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u72b6\u6001\u538b\u7f29\uff0c\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002

    \u603b\u7684\u770b\u6765\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u7b97\u6cd5\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5404\u6709\u7279\u70b9\uff1a

    • \u5206\u6cbb\u7b97\u6cd5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u51e0\u4e2a\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u7136\u540e\u9012\u5f52\u89e3\u51b3\u5b50\u95ee\u9898\uff0c\u6700\u540e\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002\u4f8b\u5982\uff0c\u5f52\u5e76\u6392\u5e8f\u5c06\u957f\u6570\u7ec4\u4e0d\u65ad\u5212\u5206\u4e3a\u4e24\u4e2a\u77ed\u5b50\u6570\u7ec4\uff0c\u518d\u5c06\u6392\u5e8f\u597d\u7684\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u6392\u5e8f\u597d\u7684\u957f\u6570\u7ec4\u3002
    • \u52a8\u6001\u89c4\u5212\u4e5f\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u5f80\u5f80\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u539f\u95ee\u9898\u7684\u89e3\u4f9d\u8d56\u4e8e\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u5b50\u95ee\u9898\u7684\u89e3\u53c8\u4f9d\u8d56\u4e8e\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\u7684\u89e3\u3002\u56e0\u6b64\uff0c\u52a8\u6001\u89c4\u5212\u901a\u5e38\u4f1a\u5f15\u5165\u8bb0\u5fc6\u5316\uff0c\u4fdd\u5b58\u5df2\u7ecf\u89e3\u51b3\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u540e\u7684\u5269\u4f59\u95ee\u9898\u770b\u4f5c\u4e3a\u4e00\u4e2a\u5b50\u95ee\u9898\u3002
    "},{"location":"chapter_graph/","title":"9. \u00a0 \u56fe","text":""},{"location":"chapter_graph/graph/","title":"9.1. \u00a0 \u56fe","text":"

    \u300c\u56fe Graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 Vertex\u300d\u548c\u300c\u8fb9 Edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002

    \\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\]

    Fig. \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb

    \u90a3\u4e48\uff0c\u56fe\u4e0e\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb\u662f\u4ec0\u4e48\uff1f\u5982\u679c\u6211\u4eec\u628a\u300c\u9876\u70b9\u300d\u770b\u4f5c\u8282\u70b9\uff0c\u628a\u300c\u8fb9\u300d\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u6307\u9488\uff0c\u5219\u53ef\u5c06\u300c\u56fe\u300d\u770b\u4f5c\u662f\u4e00\u79cd\u4ece\u300c\u94fe\u8868\u300d\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002

    "},{"location":"chapter_graph/graph/#911","title":"9.1.1. \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b","text":"

    \u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe Undirected Graph\u300d\u548c\u300c\u6709\u5411\u56fe Directed Graph\u300d\u3002

    • \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\uff1b
    • \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\uff1b

    Fig. \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe

    \u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe Connected Graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe Disconnected Graph\u300d\u3002

    • \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\uff1b
    • \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\uff1b

    Fig. \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe

    \u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u300c\u6709\u6743\u56fe Weighted Graph\u300d\u3002\u4f8b\u5982\uff0c\u5728\u738b\u8005\u8363\u8000\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002

    Fig. \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe

    "},{"location":"chapter_graph/graph/#912","title":"9.1.2. \u00a0 \u56fe\u5e38\u7528\u672f\u8bed","text":"
    • \u300c\u90bb\u63a5 Adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u4e0a\u56fe\u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002
    • \u300c\u8def\u5f84 Path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u4e0a\u56fe\u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002
    • \u300c\u5ea6 Degree\u300d\u8868\u793a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 In-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 Out-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
    "},{"location":"chapter_graph/graph/#913","title":"9.1.3. \u00a0 \u56fe\u7684\u8868\u793a","text":"

    \u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u6cd5\u5305\u62ec\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002

    "},{"location":"chapter_graph/graph/#_1","title":"\u90bb\u63a5\u77e9\u9635","text":"

    \u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 Adjacency Matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\) \u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i][j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i][j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002

    Fig. \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a

    \u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
    • \u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
    • \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) , \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002

    \u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002

    "},{"location":"chapter_graph/graph/#_2","title":"\u90bb\u63a5\u8868","text":"

    \u300c\u90bb\u63a5\u8868 Adjacency List\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u5373\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002

    Fig. \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a

    \u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002

    \u89c2\u5bdf\u4e0a\u56fe\u53ef\u53d1\u73b0\uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u300c\u94fe\u5730\u5740\u6cd5\u300d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u4e2d\u5e8f\u904d\u5386\u83b7\u53d6\u6709\u5e8f\u5e8f\u5217\uff1b\u6b64\u5916\uff0c\u8fd8\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    "},{"location":"chapter_graph/graph/#914","title":"9.1.4. \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"

    \u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u5f85\u6c42\u89e3\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002

    \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2. \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"

    \u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u300c\u8fb9\u300d\u7684\u64cd\u4f5c\u548c\u5bf9\u300c\u9876\u70b9\u300d\u7684\u64cd\u4f5c\u3002\u5728\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002

    "},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"

    \u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u6709\uff1a

    • \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
    • \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 vertices \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 adjMat \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_adjacency_matrix.java
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<Integer> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new ArrayList<>();\nthis.adjMat = new ArrayList<>();\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (int[] e : edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<Integer> newRow = new ArrayList<>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.add(0);\n}\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<Integer> row : adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfBoundsException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<Integer> row : adjMat) {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat.get(i).set(j, 1);\nadjMat.get(j).set(i, 1);\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\nadjMat.get(i).set(j, 0);\nadjMat.get(j).set(i, 0);\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nSystem.out.print(\"\u9876\u70b9\u5217\u8868 = \");\nSystem.out.println(vertices);\nSystem.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
    graph_adjacency_matrix.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvector<int> vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(const vector<int> &vertices, const vector<vector<int>> &edges) {\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const vector<int> &edge : edges) {\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() const {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.push_back(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nadjMat.emplace_back(vector<int>(n, 0));\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (vector<int> &row : adjMat) {\nrow.push_back(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.erase(vertices.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.erase(adjMat.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (vector<int> &row : adjMat) {\nrow.erase(row.begin() + index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid print() {\ncout << \"\u9876\u70b9\u5217\u8868 = \";\nprintVector(vertices);\ncout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\nprintVectorMatrix(adjMat);\n}\n};\n
    graph_adjacency_matrix.py
    class GraphAdjMat:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n# \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices: list[int] = []\n# \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadj_mat: list[list[int]] = []\ndef __init__(self, vertices: list[int], edges: list[list[int]]) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.vertices: list[int] = []\nself.adj_mat: list[list[int]] = []\n# \u6dfb\u52a0\u9876\u70b9\nfor val in vertices:\nself.add_vertex(val)\n# \u6dfb\u52a0\u8fb9\n# \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges:\nself.add_edge(e[0], e[1])\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.vertices)\ndef add_vertex(self, val: int) -> None:\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nn = self.size()\n# \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.append(val)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnew_row = [0] * n\nself.adj_mat.append(new_row)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in self.adj_mat:\nrow.append(0)\ndef remove_vertex(self, index: int) -> None:\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif index >= self.size():\nraise IndexError()\n# \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in self.adj_mat:\nrow.pop(index)\ndef add_edge(self, i: int, j: int) -> None:\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\n# \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1\nself.adj_mat[j][i] = 1\ndef remove_edge(self, i: int, j: int) -> None:\n\"\"\"\u5220\u9664\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\nself.adj_mat[i][j] = 0\nself.adj_mat[j][i] = 0\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\nprint(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\nprint(\"\u90bb\u63a5\u77e9\u9635 =\")\nprint_matrix(self.adj_mat)\n
    graph_adjacency_matrix.go
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices []int\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat [][]int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n// \u6dfb\u52a0\u9876\u70b9\nn := len(vertices)\nadjMat := make([][]int, n)\nfor i := range adjMat {\nadjMat[i] = make([]int, n)\n}\n// \u521d\u59cb\u5316\u56fe\ng := &graphAdjMat{\nvertices: vertices,\nadjMat:   adjMat,\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor i := range edges {\ng.addEdge(edges[i][0], edges[i][1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\nreturn len(g.vertices)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\nn := g.size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\ng.vertices = append(g.vertices, val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnewRow := make([]int, n)\ng.adjMat = append(g.adjMat, newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i], 0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\nif index >= g.size() {\nreturn\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\ng.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\ng.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\ng.adjMat[i][j] = 1\ng.adjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\ng.adjMat[i][j] = 0\ng.adjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\nfmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\nfmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\nfor i := range g.adjMat {\nfmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n}\n}\n
    graph_adjacency_matrix.js
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices, edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val) {\nconst n = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow = [];\nfor (let j = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index) {\nif (index >= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint() {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n
    graph_adjacency_matrix.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices: number[], edges: number[][]) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val: number): void {\nconst n: number = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow: number[] = [];\nfor (let j: number = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index: number): void {\nif (index >= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint(): void {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n
    graph_adjacency_matrix.c
    [class]{graphAdjMat}-[func]{}\n
    graph_adjacency_matrix.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<int> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new List<int>();\nthis.adjMat = new List<List<int>>();\n// \u6dfb\u52a0\u9876\u70b9\nforeach (int val in vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nforeach (int[] e in edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.Count;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.Add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = new List<int>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.Add(0);\n}\nadjMat.Add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nforeach (List<int> row in adjMat) {\nrow.Add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfRangeException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nforeach (List<int> row in adjMat) {\nrow.RemoveAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nConsole.Write(\"\u9876\u70b9\u5217\u8868 = \");\nPrintUtil.PrintList(vertices);\nConsole.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.PrintMatrix(adjMat);\n}\n}\n
    graph_adjacency_matrix.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nprivate var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nprivate var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(vertices: [Int], edges: [[Int]]) {\nself.vertices = []\nadjMat = []\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\naddVertex(val: val)\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges {\naddEdge(i: e[0], j: e[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc size() -> Int {\nvertices.count\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc addVertex(val: Int) {\nlet n = size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.append(val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nlet newRow = Array(repeating: 0, count: n)\nadjMat.append(newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i in adjMat.indices {\nadjMat[i].append(0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc removeVertex(index: Int) {\nif index >= size() {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i in adjMat.indices {\nadjMat[i].remove(at: index)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc addEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1\nadjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc removeEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\nadjMat[i][j] = 0\nadjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc print() {\nSwift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\nSwift.print(vertices)\nSwift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\nPrintUtil.printMatrix(matrix: adjMat)\n}\n}\n
    graph_adjacency_matrix.zig
    \n
    graph_adjacency_matrix.dart
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<int> vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(List<int> vertices, List<List<int>> edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val in vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (List<int> e in edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = List.filled(n, 0, growable: true);\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<int> row in adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow IndexError;\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<int> row in adjMat) {\nrow.removeAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow IndexError;\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow IndexError;\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printAdjMat() {\nprint(\"\u9876\u70b9\u5217\u8868 = $vertices\");\nprint(\"\u90bb\u63a5\u77e9\u9635 = \");\nprintMatrix(adjMat);\n}\n}\n
    "},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"

    \u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u6709\uff1a

    • \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
    • \u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
    • \u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
    \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u56fe\u7684\u4ee3\u7801\u793a\u4f8b\u3002\u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u6ce8\u610f\u5230\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 Vertex \u8282\u70b9\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u6709\uff1a

    • \u5982\u679c\u6211\u4eec\u9009\u62e9\u901a\u8fc7\u9876\u70b9\u503c\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u503c\u91cd\u590d\u7684\u9876\u70b9\u5c06\u65e0\u6cd5\u88ab\u533a\u5206\u3002
    • \u5982\u679c\u7c7b\u4f3c\u90bb\u63a5\u77e9\u9635\u90a3\u6837\uff0c\u4f7f\u7528\u9876\u70b9\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\u3002\u90a3\u4e48\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u5176\u4e2d \\(> i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u8fd9\u6837\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002
    • \u56e0\u6b64\u6211\u4eec\u8003\u8651\u5f15\u5165\u9876\u70b9\u7c7b Vertex \uff0c\u4f7f\u5f97\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684\u5bf9\u8c61\uff0c\u6b64\u65f6\u5220\u9664\u9876\u70b9\u65f6\u5c31\u65e0\u9700\u6539\u52a8\u5176\u4f59\u9876\u70b9\u4e86\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_adjacency_list.java
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new HashMap<>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (Vertex[] edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList.get(vet1).add(vet2);\nadjList.get(vet2).add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList.get(vet1).remove(vet2);\nadjList.get(vet2).remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.containsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.put(vet, new ArrayList<>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet))\nthrow new IllegalArgumentException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (List<Vertex> list : adjList.values()) {\nlist.remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nSystem.out.println(\"\u90bb\u63a5\u8868 =\");\nfor (Map.Entry<Vertex, List<Vertex>> pair : adjList.entrySet()) {\nList<Integer> tmp = new ArrayList<>();\nfor (Vertex vertex : pair.getValue())\ntmp.add(vertex.val);\nSystem.out.println(pair.getKey().val + \": \" + tmp + \",\");\n}\n}\n}\n
    graph_adjacency_list.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\npublic:\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nunordered_map<Vertex *, vector<Vertex *>> adjList;\n/* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\nvoid remove(vector<Vertex *> &vec, Vertex *vet) {\nfor (int i = 0; i < vec.size(); i++) {\nif (vec[i] == vet) {\nvec.erase(vec.begin() + i);\nbreak;\n}\n}\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(const vector<vector<Vertex *>> &edges) {\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const vector<Vertex *> &edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].push_back(vet2);\nadjList[vet2].push_back(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5220\u9664\u8fb9 vet1 - vet2\nremove(adjList[vet1], vet2);\nremove(adjList[vet2], vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex *vet) {\nif (adjList.count(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = vector<Vertex *>();\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex *vet) {\nif (!adjList.count(vet))\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.erase(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (auto &adj : adjList) {\nremove(adj.second, vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid print() {\ncout << \"\u90bb\u63a5\u8868 =\" << endl;\nfor (auto &adj : adjList) {\nconst auto &key = adj.first;\nconst auto &vec = adj.second;\ncout << key->val << \": \";\nprintVector(vetsToVals(vec));\n}\n}\n};\n
    graph_adjacency_list.py
    class GraphAdjList:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\ndef __init__(self, edges: list[list[Vertex]]) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nself.adj_list = dict[Vertex, Vertex]()\n# \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges:\nself.add_vertex(edge[0])\nself.add_vertex(edge[1])\nself.add_edge(edge[0], edge[1])\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.adj_list)\ndef add_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list[vet1].append(vet2)\nself.adj_list[vet2].append(vet1)\ndef remove_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\"\u5220\u9664\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list[vet1].remove(vet2)\nself.adj_list[vet2].remove(vet1)\ndef add_vertex(self, vet: Vertex) -> None:\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nif vet in self.adj_list:\nreturn\n# \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list[vet] = []\ndef remove_vertex(self, vet: Vertex) -> None:\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif vet not in self.adj_list:\nraise ValueError()\n# \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.pop(vet)\n# \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor vertex in self.adj_list:\nif vet in self.adj_list[vertex]:\nself.adj_list[vertex].remove(vet)\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\nprint(\"\u90bb\u63a5\u8868 =\")\nfor vertex in self.adj_list:\ntmp = [v.val for v in self.adj_list[vertex]]\nprint(f\"{vertex.val}: {tmp},\")\n
    graph_adjacency_list.go
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList map[Vertex][]Vertex\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\ng := &graphAdjList{\nadjList: make(map[Vertex][]Vertex),\n}\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor _, edge := range edges {\ng.addVertex(edge[0])\ng.addVertex(edge[1])\ng.addEdge(edge[0], edge[1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\nreturn len(g.adjList)\n}\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\ng.adjList[vet1] = append(g.adjList[vet1], vet2)\ng.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nDeleteSliceElms(g.adjList[vet1], vet2)\nDeleteSliceElms(g.adjList[vet2], vet1)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif ok {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ng.adjList[vet] = make([]Vertex, 0)\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif !ok {\npanic(\"error\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\ndelete(g.adjList, vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor _, list := range g.adjList {\nDeleteSliceElms(list, vet)\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\nvar builder strings.Builder\nfmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\nfor k, v := range g.adjList {\nbuilder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\nfor _, vet := range v {\nbuilder.WriteString(strconv.Itoa(vet.Val) + \" \")\n}\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
    graph_adjacency_list.js
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet) {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet) {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint() {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
    graph_adjacency_list.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList: Map<Vertex, Vertex[]>;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges: Vertex[][]) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet: Vertex): void {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet: Vertex): void {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index: number = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint(): void {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList.entries()) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
    graph_adjacency_list.c
    [class]{graphAdjList}-[func]{}\n
    graph_adjacency_list.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic Dictionary<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new Dictionary<Vertex, List<Vertex>>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nforeach (Vertex[] edge in edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.Count;\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].Add(vet2);\nadjList[vet2].Add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1].Remove(vet2);\nadjList[vet2].Remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.ContainsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.Add(vet, new List<Vertex>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.ContainsKey(vet))\nthrow new InvalidOperationException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.Remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nforeach (List<Vertex> list in adjList.Values) {\nlist.Remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nConsole.WriteLine(\"\u90bb\u63a5\u8868 =\");\nforeach (KeyValuePair<Vertex, List<Vertex>> pair in adjList) {\nList<int> tmp = new List<int>();\nforeach (Vertex vertex in pair.Value)\ntmp.Add(vertex.val);\nConsole.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n}\n}\n}\n
    graph_adjacency_list.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic private(set) var adjList: [Vertex: [Vertex]]\n/* \u6784\u9020\u65b9\u6cd5 */\npublic init(edges: [[Vertex]]) {\nadjList = [:]\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\naddVertex(vet: edge[0])\naddVertex(vet: edge[1])\naddEdge(vet1: edge[0], vet2: edge[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic func size() -> Int {\nadjList.count\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic func addEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]?.append(vet2)\nadjList[vet2]?.append(vet1)\n}\n/* \u5220\u9664\u8fb9 */\npublic func removeEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]?.removeAll(where: { $0 == vet2 })\nadjList[vet2]?.removeAll(where: { $0 == vet1 })\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic func addVertex(vet: Vertex) {\nif adjList[vet] != nil {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = []\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic func removeVertex(vet: Vertex) {\nif adjList[vet] == nil {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.removeValue(forKey: vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor key in adjList.keys {\nadjList[key]?.removeAll(where: { $0 == vet })\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic func print() {\nSwift.print(\"\u90bb\u63a5\u8868 =\")\nfor pair in adjList {\nvar tmp: [Int] = []\nfor vertex in pair.value {\ntmp.append(vertex.val)\n}\nSwift.print(\"\\(pair.key.val): \\(tmp),\")\n}\n}\n}\n
    graph_adjacency_list.zig
    [class]{GraphAdjList}-[func]{}\n
    graph_adjacency_list.dart
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList = {};\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(List<List<Vertex>> edges) {\nfor (List<Vertex> edge in edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.length;\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]!.add(vet2);\nadjList[vet2]!.add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]!.remove(vet2);\nadjList[vet2]!.remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex vet) {\nif (adjList.containsKey(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = [];\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet)) {\nthrow ArgumentError;\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nadjList.forEach((key, value) {\nvalue.remove(vet);\n});\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid printAdjList() {\nprint(\"\u90bb\u63a5\u8868 =\");\nadjList.forEach((key, value) {\nList<int> tmp = [];\nfor (Vertex vertex in value) {\ntmp.add(vertex.val);\n}\nprint(\"${key.val}: $tmp,\");\n});\n}\n}\n
    "},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3. \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"

    \u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u4e0b\u8868\u4e3a\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\u3002

    \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\)

    \u89c2\u5bdf\u4e0a\u8868\uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002

    "},{"location":"chapter_graph/graph_traversal/","title":"9.3. \u00a0 \u56fe\u7684\u904d\u5386","text":"

    \u56fe\u4e0e\u6811\u7684\u5173\u7cfb

    \u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u5efa\u8bae\u4f60\u5728\u5b66\u4e60\u672c\u7ae0\u8282\u65f6\u878d\u4f1a\u8d2f\u901a\u4e24\u8005\u7684\u6982\u5ff5\u4e0e\u5b9e\u73b0\u65b9\u6cd5\u3002

    \u300c\u56fe\u300d\u548c\u300c\u6811\u300d\u90fd\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u641c\u7d22\u7b97\u6cd5\u300d\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002

    \u4e0e\u6811\u7c7b\u4f3c\uff0c\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u53ef\u5206\u4e3a\u4e24\u79cd\uff0c\u5373\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 Breadth-First Traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u4e5f\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 Depth-First Search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS\u3002

    "},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1. \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\u5f00\u59cb\u8bbf\u95ee\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002

    Fig. \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386

    "},{"location":"chapter_graph/graph_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    BFS \u901a\u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002

    1. \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 startVet \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\uff1b
    2. \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\uff1b
    3. \u5faa\u73af\u6b65\u9aa4 2. \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6210\u540e\u7ed3\u675f\uff1b

    \u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_bfs.java
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>() {{ add(startVet); }};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new LinkedList<>() {{ offer(startVet); }};\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.isEmpty()) {\nVertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.cpp
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex *> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex *> visited = {startVet};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nqueue<Vertex *> que;\nque.push(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.empty()) {\nVertex *vet = que.front();\nque.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (auto adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.py
    def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n\"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]([start_vet])\n# \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nque = deque[Vertex]([start_vet])\n# \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile len(que) > 0:\nvet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adj_vet in graph.adj_list[vet]:\nif adj_vet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n
    graph_bfs.go
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\nvisited[startVet] = struct{}{}\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\nqueue := make([]Vertex, 0)\nqueue = append(queue, startVet)\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nfor len(queue) > 0 {\n// \u961f\u9996\u9876\u70b9\u51fa\u961f\nvet := queue[0]\nqueue = queue[1:]\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres = append(res, vet)\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nif !isExist {\nqueue = append(queue, adjVet)\nvisited[adjVet] = struct{}{}\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_bfs.js
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.ts
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.c
    [class]{}-[func]{graphBFS}\n
    graph_bfs.cs
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new List<Vertex>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet<Vertex> visited = new HashSet<Vertex>() { startVet };\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new Queue<Vertex>();\nque.Enqueue(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.Count > 0) {\nVertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.swift
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = [startVet]\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nvar que: [Vertex] = [startVet]\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.isEmpty {\nlet vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_bfs.zig
    [class]{}-[func]{graphBFS}\n
    graph_bfs.dart
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = {};\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = Queue();\nque.add(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.isNotEmpty) {\nVertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n

    \u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u4ee5\u4e0b\u52a8\u753b\u56fe\u793a\u6765\u52a0\u6df1\u7406\u89e3\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f

    \u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u662f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u7684\u3002\u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\) , \\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u3001\u9876\u70b9 \\(2\\) , \\(4\\) , \\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002

    "},{"location":"chapter_graph/graph_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \uff0c\u961f\u5217 que \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2. \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002

    Fig. \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386

    "},{"location":"chapter_graph/graph_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    \u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934 + \u56de\u6eaf\u201d\u7684\u7b97\u6cd5\u5f62\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e BFS \u7c7b\u4f3c\uff0c\u5728 DFS \u4e2d\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_dfs.java
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.cpp
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &graph, unordered_set<Vertex *> &visited, vector<Vertex *> &res, Vertex *vet) {\nres.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex *adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphDFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex *> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex *> visited;\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.py
    def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adj_list[vet]:\nif adjVet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n# \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet)\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]()\ndfs(graph, visited, res, start_vet)\nreturn res\n
    graph_dfs.go
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n// append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n*res = append(*res, vet)\nvisited[vet] = struct{}{}\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nif !isExist {\ndfs(g, visited, res, adjVet)\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\ndfs(g, visited, &res, startVet)\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_dfs.js
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.ts
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\ngraph: GraphAdjList,\nvisited: Set<Vertex>,\nres: Vertex[],\nvet: Vertex\n): void {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
    graph_dfs.cs
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, HashSet<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9                             \n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new List<Vertex>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet<Vertex> visited = new HashSet<Vertex>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.swift
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = []\ndfs(graph: graph, visited: &visited, res: &res, vet: startVet)\nreturn res\n}\n
    graph_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
    graph_dfs.dart
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\nGraphAdjList graph,\nSet<Vertex> visited,\nList<Vertex> res,\nVertex vet,\n) {\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = {};\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5176\u4e2d\uff1a

    • \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\uff1b
    • \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u9012\u5f52\u65b9\u6cd5\u7684\u4f4d\u7f6e\uff1b

    \u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6a21\u62df\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f

    \u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002

    \u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u4e0d\u540c\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002

    "},{"location":"chapter_graph/graph_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_graph/summary/","title":"9.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u88ab\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
    • \u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
    • \u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
    • \u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002
    • \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
    • \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
    • \u4ece\u7b97\u6cd5\u601d\u60f3\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
    • \u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
    • \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
    • \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
    • \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_graph/summary/#941-q-a","title":"9.4.1. \u00a0 Q & A","text":"

    \u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f

    \u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u8ba4\u4e3a\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002

    \u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f

    \u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002

    \u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f

    \u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u4f1a\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\u3001\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\u7b49\uff0c\u8fd9\u6837\u53ef\u4ee5\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002

    "},{"location":"chapter_hashing/","title":"6. \u00a0 \u6563\u5217\u8868","text":""},{"location":"chapter_hashing/hash_algorithm/","title":"6.3. \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"

    \u5728\u4e0a\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u4e86\u89e3\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\uff0c\u4ee5\u53ca\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\uff0c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5730\u5740\u6cd5\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u4f46\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002

    \u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u4f8b\u5982\u5bf9\u4e8e\u94fe\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5e73\u5747\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u597d\u7684\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u5168\u90e8\u952e\u503c\u90fd\u88ab\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002

    Fig. \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u4e0e\u6700\u5dee\u60c5\u51b5

    \u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u662f\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u7684\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a

    index = hash(key) % capacity\n

    \u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf capacity \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 hash() \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u51cf\u5c0f\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u9700\u8981\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 hash() \u7684\u8bbe\u8ba1\u4e0a\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1. \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"

    \u4e3a\u4e86\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5305\u542b\u4ee5\u4e0b\u7279\u70b9\uff1a

    • \u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002
    • \u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002
    • \u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5e73\u5747\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5e73\u5747\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002

    \u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\uff0c\u5305\u62ec\uff1a

    • \u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002
    • \u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u7684\u3002

    \u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u6ee1\u8db3\u66f4\u9ad8\u7684\u5b89\u5168\u6807\u51c6\uff0c\u4ee5\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u5305\u62ec\uff1a

    • \u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u5176\u56f0\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002
    • \u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002

    \u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 key \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 key % 100 \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\uff0c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 key \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 key \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2. \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"

    \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u590d\u6742\u4e14\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u4e00\u4e9b\u7b80\u5355\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4ee5\u5b57\u7b26\u4e32\u54c8\u5e0c\u4e3a\u4f8b\uff1a

    • \u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002
    • \u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e86\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002
    • \u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002
    • \u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart simple_hash.java
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (31 * hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash ^= (int) c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = ((hash << 4) ^ (hash >> 28) ^ (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n
    simple_hash.cpp
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (31 * hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\ncout<<(int)c<<endl;\nhash ^= (int)c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n
    simple_hash.py
    def add_hash(key: str) -> int:\n\"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash += ord(c)\nreturn hash % modulus\ndef mul_hash(key: str) -> int:\n\"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = 31 * hash + ord(c)\nreturn hash % modulus\ndef xor_hash(key: str) -> int:\n\"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash ^= ord(c)\nreturn hash % modulus\ndef rot_hash(key: str) -> int:\n\"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = (hash << 4) ^ (hash >> 28) ^ ord(c)\nreturn hash % modulus\n
    simple_hash.go
    /* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (31*hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\nhash := 0\nmodulus := 1000000007\nfor _, b := range []byte(key) {\nfmt.Println(int(b))\nhash ^= int(b)\nhash = (31*hash + int(b)) % modulus\n}\nreturn hash & modulus\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ int64(b)) % modulus\n}\nreturn int(hash)\n}\n
    simple_hash.js
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.ts
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.c
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.cs
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (31 * hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash ^= c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ c) % MODULUS;\n}\nreturn (int)hash;\n}\n
    simple_hash.swift
    /* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (31 * hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash ^= Int(scalar.value)\n}\n}\nreturn hash & MODULUS\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = ((hash << 4) ^ (hash >> 28) ^ Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n
    simple_hash.zig
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.dart
    [class]{}-[func]{add_hash}\n[class]{}-[func]{mul_hash}\n[class]{}-[func]{xor_hash}\n[class]{}-[func]{rot_hash}\n

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002

    \u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u5f53\u6211\u4eec\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\u65f6\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4f1a\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002

    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u3002\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 key \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\) , \\(3\\) , \\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002

    \\[ \\begin{aligned} \\text{modulus} & = 9 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\cdots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\cdots \\} \\end{aligned} \\]

    \u5982\u679c\u8f93\u5165 key \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 modulus \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e key \u548c modulus \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002

    \\[ \\begin{aligned} \\text{modulus} & = 13 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\cdots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\cdots \\} \\end{aligned} \\]

    \u503c\u5f97\u5f3a\u8c03\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 key \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u662f\u53ef\u4ee5\u7684\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 key \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002

    \u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u5927\u4e00\u4e9b\uff0c\u4ee5\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3. \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"

    \u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002

    \u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5, SHA-1, SHA-2, SHA3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002

    \u76f4\u81f3\u76ee\u524d\uff0cMD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u88ab\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002

    MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256 / 512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4. \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"

    \u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 key \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 hash() \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\uff0c\u5305\u62ec\uff1a

    • \u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002
    • \u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u8bf7\u81ea\u884c\u5b66\u4e60\u3002
    • \u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002
    • \u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002

    Tip

    \u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart built_in_hash.java
    int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -727081396\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n
    built_in_hash.cpp
    int num = 3;\nsize_t hashNum = hash<int>()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbool bol = true;\nsize_t hashBol = hash<bool>()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndouble dec = 3.14159;\nsize_t hashDec = hash<double>()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash<string>()(str);\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n
    built_in_hash.py
    num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndec = 3.14159\nhash_dec = hash(dec) \n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 <ListNode object at 0x1058fd810> \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n
    built_in_hash.go
    \n
    built_in_hash.js
    \n
    built_in_hash.ts
    \n
    built_in_hash.c
    \n
    built_in_hash.cs
    int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\nobject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\nListNode obj = new ListNode(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n
    built_in_hash.swift
    let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n
    built_in_hash.zig
    \n
    built_in_hash.dart
    int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 468167534\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n

    \u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 key \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a key \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 value \u4e86\u3002

    \u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u4f8b\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\uff0c\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u9ed8\u8ba4\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002

    \u5411\u54c8\u5e0c\u51fd\u6570\u52a0\u76d0

    Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u56e0\u6b64\u5728\u4e0d\u540c\u7684 Python \u8fd0\u884c\u5b9e\u4f8b\u4e2d\uff0c\u540c\u4e00\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u4e0d\u540c\u7684\u3002\u6b64\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002

    "},{"location":"chapter_hashing/hash_collision/","title":"6.2. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"

    \u4e0a\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6570\u7ec4\u7d22\u5f15\u3002

    \u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u65f6\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u6362\u4e00\u79cd\u601d\u8def\uff1a

    1. \u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u5b58\u5728\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002
    2. \u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002

    \u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u94fe\u5f0f\u5730\u5740\u548c\u5f00\u653e\u5bfb\u5740\u3002

    "},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1. \u00a0 \u94fe\u5f0f\u5730\u5740","text":"

    \u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 Separate Chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002

    Fig. \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868

    \u94fe\u5f0f\u5730\u5740\u4e0b\uff0c\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u5305\u62ec\uff1a

    • \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 key \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6570\u7ec4\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 key \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002
    • \u6dfb\u52a0\u5143\u7d20\uff1a\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u5373\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002
    • \u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\uff0c\u5e76\u5c06\u5176\u5220\u9664\u3002

    \u8be5\u65b9\u6cd5\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\uff0c\u5305\u62ec\uff1a

    • \u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u7531\u4e8e\u94fe\u8868\u6216\u4e8c\u53c9\u6811\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\uff1b
    • \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\uff1b

    \u4ee5\u4e0b\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\uff1a

    • \u4e3a\u4e86\u4f7f\u5f97\u4ee3\u7801\u5c3d\u91cf\u7b80\u77ed\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002
    • \u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5177\u4f53\u6765\u770b\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3 \\(2\\) \u500d\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map_chaining.java
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = new ArrayList<>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.add(new ArrayList<>());\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString get(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair : bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair : bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = new Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (Pair pair : bucket) {\nif (pair.key == key)\nbucket.remove(pair);\n}\nsize--;\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new ArrayList<>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.add(new ArrayList<>());\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List<Pair> bucket : bucketsTmp) {\nfor (Pair pair : bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (List<Pair> bucket : buckets) {\nList<String> res = new ArrayList<>();\nfor (Pair pair : bucket) {\nres.add(pair.key + \" -> \" + pair.val);\n}\nSystem.out.println(res);\n}\n}\n}\n
    hash_map_chaining.cpp
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nprivate:\nint size;                       // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;                // \u6269\u5bb9\u500d\u6570\nvector<vector<Pair *>> buckets; // \u6876\u6570\u7ec4\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3), extendRatio(2) {\nbuckets.resize(capacity);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double)size / (double)capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair *pair : buckets[index]) {\nif (pair->key == key) {\nreturn pair->val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nullptr\nreturn nullptr;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair *pair : buckets[index]) {\nif (pair->key == key) {\npair->val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].push_back(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nauto &bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (int i = 0; i < bucket.size(); i++) {\nif (bucket[i]->key == key) {\nPair *tmp = bucket[i];\nbucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\ndelete tmp;                       // \u91ca\u653e\u5185\u5b58\nsize--;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector<vector<Pair *>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets.clear();\nbuckets.resize(capacity);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (auto &bucket : bucketsTmp) {\nfor (Pair *pair : bucket) {\nput(pair->key, pair->val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (auto &bucket : buckets) {\ncout << \"[\";\nfor (Pair *pair : bucket) {\ncout << pair->key << \" -> \" << pair->val << \", \";\n}\ncout << \"]\\n\";\n}\n}\n};\n
    hash_map_chaining.py
    class HashMapChaining:\n\"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2 / 3  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -> float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket:\nif pair.key == key:\nreturn pair.val\n# \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\nreturn None\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() > self.load_thres:\nself.extend()\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket:\nif pair.key == key:\npair.val = val\nreturn\n# \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\npair = Pair(key, val)\nbucket.append(pair)\nself.size += 1\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor pair in bucket:\nif pair.key == key:\nbucket.remove(pair)\nself.size -= 1\nreturn\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [[] for _ in range(self.capacity)]\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in buckets:\nfor pair in bucket:\nself.put(pair.key, pair.val)\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor bucket in self.buckets:\nres = []\nfor pair in bucket:\nres.append(str(pair.key) + \" -> \" + pair.val)\nprint(res)\n
    hash_map_chaining.go
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\nsize        int      // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int      // \u6269\u5bb9\u500d\u6570\nbuckets     [][]pair // \u6876\u6570\u7ec4\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\nbuckets := make([][]pair, 4)\nfor i := 0; i < 4; i++ {\nbuckets[i] = make([]pair, 0)\n}\nreturn &hashMapChaining{\nsize:        0,\ncapacity:    4,\nloadThres:   2 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\nreturn float64(m.size / m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\nidx := m.hashFunc(key)\nbucket := m.buckets[idx]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor _, p := range bucket {\nif p.key == key {\nreturn p.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() > m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor _, p := range m.buckets[idx] {\nif p.key == key {\np.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\np := pair{\nkey: key,\nval: val,\n}\nm.buckets[idx] = append(m.buckets[idx], p)\nm.size += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor i, p := range m.buckets[idx] {\nif p.key == key {\n// \u5207\u7247\u5220\u9664\nm.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\nbreak\n}\n}\nm.size -= 1\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([][]pair, len(m.buckets))\nfor i := 0; i < len(m.buckets); i++ {\ntmpBuckets[i] = make([]pair, len(m.buckets[i]))\ncopy(tmpBuckets[i], m.buckets[i])\n}\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([][]pair, m.capacity)\nfor i := 0; i < m.capacity; i++ {\nm.buckets[i] = make([]pair, 0)\n}\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, bucket := range tmpBuckets {\nfor _, p := range bucket {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\nvar builder strings.Builder\nfor _, bucket := range m.buckets {\nbuilder.WriteString(\"[\")\nfor _, p := range bucket {\nbuilder.WriteString(strconv.Itoa(p.key) + \" -> \" + p.val + \" \")\n}\nbuilder.WriteString(\"]\")\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
    hash_map_chaining.js
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.ts
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.c
    [class]{hashMapChaining}-[func]{}\n
    hash_map_chaining.cs
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = new List<List<Pair>>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.Add(new List<Pair>());\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate double loadFactor() {\nreturn (double)size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string get(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].Add(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nforeach (Pair pair in buckets[index].ToList()) { if (pair.key == key) {\nbuckets[index].Remove(pair);\n}\n}\nsize--;   }\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new List<List<Pair>>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.Add(new List<Pair>());\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (List<Pair> bucket in bucketsTmp) {\nforeach (Pair pair in bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (List<Pair> bucket in buckets) {\nList<string> res = new List<string>();\nforeach (Pair pair in bucket) {\nres.Add(pair.key + \" -> \" + pair.val);\n}\nforeach (string kv in res) {\nConsole.WriteLine(kv);\n}\n}\n}\n}\n
    hash_map_chaining.swift
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [[Pair]] // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2 / 3\nextendRatio = 2\nbuckets = Array(repeating: [], count: capacity)\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -> Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -> Double {\nDouble(size / capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket {\nif pair.key == key {\nreturn pair.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() > loadThres {\nextend()\n}\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket {\nif pair.key == key {\npair.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nlet pair = Pair(key: key, val: val)\nbuckets[index].append(pair)\nsize += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (pairIndex, pair) in bucket.enumerated() {\nif pair.key == key {\nbuckets[index].remove(at: pairIndex)\n}\n}\nsize -= 1\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: [], count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in bucketsTmp {\nfor pair in bucket {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor bucket in buckets {\nlet res = bucket.map { \"\\($0.key) -> \\($0.val)\" }\nSwift.print(res)\n}\n}\n}\n
    hash_map_chaining.zig
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.dart
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nlate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nlate int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nlate double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nlate int extendRatio; // \u6269\u5bb9\u500d\u6570\nlate List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = List.generate(capacity, (_) => []);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair in bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair in bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nbucket.removeWhere((Pair pair) => pair.key == key);\nsize--;\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = List.generate(capacity, (_) => []);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List<Pair> bucket in bucketsTmp) {\nfor (Pair pair in bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (List<Pair> bucket in buckets) {\nList<String> res = [];\nfor (Pair pair in bucket) {\nres.add(\"${pair.key} -> ${pair.val}\");\n}\nprint(res);\n}\n}\n}\n

    Tip

    \u4e3a\u4e86\u63d0\u9ad8\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u300cAVL \u6811\u300d\u6216\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002

    "},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2. \u00a0 \u5f00\u653e\u5bfb\u5740","text":"

    \u300c\u5f00\u653e\u5bfb\u5740 Open Addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u3002

    "},{"location":"chapter_hashing/hash_collision/#_1","title":"\u7ebf\u6027\u63a2\u6d4b","text":"

    \u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u67e5\u627e\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5bf9\u5e94\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u65b9\u6cd5\u4e3a\uff1a

    • \u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6570\u7ec4\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u4f4d\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002
    • \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de value \u5373\u53ef\uff1b\u6216\u8005\u82e5\u9047\u5230\u7a7a\u4f4d\uff0c\u8bf4\u660e\u76ee\u6807\u952e\u503c\u5bf9\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002

    Fig. \u7ebf\u6027\u63a2\u6d4b

    \u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5b58\u5728\u4ee5\u4e0b\u7f3a\u9677\uff1a

    • \u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u67e5\u627e\u5176\u4ed6\u5143\u7d20\u65f6\uff0c\u8be5\u7a7a\u4f4d\u53ef\u80fd\u5bfc\u81f4\u7a0b\u5e8f\u8bef\u5224\u5143\u7d20\u4e0d\u5b58\u5728\u3002\u56e0\u6b64\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u6807\u8bb0\u5df2\u5220\u9664\u5143\u7d20\u3002
    • \u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u6570\u7ec4\u5185\u8fde\u7eed\u88ab\u5360\u7528\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8fd9\u4e00\u4f4d\u7f6e\u7684\u201c\u805a\u5806\u751f\u957f\u201d\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u964d\u4f4e\u3002

    \u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u4e3a\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u91cd\u70b9\u5305\u62ec\uff1a

    • \u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u56fa\u5b9a\u7684\u952e\u503c\u5bf9\u5b9e\u4f8b removed \u6765\u6807\u8bb0\u5df2\u5220\u9664\u5143\u7d20\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u4e00\u4e2a\u6876\u4e3a \\(\\text{None}\\) \u6216 removed \u65f6\uff0c\u8fd9\u4e2a\u6876\u90fd\u662f\u7a7a\u7684\uff0c\u53ef\u7528\u4e8e\u653e\u7f6e\u952e\u503c\u5bf9\u3002
    • \u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u662f\u53ef\u4ee5\u518d\u6b21\u88ab\u4f7f\u7528\u7684\u3002\u5f53\u63d2\u5165\u5143\u7d20\u65f6\uff0c\u82e5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u4e86\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7d22\u5f15\uff0c\u5219\u53ef\u5c06\u8be5\u5143\u7d20\u653e\u7f6e\u5230\u8be5\u7d22\u5f15\u3002
    • \u5728\u7ebf\u6027\u63a2\u6d4b\u65f6\uff0c\u6211\u4eec\u4ece\u5f53\u524d\u7d22\u5f15 index \u5411\u540e\u904d\u5386\uff1b\u800c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u9700\u8981\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map_open_addressing.java
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nprivate int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nprivate double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nprivate int extendRatio; // \u6269\u5bb9\u500d\u6570\nprivate Pair[] buckets; // \u6876\u6570\u7ec4\nprivate Pair removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new Pair[capacity];\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\npublic int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\npublic double loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (buckets[j] == null)\nreturn null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j].key == key && buckets[j] != removed)\nreturn buckets[j].val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == null || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j].key == key) {\nbuckets[j].val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j].key == key) {\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\npublic void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair pair : bucketsTmp) {\nif (pair != null && pair != removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair pair : buckets) {\nif (pair != null) {\nSystem.out.println(pair.key + \" -> \" + pair.val);\n} else {\nSystem.out.println(\"null\");\n}\n}\n}\n}\n
    hash_map_open_addressing.cpp
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate:\nint size;               // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;           // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres;       // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;        // \u6269\u5bb9\u500d\u6570\nvector<Pair *> buckets; // \u6876\u6570\u7ec4\nPair *removed;          // \u5220\u9664\u6807\u8bb0\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() {\n// \u6784\u9020\u65b9\u6cd5\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = vector<Pair *>(capacity, nullptr);\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn static_cast<double>(size) / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de nullptr\nif (buckets[j] == nullptr)\nreturn nullptr;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j]->key == key && buckets[j] != removed)\nreturn buckets[j]->val;\n}\nreturn nullptr;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres)\nextend();\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == nullptr || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j]->key == key) {\nbuckets[j]->val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == nullptr)\nreturn;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j]->key == key) {\ndelete buckets[j]; // \u91ca\u653e\u5185\u5b58\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector<Pair *> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = vector<Pair *>(capacity, nullptr);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair *pair : bucketsTmp) {\nif (pair != nullptr && pair != removed) {\nput(pair->key, pair->val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (auto &pair : buckets) {\nif (pair != nullptr) {\ncout << pair->key << \" -> \" << pair->val << endl;\n} else {\ncout << \"nullptr\" << endl;\n}\n}\n}\n};\n
    hash_map_open_addressing.py
    class HashMapOpenAddressing:\n\"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2 / 3  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\nself.removed = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -> float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de None\nif self.buckets[j] is None:\nreturn None\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif self.buckets[j].key == key and self.buckets[j] != self.removed:\nreturn self.buckets[j].val\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() > self.load_thres:\nself.extend()\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif self.buckets[j] in [None, self.removed]:\nself.buckets[j] = Pair(key, val)\nself.size += 1\nreturn\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif self.buckets[j].key == key:\nself.buckets[j].val = val\nreturn\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif self.buckets[j] is None:\nreturn\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif self.buckets[j].key == key:\nself.buckets[j] = self.removed\nself.size -= 1\nreturn\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets_tmp = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [None] * self.capacity\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in buckets_tmp:\nif pair not in [None, self.removed]:\nself.put(pair.key, pair.val)\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\nelse:\nprint(\"None\")\n
    hash_map_open_addressing.go
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\nsize        int     // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int     // \u6269\u5bb9\u500d\u6570\nbuckets     []pair  // \u6876\u6570\u7ec4\nremoved     pair    // \u5220\u9664\u6807\u8bb0\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\nbuckets := make([]pair, 4)\nreturn &hashMapOpenAddressing{\nsize:        0,\ncapacity:    4,\nloadThres:   2 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\nremoved: pair{\nkey: -1,\nval: \"-1\",\n},\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\nreturn float64(m.size) / float64(m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + 1) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif m.buckets[j] == (pair{}) {\nreturn \"\"\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif m.buckets[j].key == key && m.buckets[j] != m.removed {\nreturn m.buckets[j].val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() > m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + i) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\nm.buckets[j] = pair{\nkey: key,\nval: val,\n}\nm.size += 1\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif m.buckets[j].key == key {\nm.buckets[j].val = val\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + 1) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif m.buckets[j] == (pair{}) {\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif m.buckets[j].key == key {\nm.buckets[j] = m.removed\nm.size -= 1\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([]pair, len(m.buckets))\ncopy(tmpBuckets, m.buckets)\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([]pair, m.capacity)\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, p := range tmpBuckets {\nif p != (pair{}) && p != m.removed {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\nfor _, p := range m.buckets {\nif p != (pair{}) {\nfmt.Println(strconv.Itoa(p.key) + \" -> \" + p.val)\n} else {\nfmt.Println(\"nil\")\n}\n}\n}\n
    hash_map_open_addressing.js
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.ts
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.c
    [class]{hashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.cs
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nPair[] buckets; // \u6876\u6570\u7ec4\nPair removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new Pair[capacity];\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate double loadFactor() {\nreturn (double)size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (buckets[j] == null)\nreturn null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j].key == key && buckets[j] != removed)\nreturn buckets[j].val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == null || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j].key == key) {\nbuckets[j].val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j].key == key) {\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (Pair pair in bucketsTmp) {\nif (pair != null && pair != removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (Pair pair in buckets) {\nif (pair != null) {\nConsole.WriteLine(pair.key + \" -> \" + pair.val);\n} else {\nConsole.WriteLine(\"null\");\n}\n}\n}\n}\n
    hash_map_open_addressing.swift
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [Pair?] // \u6876\u6570\u7ec4\nvar removed: Pair // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2 / 3\nextendRatio = 2\nbuckets = Array(repeating: nil, count: capacity)\nremoved = Pair(key: -1, val: \"-1\")\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -> Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -> Double {\nDouble(size / capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, to: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de nil\nif buckets[j] == nil {\nreturn nil\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif buckets[j]?.key == key, buckets[j] != removed {\nreturn buckets[j]?.val\n}\n}\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() > loadThres {\nextend()\n}\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, through: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif buckets[j] == nil || buckets[j] == removed {\nbuckets[j] = Pair(key: key, val: val)\nsize += 1\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif buckets[j]?.key == key {\nbuckets[j]?.val = val\nreturn\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, to: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif buckets[j] == nil {\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif buckets[j]?.key == key {\nbuckets[j] = removed\nsize -= 1\nreturn\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: nil, count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in bucketsTmp {\nif let pair, pair != removed {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in buckets {\nif let pair {\nSwift.print(\"\\(pair.key) -> \\(pair.val)\")\n} else {\nSwift.print(\"null\")\n}\n}\n}\n}\n
    hash_map_open_addressing.zig
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.dart
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nlate int _size; // \u952e\u503c\u5bf9\u6570\u91cf\nlate int _capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nlate double _loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nlate int _extendRatio; // \u6269\u5bb9\u500d\u6570\nlate List<Pair?> _buckets; // \u6876\u6570\u7ec4\nlate Pair _removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() {\n_size = 0;\n_capacity = 4;\n_loadThres = 2.0 / 3.0;\n_extendRatio = 2;\n_buckets = List.generate(_capacity, (index) => null);\n_removed = Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % _capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn _size / _capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (_buckets[j] == null) return null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (_buckets[j]!.key == key && _buckets[j] != _removed)\nreturn _buckets[j]!.val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > _loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (_buckets[j] == null || _buckets[j] == _removed) {\n_buckets[j] = new Pair(key, val);\n_size += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (_buckets[j]!.key == key) {\n_buckets[j]!.val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (_buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (_buckets[j]!.key == key) {\n_buckets[j] = _removed;\n_size -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<Pair?> bucketsTmp = _buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n_capacity *= _extendRatio;\n_buckets = List.generate(_capacity, (index) => null);\n_size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair? pair in bucketsTmp) {\nif (pair != null && pair != _removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (Pair? pair in _buckets) {\nif (pair != null) {\nprint(\"${pair.key} -> ${pair.val}\");\n} else {\nprint(null);\n}\n}\n}\n}\n
    "},{"location":"chapter_hashing/hash_collision/#_2","title":"\u591a\u6b21\u54c8\u5e0c","text":"

    \u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u662f\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) , \\(f_2(x)\\) , \\(f_3(x)\\) , \\(\\cdots\\) \u8fdb\u884c\u63a2\u6d4b\u3002

    • \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002
    • \u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u6216\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002

    \u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u589e\u52a0\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002

    \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9

    Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u88ab\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002

    Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002

    Golang \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002

    "},{"location":"chapter_hashing/hash_map/","title":"6.1. \u00a0 \u54c8\u5e0c\u8868","text":"

    \u300c\u54c8\u5e0c\u8868 Hash Table\u300d\u901a\u8fc7\u5efa\u7acb\u952e key \u4e0e\u503c value \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a key \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684 value \u3002

    \u4ee5\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5b66\u751f\u7684\u6570\u636e\u5e93\u4e3a\u4f8b\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002

    Fig. \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a

    \u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5176\u4e2d\uff1a

    • \u67e5\u8be2\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
    • \u6dfb\u52a0\u5143\u7d20\u4ec5\u9700\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
    • \u5220\u9664\u5143\u7d20\u9700\u8981\u5148\u67e5\u8be2\u518d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
    \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\)

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u5e38\u7528\u4e8e\u5bf9\u67e5\u627e\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u7684\u573a\u666f\u3002

    "},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1. \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map.java
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   map.put(15937, \"\u5c0f\u5570\");   map.put(16750, \"\u5c0f\u7b97\");   map.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
    hash_map.cpp
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
    hash_map.py
    # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nmapp: Dict = {}\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = mapp[15937]\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmapp.pop(10583)\n
    hash_map.go
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nmapp := make(map[int]string)\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := mapp[15937]\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(mapp, 10583)\n
    hash_map.js
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new ArrayHashMap();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
    hash_map.ts
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
    hash_map.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
    hash_map.cs
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, String> map = new ();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.Add(12836, \"\u5c0f\u54c8\");\nmap.Add(15937, \"\u5c0f\u5570\");\nmap.Add(16750, \"\u5c0f\u7b97\");\nmap.Add(13276, \"\u5c0f\u6cd5\");\nmap.Add(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
    hash_map.swift
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
    hash_map.zig
    \n
    hash_map.dart
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<int, String> map = {};\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n

    \u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map.java
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\nSystem.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\nSystem.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\nSystem.out.println(val);\n}\n
    hash_map.cpp
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\ncout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (auto key: map) {\ncout << key.first << endl;\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (auto val: map) {\ncout << val.second << endl;\n}\n
    hash_map.py
    # \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in mapp.items():\nprint(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in mapp.keys():\nprint(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in mapp.values():\nprint(value)\n
    hash_map_test.go
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range mapp {\nfmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range mapp {\nfmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range mapp {\nfmt.Println(value)\n}\n
    hash_map.js
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
    hash_map.ts
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
    hash_map.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
    hash_map.cs
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map)\n{\nConsole.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys)\n{\nConsole.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (String val in map.Values)\n{\nConsole.WriteLine(val);\n}\n
    hash_map.swift
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\nprint(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\nprint(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\nprint(value)\n}\n
    hash_map.zig
    \n
    hash_map.dart
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nmap.forEach((key, value) {\nprint('$key -> $value');\n});\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\nprint(key);\n});\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\nprint(value);\n});\n
    "},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2. \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"

    \u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 Bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 key \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 value \u3002

    \u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e key \u6765\u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 Hash Function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 key \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a key \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 key \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002

    \u8f93\u5165\u4e00\u4e2a key \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\uff0c\u901a\u8fc7\u54c8\u5e0c\u7b97\u6cd5 hash() \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\uff1b\u63a5\u4e0b\u6765\uff0c\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09capacity \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 key \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 index \u3002

    index = hash(key) % capacity\n

    \u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 index \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 value \u3002

    \u8bbe\u6570\u7ec4\u957f\u5ea6 capacity = 100 \u3001\u54c8\u5e0c\u7b97\u6cd5 hash(key) = key \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a key % 100 \u3002\u4e0b\u56fe\u4ee5 key \u5b66\u53f7\u548c value \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002

    Fig. \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406

    \u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 key \u548c value \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b Pair \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_hash_map.java
    /* \u952e\u503c\u5bf9 */\nclass Pair {\npublic int key;\npublic String val;\npublic Pair(int key, String val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Pair> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new ArrayList<>();\nfor (int i = 0; i < 100; i++) {\nbuckets.add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\nPair pair = buckets.get(index);\nif (pair == null)\nreturn null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\nPair pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets.set(index, pair);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets.set(index, null);\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Pair> pairSet() {\nList<Pair> pairSet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\npairSet.add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<Integer> keySet() {\nList<Integer> keySet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\nkeySet.add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet() {\nList<String> valueSet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\nvalueSet.add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair kv : pairSet()) {\nSystem.out.println(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
    array_hash_map.cpp
    /* \u952e\u503c\u5bf9 */\nstruct Pair {\npublic:\nint key;\nstring val;\nPair(int key, string val) {\nthis->key = key;\nthis->val = val;\n}\n};\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate:\nvector<Pair *> buckets;\npublic:\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = vector<Pair *>(100);\n}\n~ArrayHashMap() {\n// \u91ca\u653e\u5185\u5b58\nfor (const auto &bucket : buckets) {\ndelete bucket;\n}\nbuckets.clear();\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\nPair *pair = buckets[index];\nif (pair == nullptr)\nreturn nullptr;\nreturn pair->val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\nPair *pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\ndelete buckets[index];\nbuckets[index] = nullptr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvector<Pair *> pairSet() {\nvector<Pair *> pairSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\npairSet.push_back(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvector<int> keySet() {\nvector<int> keySet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nkeySet.push_back(pair->key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvector<string> valueSet() {\nvector<string> valueSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nvalueSet.push_back(pair->val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Pair *kv : pairSet()) {\ncout << kv->key << \" -> \" << kv->val << endl;\n}\n}\n};\n
    array_hash_map.py
    class Pair:\n\"\"\"\u952e\u503c\u5bf9\"\"\"\ndef __init__(self, key: int, val: str):\nself.key = key\nself.val = val\nclass ArrayHashMap:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets: list[Pair | None] = [None] * 100\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nindex = key % 100\nreturn index\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\npair: Pair = self.buckets[index]\nif pair is None:\nreturn None\nreturn pair.val\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\npair = Pair(key, val)\nindex: int = self.hash_func(key)\nself.buckets[index] = pair\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\n# \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None\ndef entry_set(self) -> list[Pair]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\nresult: list[Pair] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair)\nreturn result\ndef key_set(self) -> list[int]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.key)\nreturn result\ndef value_set(self) -> list[str]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.val)\nreturn result\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\n
    array_hash_map.go
    /* \u952e\u503c\u5bf9 */\ntype pair struct {\nkey int\nval string\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\nbuckets []*pair\n}\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets := make([]*pair, 100)\nreturn &arrayHashMap{buckets: buckets}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\nindex := key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\nindex := a.hashFunc(key)\npair := a.buckets[index]\nif pair == nil {\nreturn \"Not Found\"\n}\nreturn pair.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\npair := &pair{key: key, val: val}\nindex := a.hashFunc(key)\na.buckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\nindex := a.hashFunc(key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\na.buckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\nvar pairs []*pair\nfor _, pair := range a.buckets {\nif pair != nil {\npairs = append(pairs, pair)\n}\n}\nreturn pairs\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\nvar keys []int\nfor _, pair := range a.buckets {\nif pair != nil {\nkeys = append(keys, pair.key)\n}\n}\nreturn keys\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\nvar values []string\nfor _, pair := range a.buckets {\nif pair != nil {\nvalues = append(values, pair.val)\n}\n}\nreturn values\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\nfor _, pair := range a.buckets {\nif pair != nil {\nfmt.Println(pair.key, \"->\", pair.val)\n}\n}\n}\n
    array_hash_map.js
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\nconstructor(key, val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n#buckets;\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.#buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nlet index = this.#hashFunc(key);\nlet pair = this.#buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nset(key, val) {\nlet index = this.#hashFunc(key);\nthis.#buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\ndelete(key) {\nlet index = this.#hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.#buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nentries() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nkeys() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvalues() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nif (!pair) continue;\nconsole.info(`${pair.key} -> ${pair.val}`);\n}\n}\n}\n
    array_hash_map.ts
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\npublic key: number;\npublic val: string;\nconstructor(key: number, val: string) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate readonly buckets: (Pair | null)[];\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic get(key: number): string | null {\nlet index = this.hashFunc(key);\nlet pair = this.buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic set(key: number, val: string) {\nlet index = this.hashFunc(key);\nthis.buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic delete(key: number) {\nlet index = this.hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic entries(): (Pair | null)[] {\nlet arr: (Pair | null)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic keys(): (number | undefined)[] {\nlet arr: (number | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic values(): (string | undefined)[] {\nlet arr: (string | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic print() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nif (!pair) continue;\nconsole.info(`${pair.key} -> ${pair.val}`);\n}\n}\n}\n
    array_hash_map.c
    /* \u952e\u503c\u5bf9 int->string */\nstruct pair {\nint key;\nchar *val;\n};\ntypedef struct pair pair;\n[class]{arrayHashMap}-[func]{}\n
    array_hash_map.cs
    /* \u952e\u503c\u5bf9 int->string */\nclass Pair {\npublic int key;\npublic string val;\npublic Pair(int key, string val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Pair?> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new();\nfor (int i = 0; i < 100; i++) {\nbuckets.Add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string? get(int key) {\nint index = hashFunc(key);\nPair? pair = buckets[index];\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\nPair pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Pair> pairSet() {\nList<Pair> pairSet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\npairSet.Add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<int> keySet() {\nList<int> keySet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\nkeySet.Add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<string> valueSet() {\nList<string> valueSet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\nvalueSet.Add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (Pair kv in pairSet()) {\nConsole.WriteLine(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
    array_hash_map.swift
    /* \u952e\u503c\u5bf9 */\nclass Pair {\nvar key: Int\nvar val: String\ninit(key: Int, val: String) {\nself.key = key\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate var buckets: [Pair?] = []\ninit() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nfor _ in 0 ..< 100 {\nbuckets.append(nil)\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate func hashFunc(key: Int) -> Int {\nlet index = key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet pair = buckets[index]\nreturn pair?.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\nlet pair = Pair(key: key, val: val)\nlet index = hashFunc(key: key)\nbuckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nfunc pairSet() -> [Pair] {\nvar pairSet: [Pair] = []\nfor pair in buckets {\nif let pair = pair {\npairSet.append(pair)\n}\n}\nreturn pairSet\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc keySet() -> [Int] {\nvar keySet: [Int] = []\nfor pair in buckets {\nif let pair = pair {\nkeySet.append(pair.key)\n}\n}\nreturn keySet\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc valueSet() -> [String] {\nvar valueSet: [String] = []\nfor pair in buckets {\nif let pair = pair {\nvalueSet.append(pair.val)\n}\n}\nreturn valueSet\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in pairSet() {\nSwift.print(\"\\(pair.key) -> \\(pair.val)\")\n}\n}\n}\n
    array_hash_map.zig
    // \u952e\u503c\u5bf9\nconst Pair = struct {\nkey: usize = undefined,\nval: []const u8 = undefined,\npub fn init(key: usize, val: []const u8) Pair {\nreturn Pair {\n.key = key,\n.val = val,\n};\n}\n};\n// \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\nreturn struct {\nbucket: ?std.ArrayList(?T) = null,\nmem_allocator: std.mem.Allocator = undefined,\nconst Self = @This();\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nself.mem_allocator = allocator;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\nself.bucket = std.ArrayList(?T).init(self.mem_allocator);\nvar i: i32 = 0;\nwhile (i < 100) : (i += 1) {\ntry self.bucket.?.append(null);\n}\n}\n// \u6790\u6784\u51fd\u6570\npub fn deinit(self: *Self) void {\nif (self.bucket != null) self.bucket.?.deinit();\n}\n// \u54c8\u5e0c\u51fd\u6570\nfn hashFunc(key: usize) usize {\nvar index = key % 100;\nreturn index;\n}\n// \u67e5\u8be2\u64cd\u4f5c\npub fn get(self: *Self, key: usize) []const u8 {\nvar index = hashFunc(key);\nvar pair = self.bucket.?.items[index];\nreturn pair.?.val;\n}\n// \u6dfb\u52a0\u64cd\u4f5c\npub fn put(self: *Self, key: usize, val: []const u8) !void {\nvar pair = Pair.init(key, val);\nvar index = hashFunc(key);\nself.bucket.?.items[index] = pair;\n}\n// \u5220\u9664\u64cd\u4f5c\npub fn remove(self: *Self, key: usize) !void {\nvar index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nself.bucket.?.items[index] = null;\n}       // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\npub fn pairSet(self: *Self) !std.ArrayList(T) {\nvar entry_set = std.ArrayList(T).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry entry_set.append(item.?);\n}\nreturn entry_set;\n}  // \u83b7\u53d6\u6240\u6709\u952e\npub fn keySet(self: *Self) !std.ArrayList(usize) {\nvar key_set = std.ArrayList(usize).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry key_set.append(item.?.key);\n}\nreturn key_set;\n}  // \u83b7\u53d6\u6240\u6709\u503c\npub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\nvar value_set = std.ArrayList([]const u8).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry value_set.append(item.?.val);\n}\nreturn value_set;\n}\n// \u6253\u5370\u54c8\u5e0c\u8868\npub fn print(self: *Self) !void {\nvar entry_set = try self.pairSet();\ndefer entry_set.deinit();\nfor (entry_set.items) |item| {\nstd.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n}\n}\n};\n}\n
    array_hash_map.dart
    /* \u952e\u503c\u5bf9 */\nclass Pair {\nint key;\nString val;\nPair(this.key, this.val);\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nlate List<Pair?> _buckets;\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n_buckets = List.filled(100, null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint _hashFunc(int key) {\nfinal int index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nfinal int index = _hashFunc(key);\nfinal Pair? pair = _buckets[index];\nif (pair == null) {\nreturn null;\n}\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\nfinal Pair pair = Pair(key, val);\nfinal int index = _hashFunc(key);\n_buckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nfinal int index = _hashFunc(key);\n_buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nList<Pair> pairSet() {\nList<Pair> pairSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\npairSet.add(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nList<int> keySet() {\nList<int> keySet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nkeySet.add(pair.key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nList<String> values() {\nList<String> valueSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nvalueSet.add(pair.val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (final Pair kv in pairSet()) {\nprint(\"${kv.key} -> ${kv.val}\");\n}\n}\n}\n
    "},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3. \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"

    \u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u8f93\u5165\u7a7a\u95f4\uff08key \u8303\u56f4\uff09\u6620\u5c04\u5230\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\uff09\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002

    \u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 key \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a

    12386 % 100 = 36\n20386 % 100 = 36\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002

    Fig. \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b

    \u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a key \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 (136, A) \u548c (236, D) \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002

    Fig. \u54c8\u5e0c\u8868\u6269\u5bb9

    \u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\u3002\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf capacity \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u56e0\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002

    \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u300c\u8d1f\u8f7d\u56e0\u5b50 Load Factor\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u4e3a\u4e86\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u88ab\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u5bb9\u91cf\u6269\u5c55\u4e3a\u539f\u5148\u7684 \\(2\\) \u500d\u3002

    "},{"location":"chapter_hashing/summary/","title":"6.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u8f93\u5165 key \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 value \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002
    • \u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002
    • \u54c8\u5e0c\u51fd\u6570\u5c06 key \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 value \u3002
    • \u4e24\u4e2a\u4e0d\u540c\u7684 key \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002
    • \u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002
    • \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002
    • \u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002
    • \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
    • \u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 HashMap \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 Dict \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002
    • \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002
    • \u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002
    • \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5, SHA-1, SHA-2, SHA3 \u7b49\u3002MD5 \u5e38\u7528\u8bed\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002
    • \u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002
    "},{"location":"chapter_hashing/summary/#641-q-a","title":"6.4.1. \u00a0 Q & A","text":"

    \u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(O(n)\\) \uff1f

    \u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u7684\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002

    \u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86

    \u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002

    \u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u4ed6\u4eec\u66f4\u9ad8\u5462\uff1f

    \u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u7684\u5185\u5b58\u662f\u672a\u4f7f\u7528\u7684\uff0c

    \u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002

    \u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002

    \u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u5bf9\u4e8e\u6807\u8bb0\u5df2\u5220\u9664\u7684\u7a7a\u95f4\uff0c\u8fd9\u4e2a\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f

    \u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u662f\u53ef\u4ee5\u518d\u6b21\u88ab\u4f7f\u7528\u7684\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u4e86\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u7684\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002

    \u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f

    \u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 key \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002

    \u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f

    \u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5165\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c key \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a key \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002

    "},{"location":"chapter_heap/","title":"8. \u00a0 \u5806","text":""},{"location":"chapter_heap/build_heap/","title":"8.2. \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"

    \u5982\u679c\u6211\u4eec\u60f3\u8981\u6839\u636e\u8f93\u5165\u5217\u8868\u751f\u6210\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u300c\u5efa\u5806\u300d\u3002

    "},{"location":"chapter_heap/build_heap/#821","title":"8.2.1. \u00a0 \u501f\u52a9\u5165\u5806\u65b9\u6cd5\u5b9e\u73b0","text":"

    \u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u501f\u52a9\u201c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\uff0c\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u5c06\u5217\u8868\u5143\u7d20\u4f9d\u6b21\u6dfb\u52a0\u5230\u5806\u4e2d\u3002

    \u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u5165\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u5728\u4f9d\u6b21\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5806\u7684\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8be5\u65b9\u6cd5\u7684\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002

    "},{"location":"chapter_heap/build_heap/#822","title":"8.2.2. \u00a0 \u57fa\u4e8e\u5806\u5316\u64cd\u4f5c\u5b9e\u73b0","text":"

    \u6709\u8da3\u7684\u662f\uff0c\u5b58\u5728\u4e00\u79cd\u66f4\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u4ec5\u4e3a \\(O(n)\\) \u3002\u6211\u4eec\u5148\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u7136\u540e\u8fed\u4ee3\u5730\u5bf9\u5404\u4e2a\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002\u5f53\u7136\uff0c\u6211\u4eec\u4e0d\u9700\u8981\u5bf9\u53f6\u8282\u70b9\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff0c\u56e0\u4e3a\u5b83\u4eec\u6ca1\u6709\u5b50\u8282\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new ArrayList<>(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.cpp
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.py
    def __init__(self, nums: list[int]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n# \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nself.max_heap = nums\n# \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(self.parent(self.size() - 1), -1, -1):\nself.sift_down(i)\n
    my_heap.go
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nh := &maxHeap{data: nums}\nfor i := len(h.data) - 1; i >= 0; i-- {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nh.siftDown(i)\n}\nreturn h\n}\n
    my_heap.js
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.#maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.#parent(this.size() - 1); i >= 0; i--) {\nthis.#siftDown(i);\n}\n}\n
    my_heap.ts
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.parent(this.size() - 1); i >= 0; i--) {\nthis.siftDown(i);\n}\n}\n
    my_heap.c
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nmaxHeap *newMaxHeap(int nums[], int size) {\n// \u6240\u6709\u5143\u7d20\u5165\u5806\nmaxHeap *h = (maxHeap *)malloc(sizeof(maxHeap));\nh->size = size;\nmemcpy(h->data, nums, size * sizeof(int));\nfor (int i = size - 1; i >= 0; i--) {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nsiftDown(h, i);\n}\nreturn h;\n}\n
    my_heap.cs
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new List<int>(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar size = parent(this.size() - 1);\nfor (int i = size; i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.swift
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\nsiftDown(i: i)\n}\n}\n
    my_heap.zig
    // \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\nif (self.max_heap != null) return;\nself.max_heap = std.ArrayList(T).init(allocator);\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\ntry self.max_heap.?.appendSlice(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar i: usize = parent(self.size() - 1) + 1;\nwhile (i > 0) : (i -= 1) {\ntry self.siftDown(i - 1);\n}\n}\n
    my_heap.dart
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n_maxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = _parent(size() - 1); i >= 0; i--) {\n_siftDown(i);\n}\n}\n
    "},{"location":"chapter_heap/build_heap/#823","title":"8.2.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u4e3a\u4ec0\u4e48\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \uff1f\u6211\u4eec\u6765\u5c55\u5f00\u63a8\u7b97\u4e00\u4e0b\u3002

    • \u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\uff0c\u5728\u6392\u9664\u53f6\u8282\u70b9\u540e\uff0c\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1)/2\\) \uff0c\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    • \u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(O(\\log n)\\) \uff1b

    \u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u7279\u6027\u3002

    \u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u8be6\u7ec6\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u51cf\u5c0f\u8ba1\u7b97\u96be\u5ea6\uff0c\u6211\u4eec\u5047\u8bbe\u6811\u662f\u4e00\u4e2a\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002\u8bbe\u4e8c\u53c9\u6811\uff08\u5373\u5806\uff09\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \u3002\u4e0a\u6587\u63d0\u5230\uff0c\u8282\u70b9\u5806\u5316\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf

    \u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002

    \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{(h-1)}\\times1 \\]

    \u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230

    \\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\cdots + 2^{h}\\times1 \\newline \\end{aligned} \\]

    \u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u4ee4\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97

    \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\cdots + 2^{h-1} + 2^h \\]

    \u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a

    \\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h \\newline & = O(2^h) \\end{aligned} \\]

    \u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002

    "},{"location":"chapter_heap/heap/","title":"8.1. \u00a0 \u5806","text":"

    \u300c\u5806 Heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff1a

    • \u300c\u5927\u9876\u5806 Max Heap\u300d\uff0c\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\uff1b
    • \u300c\u5c0f\u9876\u5806 Min Heap\u300d\uff0c\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\uff1b

    Fig. \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806

    \u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002
    • \u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u300c\u5806\u9876\u300d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u300c\u5806\u5e95\u300d\u3002
    • \u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u5373\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002
    "},{"location":"chapter_heap/heap/#811","title":"8.1.1. \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"

    \u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 Priority Queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002

    \u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4f5c\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u4f18\u5148\u961f\u5217\u300d\u548c\u300c\u5806\u300d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u4f7f\u7528\u300c\u5806\u300d\u6765\u547d\u540d\u3002

    \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\)

    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002

    Tip

    \u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539 Comparator \u6765\u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart heap.java
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = heap.poll();  // 5\npeek = heap.poll();  // 4\npeek = heap.poll();  // 3\npeek = heap.poll();  // 2\npeek = heap.poll();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
    heap.cpp
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
    heap.py
    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: List[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
    heap.go
    // Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n// Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n// \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n*h = append(*h, x.(int))\n}\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n// \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\nlast := (*h)[len(*h)-1]\n*h = (*h)[:len(*h)-1]\nreturn last\n}\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\nreturn len(*h)\n}\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n// \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\nreturn (*h)[i].(int) > (*h)[j].(int)\n}\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n(*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\nreturn (*h)[0]\n}\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nmaxHeap := &intHeap{}\nheap.Init(maxHeap)\n/* \u5143\u7d20\u5165\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\nheap.Push(maxHeap, 1)\nheap.Push(maxHeap, 3)\nheap.Push(maxHeap, 2)\nheap.Push(maxHeap, 4)\nheap.Push(maxHeap, 5)\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\ntop := maxHeap.Top()\nfmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\nheap.Pop(maxHeap) // 5\nheap.Pop(maxHeap) // 4\nheap.Pop(maxHeap) // 3\nheap.Pop(maxHeap) // 2\nheap.Pop(maxHeap) // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nsize := len(*maxHeap)\nfmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(*maxHeap) == 0\nfmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
    heap.js
    // JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.ts
    // TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.cs
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new PriorityQueue<int, int>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((x, y) => y - x));\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), });\n
    heap.swift
    // Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.zig
    \n
    heap.dart
    // Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    "},{"location":"chapter_heap/heap/#812","title":"8.1.2. \u00a0 \u5806\u7684\u5b9e\u73b0","text":"

    \u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002

    "},{"location":"chapter_heap/heap/#_1","title":"\u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"

    \u6211\u4eec\u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u4e2d\u5b66\u4e60\u5230\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002

    \u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002

    \u5177\u4f53\u800c\u8a00\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u53d6\u6574\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002

    Fig. \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8

    \u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.cpp
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n
    my_heap.py
    def left(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n
    my_heap.go
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n// \u5411\u4e0b\u6574\u9664\nreturn (i - 1) / 2\n}\n
    my_heap.js
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.ts
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.c
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(maxHeap *h, int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(maxHeap *h, int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(maxHeap *h, int i) {\nreturn (i - 1) / 2;\n}\n
    my_heap.cs
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.swift
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.zig
    // \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\nreturn 2 * i + 1;\n}\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\nreturn 2 * i + 2;\n}\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n// return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\nreturn @divFloor(i - 1, 2);\n}\n
    my_heap.dart
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\nreturn (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n
    "},{"location":"chapter_heap/heap/#_2","title":"\u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"

    \u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap.get(0);\n}\n
    my_heap.cpp
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
    my_heap.py
    def peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\nreturn self.max_heap[0]\n
    my_heap.go
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\nreturn h.data[0]\n}\n
    my_heap.js
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\nreturn this.#maxHeap[0];\n}\n
    my_heap.ts
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\nreturn this.maxHeap[0];\n}\n
    my_heap.c
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(maxHeap *h) {\nreturn h->data[0];\n}\n
    my_heap.cs
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
    my_heap.swift
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nmaxHeap[0]\n}\n
    my_heap.zig
    // \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\nreturn self.max_heap.?.items[0];\n}
    my_heap.dart
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn _maxHeap[0];\n}\n
    "},{"location":"chapter_heap/heap/#_3","title":"\u5143\u7d20\u5165\u5806","text":"

    \u7ed9\u5b9a\u5143\u7d20 val \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\u3002\u56e0\u6b64\uff0c\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 Heapify\u300d\u3002

    \u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002

    <1><2><3><4><5><6><7><8><9>

    \u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.cpp
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.push_back(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(maxHeap[i], maxHeap[p]);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.py
    def push(self, val: int):\n\"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n# \u6dfb\u52a0\u8282\u70b9\nself.max_heap.append(val)\n# \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1)\ndef sift_up(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\nwhile True:\n# \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np = self.parent(i)\n# \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 or self.max_heap[i] <= self.max_heap[p]:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, p)\n# \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n
    my_heap.go
    /* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n// \u6dfb\u52a0\u8282\u70b9\nh.data = append(h.data, val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nh.siftUp(len(h.data) - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\nfor true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np := h.parent(i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || h.data[i].(int) <= h.data[p].(int) {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
    my_heap.js
    /* \u5143\u7d20\u5165\u5806 */\npush(val) {\n// \u6dfb\u52a0\u8282\u70b9\nthis.#maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.#siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.#parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.ts
    /* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n// \u6dfb\u52a0\u8282\u70b9\nthis.maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.c
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(maxHeap *h, int val) {\n// \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\nif (h->size == MAX_SIZE) {\nprintf(\"heap is full!\");\nreturn;\n}\n// \u6dfb\u52a0\u8282\u70b9\nh->data[h->size] = val;\nh->size++;\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(h, h->size - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(maxHeap *h, int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(h, i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || h->data[i] <= h->data[p]) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(h, i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.cs
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.Add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.swift
    /* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.append(val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(i: size() - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\nvar i = i\nwhile true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nlet p = parent(i: i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || maxHeap[i] <= maxHeap[p] {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
    my_heap.zig
    // \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n// \u6dfb\u52a0\u8282\u70b9\ntry self.max_heap.?.append(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\ntry self.siftUp(self.size() - 1);\n}  // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nvar p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.dart
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\n_maxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\n_siftUp(size() - 1);\n}\n[class]{MaxHeap}-[func]{siftUp}\n
    "},{"location":"chapter_heap/heap/#_4","title":"\u5806\u9876\u5143\u7d20\u51fa\u5806","text":"

    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u53d6\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\uff1a

    1. \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u5373\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\uff1b
    2. \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\uff1b
    3. \u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\uff1b

    \u987e\u540d\u601d\u4e49\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u4ece\u5e95\u81f3\u9876\u5806\u5316\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\uff1b\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002

    <1><2><3><4><5><6><7><8><9><10>

    \u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.remove(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap.get(l) > maxHeap.get(ma))\nma = l;\nif (r < size() && maxHeap.get(r) > maxHeap.get(ma))\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.cpp
    /* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n// \u5224\u7a7a\u5904\u7406\nif (empty()) {\nthrow out_of_range(\"\u5806\u4e3a\u7a7a\");\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap[0], maxHeap[size() - 1]);\n// \u5220\u9664\u8282\u70b9\nmaxHeap.pop_back();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\nswap(maxHeap[i], maxHeap[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.py
    def pop(self) -> int:\n\"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n# \u5224\u7a7a\u5904\u7406\nif self.is_empty():\nraise IndexError(\"\u5806\u4e3a\u7a7a\")\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1)\n# \u5220\u9664\u8282\u70b9\nval = self.max_heap.pop()\n# \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0)\n# \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\ndef sift_down(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl, r, ma = self.left(i), self.right(i), i\nif l < self.size() and self.max_heap[l] > self.max_heap[ma]:\nma = l\nif r < self.size() and self.max_heap[r] > self.max_heap[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, ma)\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n
    my_heap.go
    /* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n// \u5224\u7a7a\u5904\u7406\nif h.isEmpty() {\nfmt.Println(\"error\")\nreturn nil\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nh.swap(0, h.size()-1)\n// \u5220\u9664\u8282\u70b9\nval := h.data[len(h.data)-1]\nh.data = h.data[:len(h.data)-1]\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nh.siftDown(0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nl, r, max := h.left(i), h.right(i), i\nif l < h.size() && h.data[l].(int) > h.data[max].(int) {\nmax = l\n}\nif r < h.size() && h.data[r].(int) > h.data[max].(int) {\nmax = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif max == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, max)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max\n}\n}\n
    my_heap.js
    /* \u5143\u7d20\u51fa\u5806 */\npop() {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.#swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.#maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.#siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.#left(i),\nr = this.#right(i);\nlet ma = i;\nif (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\nif (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.ts
    /* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new RangeError('Heap is empty.');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.left(i),\nr = this.right(i);\nlet ma = i;\nif (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\nif (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.c
    /* \u5143\u7d20\u51fa\u5806 */\nint pop(maxHeap *h) {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty(h)) {\nprintf(\"heap is empty!\");\nreturn INT_MAX;\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(h, 0, size(h) - 1);\n// \u5220\u9664\u8282\u70b9\nint val = h->data[h->size - 1];\nh->size--;\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(h, 0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(maxHeap *h, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nint l = left(h, i);\nint r = right(h, i);\nint max = i;\nif (l < size(h) && h->data[l] > h->data[max]) {\nmax = l;\n}\nif (r < size(h) && h->data[r] > h->data[max]) {\nmax = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (max == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(h, i, max);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max;\n}\n}\n
    my_heap.cs
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfRangeException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.Last();\nmaxHeap.RemoveAt(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.swift
    /* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n// \u5224\u7a7a\u5904\u7406\nif isEmpty() {\nfatalError(\"\u5806\u4e3a\u7a7a\")\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(i: 0, j: size() - 1)\n// \u5220\u9664\u8282\u70b9\nlet val = maxHeap.remove(at: size() - 1)\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(i: 0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = left(i: i)\nlet r = right(i: i)\nvar ma = i\nif l < size(), maxHeap[l] > maxHeap[ma] {\nma = l\n}\nif r < size(), maxHeap[r] > maxHeap[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n
    my_heap.zig
    // \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n// \u5224\u65ad\u5904\u7406\nif (self.isEmpty()) unreachable;\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\ntry self.swap(0, self.size() - 1);\n// \u5220\u9664\u8282\u70b9\nvar val = self.max_heap.?.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\ntry self.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n} // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nvar l = left(i);\nvar r = right(i);\nvar ma = i;\nif (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\nif (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.dart
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n_swap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = _maxHeap.removeLast();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\n_siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n[class]{MaxHeap}-[func]{siftDown}\n
    "},{"location":"chapter_heap/heap/#813","title":"8.1.3. \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"
    • \u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002
    • \u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u540e\u7eed\u7684\u5806\u6392\u5e8f\u7ae0\u8282\u3002
    • \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002
    "},{"location":"chapter_heap/summary/","title":"8.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002
    • \u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002
    • \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) \u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
    • \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002
    • \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
    • Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002
    "},{"location":"chapter_heap/summary/#841-q-a","title":"8.4.1. \u00a0 Q & A","text":"

    \u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f

    \u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u9732\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4e0d\u6070\u5f53\u7684\u4f7f\u7528\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u9732\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002

    "},{"location":"chapter_heap/top_k/","title":"8.3. \u00a0 Top-K \u95ee\u9898","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u65e0\u5e8f\u6570\u7ec4 nums \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002

    \u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002

    "},{"location":"chapter_heap/top_k/#831","title":"8.3.1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"

    \u6211\u4eec\u53ef\u4ee5\u8fdb\u884c \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\) , \\(2\\) , \\(\\cdots\\) , \\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002

    \u8be5\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002

    Fig. \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20

    Tip

    \u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\uff0c\u7b49\u4ef7\u4e8e\u300c\u9009\u62e9\u6392\u5e8f\u300d\u7b97\u6cd5\u3002

    "},{"location":"chapter_heap/top_k/#832","title":"8.3.2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"

    \u6211\u4eec\u53ef\u4ee5\u5bf9\u6570\u7ec4 nums \u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002

    \u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u8981\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002

    Fig. \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20

    "},{"location":"chapter_heap/top_k/#833","title":"8.3.3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\uff1b
    2. \u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\uff1b
    3. \u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\uff1b
    4. \u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1b
    <1><2><3><4><5><6><7><8><9>

    \u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002

    \u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart top_k.java
    /* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue<Integer> topKHeap(int[] nums, int k) {\nQueue<Integer> heap = new PriorityQueue<Integer>();\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i < k; i++) {\nheap.add(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i < nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] > heap.peek()) {\nheap.poll();\nheap.add(nums[i]);\n}\n}\nreturn heap;\n}\n
    top_k.cpp
    /* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue<int, vector<int>, greater<int>> topKHeap(vector<int> &nums, int k) {\npriority_queue<int, vector<int>, greater<int>> heap;\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i < k; i++) {\nheap.push(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i < nums.size(); i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] > heap.top()) {\nheap.pop();\nheap.push(nums[i]);\n}\n}\nreturn heap;\n}\n
    top_k.py
    def top_k_heap(nums: list[int], k: int) -> list[int]:\n\"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\nheap = []\n# \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor i in range(k):\nheapq.heappush(heap, nums[i])\n# \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor i in range(k, len(nums)):\n# \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif nums[i] > heap[0]:\nheapq.heappop(heap)\nheapq.heappush(heap, nums[i])\nreturn heap\n
    top_k.go
    [class]{maxHeap}-[func]{topKHeap}\n
    top_k.js
    [class]{}-[func]{topKHeap}\n
    top_k.ts
    [class]{}-[func]{topKHeap}\n
    top_k.c
    [class]{maxHeap}-[func]{topKHeap}\n
    top_k.cs
    [class]{top_k}-[func]{topKHeap}\n
    top_k.swift
    [class]{}-[func]{topKHeap}\n
    top_k.zig
    [class]{}-[func]{topKHeap}\n
    top_k.dart
    [class]{}-[func]{top_k_heap}\n
    "},{"location":"chapter_introduction/","title":"1. \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":""},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1. \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"

    \u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u4e8e\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002

    \u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\uff0c\u6211\u5c06\u4e3e\u5373\u4e2a\u5177\u4f53\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002

    \u4f8b\u4e00\uff1a\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002

    \u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u79ef\u6728\u7684\u5404\u79cd\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u800c\u7ec4\u88c5\u8bf4\u660e\u4e66\u4e0a\u7684\u4e00\u7cfb\u5217\u6b65\u9aa4\u5219\u662f\u7b97\u6cd5\u3002

    Fig. \u62fc\u88c5\u79ef\u6728

    \u4f8b\u4e8c\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u7684\u82f1\u6587\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u8fd9\u6837\u64cd\u4f5c\uff1a

    1. \u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff08\u5047\u8bbe\u4e3a \\(m\\) \uff09\uff1b
    2. \u7531\u4e8e\u5728\u82f1\u6587\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\uff1b
    3. \u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 1-2 \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002
    <1><2><3><4><5>

    \u67e5\u9605\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u300c\u6570\u7ec4\u300d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u662f\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7b97\u6cd5\u3002

    \u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u6536\u94f6\u5458\u4ed8\u4e86 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u7ed9\u6211\u4eec\u627e \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u4ee5\u4e0b\u601d\u8003\uff1a

    1. \u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) , \\(5\\) , \\(10\\) , \\(20\\) \u5143\u3002
    2. \u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002
    3. \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002
    4. \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002
    5. \u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002

    \u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u3002

    Fig. \u8d27\u5e01\u627e\u96f6

    \u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002

    Tip

    \u9605\u8bfb\u81f3\u6b64\uff0c\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u90a3\u4e48\u592a\u597d\u4e86\uff01\u56e0\u4e3a\u8fd9\u6b63\u662f\u672c\u4e66\u5b58\u5728\u7684\u610f\u4e49\u3002\u63a5\u4e0b\u6765\uff0c\u8fd9\u672c\u4e66\u5c06\u4e00\u6b65\u6b65\u5f15\u5bfc\u4f60\u6df1\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002

    "},{"location":"chapter_introduction/summary/","title":"1.3. \u00a0 \u5c0f\u7ed3","text":"
    • \u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002
    • \u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
    • \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
    • \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u5219\u662f\u53d1\u6325\u6570\u636e\u7ed3\u6784\u4f5c\u7528\u7684\u821e\u53f0\u3002
    • \u4e50\u9ad8\u79ef\u6728\u5bf9\u5e94\u4e8e\u6570\u636e\uff0c\u79ef\u6728\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002
    "},{"location":"chapter_introduction/what_is_dsa/","title":"1.2. \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1. \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"

    \u300c\u7b97\u6cd5 Algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\u3002\u7b97\u6cd5\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
    • \u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
    • \u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002
    "},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"

    \u300c\u6570\u636e\u7ed3\u6784 Data Structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u4e3a\u4e86\u63d0\u9ad8\u6570\u636e\u5b58\u50a8\u548c\u64cd\u4f5c\u6027\u80fd\uff0c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u76ee\u6807\u5305\u62ec\uff1a

    • \u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u51cf\u5c11\uff0c\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
    • \u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
    • \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u4f7f\u5f97\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002

    \u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\uff0c\u8fd9\u610f\u5473\u7740\u8981\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u4f18\u52bf\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4f8b\u5982\uff0c\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\uff1b\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    "},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3. \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"

    \u300c\u6570\u636e\u7ed3\u6784\u300d\u4e0e\u300c\u7b97\u6cd5\u300d\u9ad8\u5ea6\u76f8\u5173\u4e14\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\uff1a

    • \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u7528\u4e8e\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002
    • \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u901a\u8fc7\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
    • \u7279\u5b9a\u7b97\u6cd5\u901a\u5e38\u6709\u5bf9\u5e94\u6700\u4f18\u7684\u6570\u636e\u7ed3\u6784\u3002\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u8fdb\u884c\u5b9e\u73b0\uff0c\u4f46\u6700\u7ec8\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\u3002

    Fig. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb

    \u7c7b\u6bd4\u300cLEGO \u4e50\u9ad8\u300d\u548c\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\uff0c\u5219\u5bf9\u5e94\u5173\u7cfb\u5982\u4e0b\u8868\u6240\u793a\u3002

    \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 LEGO \u4e50\u9ad8 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b

    \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002

    \u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0

    \u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\u7b80\u79f0\u4e3a\u300c\u7b97\u6cd5\u300d\u3002\u4f8b\u5982\uff0c\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u5bdf\u4e86\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002

    "},{"location":"chapter_preface/","title":"0. \u00a0 \u524d\u8a00","text":""},{"location":"chapter_preface/about_the_book/","title":"0.1. \u00a0 \u5173\u4e8e\u672c\u4e66","text":"

    \u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002

    • \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\uff1b
    • \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00\uff1b
    • \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\uff1b
    "},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1. \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"

    \u82e5\u60a8\u662f\u300c\u7b97\u6cd5\u521d\u5b66\u8005\u300d\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u8fd9\u672c\u4e66\u6b63\u662f\u4e3a\u60a8\u91cf\u8eab\u5b9a\u5236\uff01

    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u8001\u624b\u300d\uff0c\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u60a8\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002

    \u82e5\u60a8\u662f\u300c\u7b97\u6cd5\u5927\u795e\u300d\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u60a8\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002

    \u524d\u7f6e\u6761\u4ef6

    \u60a8\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002

    "},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2. \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"

    \u672c\u4e66\u4e3b\u8981\u5185\u5bb9\u5305\u62ec\uff1a

    • \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3001\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u4f30\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5305\u62ec\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
    • \u6570\u636e\u7ed3\u6784\uff1a\u5e38\u89c1\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u5728\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u5f62\u5f0f\u3001\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6d89\u53ca\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u6563\u5217\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
    • \u7b97\u6cd5\uff1a\u67e5\u627e\u7b97\u6cd5\u3001\u6392\u5e8f\u7b97\u6cd5\u3001\u641c\u7d22\u4e0e\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u5206\u6cbb\u7b97\u6cd5\u7b49\uff0c\u5185\u5bb9\u6db5\u76d6\u5b9a\u4e49\u3001\u5e94\u7528\u573a\u666f\u3001\u4f18\u7f3a\u70b9\u3001\u65f6\u7a7a\u6548\u7387\u3001\u5b9e\u73b0\u65b9\u6cd5\u3001\u793a\u4f8b\u9898\u76ee\u7b49\u3002

    Fig. Hello \u7b97\u6cd5\u5185\u5bb9\u7ed3\u6784

    "},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3. \u00a0 \u81f4\u8c22","text":"

    \u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a

    • \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u60a8\u9f13\u52b1\u6211\u201c\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\u3002
    • \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\u3002
    • \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\u3002
    • \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c LOGO\uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\u3002
    • \u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u6770\u51fa\u7684\u5f00\u6e90\u9879\u76ee Material-for-MkDocs \u3002

    \u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01

    \u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u8457\u4f5c\u3002

    \u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4e9b\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002

    "},{"location":"chapter_preface/suggestions/","title":"0.2. \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"

    Tip

    \u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u60a8\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002

    "},{"location":"chapter_preface/suggestions/#021","title":"0.2.1. \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"
    • \u6807\u9898\u540e\u6807\u6ce8 * \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002
    • \u6587\u7ae0\u4e2d\u7684\u91cd\u8981\u540d\u8bcd\u4f1a\u7528 \u300c \u300d \u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982 \u300c\u6570\u7ec4 Array\u300d \u3002\u8bf7\u52a1\u5fc5\u8bb0\u4f4f\u8fd9\u4e9b\u540d\u8bcd\uff0c\u5305\u62ec\u82f1\u6587\u7ffb\u8bd1\uff0c\u4ee5\u4fbf\u540e\u7eed\u9605\u8bfb\u6587\u732e\u65f6\u4f7f\u7528\u3002
    • \u52a0\u7c97\u7684\u6587\u5b57 \u8868\u793a\u91cd\u70b9\u5185\u5bb9\u6216\u603b\u7ed3\u6027\u8bed\u53e5\uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
    • \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 \u201c\u53cc\u5f15\u53f7\u201d \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002
    • \u6d89\u53ca\u5230\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002
    • \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    \"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    // \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n// \u591a\u884c\n// \u6ce8\u91ca\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    "},{"location":"chapter_preface/suggestions/#022","title":"0.2.2. \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"

    \u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u56e0\u6b64\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002

    \u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5982\u679c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u5efa\u8bae\u4ee5\u56fe\u4e3a\u4e3b\u7ebf\uff0c\u4ee5\u6587\u5b57\uff08\u901a\u5e38\u4f4d\u4e8e\u56fe\u50cf\u4e0a\u65b9\uff09\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002

    Fig. \u52a8\u753b\u56fe\u89e3\u793a\u4f8b

    "},{"location":"chapter_preface/suggestions/#023","title":"0.2.3. \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"

    \u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728GitHub \u4ed3\u5e93\uff0c\u6e90\u4ee3\u7801\u5305\u542b\u8be6\u7ec6\u6ce8\u91ca\uff0c\u5e76\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u76f4\u63a5\u8fd0\u884c\u3002

    \u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u4f60\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002\u5982\u679c\u65f6\u95f4\u5145\u88d5\uff0c\u5efa\u8bae\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u4e0e\u4ec5\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002

    Fig. \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b

    \u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002

    \u7b2c\u4e8c\u6b65\uff1a\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\u3002

    git clone https://github.com/krahets/hello-algo.git\n

    \u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201cDownload ZIP\u201d\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002

    Fig. \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801

    \u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u679c\u4ee3\u7801\u5757\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\uff0c\u5219\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 codes \u6587\u4ef6\u5939\u4e2d\u627e\u5230\u76f8\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002

    Fig. \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6

    "},{"location":"chapter_preface/suggestions/#024","title":"0.2.4. \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"

    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u201c\u60ef\u7740\u201d\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5176\u4ed6\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u5f97\u5230\u56de\u590d\u3002

    \u540c\u65f6\uff0c\u4e5f\u5e0c\u671b\u60a8\u80fd\u5728\u8bc4\u8bba\u533a\u591a\u82b1\u4e9b\u65f6\u95f4\u3002\u4e00\u65b9\u9762\uff0c\u60a8\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u8fd9\u5c06\u6709\u52a9\u4e8e\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5e0c\u671b\u60a8\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\u3001\u5206\u4eab\u60a8\u7684\u89c1\u89e3\uff0c\u8ba9\u5927\u5bb6\u5171\u540c\u5b66\u4e60\u548c\u8fdb\u6b65\u3002

    Fig. \u8bc4\u8bba\u533a\u793a\u4f8b

    "},{"location":"chapter_preface/suggestions/#025","title":"0.2.5. \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"

    \u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\uff1a

    1. \u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u5185\u5bb9\u3002
    2. \u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u5251\u6307 Offer\u548cLeetCode Hot 100\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3-5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002
    3. \u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002

    \u4f5c\u4e3a\u4e00\u672c\u5165\u95e8\u6559\u7a0b\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u7b2c\u4e00\u9636\u6bb5\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u7b2c\u4e8c\u548c\u7b2c\u4e09\u9636\u6bb5\u7684\u5b66\u4e60\u3002

    Fig. \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf

    "},{"location":"chapter_preface/summary/","title":"0.3. \u00a0 \u5c0f\u7ed3","text":"
    • \u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u60a8\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u5185\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002
    • \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
    • \u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u7c4d\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
    • \u4e66\u5185\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002
    • \u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u6253\u4ee3\u7801\u3002
    • \u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002
    "},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"

    [1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).

    [2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).

    [3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.

    [4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.

    [5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.

    [6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.

    [7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.

    [8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition).

    [9] Aston Zhang, et al. Dive into Deep Learning.

    "},{"location":"chapter_searching/","title":"10. \u00a0 \u641c\u7d22","text":""},{"location":"chapter_searching/binary_search/","title":"10.1. \u00a0 \u4e8c\u5206\u67e5\u627e","text":"

    \u300c\u4e8c\u5206\u67e5\u627e Binary Search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u51cf\u5c11\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\uff0c\u6570\u7ec4\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 target \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002

    \u5bf9\u4e8e\u4e0a\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u5176\u4e2d\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u201c\u95ed\u533a\u95f4\u201d\uff0c\u5373\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002

    \u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u4e2a\u6b65\u9aa4\uff1a

    1. \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\space \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002
    2. \u5224\u65ad nums[m] \u548c target \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a
      1. \u5f53 nums[m] < target \u65f6\uff0c\u8bf4\u660e target \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \uff1b
      2. \u5f53 nums[m] > target \u65f6\uff0c\u8bf4\u660e target \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \uff1b
      3. \u5f53 nums[m] = target \u65f6\uff0c\u8bf4\u660e\u627e\u5230 target \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \uff1b

    \u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002

    <0><1><2><3><4><5><6><7>

    \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f int \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa int \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int> &nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.size() - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.py
    def binary_search(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n# \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j = 0, len(nums) - 1\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j:\n# \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u9700\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
    binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j := 0, len(nums)-1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nfor i <= j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] < target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
    binary_search.c
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.Length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i = 0\nvar j = nums.count - 1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i: usize = 0;\nvar j: usize = nums.items.len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.dart
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List<int> nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n

    \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u533a\u95f4\uff0c\u56e0\u6b64\u4e8c\u5206\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002\u6307\u9488 i , j \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002

    "},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1. \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"

    \u9664\u4e86\u4e0a\u8ff0\u7684\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j]\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(vector<int> &nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.size();\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.py
    def binary_search_lcro(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\"\"\"\n# \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j = 0, len(nums)\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\nelif nums[m] > target:\nj = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
    binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j := 0, len(nums)\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nfor i < j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearchLCRO(nums, target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] < target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nelse return m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
    binary_search.c
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.Length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i = 0\nvar j = nums.count\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i: usize = 0;\nvar j: usize = nums.items.len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.dart
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List<int> nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002

    \u5728\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u6cd5\u4e2d\uff0c\u7531\u4e8e\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u6307\u9488 \\(i\\) \u548c \\(j\\) \u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002

    Fig. \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49

    "},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2. \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"

    \u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\uff1a

    • \u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
    • \u4e8c\u5206\u67e5\u627e\u65e0\u9700\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002

    \u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
    • \u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700\u8981 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
    "},{"location":"chapter_searching/binary_search_edge/","title":"10.2. \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":"

    \u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u9898\u76ee\u89c4\u5b9a\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\u3002\u5982\u679c\u76ee\u6807\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u591a\u6b21\u51fa\u73b0\uff0c\u4e0a\u8282\u4ecb\u7ecd\u7684\u65b9\u6cd5\u53ea\u80fd\u4fdd\u8bc1\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a\u76ee\u6807\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u7d22\u5f15\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11\u76ee\u6807\u5143\u7d20\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 nums \uff0c\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 target \u5728\u6570\u7ec4\u4e2d\u9996\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002

    "},{"location":"chapter_searching/binary_search_edge/#1021","title":"10.2.1. \u00a0 \u7ebf\u6027\u65b9\u6cd5","text":"

    \u4e3a\u4e86\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u8fb9\u7684 target \uff0c\u6211\u4eec\u53ef\u4ee5\u5206\u4e3a\u4e24\u6b65\uff1a

    1. \u8fdb\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5b9a\u4f4d\u5230\u4efb\u610f\u4e00\u4e2a target \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \uff1b
    2. \u4ee5\u7d22\u5f15 \\(k\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u627e\u5230\u6700\u5de6\u8fb9\u7684 target \u8fd4\u56de\u5373\u53ef\u3002

    Fig. \u7ebf\u6027\u67e5\u627e\u6700\u5de6\u8fb9\u7684\u5143\u7d20

    \u8fd9\u4e2a\u65b9\u6cd5\u867d\u7136\u6709\u6548\uff0c\u4f46\u7531\u4e8e\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5f53\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 target \u65f6\u6548\u7387\u8f83\u4f4e\u3002

    "},{"location":"chapter_searching/binary_search_edge/#1022","title":"10.2.2. \u00a0 \u4e8c\u5206\u65b9\u6cd5","text":"

    \u8003\u8651\u4ec5\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u89e3\u51b3\u8be5\u95ee\u9898\u3002\u6574\u4f53\u7b97\u6cd5\u6d41\u7a0b\u4e0d\u53d8\uff0c\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad target \u548c nums[m] \u5927\u5c0f\u5173\u7cfb\uff1a

    • \u5f53 nums[m] < target \u6216 nums[m] > target \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 target \uff0c\u56e0\u6b64\u91c7\u53d6\u4e0e\u4e0a\u8282\u4ee3\u7801\u76f8\u540c\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 target \u9760\u8fd1\u3002
    • \u5f53 nums[m] == target \u65f6\uff0c\u8bf4\u660e\u201c\u5c0f\u4e8e target \u7684\u5143\u7d20\u201d\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e target \u7684\u5143\u7d20\u9760\u8fd1\u3002

    \u4e8c\u5206\u67e5\u627e\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 target \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(i\\) \u5373\u53ef\u3002

    <1><2><3><4><5><6><7><8>

    \u6ce8\u610f\uff0c\u6570\u7ec4\u53ef\u80fd\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20 target \u3002\u56e0\u6b64\u5728\u51fd\u6570\u8fd4\u56de\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u5224\u65ad nums[i] \u4e0e target \u662f\u5426\u76f8\u7b49\uff0c\u4ee5\u53ca\u7d22\u5f15 \\(i\\) \u662f\u5426\u8d8a\u754c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_edge.java
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.length || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.cpp
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(vector<int> &nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.size() || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.py
    def binary_search_left_edge(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i <= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nj = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\nif i == len(nums) or nums[i] != target:\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i\n
    binary_search_edge.go
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i <= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] < target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n}\n}\nif i == len(nums) || nums[i] != target {\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\nreturn i\n}\n
    binary_search_edge.js
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchLeftEdge(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif (i == nums.length || nums[i] != target) {\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i;\n}\n
    binary_search_edge.ts
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchLeftEdge(nums: number[], target: number): number {\nlet i = 0, j = nums.length - 1;  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2);  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif (i == nums.length || nums[i] != target) {\nreturn -1;  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i;\n}\n
    binary_search_edge.c
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int *nums, int size, int target) {\nint i = 0, j = size - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == size || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.cs
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.Length || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.swift
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nvar i = 0\nvar j = nums.count - 1\nwhile i <= j {\nlet m = i + (j - 1) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] > target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif i == nums.count || nums[i] != target {\nreturn -1 // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i\n}\n
    binary_search_edge.zig
    [class]{}-[func]{binarySearchLeftEdge}\n
    binary_search_edge.dart
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(List<int> nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u95f4\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.length || nums[i] != target) return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    "},{"location":"chapter_searching/binary_search_edge/#1023","title":"10.2.3. \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"

    \u7c7b\u4f3c\u5730\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u8fb9\u7684 target \u3002\u5f53 nums[m] == target \u65f6\uff0c\u8bf4\u660e\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c i = m + 1 \uff0c\u4f7f\u5f97\u6307\u9488 \\(i\\) \u5411\u5927\u4e8e target \u7684\u5143\u7d20\u9760\u8fd1\u3002

    \u5b8c\u6210\u4e8c\u5206\u540e\uff0c\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u6700\u53f3\u8fb9\u7684 target \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(j\\) \u5373\u53ef\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_edge.java
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.cpp
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(vector<int> &nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.py
    def binary_search_right_edge(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i <= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\ni = m + 1  # \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\nif j < 0 or nums[j] != target:\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j\n
    binary_search_edge.go
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchRightEdge(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i <= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] < target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n}\n}\nif j < 0 || nums[j] != target {\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\nreturn j\n}\n
    binary_search_edge.js
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchRightEdge(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif (j < 0 || nums[j] != target) {\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j;\n}\n
    binary_search_edge.ts
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchRightEdge(nums: number[], target: number): number {\nlet i = 0, j = nums.length - 1;  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2);  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1;  // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif (j < 0 || nums[j] != target) {\nreturn -1;  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j;\n}\n
    binary_search_edge.c
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int *nums, int size, int target) {\nint i = 0, j = size - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.cs
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.swift
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nvar i = 0\nvar j = nums.count - 1\nwhile i <= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] > target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1 // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif j < 0 || nums[j] != target {\nreturn -1 // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j\n}\n
    binary_search_edge.zig
    [class]{}-[func]{binarySearchRightEdge}\n
    binary_search_edge.dart
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(List<int> nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u95f4\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target) return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n

    \u89c2\u5bdf\u4e0b\u56fe\uff0c\u641c\u7d22\u6700\u53f3\u8fb9\u5143\u7d20\u65f6\u6307\u9488 \\(j\\) \u7684\u4f5c\u7528\u4e0e\u641c\u7d22\u6700\u5de6\u8fb9\u5143\u7d20\u65f6\u6307\u9488 \\(i\\) \u7684\u4f5c\u7528\u4e00\u81f4\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u641c\u7d22\u6700\u5de6\u8fb9\u5143\u7d20\u548c\u6700\u53f3\u8fb9\u5143\u7d20\u7684\u5b9e\u73b0\u662f\u955c\u50cf\u5bf9\u79f0\u7684\u3002

    Fig. \u67e5\u627e\u6700\u5de6\u8fb9\u548c\u6700\u53f3\u8fb9\u5143\u7d20\u7684\u5bf9\u79f0\u6027

    Tip

    \u4ee5\u4e0a\u4ee3\u7801\u91c7\u53d6\u7684\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.3. \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"

    \u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 target \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a target \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/#1031","title":"10.3.1. \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"

    \u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a target \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002

    Fig. \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart two_sum.java
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
    two_sum.cpp
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int> &nums, int target) {\nint size = nums.size();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn {i, j};\n}\n}\nreturn {};\n}\n
    two_sum.py
    def two_sum_brute_force(nums: list[int], target: int) -> list[int]:\n\"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n# \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in range(len(nums) - 1):\nfor j in range(i + 1, len(nums)):\nif nums[i] + nums[j] == target:\nreturn [i, j]\nreturn []\n
    two_sum.go
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\nsize := len(nums)\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i := 0; i < size-1; i++ {\nfor j := i + 1; i < size; j++ {\nif nums[i]+nums[j] == target {\nreturn []int{i, j}\n}\n}\n}\nreturn nil\n}\n
    two_sum.js
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
    two_sum.ts
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
    two_sum.c
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\nfor (int i = 0; i < numsSize; ++i) {\nfor (int j = i + 1; j < numsSize; ++j) {\nif (nums[i] + nums[j] == target) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = i, res[1] = j;\n*returnSize = 2;\nreturn res;\n}\n}\n}\n*returnSize = 0;\nreturn NULL;\n}\n
    two_sum.cs
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.Length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
    two_sum.swift
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in nums.indices.dropLast() {\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[i] + nums[j] == target {\nreturn [i, j]\n}\n}\n}\nreturn [0]\n}\n
    two_sum.zig
    // \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\nvar size: usize = nums.len;\nvar i: usize = 0;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nwhile (i < size - 1) : (i += 1) {\nvar j = i + 1;\nwhile (j < size) : (j += 1) {\nif (nums[i] + nums[j] == target) {\nreturn [_]i32{@intCast(i32, i), @intCast(i32, j)};\n}\n}\n}\nreturn null;\n}\n
    two_sum.dart
    /* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList<int> twoSumBruteForce(List<int> nums, int target) {\nint size = nums.length;\nfor (var i = 0; i < size - 1; i++) {\nfor (var j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target) return [i, j];\n}\n}\nreturn [0];\n}\n

    \u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/#1032","title":"10.3.2. \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"

    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\uff1a

    1. \u5224\u65ad\u6570\u5b57 target - nums[i] \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1b
    2. \u5c06\u952e\u503c\u5bf9 nums[i] \u548c\u7d22\u5f15 i \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\uff1b
    <1><2><3>

    \u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart two_sum.java
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nMap<Integer, Integer> dic = new HashMap<>();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn new int[] { dic.get(target - nums[i]), i };\n}\ndic.put(nums[i], i);\n}\nreturn new int[0];\n}\n
    two_sum.cpp
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int> &nums, int target) {\nint size = nums.size();\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nunordered_map<int, int> dic;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.find(target - nums[i]) != dic.end()) {\nreturn {dic[target - nums[i]], i};\n}\ndic.emplace(nums[i], i);\n}\nreturn {};\n}\n
    two_sum.py
    def two_sum_hash_table(nums: list[int], target: int) -> list[int]:\n\"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n# \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\ndic = {}\n# \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in range(len(nums)):\nif target - nums[i] in dic:\nreturn [dic[target - nums[i]], i]\ndic[nums[i]] = i\nreturn []\n
    two_sum.go
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nhashTable := map[int]int{}\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor idx, val := range nums {\nif preIdx, ok := hashTable[target-val]; ok {\nreturn []int{preIdx, idx}\n}\nhashTable[val] = idx\n}\nreturn nil\n}\n
    two_sum.js
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m = {};\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nif (m[target - nums[i]] !== undefined) {\nreturn [m[target-nums[i]], i];\n} else {\nm[nums[i]] = i;\n}\n}\nreturn [];\n}\n
    two_sum.ts
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m: Map<number, number> = new Map();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nlet index = m.get(target - nums[i]);\nif (index !== undefined) {\nreturn [index, i];\n} else {\nm.set(nums[i], i);\n}\n}\nreturn [];\n}\n
    two_sum.c
    /* \u54c8\u5e0c\u8868 */\nstruct hashTable {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n};\ntypedef struct hashTable hashTable;\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nhashTable *find(hashTable *h, int key) {\nhashTable *tmp;\nHASH_FIND_INT(h, &key, tmp);\nreturn tmp;\n}\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(hashTable *h, int key, int val) {\nhashTable *t = find(h, key);\nif (t == NULL) {\nhashTable *tmp = malloc(sizeof(hashTable));\ntmp->key = key, tmp->val = val;\nHASH_ADD_INT(h, key, tmp);\n} else {\nt->val = val;\n}\n}\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\nhashTable *hashtable = NULL;\nfor (int i = 0; i < numsSize; i++) {\nhashTable *t = find(hashtable, target - nums[i]);\nif (t != NULL) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = t->val, res[1] = i;\n*returnSize = 2;\nreturn res;\n}\ninsert(hashtable, nums[i], i);\n}\n*returnSize = 0;\nreturn NULL;\n}\n
    two_sum.cs
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.Length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nDictionary<int, int> dic = new();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.ContainsKey(target - nums[i])) {\nreturn new int[] { dic[target - nums[i]], i };\n}\ndic.Add(nums[i], i);\n}\nreturn new int[0];\n}\n
    two_sum.swift
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic: [Int: Int] = [:]\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in nums.indices {\nif let j = dic[target - nums[i]] {\nreturn [j, i]\n}\ndic[nums[i]] = i\n}\nreturn [0]\n}\n
    two_sum.zig
    // \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\nvar size: usize = nums.len;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\ndefer dic.deinit();\nvar i: usize = 0;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nwhile (i < size) : (i += 1) {\nif (dic.contains(target - nums[i])) {\nreturn [_]i32{dic.get(target - nums[i]).?, @intCast(i32, i)};\n}\ntry dic.put(nums[i], @intCast(i32, i));\n}\nreturn null;\n}\n
    two_sum.dart
    /* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList<int> twoSumHashTable(List<int> nums, int target) {\nint size = nums.length;\nMap<int, int> dic = HashMap();\nfor (var i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn [dic[target - nums[i]]!, i];\n}\ndic.putIfAbsent(nums[i], () => i);\n}\nreturn [0];\n}\n

    \u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u4f4e\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002

    \u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.4. \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"

    \u300c\u641c\u7d22\u7b97\u6cd5 Searching Algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002

    \u6839\u636e\u5b9e\u73b0\u601d\u8def\uff0c\u641c\u7d22\u7b97\u6cd5\u603b\u4f53\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a

    • \u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002
    • \u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002

    \u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1041","title":"10.4.1. \u00a0 \u66b4\u529b\u641c\u7d22","text":"

    \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    • \u300c\u7ebf\u6027\u641c\u7d22\u300d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
    • \u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u662f\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\u4e3a\u6b62\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002

    \u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u9700\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002

    \u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1042","title":"10.4.2. \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"

    \u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    • \u300c\u4e8c\u5206\u67e5\u627e\u300d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002
    • \u300c\u54c8\u5e0c\u67e5\u627e\u300d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002
    • \u300c\u6811\u67e5\u627e\u300d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    \u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002

    \u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u652f\u3002

    Note

    \u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u5173\u6ce8\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1043","title":"10.4.3. \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"

    \u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u5728\u8be5\u6570\u636e\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u4e0b\u56fe\u6240\u793a\u3002

    Fig. \u591a\u79cd\u641c\u7d22\u7b56\u7565

    \u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u4e0b\u8868\u6240\u793a\u3002

    \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f

    \u9664\u4e86\u4ee5\u4e0a\u8868\u683c\u5185\u5bb9\uff0c\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002

    \u7ebf\u6027\u641c\u7d22

    • \u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u9700\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002
    • \u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002
    • \u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002

    \u4e8c\u5206\u67e5\u627e

    • \u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
    • \u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002
    • \u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002

    \u54c8\u5e0c\u67e5\u627e

    • \u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
    • \u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002
    • \u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002
    • \u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002

    \u6811\u67e5\u627e

    • \u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u79bb\u6563\u5b58\u50a8\u7684\u3002
    • \u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002
    • \u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
    • \u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002
    "},{"location":"chapter_searching/summary/","title":"10.5. \u00a0 \u5c0f\u7ed3","text":"
    • \u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u4e8e\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u5b9e\u73b0\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
    • \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u9700\u5bf9\u6570\u636e\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002
    • \u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002
    • \u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002
    • \u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u5408\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u9700\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002
    • \u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002
    "},{"location":"chapter_sorting/","title":"11. \u00a0 \u6392\u5e8f","text":""},{"location":"chapter_sorting/bubble_sort/","title":"11.3. \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"

    \u300c\u5192\u6ce1\u6392\u5e8f Bubble Sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002

    \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u4e0a\u8ff0\u8fc7\u7a0b\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 > \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u5b83\u4fe9\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002

    <1><2><3><4><5><6><7>

    "},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u4e3a\uff1a

    1. \u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c
    2. \u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
    3. \u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
    4. \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u9700\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002

    Fig. \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\n}\n}\n}\n}\n
    bubble_sort.py
    def bubble_sort(nums: list[int]) -> None:\n\"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\n
    bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\n}\n}\n}\n}\n
    bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.c
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i < size - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < size - 1 - i; j++) {\nif (nums[j] > nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\n}\n}\n}\n}\n
    bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\n}\n}\n}\n}\n
    bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.dart
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2. \u00a0 \u6548\u7387\u4f18\u5316","text":"

    \u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d flag \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002

    \u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\nboolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.py
    def bubble_sort_with_flag(nums: list[int]) -> None:\n\"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\nflag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\nflag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\nif not flag:\nbreak  # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
    bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\nflag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
    bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.c
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i < size - 1; i++) {\nbool flag = false;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < size - 1 - i; j++) {\nif (nums[j] > nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\nflag = true;\n}\n}\nif (!flag)\nbreak;\n}\n}\n
    bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\nvar flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
    bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;\n}\n}\nif (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.dart
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\) , \\(n - 2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \uff0c\u603b\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \u3002\u5728\u5f15\u5165 flag \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002
    "},{"location":"chapter_sorting/bucket_sort/","title":"11.8. \u00a0 \u6876\u6392\u5e8f","text":"

    \u524d\u8ff0\u7684\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u6c34\u5e73\u3002

    \u300c\u6876\u6392\u5e8f Bucket Sort\u300d\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002

    "},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\uff1b
    2. \u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u672c\u6587\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\uff1b
    3. \u6309\u7167\u6876\u7684\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\uff0c\u5408\u5e76\u7ed3\u679c\uff1b

    Fig. \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bucket_sort.java
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length / 2;\nList<List<Float>> buckets = new ArrayList<>();\nfor (int i = 0; i < k; i++) {\nbuckets.add(new ArrayList<>());\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int) (num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets.get(i).add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List<Float> bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nCollections.sort(bucket);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List<Float> bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.cpp
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(vector<float> &nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.size() / 2;\nvector<vector<float>> buckets(k);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = num * k;\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].push_back(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (vector<float> &bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort(bucket.begin(), bucket.end());\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (vector<float> &bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.py
    def bucket_sort(nums: list[float]) -> None:\n\"\"\"\u6876\u6392\u5e8f\"\"\"\n# \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk = len(nums) // 2\nbuckets = [[] for _ in range(k)]\n# 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums:\n# \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni = int(num * k)\n# \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n# 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f5\nfor bucket in buckets:\n# \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort()\n# 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni = 0\nfor bucket in buckets:\nfor num in bucket:\nnums[i] = num\ni += 1\n
    bucket_sort.go
    /* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk := len(nums) / 2\nbuckets := make([][]float64, k)\nfor i := 0; i < k; i++ {\nbuckets[i] = make([]float64, 0)\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor _, num := range nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni := int(num * float64(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i] = append(buckets[i], num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i := 0; i < k; i++ {\n// \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort.Float64s(buckets[i])\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni := 0\nfor _, bucket := range buckets {\nfor _, num := range bucket {\nnums[i] = num\ni++\n}\n}\n}\n
    bucket_sort.js
    /* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets = [];\nfor (let i = 0; i < k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) => a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.ts
    /* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets: number[][] = [];\nfor (let i = 0; i < k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) => a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.c
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = size / 2;\nfloat **buckets = calloc(k, sizeof(float *));\nfor (int i = 0; i < k; i++) {\n// \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\nbuckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (int i = 0; i < size; i++) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint bucket_idx = nums[i] * k;\nint j = 0;\n// \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\nwhile (buckets[bucket_idx][j] > 0 && buckets[bucket_idx][j] < nums[i]) {\nj++;\n}\nfloat temp = nums[i];\nwhile (j < ARRAY_SIZE && buckets[bucket_idx][j] > 0) {\nswap(&temp, &buckets[bucket_idx][j]);\nj++;\n}\nbuckets[bucket_idx][j] = temp;\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (int i = 0; i < k; i++) {\nqsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nfor (int i = 0, j = 0; j < k; j++) {\nfor (int l = 0; l < ARRAY_SIZE; l++) {\nif (buckets[j][l] > 0) {\nnums[i++] = buckets[j][l];\n}\n}\n}\n// \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\nfor (int i = 0; i < k; i++) {\nfree(buckets[i]);\n}\nfree(buckets);\n}\n
    bucket_sort.cs
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.Length / 2;\nList<List<float>> buckets = new List<List<float>>();\nfor (int i = 0; i < k; i++) {\nbuckets.Add(new List<float>());\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nforeach (float num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int) (num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].Add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nforeach (List<float> bucket in buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.Sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint j = 0;\nforeach (List<float> bucket in buckets) {\nforeach (float num in bucket) {\nnums[j++] = num;\n}\n}\n}\n
    bucket_sort.swift
    /* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nlet k = nums.count / 2\nvar buckets = (0 ..< k).map { _ in [Double]() }\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nlet i = Int(num * Double(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i in buckets.indices {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbuckets[i].sort()\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nvar i = nums.startIndex\nfor bucket in buckets {\nfor num in bucket {\nnums[i] = num\nnums.formIndex(after: &i)\n}\n}\n}\n
    bucket_sort.zig
    [class]{}-[func]{bucketSort}\n
    bucket_sort.dart
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(List<double> nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length ~/ 2;\nList<List<double>> buckets = List.generate(k, (index) => []);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (double num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (num * k).toInt();\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List<double> bucket in buckets) {\nbucket.sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List<double> bucket in buckets) {\nfor (double num in bucket) {\nnums[i++] = num;\n}\n}\n}\n

    \u6876\u6392\u5e8f\u7684\u9002\u7528\u573a\u666f\u662f\u4ec0\u4e48\uff1f

    \u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002

    "},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386 \\(n\\) \u4e2a\u6876\uff0c\u82b1\u8d39 \\(O(k)\\) \u65f6\u95f4\u3002
    • \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u574f\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002
    "},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3. \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"

    \u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4efd\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002

    \u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u4e2a\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u4e2a\u9012\u5f52\u6811\uff0c\u4f7f\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002

    Fig. \u9012\u5f52\u5212\u5206\u6876

    \u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002

    Fig. \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876

    "},{"location":"chapter_sorting/counting_sort/","title":"11.9. \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"

    \u300c\u8ba1\u6570\u6392\u5e8f Counting Sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002

    "},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1. \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"

    \u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\u3002\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u6570\u7ec4\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 counter \uff1b
    2. \u501f\u52a9 counter \u7edf\u8ba1 nums \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d counter[num] \u5bf9\u5e94\u6570\u5b57 num \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 nums\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a num\uff09\uff0c\u6bcf\u8f6e\u5c06 counter[num] \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002
    3. \u7531\u4e8e counter \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u88ab\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 counter \uff0c\u6839\u636e\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5c06\u5b83\u4eec\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 nums \u5373\u53ef\u3002

    Fig. \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int> &nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.py
    def counting_sort_naive(nums: list[int]) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = 0\nfor num in nums:\nm = max(m, num)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\ni = 0\nfor num in range(m + 1):\nfor _ in range(counter[num]):\nnums[i] = num\ni += 1\n
    counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nfor i, num := 0, 0; num < m+1; num++ {\nfor j := 0; j < counter[num]; j++ {\nnums[i] = num\ni++\n}\n}\n}\n
    counting_sort.js
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num < m + 1; num++) {\nfor (let j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.ts
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array<number>(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num < m + 1; num++) {\nfor (let j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.c
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i < size; i++) {\nif (nums[i] > m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = malloc(sizeof(int) * m);\nfor (int i = 0; i < size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.cs
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nvar i = 0\nfor num in stride(from: 0, to: m + 1, by: 1) {\nfor _ in stride(from: 0, to: counter[num], by: 1) {\nnums[i] = num\ni += 1\n}\n}\n}\n
    counting_sort.zig
    [class]{}-[func]{countingSortNaive}\n
    counting_sort.dart
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List<int> nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num in nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nList<int> counter = List.filled(m + 1, 0);\nfor (int num in nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n

    \u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb

    \u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 counter \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u662f\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002

    "},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2. \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"

    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u53d1\u73b0\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 3. \u5c31\u5931\u6548\u4e86\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u8981\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002

    \u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 counter \u7684\u300c\u524d\u7f00\u548c\u300d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 i \u5904\u7684\u524d\u7f00\u548c prefix[i] \u7b49\u4e8e\u6570\u7ec4\u524d i \u4e2a\u5143\u7d20\u4e4b\u548c\uff0c\u5373

    \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\]

    \u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0cprefix[num] - 1 \u4ee3\u8868\u5143\u7d20 num \u5728\u7ed3\u679c\u6570\u7ec4 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 nums \u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\uff1a

    1. \u5c06 num \u586b\u5165\u6570\u7ec4 res \u7684\u7d22\u5f15 prefix[num] - 1 \u5904\uff1b
    2. \u4ee4\u524d\u7f00\u548c prefix[num] \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\uff1b

    \u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 res \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 res \u8986\u76d6\u539f\u6570\u7ec4 nums \u5373\u53ef\u3002

    <1><2><3><4><5><6><7><8>

    \u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int> &nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.size();\nvector<int> res(n);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums = res;\n}\n
    counting_sort.py
    def counting_sort(nums: list[int]) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = max(nums)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n# \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in range(m):\ncounter[i + 1] += counter[i]\n# 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n# \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn = len(nums)\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nnum = nums[i]\nres[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n# \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\n
    counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i := 0; i < m; i++ {\ncounter[i+1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn := len(nums)\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nnum := nums[i]\n// \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\nres[counter[num]-1] = num\n// \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\ncounter[num]--\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\ncopy(nums, res)\n}\n
    counting_sort.js
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res = new Array(n);\nfor (let i = n - 1; i >= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.ts
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array<number>(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res: number[] = new Array<number>(n);\nfor (let i = n - 1; i >= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.c
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i < size; i++) {\nif (nums[i] > m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = malloc(sizeof(int) * m);\nfor (int i = 0; i < size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint *res = malloc(sizeof(int) * size);\nfor (int i = size - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nmemcpy(nums, res, size * sizeof(int));\n}\n
    counting_sort.cs
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.Length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in stride(from: 0, to: m, by: 1) {\ncounter[i + 1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nvar res = Array(repeating: 0, count: nums.count)\nfor i in stride(from: nums.count - 1, through: 0, by: -1) {\nlet num = nums[i]\nres[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in stride(from: 0, to: nums.count, by: 1) {\nnums[i] = res[i]\n}\n}\n
    counting_sort.zig
    [class]{}-[func]{countingSort}\n
    counting_sort.dart
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List<int> nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num in nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nList<int> counter = List.filled(m + 1, 0);\nfor (int num in nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nList<int> res = List.filled(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums.setAll(0, res);\n}\n
    "},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 nums \u548c\u904d\u5386 counter \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 res \u548c counter \u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 res \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 nums \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 nums \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002
    "},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4. \u00a0 \u5c40\u9650\u6027","text":"

    \u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u5de5\u4f5c\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002

    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u8981\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u88ab\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u5373\u53ef\u3002

    \u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002

    "},{"location":"chapter_sorting/heap_sort/","title":"11.7. \u00a0 \u5806\u6392\u5e8f","text":"

    Tip

    \u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u300c\u5806\u300d\u7ae0\u8282\u3002

    \u300c\u5806\u6392\u5e8f Heap Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\uff1a

    1. \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
    2. \u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002

    \u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002

    "},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
    2. \u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002
    3. \u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002
    4. \u5faa\u73af\u6267\u884c\u7b2c 2. \u548c 3. \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002

    \u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c 2. \u548c 3. \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002

    <1><2><3><4><5><6><7><8><9><10><11><12>

    \u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u5806\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316\uff08Sift Down\uff09\u7684\u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 Sift Down \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart heap_sort.java
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length / 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.cpp
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector<int> &nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector<int> &nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.size() / 2 - 1; i >= 0; --i) {\nsiftDown(nums, nums.size(), i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.size() - 1; i > 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.py
    def sift_down(nums: list[int], n: int, i: int):\n\"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl = 2 * i + 1\nr = 2 * i + 2\nma = i\nif l < n and nums[l] > nums[ma]:\nma = l\nif r < n and nums[r] > nums[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nnums[i], nums[ma] = nums[ma], nums[i]\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\ndef heap_sort(nums: list[int]):\n\"\"\"\u5806\u6392\u5e8f\"\"\"\n# \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(len(nums) // 2 - 1, -1, -1):\nsift_down(nums, len(nums), i)\n# \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in range(len(nums) - 1, 0, -1):\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums[0], nums[i] = nums[i], nums[0]\n# \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsift_down(nums, i, 0)\n
    heap_sort.go
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl := 2*i + 1\nr := 2*i + 2\nma := i\nif l < n && (*nums)[l] > (*nums)[ma] {\nma = l\n}\nif r < n && (*nums)[r] > (*nums)[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i := len(*nums)/2 - 1; i >= 0; i-- {\nsiftDown(nums, len(*nums), i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i := len(*nums) - 1; i > 0; i-- {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0)\n}\n}\n
    heap_sort.js
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l < n && nums[l] > nums[ma]) {\nma = l;\n}\nif (r < n && nums[r] > nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.ts
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l < n && nums[l] > nums[ma]) {\nma = l;\n}\nif (r < n && nums[r] > nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.c
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\nwhile (1) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = n / 2 - 1; i >= 0; --i) {\nsiftDown(nums, n, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = n - 1; i > 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.cs
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(nums[ma], nums[i]) = (nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.Length / 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.Length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(nums[i], nums[0]) = (nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.swift
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1\nlet r = 2 * i + 2\nvar ma = i\nif l < n, nums[l] > nums[ma] {\nma = l\n}\nif r < n, nums[r] > nums[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nnums.swapAt(i, ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\nsiftDown(nums: &nums, n: nums.count, i: i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums.swapAt(0, i)\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums: &nums, n: i, i: 0)\n}\n}\n
    heap_sort.zig
    [class]{}-[func]{siftDown}\n[class]{}-[func]{heapSort}\n
    heap_sort.dart
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List<int> nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma]) ma = l;\nif (r < n && nums[r] > nums[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List<int> nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length ~/ 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    "},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002
    "},{"location":"chapter_sorting/insertion_sort/","title":"11.4. \u00a0 \u63d2\u5165\u6392\u5e8f","text":"

    \u300c\u63d2\u5165\u6392\u5e8f Insertion Sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002

    \u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002

    \u56de\u5fc6\u6570\u7ec4\u7684\u5143\u7d20\u63d2\u5165\u64cd\u4f5c\uff0c\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a base \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 base \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u5c06 base \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002

    Fig. \u5355\u6b21\u63d2\u5165\u64cd\u4f5c

    "},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
    2. \u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    3. \u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    4. \u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002

    Fig. \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart insertion_sort.java
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.cpp
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.size(); i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.py
    def insertion_sort(nums: list[int]) -> None:\n\"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n# \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\nfor i in range(1, len(nums)):\nbase = nums[i]\nj = i - 1\n# \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0 and nums[j] > base:\nnums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\nnums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
    insertion_sort.go
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := 1; i < len(nums); i++ {\nbase := nums[i]\nj := i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nfor j >= 0 && nums[j] > base {\nnums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--\n}\nnums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.js
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i < nums.length; i++) {\nlet base = nums[i],\nj = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.ts
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i < nums.length; i++) {\nconst base = nums[i];\nlet j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.c
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < size; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\n// \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nnums[j + 1] = nums[j];\nj--;\n}\n// \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\nnums[j + 1] = base;\n}\n}\n
    insertion_sort.cs
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.Length; i++) {\nint bas = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > bas) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.swift
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor i in stride(from: 1, to: nums.count, by: 1) {\nlet base = nums[i]\nvar j = i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0, nums[j] > base {\nnums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\n}\nnums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.zig
    // \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nvar i: usize = 1;\nwhile (i < nums.len) : (i += 1) {\nvar base = nums[i];\nvar j: usize = i;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 1 and nums[j - 1] > base) : (j -= 1) {\nnums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n}\nnums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.dart
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    "},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\) , \\(n-2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\(\\frac{(n - 1) n}{2}\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002
    "},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3. \u00a0 \u63d2\u5165\u6392\u5e8f\u4f18\u52bf","text":"

    \u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u6bd4\u5feb\u901f\u6392\u5e8f\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002

    \u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u4f5c\u7528\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u56e0\u7d20\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u90fd\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002

    \u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\u3002\u8fd9\u662f\u56e0\u4e3a\uff1a

    • \u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002
    • \u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002
    • \u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002
    "},{"location":"chapter_sorting/merge_sort/","title":"11.6. \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"

    \u300c\u5f52\u5e76\u6392\u5e8f Merge Sort\u300d\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u5b9e\u73b0\u6392\u5e8f\uff0c\u5305\u542b\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u4e24\u4e2a\u9636\u6bb5\uff1a

    1. \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\uff1b
    2. \u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\uff1b

    Fig. \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5

    "},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u957f\u5ea6\u4e3a 1 \uff1b

    1. \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 mid \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [left, mid] \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [mid + 1, right] \uff09\uff1b
    2. \u9012\u5f52\u6267\u884c\u6b65\u9aa4 1. \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5212\u5206\uff1b

    \u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002

    <1><2><3><4><5><6><7><8><9><10>

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u9012\u5f52\u987a\u5e8f\u4e0e\u4e8c\u53c9\u6811\u7684\u540e\u5e8f\u904d\u5386\u76f8\u540c\uff0c\u5177\u4f53\u6765\u770b\uff1a

    • \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002
    • \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart merge_sort.java
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = Arrays.copyOfRange(nums, left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn;                      // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.cpp
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(vector<int> &nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvector<int> tmp(nums.begin() + left, nums.begin() + right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int> &nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.py
    def merge(nums: list[int], left: int, mid: int, right: int) -> None:\n\"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n# \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n# \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\n# \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\ntmp = list(nums[left : right + 1])\n# \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleft_start = 0\nleft_end = mid - left\n# \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nright_start = mid + 1 - left\nright_end = right - left\n# i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni = left_start\nj = right_start\n# \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in range(left, right + 1):\n# \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > left_end:\nnums[k] = tmp[j]\nj += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelif j > right_end or tmp[i] <= tmp[j]:\nnums[k] = tmp[i]\ni += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse:\nnums[k] = tmp[j]\nj += 1\ndef merge_sort(nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif left >= right:\nreturn  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n# \u5212\u5206\u9636\u6bb5\nmid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n# \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n
    merge_sort.go
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums []int, left, mid, right int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy \u6a21\u5757\ntmp := make([]int, right-left+1)\nfor i := left; i <= right; i++ {\ntmp[i-left] = nums[i]\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleftStart, leftEnd := left-left, mid-left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nrightStart, rightEnd := mid+1-left, right-left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni, j := leftStart, rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k := left; k <= right; k++ {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj++\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni++\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j]\nj++\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right {\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nmid := (left + right) / 2\nmergeSort(nums, left, mid)\nmergeSort(nums, mid+1, right)\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n}\n
    merge_sort.js
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums, left, mid, right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left,\nleftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left,\nrightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart,\nj = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\nif (i > leftEnd) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nnums[k] = tmp[i++];\n} else {\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.ts
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left,\nleftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left,\nrightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart,\nj = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\nif (i > leftEnd) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.c
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int *nums, int left, int mid, int right) {\nint index;\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint tmp[right + 1 - left];\nfor (index = left; index < right + 1; index++) {\ntmp[index - left] = nums[index];\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.cs
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = nums[left..(right + 1)];\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.swift
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = Array(nums[left ..< (right + 1)])\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left\nlet leftEnd = mid - left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left\nlet rightEnd = right - left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart\nvar j = rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in left ... right {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse {\nnums[k] = tmp[j]\nj += 1\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums: &nums, left: left, mid: mid, right: right)\n}\n
    merge_sort.zig
    // \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\ndefer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar tmp = try mem_allocator.alloc(i32, right + 1 - left);\nstd.mem.copy(i32, tmp, nums[left..right+1]);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nvar leftStart = left - left;\nvar leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nvar rightStart = mid + 1 - left;\nvar rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart;\nvar j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nvar k = left;\nwhile (k <= right) : (k += 1) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j];\nj += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if  (j > rightEnd or tmp[i] <= tmp[j]) {\nnums[k] = tmp[i];\ni += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j];\nj += 1;\n}\n}\n}\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nvar mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\ntry mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\ntry mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\ntry merge(nums, left, mid, right);\n}\n
    merge_sort.dart
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(List<int> nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nList<int> tmp = nums.sublist(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List<int> nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n

    \u5408\u5e76\u65b9\u6cd5 merge() \u4ee3\u7801\u4e2d\u7684\u96be\u70b9\u5305\u62ec\uff1a

    • \u5728\u9605\u8bfb\u4ee3\u7801\u65f6\uff0c\u9700\u8981\u7279\u522b\u6ce8\u610f\u5404\u4e2a\u53d8\u91cf\u7684\u542b\u4e49\u3002nums \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a [left, right] \uff0c\u4f46\u7531\u4e8e tmp \u4ec5\u590d\u5236\u4e86 nums \u8be5\u533a\u95f4\u7684\u5143\u7d20\uff0c\u56e0\u6b64 tmp \u5bf9\u5e94\u533a\u95f4\u4e3a [0, right - left] \u3002
    • \u5728\u6bd4\u8f83 tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\u65f6\uff0c\u8fd8\u9700\u8003\u8651\u5b50\u6570\u7ec4\u904d\u5386\u5b8c\u6210\u540e\u7684\u7d22\u5f15\u8d8a\u754c\u95ee\u9898\uff0c\u5373 i > leftEnd \u548c j > rightEnd \u7684\u60c5\u51b5\u3002\u7d22\u5f15\u8d8a\u754c\u7684\u4f18\u5148\u7ea7\u662f\u6700\u9ad8\u7684\uff0c\u5982\u679c\u5de6\u5b50\u6570\u7ec4\u5df2\u7ecf\u88ab\u5408\u5e76\u5b8c\u4e86\uff0c\u90a3\u4e48\u4e0d\u9700\u8981\u7ee7\u7eed\u6bd4\u8f83\uff0c\u76f4\u63a5\u5408\u5e76\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u5373\u53ef\u3002
    "},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002
    "},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3. \u00a0 \u94fe\u8868\u6392\u5e8f *","text":"

    \u5f52\u5e76\u6392\u5e8f\u5728\u6392\u5e8f\u94fe\u8868\u65f6\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u7531\u4e8e\u94fe\u8868\u4ec5\u9700\u6539\u53d8\u6307\u9488\u5c31\u53ef\u5b9e\u73b0\u8282\u70b9\u7684\u589e\u5220\u64cd\u4f5c\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u9700\u521b\u5efa\u8f85\u52a9\u94fe\u8868\u3002
    • \u901a\u8fc7\u4f7f\u7528\u201c\u8fed\u4ee3\u5212\u5206\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u5212\u5206\u201d\uff0c\u53ef\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\uff1b

    \u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002

    "},{"location":"chapter_sorting/quick_sort/","title":"11.5. \u00a0 \u5feb\u901f\u6392\u5e8f","text":"

    \u300c\u5feb\u901f\u6392\u5e8f Quick Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002

    \u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u4e3a\uff1a

    1. \u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 i \u548c j \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\uff1b
    2. \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 i\uff08j\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\uff1b
    3. \u5faa\u73af\u6267\u884c\u6b65\u9aa4 2. \uff0c\u76f4\u5230 i \u548c j \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\uff1b

    \u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002

    <1><2><3><4><5><6><7><8><9>

    \u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u601d\u60f3

    \u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cpp
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int> &nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int> &nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.py
    def partition(self, nums: list[int], left: int, right: int) -> int:\n\"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
    quick_sort.go
    /* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.js
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.ts
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.c
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i < j && nums[i] <= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n
    quick_sort.cs
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.swift
    /* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\nlet tmp = nums[i]\nnums[i] = nums[j]\nnums[j] = tmp\n}\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left\nvar j = right\nwhile i < j {\nwhile i < j, nums[j] >= nums[left] {\nj -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i < j, nums[i] <= nums[left] {\ni += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.zig
    // \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\nvar tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.dart
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List<int> nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List<int> nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
    1. \u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\uff1b
    2. \u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u300c\u54e8\u5175\u5212\u5206\u300d\uff1b
    3. \u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\uff1b

    Fig. \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int> &nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.py
    def quick_sort(self, nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right:\nreturn\n# \u54e8\u5175\u5212\u5206\npivot = self.partition(nums, left, right)\n# \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nself.quick_sort(nums, left, pivot - 1)\nself.quick_sort(nums, pivot + 1, right)\n
    quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\npivot := q.partition(nums, left, right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nq.quickSort(nums, left, pivot-1)\nq.quickSort(nums, pivot+1, right)\n}\n
    quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
    quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i < j && nums[i] <= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums: &nums, left: left, right: pivot - 1)\nquickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
    quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nvar pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.dart
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List<int> nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nint pivot = _partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \u5c42\uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002
    "},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3. \u00a0 \u5feb\u6392\u4e3a\u4ec0\u4e48\u5feb\uff1f","text":"

    \u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u300c\u5f52\u5e76\u6392\u5e8f\u300d\u548c\u300c\u5806\u6392\u5e8f\u300d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002
    • \u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u300c\u5806\u6392\u5e8f\u300d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002
    • \u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u4f4e\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u300c\u63d2\u5165\u6392\u5e8f\u300d\u6bd4\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002
    "},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4. \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"

    \u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\) \u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u3002

    \u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002

    \u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cpp
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int> &nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int> &nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.py
    def median_three(self, nums: list[int], left: int, mid: int, right: int) -> int:\n\"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n# \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n# \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\nreturn left\nelif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\nreturn mid\nreturn right\ndef partition(self, nums: list[int], left: int, right: int) -> int:\n\"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed = self.median_three(nums, left, (left + right) // 2, right)\n# \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
    quick_sort.go
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n}\nreturn right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed := q.medianThree(nums, left, (left+right)/2, right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n//\u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n//\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.js
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse return right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.ts
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\nnums: number[],\nleft: number,\nmid: number,\nright: number\n): number {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\nreturn left;\n} else if (\nNumber(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])\n) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nint partitionMedian(int nums[], int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cs
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.swift
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n} else {\nreturn right\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums: &nums, i: left, j: med)\nreturn partition(nums: &nums, left: left, right: right)\n}\n
    quick_sort.zig
    // \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\nreturn left;\n} else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nvar med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.dart
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List<int> nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List<int> nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = _medianThree(nums, left, (left + right) ~/ 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n_swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5. \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"

    \u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u5012\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u7531\u4e8e\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002

    \u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int> &nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.py
    def quick_sort(self, nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right:\n# \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot = self.partition(nums, left, right)\n# \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot - left < right - pivot:\nself.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\nelse:\nself.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
    quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nfor left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot := q.partition(nums, left, right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot-left < right-pivot {\nq.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nq.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\nvar left = left\nvar right = right\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left) < (right - pivot) {\nquickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\nvar left = left_;\nvar right = right_;\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nvar pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.dart
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List<int> nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = _partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    "},{"location":"chapter_sorting/radix_sort/","title":"11.10. \u00a0 \u57fa\u6570\u6392\u5e8f","text":"

    \u4e0a\u4e00\u8282\u6211\u4eec\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002

    \u300c\u57fa\u6570\u6392\u5e8f Radix Sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002

    "},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \uff1b
    2. \u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u300c\u8ba1\u6570\u6392\u5e8f\u300d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff1b
    3. \u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 2. \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\uff1b

    Fig. \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b

    \u4e0b\u9762\u6765\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a

    \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\]

    \u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\space d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002

    \u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart radix_sort.java
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint[] counter = new int[10];\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = Integer.MIN_VALUE;\nfor (int num : nums)\nif (num > m)\nm = num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n
    radix_sort.cpp
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector<int> &nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvector<int> counter(10, 0);\nint n = nums.size();\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvector<int> res(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector<int> &nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = *max_element(nums.begin(), nums.end());\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n
    radix_sort.py
    def digit(num: int, exp: int) -> int:\n\"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n# \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num // exp) % 10\ndef counting_sort_digit(nums: list[int], exp: int) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n# \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\ncounter = [0] * 10\nn = len(nums)\n# \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in range(n):\nd = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n# \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in range(1, 10):\ncounter[i] += counter[i - 1]\n# \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nd = digit(nums[i], exp)\nj = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n# \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\ndef radix_sort(nums: list[int]) -> None:\n\"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n# \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nm = max(nums)\n# \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nexp = 1\nwhile exp <= m:\n# \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n# k = 1 -> exp = 1\n# k = 2 -> exp = 10\n# \u5373 exp = 10^(k-1)\ncounting_sort_digit(nums, exp)\nexp *= 10\n
    radix_sort.go
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\ncounter := make([]int, 10)\nn := len(nums)\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i := 0; i < n; i++ {\nd := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i := 1; i < 10; i++ {\ncounter[i] += counter[i-1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nd := digit(nums[i], exp)\nj := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i := 0; i < n; i++ {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nmax := math.MinInt\nfor _, num := range nums {\nif num > max {\nmax = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp := 1; max >= exp; exp *= 10 {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp)\n}\n}\n
    radix_sort.js
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i < n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i >= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num > m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.ts
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i < n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i >= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num > m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.c
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint *counter = (int *)malloc((sizeof(int) * 10));\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < size; i++) {\n// \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\nint d = digit(nums[i], exp);\n// \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\ncounter[d]++;\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint *res = (int *)malloc(sizeof(int) * size);\nfor (int i = size - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < size; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint max = INT32_MIN;\nfor (size_t i = 0; i < size - 1; i++) {\nif (nums[i] > max) {\nmax = nums[i];\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; max >= exp; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, size, exp);\n}\n
    radix_sort.cs
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint[] counter = new int[10];\nint n = nums.Length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = int.MinValue;\nforeach (int num in nums) {\nif (num > m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.swift
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -> Int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n(num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvar counter = Array(repeating: 0, count: 10)\nlet n = nums.count\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in nums.indices {\nlet d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in 1 ..< 10 {\ncounter[i] += counter[i - 1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = Array(repeating: 0, count: n)\nfor i in stride(from: n - 1, through: 0, by: -1) {\nlet d = digit(num: nums[i], exp: exp)\nlet j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in nums.indices {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m = Int.min\nfor num in nums {\nif num > m {\nm = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp in sequence(first: 1, next: { m >= ($0 * 10) ? $0 * 10 : nil }) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums: &nums, exp: exp)\n}\n}\n
    radix_sort.zig
    // \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn @mod(@divFloor(num, exp), 10);\n}\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n// defer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar counter = try mem_allocator.alloc(usize, 10);\n@memset(counter, 0);\nvar n = nums.len;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (nums) |num| {\nvar d = @bitCast(u32, digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nvar i: usize = 1;\nwhile (i < 10) : (i += 1) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = try mem_allocator.alloc(i32, n);\ni = n - 1;\nwhile (i >= 0) : (i -= 1) {\nvar d = @bitCast(u32, digit(nums[i], exp));\nvar j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\nif (i == 0) break;\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\ni = 0;\nwhile (i < n) : (i += 1) {\nnums[i] = res[i];\n}\n}\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m: i32 = std.math.minInt(i32);\nfor (nums) |num| {\nif (num > m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nvar exp: i32 = 1;\nwhile (exp <= m) : (exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ntry countingSortDigit(nums, exp);    }\n}
    radix_sort.dart
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num ~/ exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List<int> nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nList<int> counter = List<int>.filled(10, 0);\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nList<int> res = List<int>.filled(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) nums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List<int> nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n// dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\nint m = -1 << 63; for (int num in nums) if (num > m) m = num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n

    \u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f

    \u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a < b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a > b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u6211\u4eec\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002

    "},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"

    \u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002

    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\) \uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\) \u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 res \u548c counter \u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\u3002
    "},{"location":"chapter_sorting/selection_sort/","title":"11.2. \u00a0 \u9009\u62e9\u6392\u5e8f","text":"

    \u300c\u9009\u62e9\u6392\u5e8f Selection Sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u76f4\u63a5\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002
    2. \u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    3. \u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    4. \u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    5. \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u9700\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
    <1><2><3><4><5><6><7><8><9><10><11>

    \u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart selection_sort.java
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    selection_sort.cpp
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector<int> &nums) {\nint n = nums.size();\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nswap(nums[i], nums[k]);\n}\n}\n
    selection_sort.py
    def selection_sort(nums: list[int]):\n\"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in range(n - 1):\n# \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk = i\nfor j in range(i + 1, n):\nif nums[j] < nums[k]:\nk = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n# \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n
    selection_sort.go
    /* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\nn := len(nums)\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i := 0; i < n-1; i++ {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk := i\nfor j := i + 1; j < n; j++ {\nif nums[j] < nums[k] {\n// \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\nk = j\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n}\n}\n
    selection_sort.js
    /* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n
    selection_sort.ts
    /* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n
    selection_sort.c
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j;  // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    selection_sort.cs
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\nint n = nums.Length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n(nums[k], nums[i]) = (nums[i], nums[k]);\n}\n}\n
    selection_sort.swift
    /* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in nums.indices.dropLast() {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nvar k = i\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[j] < nums[k] {\nk = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums.swapAt(i, k)\n}\n}\n
    selection_sort.zig
    [class]{}-[func]{selectionSort}\n
    selection_sort.dart
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List<int> nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    "},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\) , \\(n - 1\\) , \\(\\cdots\\) , \\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5143\u7d20\u65f6\uff0c\u6709\u53ef\u80fd\u5c06 nums[i] \u4ea4\u6362\u81f3\u5176\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002

    Fig. \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b

    "},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1. \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"

    \u300c\u6392\u5e8f\u7b97\u6cd5 Sorting Algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u6709\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002

    \u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\uff1b\u987a\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002

    Fig. \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b

    "},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1. \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"

    \u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u5373\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u964d\u4f4e\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002

    \u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u9700\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002

    \u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002\u7a33\u5b9a\u6392\u5e8f\u662f\u4f18\u826f\u7279\u6027\uff0c\u4e5f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002

    \u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1, 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\u3002

    # \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n('A', 19)\n('B', 18)\n('C', 21)\n('D', 19)\n('E', 23)\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n('B', 18)\n('D', 19)\n('A', 19)\n('C', 21)\n('E', 23)\n

    \u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u3001\u6700\u5dee\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002

    \u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002

    \u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u4e8e\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(<\\) , \\(=\\) , \\(>\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002

    "},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2. \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"

    \u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002

    \u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002

    "},{"location":"chapter_sorting/summary/","title":"11.11. \u00a0 \u5c0f\u7ed3","text":"
    • \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002
    • \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5b83\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002
    • \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002
    • \u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
    • \u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002
    • \u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002
    • \u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002

    Fig. \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4

    • \u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002
    "},{"location":"chapter_sorting/summary/#11111-q-a","title":"11.11.1. \u00a0 Q & A","text":"

    \u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u987b\u7684\uff1f

    \u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u5728\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u4e0a\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f/

    \u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 (A, 180) (B, 185) (C, 170) (D, 170) \uff1b\u63a5\u4e0b\u6765\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u6211\u4eec\u53ef\u80fd\u5f97\u5230 (D, 170) (C, 170) (A, 180) (B, 185) \u3002

    \u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002

    \u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f

    \u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002

    \u54e8\u5175\u5212\u5206 partition() \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 nums[left] \u548c nums[i] \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <= \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d nums[left] >= nums[i] \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 i == j \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd nums[j] == nums[i] > nums[left]\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002

    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 [0, 0, 0, 0, 1] \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a [1, 0, 0, 0, 0] \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002

    \u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 nums[right] \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002

    \u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(log n\\) \uff1f

    \u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u7684\u4e00\u534a\u957f\u5ea6\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(log n\\) \u3002

    \u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n, n - 1, n - 2, ..., 2, 1\\) \uff0c\u4ece\u800c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u7684\u51fa\u73b0\u3002

    \u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f

    \u662f\u7684\u3002\u8fd9\u79cd\u60c5\u51b5\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002

    \u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f

    \u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_stack_and_queue/","title":"5. \u00a0 \u6808\u4e0e\u961f\u5217","text":""},{"location":"chapter_stack_and_queue/deque/","title":"5.3. \u00a0 \u53cc\u5411\u961f\u5217","text":"

    \u5bf9\u4e8e\u961f\u5217\uff0c\u6211\u4eec\u4ec5\u80fd\u5728\u5934\u90e8\u5220\u9664\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u7136\u800c\uff0c\u300c\u53cc\u5411\u961f\u5217 Deque\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002

    Fig. \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c

    "},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1. \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"

    \u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\)

    \u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart deque.java
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
    deque.cpp
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
    deque.py
    # \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: Deque[int] = collections.deque()\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n
    deque_test.go
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
    deque.js
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
    deque.ts
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
    deque.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n
    deque.cs
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new LinkedList<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
    deque.swift
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
    deque.zig
    \n
    deque.dart
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue<int> deque = Queue<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n
    "},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2. \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"

    \u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_stack_and_queue/deque/#_1","title":"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002

    \u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u300c\u53cc\u5411\u94fe\u8868\u300d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002

    \u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002

    LinkedListDequepushLast()pushFirst()popLast()popFirst()

    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_deque.java
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(int val) {\nthis.val = val;\nprev = next = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, boolean isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate Integer pop(boolean isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty())\nreturn null;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic Integer popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic Integer popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic Integer peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic Integer peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_deque.cpp
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\nint val;              // \u8282\u70b9\u503c\nDoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\nDoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\nDoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n}\n};\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate:\nDoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nLinkedListDeque() : front(nullptr), rear(nullptr) {\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~LinkedListDeque() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nDoublyListNode *pre, *cur = front;\nwhile (cur != nullptr) {\npre = cur;\ncur = cur->next;\ndelete pre;\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nDoublyListNode *node = new DoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront->prev = node;\nnode->next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear->next = node;\nnode->prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de -1\nif (isEmpty())\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front->val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nDoublyListNode *fNext = front->next;\nif (fNext != nullptr) {\nfNext->prev = nullptr;\nfront->next = nullptr;\ndelete front;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear->val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nDoublyListNode *rPrev = rear->prev;\nif (rPrev != nullptr) {\nrPrev->next = nullptr;\nrear->prev = nullptr;\ndelete rear;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nreturn isEmpty() ? -1 : front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nreturn isEmpty() ? -1 : rear->val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\nDoublyListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_deque.py
    class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\ndef __init__(self, val: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.val: int = val\nself.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nself.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nclass LinkedListDeque:\n\"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself.rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself.__size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.size() == 0\ndef push(self, num: int, is_front: bool) -> None:\n\"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\nnode = ListNode(num)\n# \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif self.is_empty():\nself.front = self.rear = node\n# \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelif is_front:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.prev = node\nnode.next = self.front\nself.front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.next = node\nnode.prev = self.rear\nself.rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself.__size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\ndef push_first(self, num: int) -> None:\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nself.push(num, True)\ndef push_last(self, num: int) -> None:\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nself.push(num, False)\ndef pop(self, is_front: bool) -> int:\n\"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n# \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\nif self.is_empty():\nreturn None\n# \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front:\nval: int = self.front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n# \u5220\u9664\u5934\u8282\u70b9\nfnext: ListNode | None = self.front.next\nif fnext != None:\nfnext.prev = None\nself.front.next = None\nself.front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse:\nval: int = self.rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n# \u5220\u9664\u5c3e\u8282\u70b9\nrprev: ListNode | None = self.rear.prev\nif rprev != None:\nrprev.next = None\nself.rear.prev = None\nself.rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself.__size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\ndef pop_first(self) -> int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nreturn self.pop(True)\ndef pop_last(self) -> int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nreturn self.pop(False)\ndef peek_first(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nreturn None if self.is_empty() else self.front.val\ndef peek_last(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nreturn None if self.is_empty() else self.rear.val\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\nnode = self.front\nres = [0] * self.size()\nfor i in range(self.size()):\nres[i] = node.val\nnode = node.next\nreturn res\n
    linkedlist_deque.go
    /* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\nreturn &linkedListDeque{\ndata: list.New(),\n}\n}\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\ns.data.PushFront(value)\n}\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\ns.data.PushBack(value)\n}\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_deque.js
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval; // \u8282\u70b9\u503c\nconstructor(val) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n#front; // \u5934\u8282\u70b9 front\n#rear; // \u5c3e\u8282\u70b9 rear\n#queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\nthis.#queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.#rear.next = node;\nnode.prev = this.#rear;\nthis.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.#front.prev = node;\nnode.next = this.#front;\nthis.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp = this.#rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.#rear.prev = null;\n}\nthis.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp = this.#front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.#front.next = null;\n}\nthis.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nreturn this.#queSize === 0 ? null : this.#rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nreturn this.#queSize === 0 ? null : this.#front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint() {\nconst arr = [];\nlet temp = this.#front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n
    linkedlist_deque.ts
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval: number; // \u8282\u70b9\u503c\nconstructor(val: number) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate front: ListNode; // \u5934\u8282\u70b9 front\nprivate rear: ListNode; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.front = null;\nthis.rear = null;\nthis.queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.rear.next = node;\nnode.prev = this.rear;\nthis.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.front.prev = node;\nnode.next = this.front;\nthis.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp: ListNode = this.rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.rear.prev = null;\n}\nthis.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp: ListNode = this.front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.front.next = null;\n}\nthis.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nreturn this.queSize === 0 ? null : this.rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nreturn this.queSize === 0 ? null : this.front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint(): void {\nconst arr: number[] = [];\nlet temp: ListNode = this.front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n
    linkedlist_deque.c
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct doublyListNode {\nint val;                     // \u8282\u70b9\u503c\nstruct doublyListNode *next; // \u540e\u7ee7\u8282\u70b9\nstruct doublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n};\ntypedef struct doublyListNode doublyListNode;\n/* \u6784\u9020\u51fd\u6570 */\ndoublyListNode *newDoublyListNode(int num) {\ndoublyListNode *new = (doublyListNode *)malloc(sizeof(doublyListNode));\nnew->val = num;\nnew->next = NULL;\nnew->prev = NULL;\nreturn new;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(doublyListNode *node) {\nfree(node);\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct linkedListDeque {\ndoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n};\ntypedef struct linkedListDeque linkedListDeque;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListDeque *newLinkedListDeque() {\nlinkedListDeque *deque = (linkedListDeque *)malloc(sizeof(linkedListDeque));\ndeque->front = NULL;\ndeque->rear = NULL;\ndeque->queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(linkedListDeque *deque) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i < deque->queSize && deque->front != NULL; i++) {\ndoublyListNode *tmp = deque->front;\ndeque->front = deque->front->next;\nfree(tmp);\n}\n// \u91ca\u653e deque \u7ed3\u6784\u4f53\nfree(deque);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(linkedListDeque *deque) {\nreturn deque->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(linkedListDeque *deque) {\nreturn (size(deque) == 0);\n}\n/* \u5165\u961f */\nvoid push(linkedListDeque *deque, int num, bool isFront) {\ndoublyListNode *node = newDoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411node\nif (empty(deque)) {\ndeque->front = deque->rear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\ndeque->front->prev = node;\nnode->next = deque->front;\ndeque->front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u5bf9\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\ndeque->rear->next = node;\nnode->prev = deque->rear;\ndeque->rear = node;\n}\ndeque->queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(linkedListDeque *deque, int num) {\npush(deque, num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(linkedListDeque *deque, int num) {\npush(deque, num, false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(linkedListDeque *deque) {\nassert(size(deque) && deque->front);\nreturn deque->front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(linkedListDeque *deque) {\nassert(size(deque) && deque->rear);\nreturn deque->rear->val;\n}\n/* \u51fa\u961f */\nint pop(linkedListDeque *deque, bool isFront) {\nif (empty(deque))\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\ndoublyListNode *fNext = deque->front->next;\nif (fNext) {\nfNext->prev = NULL;\ndeque->front->next = NULL;\ndelDoublyListNode(deque->front);\n}\ndeque->front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\ndoublyListNode *rPrev = deque->rear->prev;\nif (rPrev) {\nrPrev->next = NULL;\ndeque->rear->prev = NULL;\ndelDoublyListNode(deque->rear);\n}\ndeque->rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\ndeque->queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(linkedListDeque *deque) {\nreturn pop(deque, true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(linkedListDeque *deque) {\nreturn pop(deque, false);\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(linkedListDeque *deque) {\nint arr[deque->queSize];\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\ndoublyListNode *node;\nfor (i = 0, node = deque->front; i < deque->queSize; i++) {\narr[i] = node->val;\nnode = node->next;\n}\nprintArray(arr, deque->queSize);\n}\n
    linkedlist_deque.cs
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\npublic int val;       // \u8282\u70b9\u503c\npublic ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\npublic ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\npublic ListNode(int val) {\nthis.val = val;\nprev = null;\nnext = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\nprivate int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, bool isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty()) {\nfront = node;\nrear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate int? pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty()) {\nreturn null;\n}\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int? popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int? popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int? peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int? peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_deque.swift
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nvar prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\ninit(val: Int) {\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate var front: ListNode? // \u5934\u8282\u70b9 front\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9 rear\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ninit() {\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate func push(num: Int, isFront: Bool) {\nlet node = ListNode(val: num)\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif isEmpty() {\nfront = node\nrear = node\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if isFront {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront?.prev = node\nnode.next = front\nfront = node // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear?.next = node\nnode.prev = rear\nrear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\npush(num: num, isFront: true)\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\npush(num: num, isFront: false)\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate func pop(isFront: Bool) -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nlet val: Int\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif isFront {\nval = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet fNext = front?.next\nif fNext != nil {\nfNext?.prev = nil\nfront?.next = nil\n}\nfront = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet rPrev = rear?.prev\nif rPrev != nil {\nrPrev?.next = nil\nrear?.prev = nil\n}\nrear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\npop(isFront: true)\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\npop(isFront: false)\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int? {\nisEmpty() ? nil : front?.val\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int? {\nisEmpty() ? nil : rear?.val\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_deque.zig
    // \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = undefined,     // \u8282\u70b9\u503c\nnext: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nprev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\n// Initialize a list node with specific value\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\nrear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u5165\u961f\u64cd\u4f5c\npub fn push(self: *Self, num: T, is_front: bool) !void {\nvar node = try self.mem_allocator.create(ListNode(T));\nnode.init(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (self.isEmpty()) {\nself.front = node;\nself.rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n} else if (is_front) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.?.prev = node;\nnode.next = self.front;\nself.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.?.next = node;\nnode.prev = self.rear;\nself.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n} // \u961f\u9996\u5165\u961f\npub fn pushFirst(self: *Self, num: T) !void {\ntry self.push(num, true);\n} // \u961f\u5c3e\u5165\u961f\npub fn pushLast(self: *Self, num: T) !void {\ntry self.push(num, false);\n} // \u51fa\u961f\u64cd\u4f5c\npub fn pop(self: *Self, is_front: bool) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nvar val: T = undefined;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (is_front) {\nval = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nvar fNext = self.front.?.next;\nif (fNext != null) {\nfNext.?.prev = null;\nself.front.?.next = null;\n}\nself.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nvar rPrev = self.rear.?.prev;\nif (rPrev != null) {\nrPrev.?.next = null;\nself.rear.?.prev = null;\n}\nself.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n} // \u961f\u9996\u51fa\u961f\npub fn popFirst(self: *Self) T {\nreturn self.pop(true);\n} // \u961f\u5c3e\u51fa\u961f\npub fn popLast(self: *Self) T {\nreturn self.pop(false);\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peekFirst(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\npub fn peekLast(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.rear.?.val;\n}\n// \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_deque.dart
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(this.val, {this.next, this.prev});\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\nlate ListNode? _front; // \u5934\u8282\u70b9 _front\nlate ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nLinkedListDeque() {\nthis._front = null;\nthis._rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\nint size() {\nreturn this._queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nfinal ListNode node = ListNode(num);\nif (isEmpty()) {\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front\uff0c_rear \u90fd\u6307\u5411 node\n_front = _rear = node;\n} else if (isFront) {\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n_front!.prev = node;\nnode.next = _front;\n_front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n_rear!.next = node;\nnode.prev = _rear;\n_rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint? pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty()) {\nreturn null;\n}\nfinal int val;\nif (isFront) {\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nval = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode? fNext = _front!.next;\nif (fNext != null) {\nfNext.prev = null;\n_front!.next = null;\n}\n_front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nval = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode? rPrev = _rear!.prev;\nif (rPrev != null) {\nrPrev.next = null;\n_rear!.prev = null;\n}\n_rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint? popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint? popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint? peekFirst() {\nreturn _front?.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint? peekLast() {\nreturn _rear?.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList<int> toArray() {\nListNode? node = _front;\nfinal List<int> res = [];\nfor (int i = 0; i < _queSize; i++) {\nres.add(node!.val);\nnode = node.next;\n}\nreturn res;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\u3002

    ArrayDequepushLast()pushFirst()popLast()popFirst()

    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_deque.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate:\nvector<int> nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nnums.resize(capacity);\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn nums.size();\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> res(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n};\n
    array_deque.py
    class ArrayDeque:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self, capacity: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__nums: list[int] = [0] * capacity\nself.__front: int = 0\nself.__size: int = 0\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__size == 0\ndef index(self, i: int) -> int:\n\"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + self.capacity()) % self.capacity()\ndef push_first(self, num: int) -> None:\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself.__front = self.index(self.__front - 1)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself.__nums[self.__front] = num\nself.__size += 1\ndef push_last(self, num: int) -> None:\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear = self.index(self.__front + self.__size)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop_first(self) -> int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nnum = self.peek_first()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself.__front = self.index(self.__front + 1)\nself.__size -= 1\nreturn num\ndef pop_last(self) -> int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nnum = self.peek_last()\nself.__size -= 1\nreturn num\ndef peek_first(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\nreturn self.__nums[self.__front]\ndef peek_last(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n# \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast = self.index(self.__front + self.__size - 1)\nreturn self.__nums[last]\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n# \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres = []\nfor i in range(self.__size):\nres.append(self.__nums[self.index(self.__front + i)])\nreturn res\n
    array_deque.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\nreturn &arrayDeque{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + q.queCapacity) % q.queCapacity\n}\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nq.front = q.index(q.front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[q.front] = num\nq.queSize++\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear := q.index(q.front + q.queSize)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[rear] = num\nq.queSize++\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\nnum := q.peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nq.front = q.index(q.front + 1)\nq.queSize--\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\nnum := q.peekLast()\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\nif q.isEmpty() {\nreturn nil\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast := q.index(q.front + q.queSize - 1)\nreturn q.nums[last]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres := make([]int, q.queSize)\nfor i, j := 0, q.front; i < q.queSize; i++ {\nres[i] = q.nums[q.index(j)]\nj++\n}\nreturn res\n}\n
    array_deque.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\nthis.#front = 0;\nthis.#queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.#front = this.index(this.#front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.#nums[this.#front] = num;\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear = this.index(this.#front + this.#queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst() {\nconst num = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.#front = this.index(this.#front + 1);\nthis.#queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast() {\nconst num = this.peekLast();\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.#nums[this.#front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.#front + this.#queSize - 1);\nreturn this.#nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res = [];\nfor (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\nres[i] = this.#nums[this.index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = 0;\nthis.queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i: number): number {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.front = this.index(this.front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.nums[this.front] = num;\nthis.queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear: number = this.index(this.front + this.queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst(): number {\nconst num: number = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.front = this.index(this.front + 1);\nthis.queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast(): number {\nconst num: number = this.peekLast();\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.nums[this.front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.front + this.queSize - 1);\nreturn this.nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res: number[] = [];\nfor (let i = 0, j = this.front; i < this.queSize; i++, j++) {\nres[i] = this.nums[this.index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.c
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct arrayDeque {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n};\ntypedef struct arrayDeque arrayDeque;\n/* \u6784\u9020\u51fd\u6570 */\narrayDeque *newArrayDeque(int capacity) {\narrayDeque *deque = (arrayDeque *)malloc(sizeof(arrayDeque));\n// \u521d\u59cb\u5316\u6570\u7ec4\ndeque->queCapacity = capacity;\ndeque->nums = (int *)malloc(sizeof(int) * deque->queCapacity);\ndeque->front = deque->queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(arrayDeque *deque) {\nfree(deque->nums);\ndeque->queCapacity = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(arrayDeque *deque) {\nreturn deque->queCapacity;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(arrayDeque *deque) {\nreturn deque->queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(arrayDeque *deque) {\nreturn deque->queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(arrayDeque *deque, int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn ((i + capacity(deque)) % capacity(deque));\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(arrayDeque *deque, int num) {\nif (deque->queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\ndeque->front = dequeIndex(deque, deque->front - 1);\n// \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\ndeque->nums[deque->front] = num;\ndeque->queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(arrayDeque *deque, int num) {\nif (deque->queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = dequeIndex(deque, deque->front + deque->queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque->nums[rear] = num;\ndeque->queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(arrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nreturn deque->nums[deque->front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(arrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nint last = dequeIndex(deque, deque->front + deque->queSize - 1);\nreturn deque->nums[last];\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(arrayDeque *deque) {\nint num = peekFirst(deque);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\ndeque->front = dequeIndex(deque, deque->front + 1);\ndeque->queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(arrayDeque *deque) {\nint num = peekLast(deque);\ndeque->queSize--;\nreturn num;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printArrayDeque(arrayDeque *deque) {\nint arr[deque->queSize];\n// \u62f7\u8d1d\nfor (int i = 0, j = deque->front; i < deque->queSize; i++, j++) {\narr[i] = deque->nums[j % deque->queCapacity];\n}\nprintArray(arr, deque->queSize);\n}\n
    array_deque.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate readonly int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty()) {\nthrow new InvalidOperationException();\n}\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty()) {\nthrow new InvalidOperationException();\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(capacity: Int) {\nnums = Array(repeating: 0, count: capacity)\nfront = 0\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate func index(i: Int) -> Int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n(i + capacity()) % capacity()\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(i: front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num\nqueSize += 1\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = index(i: front + size())\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\nlet num = peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(i: front + 1)\nqueSize -= 1\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\nlet num = peekLast()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = index(i: front + size() - 1)\nreturn nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: size())\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[index(i: j)]\n}\nreturn res\n}\n}\n
    array_deque.zig
    [class]{ArrayDeque}-[func]{}\n
    array_deque.dart
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nlate List<int> _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nthis._nums = List.filled(capacity, 0);\nthis._front = this._queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n_front = index(_front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n_nums[_front] = num;\n_queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(_front + _queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = num;\n_queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n_front = index(_front + 1);\n_queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(_front + _queSize - 1);\nreturn _nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList<int> toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nList<int> res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i < _queSize; i++, j++) {\nres[i] = _nums[index(j)];\n}\nreturn res;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3. \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"

    \u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002

    \u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c push \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 pop \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u5373\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002

    "},{"location":"chapter_stack_and_queue/queue/","title":"5.2. \u00a0 \u961f\u5217","text":"

    \u300c\u961f\u5217 Queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\uff08First In, First Out\uff09\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u7684\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002

    \u6211\u4eec\u628a\u961f\u5217\u7684\u5934\u90e8\u79f0\u4e3a\u300c\u961f\u9996\u300d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u300c\u961f\u5c3e\u300d\uff0c\u628a\u5c06\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u961f\u300d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u961f\u300d\u3002

    Fig. \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219

    "},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1. \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"

    \u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\)

    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart queue.java
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
    queue.cpp
    /* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
    queue.py
    # \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u770b\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u5efa\u8bae\nque: Deque[int] = collections.deque()\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n
    queue_test.go
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
    queue.js
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
    queue.ts
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
    queue.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n
    queue.cs
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count() == 0;\n
    queue.swift
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
    queue.zig
    \n
    queue.dart
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue<int> queue = Queue();\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n
    "},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2. \u00a0 \u961f\u5217\u5b9e\u73b0","text":"

    \u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u53ef\u4ee5\u7528\u6765\u5b9e\u73b0\u961f\u5217\u3002

    "},{"location":"chapter_stack_and_queue/queue/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u5bf9\u4e8e\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u300c\u5934\u8282\u70b9\u300d\u548c\u300c\u5c3e\u8282\u70b9\u300d\u5206\u522b\u89c6\u4e3a\u961f\u9996\u548c\u961f\u5c3e\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u800c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002

    LinkedListQueuepush()pop()

    \u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_queue.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new IndexOutOfBoundsException();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate:\nListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;\npublic:\nLinkedListQueue() {\nfront = nullptr;\nrear = nullptr;\nqueSize = 0;\n}\n~LinkedListQueue() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(front);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode *node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == nullptr) {\nfront = node;\nrear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear->next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nListNode *tmp = front;\nfront = front->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (size() == 0)\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn front->val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_queue.py
    class LinkedListQueue:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself.__rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self.__front\ndef push(self, num: int) -> None:\n\"\"\"\u5165\u961f\"\"\"\n# \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nnode = ListNode(num)\n# \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif self.__front is None:\nself.__front = node\nself.__rear = node\n# \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse:\nself.__rear.next = node\nself.__rear = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u961f\"\"\"\nnum = self.peek()\n# \u5220\u9664\u5934\u8282\u70b9\nself.__front = self.__front.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.size() == 0:\nprint(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn False\nreturn self.__front.val\ndef to_list(self) -> list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nqueue = []\ntemp = self.__front\nwhile temp:\nqueue.append(temp.val)\ntemp = temp.next\nreturn queue\n
    linkedlist_queue.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\nreturn &linkedListQueue{\ndata: list.New(),\n}\n}\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\ns.data.PushBack(value)\n}\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_queue.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n#front; // \u5934\u8282\u70b9 #front\n#rear; // \u5c3e\u8282\u70b9 #rear\n#queSize = 0;\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.#front) {\nthis.#front = node;\nthis.#rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.#rear.next = node;\nthis.#rear = node;\n}\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nthis.#front = this.#front.next;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#front;\nconst res = new Array(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate front: ListNode | null; // \u5934\u8282\u70b9 front\nprivate rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number = 0;\nconstructor() {\nthis.front = null;\nthis.rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.front) {\nthis.front = node;\nthis.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.rear!.next = node;\nthis.rear = node;\n}\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\nif (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n// \u5220\u9664\u5934\u8282\u70b9\nthis.front = this.front.next;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.front;\nconst res = new Array<number>(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.c
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct linkedListQueue {\nListNode *front, *rear;\nint queSize;\n};\ntypedef struct linkedListQueue linkedListQueue;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListQueue *newLinkedListQueue() {\nlinkedListQueue *queue = (linkedListQueue *)malloc(sizeof(linkedListQueue));\nqueue->front = NULL;\nqueue->rear = NULL;\nqueue->queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(linkedListQueue *queue) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i < queue->queSize && queue->front != NULL; i++) {\nListNode *tmp = queue->front;\nqueue->front = queue->front->next;\nfree(tmp);\n}\n// \u91ca\u653e queue \u7ed3\u6784\u4f53\nfree(queue);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(linkedListQueue *queue) {\nreturn queue->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(linkedListQueue *queue) {\nreturn (size(queue) == 0);\n}\n/* \u5165\u961f */\nvoid push(linkedListQueue *queue, int num) {\n// \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\nListNode *node = newListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (queue->front == NULL) {\nqueue->front = node;\nqueue->rear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nqueue->rear->next = node;\nqueue->rear = node;\n}\nqueue->queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(linkedListQueue *queue) {\nassert(size(queue) && queue->front);\nreturn queue->front->val;\n}\n/* \u51fa\u961f */\nvoid pop(linkedListQueue *queue) {\nint num = peek(queue);\nListNode *tmp = queue->front;\nqueue->front = queue->front->next;\nfree(tmp);\nqueue->queSize--;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(linkedListQueue *queue) {\nint arr[queue->queSize];\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\nListNode *node;\nfor (i = 0, node = queue->front; i < queue->queSize && queue->front != queue->rear; i++) {\narr[i] = node->val;\nnode = node->next;\n}\nprintArray(arr, queue->queSize);\n}\n
    linkedlist_queue.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else if (rear != null) {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0 || front == null)\nthrow new Exception();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nif (front == null)\nreturn Array.Empty<int>();\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate var front: ListNode? // \u5934\u8282\u70b9\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9\nprivate var _size = 0\ninit() {}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nlet node = ListNode(x: num)\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif front == nil {\nfront = node\nrear = node\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear?.next = node\nrear = node\n}\n_size += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn front!.val\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_queue.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\nrear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u5165\u961f\npub fn push(self: *Self, num: T) !void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (self.front == null) {\nself.front = node;\nself.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nself.rear.?.next = node;\nself.rear = node;\n}\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nself.front = self.front.?.next;\nself.que_size -= 1;\nreturn num;\n} // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_queue.dart
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nListNode? _front; // \u5934\u8282\u70b9 _front\nListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u961f\u5217\u957f\u5ea6\nLinkedListQueue() {\n_front = null;\n_rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nfinal node = ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (_front == null) {\n_front = node;\n_rear = node;\n} else {\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n_rear!.next = node;\n_rear = node;\n}\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nfinal int num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\n_front = _front!.next;\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (_queSize == 0) {\nthrow Exception('\u961f\u5217\u4e3a\u7a7a');\n}\nreturn _front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList<int> toArray() {\nListNode? node = _front;\nfinal List<int> queue = [];\nwhile (node != null) {\nqueue.add(node.val);\nnode = node.next;\n}\nreturn queue;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/queue/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u7531\u4e8e\u6570\u7ec4\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002

    \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf front \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf queSize \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 rear = front + queSize \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 rear \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002

    \u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a [front, rear - 1]\uff0c\u8fdb\u800c\uff1a

    • \u5bf9\u4e8e\u5165\u961f\u64cd\u4f5c\uff0c\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 rear \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 queSize \u589e\u52a0 1 \uff1b
    • \u5bf9\u4e8e\u51fa\u961f\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06 front \u589e\u52a0 1 \uff0c\u5e76\u5c06 queSize \u51cf\u5c11 1 \uff1b

    \u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002

    ArrayQueuepush()pop()

    \u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0cfront \u548c rear \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u300c\u73af\u5f62\u6570\u7ec4\u300d\u3002

    \u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 front \u6216 rear \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_queue.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % capacity()];\n}\nreturn res;\n}\n}\n
    array_queue.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate:\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u961f\u5217\u957f\u5ea6\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\npublic:\nArrayQueue(int capacity) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = new int[capacity];\nqueCapacity = capacity;\nfront = queSize = 0;\n}\n~ArrayQueue() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (queSize == queCapacity) {\ncout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % queCapacity;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (empty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> arr(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\narr[i] = nums[j % queCapacity];\n}\nreturn arr;\n}\n};\n
    array_queue.py
    class ArrayQueue:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self, size: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nself.__front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nself.__size: int = 0  # \u961f\u5217\u957f\u5ea6\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__size == 0\ndef push(self, num: int) -> None:\n\"\"\"\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nraise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8F\nrear: int = (self.__front + self.__size) % self.capacity()\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u961f\"\"\"\nnum: int = self.peek()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.__front = (self.__front + 1) % self.capacity()\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn self.__nums[self.__front]\ndef to_list(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nres = [0] * self.size()\nj: int = self.__front\nfor i in range(self.size()):\nres[i] = self.__nums[(j % self.capacity())]\nj += 1\nreturn res\n
    array_queue.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\nreturn &arrayQueue{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n// \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\nif q.queSize == q.queCapacity {\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear := (q.front + q.queSize) % q.queCapacity\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nq.nums[rear] = num\nq.queSize++\n}\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\nnum := q.peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nq.front = (q.front + 1) % q.queCapacity\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\nrear := (q.front + q.queSize)\nif rear >= q.queCapacity {\nrear %= q.queCapacity\nreturn append(q.nums[q.front:], q.nums[:rear]...)\n}\nreturn q.nums[q.front:rear]\n}\n
    array_queue.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize = 0; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#queSize == 0;\n}\n/* \u5165\u961f */\npush(num) {\nif (this.size == this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.#front + this.size) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.#front = (this.#front + 1) % this.capacity;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.empty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#nums[this.#front];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.#front; i < this.size; i++, j++) {\narr[i] = this.#nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
    array_queue.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = this.queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.queSize == 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\nif (this.size == this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.front + this.queSize) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.front = (this.front + 1) % this.capacity;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.empty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.nums[this.front];\n}\n/* \u8fd4\u56de Array */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.front; i < this.size; i++, j++) {\narr[i] = this.nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
    array_queue.c
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct arrayQueue {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n};\ntypedef struct arrayQueue arrayQueue;\n/* \u6784\u9020\u51fd\u6570 */\narrayQueue *newArrayQueue(int capacity) {\narrayQueue *queue = (arrayQueue *)malloc(sizeof(arrayQueue));\n// \u521d\u59cb\u5316\u6570\u7ec4\nqueue->queCapacity = capacity;\nqueue->nums = (int *)malloc(sizeof(int) * queue->queCapacity);\nqueue->front = queue->queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(arrayQueue *queue) {\nfree(queue->nums);\nqueue->queCapacity = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(arrayQueue *queue) {\nreturn queue->queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(arrayQueue *queue) {\nreturn queue->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(arrayQueue *queue) {\nreturn queue->queSize == 0;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(arrayQueue *queue) {\nassert(size(queue) != 0);\nreturn queue->nums[queue->front];\n}\n/* \u5165\u961f */\nvoid push(arrayQueue *queue, int num) {\nif (size(queue) == capacity(queue)) {\nprintf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (queue->front + queue->queSize) % queue->queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nqueue->nums[rear] = num;\nqueue->queSize++;\n}\n/* \u51fa\u961f */\nvoid pop(arrayQueue *queue) {\nint num = peek(queue);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nqueue->front = (queue->front + 1) % queue->queCapacity;\nqueue->queSize--;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printArrayQueue(arrayQueue *queue) {\nint arr[queue->queSize];\n// \u62f7\u8d1d\nfor (int i = 0, j = queue->front; i < queue->queSize; i++, j++) {\narr[i] = queue->nums[j % queue->queCapacity];\n}\nprintArray(arr, queue->queSize);\n}\n
    array_queue.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new Exception();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % this.capacity()];\n}\nreturn res;\n}\n}\n
    array_queue.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize = 0 // \u961f\u5217\u957f\u5ea6\ninit(capacity: Int) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = Array(repeating: 0, count: capacity)\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nqueSize == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\nif size() == capacity() {\nprint(\"\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (front + queSize) % capacity()\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: queSize)\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[j % capacity()]\n}\nreturn res\n}\n}\n
    array_queue.zig
    // \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \ncap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\nfront: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.cap = cap;\nself.nums = try self.mem_allocator.alloc(T, self.cap);\n@memset(self.nums, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.cap;\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.queSize;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.queSize == 0;\n}\n// \u5165\u961f\npub fn push(self: *Self, num: T) !void {\nif (self.size() == self.capacity()) {\nstd.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nvar rear = (self.front + self.queSize) % self.capacity();\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nself.nums[rear] = num;\nself.queSize += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.capacity();\nself.queSize -= 1;\nreturn num;\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.nums[self.front];\n} // \u8fd4\u56de\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nvar j: usize = self.front;\nwhile (i < self.size()) : ({ i += 1; j += 1; }) {\nres[i] = self.nums[j % self.capacity()];\n}\nreturn res;\n}\n};\n}\n
    array_queue.dart
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nlate List<int> _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u961f\u5217\u957f\u5ea6\nArrayQueue(int capacity) {\n_nums = List.filled(capacity, 0);\n_front = _queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capaCity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (_queSize == capaCity()) {\nthrow Exception(\"\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (_front + _queSize) % capaCity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = num;\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n_front = (_front + 1) % capaCity();\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8fd4\u56de Array */\nList<int> toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nfinal List<int> res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i < _queSize; i++, j++) {\nres[i] = _nums[j % capaCity()];\n}\nreturn res;\n}\n}\n

    \u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff0c\u5373\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002

    \u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002

    "},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3. \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"
    • \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u4f9d\u6b21\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002
    • \u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\u3002\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002
    "},{"location":"chapter_stack_and_queue/stack/","title":"5.1. \u00a0 \u6808","text":"

    \u300c\u6808 Stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\uff08First In, Last Out\uff09\u539f\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002

    \u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u9700\u8981\u62ff\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u53d6\u51fa\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u6570\u636e\u7ed3\u6784\u3002

    \u5728\u6808\u4e2d\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u300c\u6808\u9876\u300d\uff0c\u5e95\u90e8\u79f0\u4e3a\u300c\u6808\u5e95\u300d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u505a\u300c\u5165\u6808\u300d\uff0c\u800c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u505a\u300c\u51fa\u6808\u300d\u3002

    Fig. \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219

    "},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1. \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"

    \u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 push() , pop() , peek() \u547d\u540d\u4e3a\u4f8b\u3002

    \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\)

    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u300c\u6570\u7ec4\u300d\u6216\u300c\u94fe\u8868\u300d\u89c6\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u901a\u8fc7\u201c\u8111\u8865\u201d\u6765\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart stack.java
    /* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
    stack.cpp
    /* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
    stack.py
    # \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: List[int] = []\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n
    stack_test.go
    /* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
    stack.js
    /* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
    stack.ts
    /* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
    stack.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n
    stack.cs
    /* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new ();\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count()==0;\n
    stack.swift
    /* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
    stack.zig
    \n
    stack.dart
    /* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList<int> stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n
    "},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2. \u00a0 \u6808\u7684\u5b9e\u73b0","text":"

    \u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002

    \u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u88ab\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002

    "},{"location":"chapter_stack_and_queue/stack/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u4f7f\u7528\u94fe\u8868\u6765\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002

    \u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002

    LinkedListStackpush()pop()

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_stack.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0; // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new IndexOutOfBoundsException();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate:\nListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize;        // \u6808\u7684\u957f\u5ea6\npublic:\nLinkedListStack() {\nstackTop = nullptr;\nstkSize = 0;\n}\n~LinkedListStack() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(stackTop);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nListNode *node = new ListNode(num);\nnode->next = stackTop;\nstackTop = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint num = top();\nListNode *tmp = stackTop;\nstackTop = stackTop->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nstkSize--;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (size() == 0)\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stackTop->val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode *node = stackTop;\nvector<int> res(size());\nfor (int i = res.size() - 1; i >= 0; i--) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_stack.py
    class LinkedListStack:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__peek: ListNode | None = None\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self.__peek\ndef push(self, val: int) -> None:\n\"\"\"\u5165\u6808\"\"\"\nnode = ListNode(val)\nnode.next = self.__peek\nself.__peek = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u6808\"\"\"\nnum: int = self.peek()\nself.__peek = self.__peek.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n# \u5224\u7a7a\u5904\u7406\nif not self.__peek:\nreturn None\nreturn self.__peek.val\ndef to_list(self) -> list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\narr = []\nnode = self.__peek\nwhile node:\narr.append(node.val)\nnode = node.next\narr.reverse()\nreturn arr\n
    linkedlist_stack.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\nreturn &linkedListStack{\ndata: list.New(),\n}\n}\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\ns.data.PushBack(value)\n}\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_stack.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n#stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n#stkSize = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.#stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num) {\nconst node = new ListNode(num);\nnode.next = this.#stackPeek;\nthis.#stackPeek = node;\nthis.#stkSize++;\n}\n/* \u51fa\u6808 */\npop() {\nconst num = this.peek();\nthis.#stackPeek = this.#stackPeek.next;\nthis.#stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek() {\nif (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#stackPeek;\nconst res = new Array(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nconst node = new ListNode(num);\nnode.next = this.stackPeek;\nthis.stackPeek = node;\nthis.stkSize++;\n}\n/* \u51fa\u6808 */\npop(): number {\nconst num = this.peek();\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nthis.stackPeek = this.stackPeek.next;\nthis.stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek(): number {\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.stackPeek;\nconst res = new Array<number>(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.c
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nstruct linkedListStack {\nListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint size;      // \u6808\u7684\u957f\u5ea6\n};\ntypedef struct linkedListStack linkedListStack;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListStack *newLinkedListStack() {\nlinkedListStack *s = malloc(sizeof(linkedListStack));\ns->top = NULL;\ns->size = 0;\nreturn s;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(linkedListStack *s) {\nwhile (s->top) {\nListNode *n = s->top->next;\nfree(s->top);\ns->top = n;\n}\nfree(s);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(linkedListStack *s) {\nassert(s);\nreturn s->size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(linkedListStack *s) {\nassert(s);\nreturn size(s) == 0;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(linkedListStack *s) {\nassert(s);\nassert(size(s) != 0);\nreturn s->top->val;\n}\n/* \u5165\u6808 */\nvoid push(linkedListStack *s, int num) {\nassert(s);\nListNode *node = (ListNode *)malloc(sizeof(ListNode));\nnode->next = s->top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\nnode->val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\ns->top = node;       // \u66f4\u65b0\u6808\u9876\ns->size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n/* \u51fa\u6808 */\nint pop(linkedListStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nassert(s);\nint val = peek(s);\nListNode *tmp = s->top;\ns->top = s->top->next;\n// \u91ca\u653e\u5185\u5b58\nfree(tmp);\ns->size--;\nreturn val;\n}\n
    linkedlist_stack.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (stackPeek == null)\nthrow new Exception();\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0 || stackPeek == null)\nthrow new Exception();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nif (stackPeek == null)\nreturn Array.Empty<int>();\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.Length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate var _size = 0 // \u6808\u7684\u957f\u5ea6\ninit() {}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nlet node = ListNode(x: num)\nnode.next = _peek\n_peek = node\n_size += 1\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n_peek = _peek?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn _peek!.val\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = _peek\nvar res = Array(repeating: 0, count: _size)\nfor i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_stack.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.stack_top = null;\nself.stk_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stk_size;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack_top.?.val;\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\nnode.next = self.stack_top;\nself.stack_top = node;\nself.stk_size += 1;\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.peek();\nself.stack_top = self.stack_top.?.next;\nself.stk_size -= 1;\nreturn num;\n} // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.stack_top;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[res.len - i - 1] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_stack.dart
    /* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint _stkSize = 0; // \u6808\u7684\u957f\u5ea6\nLinkedListStack() {\n_stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stkSize == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nfinal ListNode node = ListNode(num);\nnode.next = _stackPeek;\n_stackPeek = node;\n_stkSize++;\n}\n/* \u51fa\u6808 */\nint pop() {\nfinal int num = peek();\n_stackPeek = _stackPeek!.next;\n_stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (_stackPeek == null) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stackPeek!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\nList<int> toList() {\nListNode? node = _stackPeek;\nList<int> list = [];\nwhile (node != null) {\nlist.add(node.val);\nnode = node.next;\n}\nlist = list.reversed.toList();\nreturn list;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u5728\u57fa\u4e8e\u300c\u6570\u7ec4\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5728\u8fd9\u6837\u7684\u8bbe\u8ba1\u4e0b\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5c31\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002

    ArrayStackpush()pop()

    \u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u9700\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_stack.java
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate ArrayList<Integer> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new ArrayList<>();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.remove(size() - 1);\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.get(size() - 1);\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic Object[] toArray() {\nreturn stack.toArray();\n}\n}\n
    array_stack.cpp
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate:\nvector<int> stack;\npublic:\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn stack.empty();\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nstack.push_back(num);\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint oldTop = top();\nstack.pop_back();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (empty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stack.back();\n}\n/* \u8fd4\u56de Vector */\nvector<int> toVector() {\nreturn stack;\n}\n};\n
    array_stack.py
    class ArrayStack:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__stack: list[int] = []\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn len(self.__stack)\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__stack == []\ndef push(self, item: int) -> None:\n\"\"\"\u5165\u6808\"\"\"\nself.__stack.append(item)\ndef pop(self) -> int:\n\"\"\"\u51fa\u6808\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self.__stack.pop()\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self.__stack[-1]\ndef to_list(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nreturn self.__stack\n
    array_stack.go
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\ndata []int // \u6570\u636e\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\nreturn &arrayStack{\n// \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\ndata: make([]int, 0, 16),\n}\n}\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\nreturn len(s.data)\n}\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\nreturn s.size() == 0\n}\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n// \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\ns.data = append(s.data, v)\n}\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\nval := s.peek()\ns.data = s.data[:len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\nval := s.data[len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\nreturn s.data\n}\n
    array_stack.js
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n#stack;\nconstructor() {\nthis.#stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nthis.#stack.push(num);\n}\n/* \u51fa\u6808 */\npop() {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop() {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack[this.#stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.#stack;\n}\n}\n
    array_stack.ts
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate stack: number[];\nconstructor() {\nthis.stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nthis.stack.push(num);\n}\n/* \u51fa\u6808 */\npop(): number | undefined {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop(): number | undefined {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack[this.stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.stack;\n}\n}\n
    array_stack.c
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct arrayStack {\nint *data;\nint size;\n};\ntypedef struct arrayStack arrayStack;\n/* \u6784\u9020\u51fd\u6570 */\narrayStack *newArrayStack() {\narrayStack *s = malloc(sizeof(arrayStack));\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\ns->data = malloc(sizeof(int) * MAX_SIZE);\ns->size = 0;\nreturn s;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(arrayStack *s) {\nreturn s->size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(arrayStack *s) {\nreturn s->size == 0;\n}\n/* \u5165\u6808 */\nvoid push(arrayStack *s, int num) {\nif (s->size == MAX_SIZE) {\nprintf(\"stack is full.\\n\");\nreturn;\n}\ns->data[s->size] = num;\ns->size++;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(arrayStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nreturn s->data[s->size - 1];\n}\n/* \u51fa\u6808 */\nint pop(arrayStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nint val = peek(s);\ns->size--;\nreturn val;\n}\n
    array_stack.cs
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate List<int> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.Count();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.Add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new Exception();\nvar val = peek();\nstack.RemoveAt(size() - 1);\nreturn val;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new Exception();\nreturn stack[size() - 1];\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nreturn stack.ToArray();\n}\n}\n
    array_stack.swift
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate var stack: [Int]\ninit() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = []\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nstack.count\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nstack.isEmpty\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nstack.append(num)\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.removeLast()\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.last!\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nstack\n}\n}\n
    array_stack.zig
    // \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack: ?std.ArrayList(T) = null,     // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) void {\nif (self.stack == null) {\nself.stack = std.ArrayList(T).init(allocator);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.stack == null) return;\nself.stack.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stack.?.items.len;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack.?.items[self.size() - 1];\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\ntry self.stack.?.append(num);\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.stack.?.pop();\nreturn num;\n} // \u8fd4\u56de ArrayList\npub fn toList(self: *Self) std.ArrayList(T) {\nreturn self.stack.?;\n}\n};\n}\n
    array_stack.dart
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nlate List<int> _stack;\nArrayStack() {\n_stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stack.isEmpty;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\n_stack.add(num);\n}\n/* \u51fa\u6808 */\nint pop() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.removeLast();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.last;\n}\n/* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList<int> toArray() => _stack;\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":""},{"location":"chapter_stack_and_queue/stack/#_3","title":"\u652f\u6301\u64cd\u4f5c","text":"

    \u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002

    "},{"location":"chapter_stack_and_queue/stack/#_4","title":"\u65f6\u95f4\u6548\u7387","text":"

    \u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u662f\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002

    \u5728\u94fe\u8868\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002

    \u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff08\u5982 int , double \uff09\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a

    • \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\uff1b
    • \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\uff1b
    "},{"location":"chapter_stack_and_queue/stack/#_5","title":"\u7a7a\u95f4\u6548\u7387","text":"

    \u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u8fc7\u5b9e\u9645\u9700\u6c42\u3002\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002

    \u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002

    \u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002

    "},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4. \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"
    • \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5c06\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u300c\u540e\u9000\u300d\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u9875\u9762\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
    • \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002
    "},{"location":"chapter_stack_and_queue/summary/","title":"5.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002
    • \u4ece\u65f6\u95f4\u6548\u7387\u89d2\u5ea6\u770b\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u964d\u4f4e\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
    • \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002
    • \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002
    • \u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002
    "},{"location":"chapter_stack_and_queue/summary/#541-q-a","title":"5.4.1. \u00a0 Q & A","text":"

    \u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f

    \u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u53cc\u5411\u961f\u5217\u7ae0\u8282\u6709\u63d0\u5230\u3002

    \u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f

    \u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0cJava \u548c Python \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 C \u548c C++ \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002

    \u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f

    \u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u8005\u662f\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002

    "},{"location":"chapter_tree/","title":"7. \u00a0 \u6811","text":""},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3. \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"

    \u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 TreeNode \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u5728\u4e0a\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u5728\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002

    \u90a3\u4e48\uff0c\u80fd\u5426\u7528\u300c\u6570\u7ec4\u300d\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002

    "},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1. \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"

    \u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u7f16\u53f7\uff08\u4ece \\(0\\) \u5f00\u59cb\uff09\uff0c\u6b64\u65f6\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u7d22\u5f15\u3002

    \u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    \u6620\u5c04\u516c\u5f0f\u7684\u4f5c\u7528\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u5982\u679c\u6211\u4eec\u5c06\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u90a3\u4e48\u5bf9\u4e8e\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5176\u5b50\u8282\u70b9\u3002

    "},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2. \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"

    \u7136\u800c\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u53ea\u662f\u4e00\u4e2a\u7279\u4f8b\u3002\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\uff0c\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \uff0c\u800c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \u3002\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002\u663e\u7136\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0a\u8ff0\u7684\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002

    Fig. \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027

    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\)\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nvector<int> tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n
    # \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u76f4\u63a5\u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n
    \n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList<int?> tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n

    Fig. \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    "},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3. \u00a0 \u4f18\u52bf\u4e0e\u5c40\u9650\u6027","text":"

    \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u5b58\u5728\u4ee5\u4e0b\u4f18\u70b9\uff1a

    • \u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\uff1b
    • \u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\uff1b
    • \u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\uff1b

    \u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5177\u6709\u4e00\u4e9b\u5c40\u9650\u6027\uff1a

    • \u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002
    • \u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\uff1b
    • \u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002

    \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u8fd9\u610f\u5473\u7740\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002\u56e0\u6b64\uff0c\u5728\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \u3002

    Fig. \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    "},{"location":"chapter_tree/avl_tree/","title":"7.5. \u00a0 AVL \u6811 *","text":"

    \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u4e86\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\)\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002

    Fig. AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316

    \u518d\u4f8b\u5982\uff0c\u5728\u4ee5\u4e0b\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002

    Fig. AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316

    G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u5176 1962 \u5e74\u53d1\u8868\u7684\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002

    "},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1. \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"

    \u300cAVL \u6811\u300d\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u3002

    "},{"location":"chapter_tree/avl_tree/#_1","title":"\u8282\u70b9\u9ad8\u5ea6","text":"

    \u5728\u64cd\u4f5c AVL \u6811\u65f6\uff0c\u6211\u4eec\u9700\u8981\u83b7\u53d6\u8282\u70b9\u7684\u9ad8\u5ea6\uff0c\u56e0\u6b64\u9700\u8981\u4e3a AVL \u6811\u7684\u8282\u70b9\u7c7b\u6dfb\u52a0 height \u53d8\u91cf\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\npublic int val;        // \u8282\u70b9\u503c\npublic int height;     // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode left;  // \u5de6\u5b50\u8282\u70b9\npublic TreeNode right; // \u53f3\u5b50\u8282\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\nint val{};          // \u8282\u70b9\u503c\nint height = 0;     // \u8282\u70b9\u9ad8\u5ea6\nTreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\nTreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\nTreeNode() = default;\nexplicit TreeNode(int x) : val(x){}\n};\n
    class TreeNode:\n\"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                    # \u8282\u70b9\u503c\nself.height: int = 0                   # \u8282\u70b9\u9ad8\u5ea6\nself.left: Optional[TreeNode] = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nself.right: Optional[TreeNode] = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n
    /* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal    int       // \u8282\u70b9\u503c\nHeight int       // \u8282\u70b9\u9ad8\u5ea6\nLeft   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nRight  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval; // \u8282\u70b9\u503c\nheight; //\u8282\u70b9\u9ad8\u5ea6\nleft; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val, left, right, height) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;            // \u8282\u70b9\u503c\nheight: number;         // \u8282\u70b9\u9ad8\u5ea6\nleft: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height; this.left = left === undefined ? null : left; this.right = right === undefined ? null : right; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;\nint height;\nstruct TreeNode *left;\nstruct TreeNode *right;\n};\ntypedef struct TreeNode TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode->val = val;\nnode->height = 0;\nnode->left = NULL;\nnode->right = NULL;\nreturn node;\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\npublic int val;          // \u8282\u70b9\u503c\npublic int height;       // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode? left;   // \u5de6\u5b50\u8282\u70b9\npublic TreeNode? right;  // \u53f3\u5b50\u8282\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar height: Int // \u8282\u70b9\u9ad8\u5ea6\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\ninit(x: Int) {\nval = x\nheight = 0\n}\n}\n
    \n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nint height;      // \u8282\u70b9\u9ad8\u5ea6\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\nTreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n

    \u300c\u8282\u70b9\u9ad8\u5ea6\u300d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a 0 \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a -1 \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
    avl_tree.cpp
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == nullptr ? -1 : node->height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode->height = max(height(node->left), height(node->right)) + 1;\n}\n
    avl_tree.py
    def height(self, node: TreeNode | None) -> int:\n\"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node is not None:\nreturn node.height\nreturn -1\ndef __update_height(self, node: TreeNode | None):\n\"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = max([self.height(node.left), self.height(node.right)]) + 1\n
    avl_tree.go
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node != nil {\nreturn node.Height\n}\nreturn -1\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\nlh := t.height(node.Left)\nrh := t.height(node.Right)\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif lh > rh {\nnode.Height = lh + 1\n} else {\nnode.Height = rh + 1\n}\n}\n
    avl_tree.js
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
    avl_tree.ts
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
    avl_tree.c
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif (node != NULL) {\nreturn node->height;\n}\nreturn -1;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\nint lh = height(node->left);\nint rh = height(node->right);\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif (lh > rh) {\nnode->height = lh + 1;\n} else {\nnode->height = rh + 1;\n}\n}\n
    avl_tree.cs
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.Max(height(node.left), height(node.right)) + 1;\n}\n
    avl_tree.swift
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nnode == nil ? -1 : node!.height\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
    avl_tree.zig
    // \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n_ = self;\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn if (node == null) -1 else node.?.height;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.?.height = std.math.max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n
    avl_tree.dart
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode!.height = max(height(node.left), height(node.right)) + 1;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_2","title":"\u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"

    \u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 Balance Factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a 0 \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
    avl_tree.cpp
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == nullptr)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
    avl_tree.py
    def balance_factor(self, node: TreeNode | None) -> int:\n\"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n# \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node is None:\nreturn 0\n# \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.left) - self.height(node.right)\n
    avl_tree.go
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node == nil {\nreturn 0\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn t.height(node.Left) - t.height(node.Right)\n}\n
    avl_tree.js
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
    avl_tree.ts
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
    avl_tree.c
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == NULL) {\nreturn 0;\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
    avl_tree.cs
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
    avl_tree.swift
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nguard let node = node else { return 0 }\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node: node.left) - height(node: node.right)\n}\n
    avl_tree.zig
    // \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.?.left) - self.height(node.?.right);\n}\n
    avl_tree.dart
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n

    Note

    \u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002

    "},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2. \u00a0 AVL \u6811\u65cb\u8f6c","text":"

    AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u300c\u65cb\u8f6c Rotation\u300d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u6811\u7684\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5c5e\u6027\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u3002

    \u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u300c\u5931\u8861\u8282\u70b9\u300d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u6211\u4eec\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002

    "},{"location":"chapter_tree/avl_tree/#_3","title":"\u53f3\u65cb","text":"

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a node \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a child \uff0c\u6267\u884c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u5df2\u7ecf\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u7279\u6027\u3002

    <1><2><3><4>

    \u6b64\u5916\uff0c\u5982\u679c\u8282\u70b9 child \u672c\u8eab\u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u53f3\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u5de6\u5b50\u8282\u70b9\u3002

    Fig. \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c

    \u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\nTreeNode child = node.left;\nTreeNode grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cpp
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child = node->left;\nTreeNode *grandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.py
    def __right_rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\nchild = node.left\ngrand_child = child.right\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node\nnode.left = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n
    avl_tree.go
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\nchild := node.Left\ngrandChild := child.Right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.Right = node\nnode.Left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.js
    /* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.ts
    /* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.c
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node->left;\ngrandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cs
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\nTreeNode? child = node.left;\nTreeNode? grandChild = child?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.swift
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.left\nlet grandChild = child?.right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild?.right = node\nnode?.left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.zig
    // \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.left;\nvar grandChild = child.?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.?.right = node;\nnode.?.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.dart
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\nTreeNode? child = node!.left;\nTreeNode? grandChild = child!.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_4","title":"\u5de6\u65cb","text":"

    \u76f8\u5e94\u7684\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u300c\u5de6\u65cb\u300d\u64cd\u4f5c\u3002

    Fig. \u5de6\u65cb\u64cd\u4f5c

    \u540c\u7406\uff0c\u82e5\u8282\u70b9 child \u672c\u8eab\u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u5de6\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u53f3\u5b50\u8282\u70b9\u3002

    Fig. \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c

    \u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u677e\u5730\u4ece\u53f3\u65cb\u7684\u4ee3\u7801\u63a8\u5bfc\u51fa\u5de6\u65cb\u7684\u4ee3\u7801\u3002\u5177\u4f53\u5730\uff0c\u53ea\u9700\u5c06\u300c\u53f3\u65cb\u300d\u4ee3\u7801\u4e2d\u7684\u628a\u6240\u6709\u7684 left \u66ff\u6362\u4e3a right \uff0c\u5c06\u6240\u6709\u7684 right \u66ff\u6362\u4e3a left \uff0c\u5373\u53ef\u5f97\u5230\u300c\u5de6\u65cb\u300d\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\nTreeNode child = node.right;\nTreeNode grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cpp
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child = node->right;\nTreeNode *grandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.py
    def __left_rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\nchild = node.right\ngrand_child = child.left\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node\nnode.right = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n
    avl_tree.go
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\nchild := node.Right\ngrandChild := child.Left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.Left = node\nnode.Right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.js
    /* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.ts
    /* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.c
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node->right;\ngrandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cs
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\nTreeNode? child = node.right;\nTreeNode? grandChild = child?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.swift
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.right\nlet grandChild = child?.left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild?.left = node\nnode?.right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.zig
    // \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.right;\nvar grandChild = child.?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.?.left = node;\nnode.?.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.dart
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\nTreeNode? child = node!.right;\nTreeNode? grandChild = child!.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_5","title":"\u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"

    \u5bf9\u4e8e\u4e0b\u56fe\u4e2d\u7684\u5931\u8861\u8282\u70b9 3\uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5de6\u65cb\u540e\u53f3\u65cb\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u5de6\u65cb\u300d\uff0c\u518d\u5bf9 node \u6267\u884c\u300c\u53f3\u65cb\u300d\u3002

    Fig. \u5148\u5de6\u65cb\u540e\u53f3\u65cb

    "},{"location":"chapter_tree/avl_tree/#_6","title":"\u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"

    \u540c\u7406\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u53f3\u65cb\u540e\u5de6\u65cb\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u53f3\u65cb\u300d\uff0c\u7136\u540e\u5bf9 node \u6267\u884c\u300c\u5de6\u65cb\u300d\u3002

    Fig. \u5148\u53f3\u65cb\u540e\u5de6\u65cb

    "},{"location":"chapter_tree/avl_tree/#_7","title":"\u65cb\u8f6c\u7684\u9009\u62e9","text":"

    \u4e0b\u56fe\u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u540e\u5de6\u3001\u5148\u5de6\u540e\u53f3\u7684\u65cb\u8f6c\u64cd\u4f5c\u3002

    Fig. AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5

    \u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u4e0a\u56fe\u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002

    \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb

    \u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.cpp
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint _balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (_balanceFactor > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (_balanceFactor < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.py
    def __rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n# \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nbalance_factor = self.balance_factor(node)\n# \u5de6\u504f\u6811\nif balance_factor > 1:\nif self.balance_factor(node.left) >= 0:\n# \u53f3\u65cb\nreturn self.__right_rotate(node)\nelse:\n# \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = self.__left_rotate(node.left)\nreturn self.__right_rotate(node)\n# \u53f3\u504f\u6811\nelif balance_factor < -1:\nif self.balance_factor(node.right) <= 0:\n# \u5de6\u65cb\nreturn self.__left_rotate(node)\nelse:\n# \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = self.__right_rotate(node.right)\nreturn self.__left_rotate(node)\n# \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n
    avl_tree.go
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n// Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\nbf := t.balanceFactor(node)\n// \u5de6\u504f\u6811\nif bf > 1 {\nif t.balanceFactor(node.Left) >= 0 {\n// \u53f3\u65cb\nreturn t.rightRotate(node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.Left = t.leftRotate(node.Left)\nreturn t.rightRotate(node)\n}\n}\n// \u53f3\u504f\u6811\nif bf < -1 {\nif t.balanceFactor(node.Right) <= 0 {\n// \u5de6\u65cb\nreturn t.leftRotate(node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.Right = t.rightRotate(node.Right)\nreturn t.leftRotate(node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
    avl_tree.js
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.#rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.#leftRotate(node.left);\nreturn this.#rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.#leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.#rightRotate(node.right);\nreturn this.#leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.ts
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.leftRotate(node.left);\nreturn this.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.rightRotate(node.right);\nreturn this.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.c
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint bf = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (bf > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (bf < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.cs
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactorInt = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactorInt > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node?.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactorInt < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node?.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.swift
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balanceFactor = balanceFactor(node: node)\n// \u5de6\u504f\u6811\nif balanceFactor > 1 {\nif self.balanceFactor(node: node?.left) >= 0 {\n// \u53f3\u65cb\nreturn rightRotate(node: node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode?.left = leftRotate(node: node?.left)\nreturn rightRotate(node: node)\n}\n}\n// \u53f3\u504f\u6811\nif balanceFactor < -1 {\nif self.balanceFactor(node: node?.right) <= 0 {\n// \u5de6\u65cb\nreturn leftRotate(node: node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode?.right = rightRotate(node: node?.right)\nreturn leftRotate(node: node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
    avl_tree.zig
    // \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nvar balance_factor = self.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balance_factor > 1) {\nif (self.balanceFactor(node.?.left) >= 0) {\n// \u53f3\u65cb\nreturn self.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.?.left = self.leftRotate(node.?.left);\nreturn self.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balance_factor < -1) {\nif (self.balanceFactor(node.?.right) <= 0) {\n// \u5de6\u65cb\nreturn self.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.?.right = self.rightRotate(node.?.right);\nreturn self.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.dart
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint factor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (factor > 1) {\nif (balanceFactor(node!.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (factor < -1) {\nif (balanceFactor(node!.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3. \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#_8","title":"\u63d2\u5165\u8282\u70b9","text":"

    \u300cAVL \u6811\u300d\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\nif (node == null)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cpp
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node->val)\nnode->left = insertHelper(node->left, val);\nelse if (val > node->val)\nnode->right = insertHelper(node->right, val);\nelse\nreturn node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.py
    def insert(self, val) -> None:\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\nself.root = self.__insert_helper(self.root, val)\ndef __insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:\n\"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn TreeNode(val)\n# 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif val < node.val:\nnode.left = self.__insert_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__insert_helper(node.right, val)\nelse:\n# \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
    avl_tree.go
    /* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\nt.root = t.insertHelper(t.root, val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn NewTreeNode(val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val < node.Val {\nnode.Left = t.insertHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.insertHelper(node.Right, val)\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.js
    /* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\nthis.root = this.#insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val) node.left = this.#insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = this.#insertHelper(node.right, val);\nelse return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.ts
    /* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\nthis.root = this.insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val) {\nnode.left = this.insertHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.insertHelper(node.right, val);\n} else {\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.c
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(aVLTree *tree, int val) {\ntree->root = insertHelper(tree->root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == NULL) {\nreturn newTreeNode(val);\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node->val) {\nnode->left = insertHelper(node->left, val);\n} else if (val > node->val) {\nnode->right = insertHelper(node->right, val);\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cs
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.swift
    /* \u63d2\u5165\u8282\u70b9 */\n@discardableResult\nfunc insert(val: Int) {\nroot = insertHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn TreeNode(x: val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val < node!.val {\nnode?.left = insertHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = insertHelper(node: node?.right, val: val)\n} else {\nreturn node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.zig
    // \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\nself.root = (try self.insertHelper(self.root, val)).?;\n}\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) {\nvar tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\ntmp_node.init(val);\nreturn tmp_node;\n}\n// 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif (val < node.?.val) {\nnode.?.left = try self.insertHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = try self.insertHelper(node.?.right, val);\n} else {\nreturn node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nself.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.dart
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\nif (node == null) return TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_9","title":"\u5220\u9664\u8282\u70b9","text":"

    \u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\nif (node == null)\nreturn null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode child = node.left != null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cpp
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn nullptr;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val)\nnode->left = removeHelper(node->left, val);\nelse if (val > node->val)\nnode->right = removeHelper(node->right, val);\nelse {\nif (node->left == nullptr || node->right == nullptr) {\nTreeNode *child = node->left != nullptr ? node->left : node->right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == nullptr) {\ndelete node;\nreturn nullptr;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\ndelete node;\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node->right;\nwhile (temp->left != nullptr) {\ntemp = temp->left;\n}\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.py
    def remove(self, val: int) -> None:\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\nself.root = self.__remove_helper(self.root, val)\ndef __remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:\n\"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn None\n# 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif val < node.val:\nnode.left = self.__remove_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__remove_helper(node.right, val)\nelse:\nif node.left is None or node.right is None:\nchild = node.left or node.right\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child is None:\nreturn None\n# \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse:\nnode = child\nelse:\n# \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp = node.right\nwhile temp.left is not None:\ntemp = temp.left\nnode.right = self.__remove_helper(node.right, temp.val)\nnode.val = temp.val\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
    avl_tree.go
    /* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\nt.root = t.removeHelper(t.root, val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node.Val {\nnode.Left = t.removeHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.removeHelper(node.Right, val)\n} else {\nif node.Left == nil || node.Right == nil {\nchild := node.Left\nif node.Right != nil {\nchild = node.Right\n}\nif child == nil {\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nreturn nil\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp := node.Right\nfor temp.Left != nil {\ntemp = temp.Left\n}\nnode.Right = t.removeHelper(node.Right, temp.Val)\nnode.Val = temp.Val\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.js
    /* \u5220\u9664\u8282\u70b9 */\nremove(val) {\nthis.root = this.#removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) node.left = this.#removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = this.#removeHelper(node.right, val);\nelse {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) return null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse node = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.#removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.ts
    /* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\nthis.root = this.removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) {\nnode.left = this.removeHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.removeHelper(node.right, val);\n} else {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) {\nreturn null;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.c
    /* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeNode(aVLTree *tree, int val) {\nTreeNode *root = removeHelper(tree->root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nTreeNode *child, *grandChild;\nif (node == NULL) {\nreturn NULL;\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val) {\nnode->left = removeHelper(node->left, val);\n} else if (val > node->val) {\nnode->right = removeHelper(node->right, val);\n} else {\nif (node->left == NULL || node->right == NULL) {\nchild = node->left;\nif (node->right != NULL) {\nchild = node->right;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == NULL) {\nreturn NULL;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node->right;\nwhile (temp->left != NULL) {\ntemp = temp->left;\n}\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cs
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left != null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.swift
    /* \u5220\u9664\u8282\u70b9 */\n@discardableResult\nfunc remove(val: Int) {\nroot = removeHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node!.val {\nnode?.left = removeHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = removeHelper(node: node?.right, val: val)\n} else {\nif node?.left == nil || node?.right == nil {\nlet child = node?.left != nil ? node?.left : node?.right\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node?.right\nwhile temp?.left != nil {\ntemp = temp?.left\n}\nnode?.right = removeHelper(node: node?.right, val: temp!.val)\nnode?.val = temp!.val\n}\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.zig
    // \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\nself.root = self.removeHelper(self.root, val).?;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) return null;\n// 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif (val < node.?.val) {\nnode.?.left = self.removeHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = self.removeHelper(node.?.right, val);\n} else {\nif (node.?.left == null or node.?.right == null) {\nvar child = if (node.?.left != null) node.?.left else node.?.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null) {\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n} else {\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node.?.right;\nwhile (temp.?.left != null) {\ntemp = temp.?.left;\n}\nnode.?.right = self.removeHelper(node.?.right, temp.?.val);\nnode.?.val = temp.?.val;\n}\n}\nself.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.dart
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left ?? node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp!.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_10","title":"\u67e5\u627e\u8282\u70b9","text":"

    AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002

    "},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4. \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"
    • \u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\uff1b
    • \u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\uff1b

    \u4e3a\u4ec0\u4e48\u7ea2\u9ed1\u6811\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\uff1f

    \u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u56e0\u6b64\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5728\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4e0a\u7684\u5e73\u5747\u6548\u7387\u9ad8\u4e8e AVL \u6811\u3002

    "},{"location":"chapter_tree/binary_search_tree/","title":"7.4. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"

    \u300c\u4e8c\u53c9\u641c\u7d22\u6811 Binary Search Tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a

    1. \u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(<\\) \u6839\u8282\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\uff1b
    2. \u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 1. \uff1b

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811

    "},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_search_tree/#_1","title":"\u67e5\u627e\u8282\u70b9","text":"

    \u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c num \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 cur \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 root \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c cur.val \u548c num \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb

    • \u82e5 cur.val < num \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.right \uff1b
    • \u82e5 cur.val > num \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.left \uff1b
    • \u82e5 cur.val = num \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\uff1b
    <1><2><3><4>

    \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\nTreeNode cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.cpp
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\nTreeNode *cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur->val > num)\ncur = cur->left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.py
    def search(self, num: int) -> TreeNode | None:\n\"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\ncur: TreeNode | None = self.root\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur is not None:\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelif cur.val > num:\ncur = cur.left\n# \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse:\nbreak\nreturn cur\n
    binary_search_tree.go
    /* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\nnode := bst.root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor node != nil {\nif node.Val < num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nnode = node.Right\n} else if node.Val > num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nnode = node.Left\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn node\n}\n
    binary_search_tree.js
    /* \u67e5\u627e\u8282\u70b9 */\nfunction search(num) {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.ts
    /* \u67e5\u627e\u8282\u70b9 */\nfunction search(num: number): TreeNode | null {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val < num) {\ncur = cur.right; // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else if (cur.val > num) {\ncur = cur.left; // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\nbreak; // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.c
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(binarySearchTree *bst, int num) {\nTreeNode *cur = bst->root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\nif (cur->val < num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else if (cur->val > num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.cs
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int num) {\nTreeNode? cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur =\ncur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.swift
    /* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -> TreeNode? {\nvar cur = root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if cur!.val > num {\ncur = cur?.left\n}\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur\n}\n
    binary_search_tree.zig
    // \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\nvar cur = self.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else if (cur.?.val > num) {\ncur = cur.?.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n} else {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{search}\n
    "},{"location":"chapter_tree/binary_search_tree/#_2","title":"\u63d2\u5165\u8282\u70b9","text":"

    \u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 num \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u5206\u4e3a\u4e24\u6b65\uff1a

    1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c num \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\uff1b
    2. \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 num \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\uff1b

    \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new TreeNode(num);\nif (pre.val < num)\npre.right = node;\nelse\npre.left = node;\n}\n
    binary_search_tree.cpp
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr)\nreturn;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur->left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = new TreeNode(num);\nif (pre->val < num)\npre->right = node;\nelse\npre->left = node;\n}\n
    binary_search_tree.py
    def insert(self, num: int) -> None:\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.root is None:\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self.root, None\nwhile cur is not None:\n# \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur.val == num:\nreturn\npre = cur\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u63d2\u5165\u8282\u70b9\nnode = TreeNode(num)\nif pre.val < num:\npre.right = node\nelse:\npre.left = node\n
    binary_search_tree.go
    /* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nreturn\n}\npre = cur\nif cur.Val < num {\ncur = cur.Right\n} else {\ncur = cur.Left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nnode := NewTreeNode(num)\nif pre.Val < num {\npre.Right = node\n} else {\npre.Left = node\n}\n}\n
    binary_search_tree.js
    /* \u63d2\u5165\u8282\u70b9 */\nfunction insert(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return;\nlet cur = root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\n}\n
    binary_search_tree.ts
    /* \u63d2\u5165\u8282\u70b9 */\nfunction insert(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val === num) {\nreturn; // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = new TreeNode(num);\nif (pre!.val < num) {\npre!.right = node;\n} else {\npre!.left = node;\n}\n}\n
    binary_search_tree.c
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(binarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (bst->root == NULL)\nreturn;\nTreeNode *cur = bst->root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num) {\nreturn;\n}\npre = cur;\nif (cur->val < num) {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = newTreeNode(num);\nif (pre->val < num) {\npre->right = node;\n} else {\npre->left = node;\n}\n}\n
    binary_search_tree.cs
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new TreeNode(num);\nif (pre != null) {\nif (pre.val < num)\npre.right = node;\nelse\npre.left = node;\n}\n}\n
    binary_search_tree.swift
    /* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur!.val == num {\nreturn\n}\npre = cur\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = TreeNode(x: num)\nif pre!.val < num {\npre?.right = node\n} else {\npre?.left = node\n}\n}\n
    binary_search_tree.zig
    // \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.?.val == num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nvar node = try self.mem_allocator.create(inc.TreeNode(T));\nnode.init(num);\nif (pre.?.val < num) {\npre.?.right = node;\n} else {\npre.?.left = node;\n}\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{insert}\n

    \u4e3a\u4e86\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u5229\u7528\u8f85\u52a9\u8282\u70b9 pre \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\uff0c\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002

    \u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    "},{"location":"chapter_tree/binary_search_tree/#_3","title":"\u5220\u9664\u8282\u70b9","text":"

    \u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u5728\u5220\u9664\u64cd\u4f5c\u540e\u7ef4\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u3002\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5728\u4e8c\u53c9\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u83b7\u53d6\u5f85\u5220\u9664\u8282\u70b9\u3002\u63a5\u4e0b\u6765\uff0c\u6839\u636e\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5220\u9664\u64cd\u4f5c\u9700\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u5f85\u5220\u9664\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0\uff09

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1\uff09

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6 \\(<\\) \u6839 \\(<\\) \u53f3\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u6216\u8005\u79f0\u4e3a\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u4e3a\uff1a

    1. \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a tmp \uff1b
    2. \u5c06 tmp \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 tmp \uff1b
    <1><2><3><4>

    \u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.cpp
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr)\nreturn;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur->left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == nullptr)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur->left == nullptr || cur->right == nullptr) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur->left != nullptr ? cur->left : cur->right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre->left == cur)\npre->left = child;\nelse\npre->right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n// \u91ca\u653e\u5185\u5b58\ndelete cur;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur->right;\nwhile (tmp->left != nullptr) {\ntmp = tmp->left;\n}\nint tmpVal = tmp->val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp->val);\n// \u7528 tmp \u8986\u76d6 cur\ncur->val = tmpVal;\n}\n}\n
    binary_search_tree.py
    def remove(self, num: int) -> None:\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.root is None:\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self.root, None\nwhile cur is not None:\n# \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur.val == num:\nbreak\npre = cur\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur is None:\nreturn\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur.left is None or cur.right is None:\n# \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nchild = cur.left or cur.right\n# \u5220\u9664\u8282\u70b9 cur\nif cur != self.root:\nif pre.left == cur:\npre.left = child\nelse:\npre.right = child\nelse:\n# \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nself.root = child\n# \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse:\n# \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp: TreeNode = cur.right\nwhile tmp.left is not None:\ntmp = tmp.left\n# \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.val)\n# \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val\n
    binary_search_tree.go
    /* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nbreak\n}\npre = cur\nif cur.Val < num {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\ncur = cur.Right\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\ncur = cur.Left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\nif cur.Left == nil || cur.Right == nil {\nvar child *TreeNode = nil\n// \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\nif cur.Left != nil {\nchild = cur.Left\n} else {\nchild = cur.Right\n}\n// \u5220\u9664\u8282\u70b9 cur\nif cur != bst.root {\nif pre.Left == cur {\npre.Left = child\n} else {\npre.Right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nbst.root = child\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp := cur.Right\nfor tmp.Left != nil {\ntmp = tmp.Left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nbst.remove(tmp.Val)\n// \u7528 tmp \u8986\u76d6 cur\ncur.Val = tmp.Val\n}\n}\n
    binary_search_tree.js
    /* \u5220\u9664\u8282\u70b9 */\nfunction remove(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return;\nlet cur = root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left === cur) pre.left = child;\nelse pre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp = cur.right;\nwhile (tmp.left !== null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.ts
    /* \u5220\u9664\u8282\u70b9 */\nfunction remove(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) {\nbreak;\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) {\nreturn;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre!.left === cur) {\npre!.left = child;\n} else {\npre!.right = child;\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp = cur.right;\nwhile (tmp.left !== null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp!.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.c
    /* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeNode(binarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (bst->root == NULL)\nreturn;\nTreeNode *cur = bst->root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num)\nbreak;\npre = cur;\nif (cur->val < num) {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == NULL)\nreturn;\n// \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\nif (cur->left == NULL || cur->right == NULL) {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur->left != NULL ? cur->left : cur->right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre->left == cur) {\npre->left = child;\n} else {\npre->right = child;\n}\n} else {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur->right;\nwhile (tmp->left != NULL) {\ntmp = tmp->left;\n}\nint tmpVal = tmp->val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremoveNode(bst, tmp->val);\n// \u7528 tmp \u8986\u76d6 cur\ncur->val = tmpVal;\n}\n}\n
    binary_search_tree.cs
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null || pre == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode? child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode? tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.swift
    /* \u5220\u9664\u8282\u70b9 */\n@discardableResult\nfunc remove(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur!.val == num {\nbreak\n}\npre = cur\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur?.left == nil || cur?.right == nil {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur?.left != nil ? cur?.left : cur?.right\n// \u5220\u9664\u8282\u70b9 cur\nif cur !== root {\nif pre?.left === cur {\npre?.left = child\n} else {\npre?.right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur?.right\nwhile tmp?.left != nil {\ntmp = tmp?.left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(num: tmp!.val)\n// \u7528 tmp \u8986\u76d6 cur\ncur?.val = tmp!.val\n}\n}\n
    binary_search_tree.zig
    // \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.?.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.?.left == null or cur.?.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nvar child = if (cur.?.left != null) cur.?.left else cur.?.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre.?.left == cur) {\npre.?.left = child;\n} else {\npre.?.right = child;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur.?.right;\nwhile (tmp.?.left != null) {\ntmp = tmp.?.left;\n}\nvar tmp_val = tmp.?.val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.?.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.?.val = tmp_val;\n}\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{remove}\n
    "},{"location":"chapter_tree/binary_search_tree/#_4","title":"\u6392\u5e8f","text":"

    \u6211\u4eec\u77e5\u9053\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002

    \u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u9700\u989d\u5916\u6392\u5e8f\uff0c\u975e\u5e38\u9ad8\u6548\u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217

    "},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"

    \u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002

    \u89c2\u5bdf\u53ef\u77e5\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u8868\u73b0\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u7684\u6570\u636e\u9002\u7528\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002

    \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\)

    \u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002

    \u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\u4e0e\u9000\u5316

    "},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"
    • \u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
    • \u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
    • \u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002
    "},{"location":"chapter_tree/binary_tree/","title":"7.1. \u00a0 \u4e8c\u53c9\u6811","text":"

    \u300c\u4e8c\u53c9\u6811 Binary Tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;          // \u8282\u70b9\u503c\nTreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n
    class TreeNode:\n\"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                   # \u8282\u70b9\u503c\nself.left: Optional[TreeNode] = None  # \u5de6\u5b50\u8282\u70b9\u6307\u9488\nself.right: Optional[TreeNode] = None # \u53f3\u5b50\u8282\u70b9\u6307\u9488\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal   int\nLeft  *TreeNode\nRight *TreeNode\n}\n/* \u8282\u70b9\u521d\u59cb\u5316\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\nreturn &TreeNode{\nLeft:  nil,\nRight: nil,\nVal:   v,\n}\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nfunction TreeNode(val, left, right) {\nthis.val = (val === undefined ? 0 : val); // \u8282\u70b9\u503c\nthis.left = (left === undefined ? null : left); // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nthis.right = (right === undefined ? null : right); // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;\nleft: TreeNode | null;\nright: TreeNode | null;\nconstructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nthis.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n}\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;                // \u8282\u70b9\u503c\nint height;             // \u8282\u70b9\u9ad8\u5ea6\nstruct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nstruct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n};\ntypedef struct TreeNode TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode->val = val;\nnode->height = 0;\nnode->left = NULL;\nnode->right = NULL;\nreturn node;\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;          // \u8282\u70b9\u503c\nTreeNode? left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode? right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u6307\u9488\ninit(x: Int) {\nval = x\n}\n}\n
    \n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(this.val, [this.left, this.right]);\n}\n

    \u8282\u70b9\u7684\u4e24\u4e2a\u6307\u9488\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9\u300d\uff0c\u540c\u65f6\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811\u300d\u3002

    \u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u4f8b\u5982\uff0c\u5728\u4ee5\u4e0b\u793a\u4f8b\u4e2d\uff0c\u82e5\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002

    Fig. \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811

    "},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1. \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"

    \u4e8c\u53c9\u6811\u6d89\u53ca\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5efa\u8bae\u5c3d\u91cf\u7406\u89e3\u5e76\u8bb0\u4f4f\u3002

    • \u300c\u6839\u8282\u70b9 Root Node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\uff1b
    • \u300c\u53f6\u8282\u70b9 Leaf Node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \uff1b
    • \u8282\u70b9\u7684\u300c\u5c42 Level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \uff1b
    • \u8282\u70b9\u7684\u300c\u5ea6 Degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u8303\u56f4\u662f 0, 1, 2 \uff1b
    • \u300c\u8fb9 Edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u6307\u9488\uff1b
    • \u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b
    • \u8282\u70b9\u7684\u300c\u6df1\u5ea6 Depth\u300d \uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b
    • \u8282\u70b9\u7684\u300c\u9ad8\u5ea6 Height\u300d\uff1a\u4ece\u6700\u8fdc\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b

    Fig. \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed

    \u9ad8\u5ea6\u4e0e\u6df1\u5ea6\u7684\u5b9a\u4e49

    \u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u300c\u9ad8\u5ea6\u300d\u548c\u300c\u6df1\u5ea6\u300d\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002

    "},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2. \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":"

    \u521d\u59cb\u5316\u4e8c\u53c9\u6811\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree.java
    // \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.cpp
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
    binary_tree.py
    # \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
    binary_tree.go
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
    binary_tree.js
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.ts
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.c
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
    binary_tree.cs
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.swift
    // \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
    binary_tree.zig
    \n
    binary_tree.dart
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n

    \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u3002

    Fig. \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree.java
    TreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.cpp
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
    binary_tree.py
    # \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
    binary_tree.go
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n
    binary_tree.js
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.ts
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.c
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
    binary_tree.cs
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.swift
    let P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
    binary_tree.zig
    \n
    binary_tree.dart
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n

    Note

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002

    "},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3. \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#_1","title":"\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 Perfect Binary Tree\u300d\u9664\u4e86\u6700\u5e95\u5c42\u5916\uff0c\u5176\u4f59\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002

    Tip

    \u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\uff0c\u8bf7\u6ce8\u610f\u533a\u5206\u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_2","title":"\u5b8c\u5168\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u5168\u4e8c\u53c9\u6811 Complete Binary Tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002

    Fig. \u5b8c\u5168\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_3","title":"\u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 Full Binary Tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002

    Fig. \u5b8c\u6ee1\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_4","title":"\u5e73\u8861\u4e8c\u53c9\u6811","text":"

    \u300c\u5e73\u8861\u4e8c\u53c9\u6811 Balanced Binary Tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002

    Fig. \u5e73\u8861\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4. \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"

    \u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u300c\u94fe\u8868\u300d\u3002

    • \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\uff1b
    • \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \uff1b

    Fig. \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u4e0e\u6700\u5dee\u7ed3\u6784

    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5728\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u6216\u6781\u5c0f\u503c\u3002

    \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u6811\u7684\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u65f6\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2. \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"

    \u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002

    \u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002

    "},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1. \u00a0 \u5c42\u5e8f\u904d\u5386","text":"

    \u300c\u5c42\u5e8f\u904d\u5386 Level-Order Traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002

    \u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Traversal\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u641c\u7d22\u65b9\u5f0f\u3002

    Fig. \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386

    "},{"location":"chapter_tree/binary_tree_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree_bfs.java
    /* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode> queue = new LinkedList<>() {{ add(root); }};\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<Integer> list = new ArrayList<>();\nwhile (!queue.isEmpty()) {\nTreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\nlist.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.cpp
    /* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode *root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue<TreeNode *> queue;\nqueue.push(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvector<int> vec;\nwhile (!queue.empty()) {\nTreeNode *node = queue.front();\nqueue.pop();              // \u961f\u5217\u51fa\u961f\nvec.push_back(node->val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node->left != nullptr)\nqueue.push(node->left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node->right != nullptr)\nqueue.push(node->right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn vec;\n}\n
    binary_tree_bfs.py
    def level_order(root: TreeNode | None) -> list[int]:\n\"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n# \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue: deque[TreeNode] = deque()\nqueue.append(root)\n# \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nres = []\nwhile queue:\nnode: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\nres.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\nif node.left is not None:\nqueue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif node.right is not None:\nqueue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\nreturn res\n
    binary_tree_bfs.go
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []int {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue := list.New()\nqueue.PushBack(root)\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nnums := make([]int, 0)\nfor queue.Len() > 0 {\n// \u961f\u5217\u51fa\u961f\nnode := queue.Remove(queue.Front()).(*TreeNode)\n// \u4fdd\u5b58\u8282\u70b9\u503c\nnums = append(nums, node.Val)\nif node.Left != nil {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Left)\n}\nif node.Right != nil {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Right)\n}\n}\nreturn nums\n}\n
    binary_tree_bfs.js
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list = [];\nwhile (queue.length) {\nlet node = queue.shift(); // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.ts
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list: number[] = [];\nwhile (queue.length) {\nlet node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) {\nqueue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right) {\nqueue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list;\n}\n
    binary_tree_bfs.c
    /* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n/* \u8f85\u52a9\u961f\u5217 */\nint front, rear;\nint index, *arr;\nTreeNode *node;\nTreeNode **queue;\n/* \u8f85\u52a9\u961f\u5217 */\nqueue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_NODE_SIZE);\n// \u961f\u5217\u6307\u9488\nfront = 0, rear = 0;\n// \u52a0\u5165\u6839\u8282\u70b9\nqueue[rear++] = root;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n/* \u8f85\u52a9\u6570\u7ec4 */\narr = (int *)malloc(sizeof(int) * MAX_NODE_SIZE);\n// \u6570\u7ec4\u6307\u9488\nindex = 0;\nwhile (front < rear) {\n// \u961f\u5217\u51fa\u961f\nnode = queue[front++];\n// \u4fdd\u5b58\u8282\u70b9\u503c\narr[index++] = node->val;\nif (node->left != NULL) {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node->left;\n}\nif (node->right != NULL) {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node->right;\n}\n}\n// \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n*size = index;\narr = realloc(arr, sizeof(int) * (*size));\n// \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\nfree(queue);\nreturn arr;\n}\n
    binary_tree_bfs.cs
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode> queue = new();\nqueue.Enqueue(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> list = new();\nwhile (queue.Count != 0) {\nTreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\nlist.Add(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.swift
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nvar queue: [TreeNode] = [root]\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list: [Int] = []\nwhile !queue.isEmpty {\nlet node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\nlist.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\nif let left = node.left {\nqueue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif let right = node.right {\nqueue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list\n}\n
    binary_tree_bfs.zig
    // \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst L = std.TailQueue(*inc.TreeNode(T));\nvar queue = L{};\nvar root_node = try mem_allocator.create(L.Node);\nroot_node.data = root;\nqueue.append(root_node); // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list = std.ArrayList(T).init(std.heap.page_allocator);\nwhile (queue.len > 0) {\nvar queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\nvar node = queue_node.data;\ntry list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.left.?;\nqueue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.right.?;\nqueue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}        }\nreturn list;\n}\n
    binary_tree_bfs.dart
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode? root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode?> queue = Queue();\nqueue.add(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> res = [];\nwhile (queue.isNotEmpty) {\nTreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\nres.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn res;\n}\n
    "},{"location":"chapter_tree/binary_tree_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\(\\frac{n + 1}{2}\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2. \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"

    \u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5de6\u4fa7\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u793a\u610f\u56fe\uff0c\u53f3\u4e0a\u65b9\u662f\u5bf9\u5e94\u7684\u9012\u5f52\u5b9e\u73b0\u4ee3\u7801\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u4e2a\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386

    \u4f4d\u7f6e \u542b\u4e49 \u6b64\u5904\u8bbf\u95ee\u8282\u70b9\u65f6\u5bf9\u5e94 \u6a59\u8272\u5706\u5708\u5904 \u521a\u8fdb\u5165\u6b64\u8282\u70b9\uff0c\u5373\u5c06\u8bbf\u95ee\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u6811 \u524d\u5e8f\u904d\u5386 Pre-Order Traversal \u84dd\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\uff0c\u5373\u5c06\u8bbf\u95ee\u53f3\u5b50\u6811 \u4e2d\u5e8f\u904d\u5386 In-Order Traversal \u7d2b\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff0c\u5373\u5c06\u8fd4\u56de \u540e\u5e8f\u904d\u5386 Post-Order Traversal"},{"location":"chapter_tree/binary_tree_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree_dfs.java
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.add(root.val);\n}\n
    binary_tree_dfs.cpp
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nvec.push_back(root->val);\npreOrder(root->left);\npreOrder(root->right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left);\nvec.push_back(root->val);\ninOrder(root->right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root->left);\npostOrder(root->right);\nvec.push_back(root->val);\n}\n
    binary_tree_dfs.py
    def pre_order(root: TreeNode | None) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nres.append(root.val)\npre_order(root=root.left)\npre_order(root=root.right)\ndef in_order(root: TreeNode | None) -> None:\n\"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\nin_order(root=root.left)\nres.append(root.val)\nin_order(root=root.right)\ndef post_order(root: TreeNode | None) -> None:\n\"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npost_order(root=root.left)\npost_order(root=root.right)\nres.append(root.val)\n
    binary_tree_dfs.go
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nnums = append(nums, node.Val)\npreOrder(node.Left)\npreOrder(node.Right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.Left)\nnums = append(nums, node.Val)\ninOrder(node.Right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(node.Left)\npostOrder(node.Right)\nnums = append(nums, node.Val)\n}\n
    binary_tree_dfs.js
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
    binary_tree_dfs.ts
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
    binary_tree_dfs.c
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\narr[(*size)++] = root->val;\npreOrder(root->left, size);\npreOrder(root->right, size);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left, size);\narr[(*size)++] = root->val;\ninOrder(root->right, size);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root->left, size);\npostOrder(root->right, size);\narr[(*size)++] = root->val;\n}\n
    binary_tree_dfs.cs
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.Add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.Add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.Add(root.val);\n}\n
    binary_tree_dfs.swift
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.append(root.val)\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root: root.left)\nlist.append(root.val)\ninOrder(root: root.right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root: root.left)\npostOrder(root: root.right)\nlist.append(root.val)\n}\n
    binary_tree_dfs.zig
    // \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\ntry list.append(root.?.val);\ntry preOrder(T, root.?.left);\ntry preOrder(T, root.?.right);\n}\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ntry inOrder(T, root.?.left);\ntry list.append(root.?.val);\ntry inOrder(T, root.?.right);\n}\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\ntry postOrder(T, root.?.left);\ntry postOrder(T, root.?.right);\ntry list.append(root.?.val);\n}\n
    binary_tree_dfs.dart
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(node.val);\npreOrder(node.left);\npreOrder(node.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.left);\nlist.add(node.val);\ninOrder(node.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(node.left);\npostOrder(node.right);\nlist.add(node.val);\n}\n

    Note

    \u6211\u4eec\u4e5f\u53ef\u4ee5\u4ec5\u57fa\u4e8e\u5faa\u73af\u5b9e\u73b0\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002

    \u9012\u5f52\u8fc7\u7a0b\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u76f8\u53cd\u7684\u90e8\u5206\u3002\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\uff1b\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u8be5\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e3a\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    "},{"location":"chapter_tree/binary_tree_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002

    "},{"location":"chapter_tree/summary/","title":"7.6. \u00a0 \u5c0f\u7ed3","text":"
    • \u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002
    • \u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002
    • \u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002
    • \u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002
    • \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002
    • \u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002
    • \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5206\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002
    • \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002
    • \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
    • AVL \u6811\uff0c\u4e5f\u79f0\u4e3a\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\uff0c\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002
    • AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
    "},{"location":"chapter_tree/summary/#761-q-a","title":"7.6.1. \u00a0 Q & A","text":"

    \u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f

    \u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\u3002

    \u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u90fd\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f

    \u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002

    \u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f

    DFS \u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\u548c\u8bbf\u95ee\u6570\u7ec4\u7684\u987a\u5e8f\u7c7b\u4f3c\uff0c\u662f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u65b9\u6cd5\uff0c\u5229\u7528\u8fd9\u4e09\u79cd\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u7ed3\u70b9\u5927\u5c0f\u6ee1\u8db3 \u5de6\u5b50\u7ed3\u70b9\u503c < \u6839\u7ed3\u70b9\u503c < \u53f3\u5b50\u7ed3\u70b9\u503c \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167 \u5de6->\u6839->\u53f3 \u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002

    \u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 node , child , grand_child \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 node \u7684\u7236\u8282\u70b9\u548c node \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f

    \u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c right_rotate(root) \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 return child \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002

    \u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 private \u548c public \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 height() \u51fd\u6570\u548c updateHeight() \u51fd\u6570\u5206\u522b\u653e\u5728 public \u548c private \u4e2d\u5462\uff1f

    \u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a private \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 updateHeight() \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c height() \u662f\u8bbf\u95ee\u7ed3\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e vector.size() \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 public \u4ee5\u4fbf\u4f7f\u7528\u3002

    \u8bf7\u95ee\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u4e2a\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f

    \u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u662f build_tree() \uff0c\u5df2\u5728\u6e90\u4ee3\u7801\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u7528\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002

    "}]} \ No newline at end of file +{"config":{"lang":["ja"],"separator":"[\\s\\-\uff0c\u3002]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"\u300a Hello \u7b97\u6cd5 \u300b

    \u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b

    \u63a8\u8350\u8bed

    Quote

    \u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d

    \u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388

    Quote

    \u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d

    \u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6

    \u5168\u4e66\u52a8\u753b\u56fe\u89e3

    \u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u7535\u8111\u3001\u5e73\u677f\u3001\u624b\u673a\u5168\u7ec8\u7aef\u9605\u8bfb

    \"A picture is worth a thousand words.\"

    \u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d

    \u4ee3\u7801\u4e00\u952e\u8fd0\u884c

    \u63d0\u4f9b\u5404\u4e2a\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u7b80\u6d01\u5b9e\u73b0\u4e0e\u6d4b\u8bd5\u6837\u4f8b\uff0c\u7686\u53ef\u76f4\u63a5\u8fd0\u884c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00

    \"Talk is cheap. Show me the code.\"

    \u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d

    \u53ef\u8ba8\u8bba\u4e0e\u63d0\u95ee

    \u9f13\u52b1\u5c0f\u4f19\u4f34\u4eec\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u6210\u957f\u63d0\u95ee\u4e0e\u8bc4\u8bba\u4e00\u822c\u80fd\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d

    \u201c\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\uff0c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d

    \u4e00\u8d77\u52a0\u6cb9\uff01

    \u5e8f

    \u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u300a\u5251\u6307 Offer\u300b\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u670b\u53cb\u7684\u559c\u7231\u4e0e\u652f\u6301\u3002\u5728\u6b64\u671f\u95f4\uff0c\u6211\u56de\u7b54\u4e86\u4f17\u591a\u8bfb\u8005\u7684\u8bc4\u8bba\u95ee\u9898\uff0c\u5176\u4e2d\u6700\u5e38\u89c1\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u5b66\u4e60\u7b97\u6cd5\u201d\u3002\u6211\u9010\u6e10\u4e5f\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002

    \u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\uff0c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5730\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e66\u7c4d\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u9762\u8bd5\u5df2\u7ecf\u5360\u636e\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u539a\u91cd\u7684\u4e66\u7c4d\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002

    \u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u6b64\u95ee\u9898\u7684\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u867d\u7136\u4e0d\u4e00\u5b9a\u6b63\u786e\uff0c\u4f46\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u8fd9\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5e94\u5bf9\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002

    \u4f5c\u8005\u7b80\u4ecb

    \u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u5176 LeetBook\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 24 \u4e07\u672c\u3002

    \u81f4\u8c22

    \u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u6210\u957f\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u662f\u4ed6\u4eec\u65e0\u79c1\u5949\u732e\u4f7f\u8fd9\u672c\u4e66\u8d8a\u53d8\u8d8a\u597d\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a

    \u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 Gonglja, gvenusleo, justin\u2010tse, krahets, nuomi1, Reanon, sjinzh \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002

    GongljaC / C++ gvenusleoDart hpstoryC# justin-tseJS / TS krahetsJava / Python nuomi1Swift ReanonGo / C sjinzhRust / Zig"},{"location":"chapter_appendix/contribution/","title":"14.2. \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"

    \u5f00\u6e90\u7684\u9b45\u529b

    \u7eb8\u8d28\u4e66\u7c4d\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u9700\u8981\u6570\u5e74\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002\u4f46\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002

    \u7531\u4e8e\u4f5c\u8005\u80fd\u529b\u6709\u9650\uff0c\u4e66\u4e2d\u96be\u514d\u5b58\u5728\u4e00\u4e9b\u9057\u6f0f\u548c\u9519\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u60a8\u53d1\u73b0\u4e86\u7b14\u8bef\u3001\u5931\u6548\u94fe\u63a5\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u534f\u52a9\u6211\u4eec\u8fdb\u884c\u4fee\u6b63\uff0c\u4ee5\u5e2e\u52a9\u5176\u4ed6\u8bfb\u8005\u83b7\u5f97\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u8d44\u6e90\u3002\u6240\u6709\u64b0\u7a3f\u4eba\u5c06\u5728\u4ed3\u5e93\u548c\u7f51\u7ad9\u4e3b\u9875\u4e0a\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\uff01

    "},{"location":"chapter_appendix/contribution/#1421","title":"14.2.1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"

    \u5728\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u6709\u4e00\u4e2a\u300c\u7f16\u8f91\u300d\u56fe\u6807\uff0c\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\uff1a

    1. \u70b9\u51fb\u7f16\u8f91\u6309\u94ae\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\uff1b
    2. \u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u5e76\u786e\u4fdd\u5185\u5bb9\u6b63\u786e\uff0c\u540c\u65f6\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\uff1b
    3. \u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\uff1b\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002

    Fig. \u9875\u9762\u7f16\u8f91\u6309\u952e

    \u7531\u4e8e\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u56e0\u6b64\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u56fe\u7247\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002

    "},{"location":"chapter_appendix/contribution/#1422","title":"14.2.2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"

    \u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd Pull Request \u5de5\u4f5c\u6d41\u7a0b\uff1a

    1. \u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\uff1b
    2. \u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 git clone \u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\uff1b
    3. \u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u901a\u8fc7\u8fd0\u884c\u6d4b\u8bd5\u4ee5\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\uff1b
    4. \u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\uff1b
    5. \u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\uff1b
    "},{"location":"chapter_appendix/contribution/#1423-docker","title":"14.2.3. \u00a0 Docker \u90e8\u7f72","text":"

    \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7 Docker \u6765\u90e8\u7f72\u672c\u9879\u76ee\u3002\u6267\u884c\u4ee5\u4e0b\u811a\u672c\uff0c\u7a0d\u7b49\u7247\u523b\u540e\uff0c\u5373\u53ef\u4f7f\u7528\u6d4f\u89c8\u5668\u6253\u5f00 http://localhost:8000 \u6765\u8bbf\u95ee\u672c\u9879\u76ee\u3002

    git clone https://github.com/krahets/hello-algo.git\ncd hello-algo\ndocker-compose up -d\n

    \u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\u3002

    docker-compose down\n
    "},{"location":"chapter_appendix/installation/","title":"14.1. \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1411-vscode","title":"14.1.1. \u00a0 VSCode","text":"

    \u672c\u4e66\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730 IDE \uff0c\u4e0b\u8f7d\u5e76\u5b89\u88c5 VSCode \u3002

    "},{"location":"chapter_appendix/installation/#1412-java","title":"14.1.2. \u00a0 Java \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 > JDK 9\uff09\u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 java \uff0c\u5b89\u88c5 Extension Pack for Java \u3002
    "},{"location":"chapter_appendix/installation/#1413-cc","title":"14.1.3. \u00a0 C/C++ \u73af\u5883","text":"
    1. Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff0cMacOS \u81ea\u5e26 Clang \u65e0\u9700\u5b89\u88c5\u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c++ \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002
    3. \uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 Clang_format_fallback Style \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a { BasedOnStyle: Microsoft, BreakBeforeBraces: Attach } \u3002
    "},{"location":"chapter_appendix/installation/#1414-python","title":"14.1.4. \u00a0 Python \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 python \uff0c\u5b89\u88c5 Python Extension Pack \u3002
    3. \uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 pip install black \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
    "},{"location":"chapter_appendix/installation/#1415-go","title":"14.1.5. \u00a0 Go \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 go \uff0c\u5b89\u88c5 Go \u3002
    3. \u5feb\u6377\u952e Ctrl + Shift + P \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 Go: Install/Update Tools \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002
    "},{"location":"chapter_appendix/installation/#1416-javascript","title":"14.1.6. \u00a0 JavaScript \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 javascript \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002
    3. \uff08\u53ef\u9009\uff09\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 Prettier \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002
    "},{"location":"chapter_appendix/installation/#1417-c","title":"14.1.7. \u00a0 C# \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 6.0 \uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 c# \uff0c\u5b89\u88c5 c# \u3002
    "},{"location":"chapter_appendix/installation/#1418-swift","title":"14.1.8. \u00a0 Swift \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift\uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 swift \uff0c\u5b89\u88c5 Swift for Visual Studio Code\u3002
    "},{"location":"chapter_appendix/installation/#1419-rust","title":"14.1.9. \u00a0 Rust \u73af\u5883","text":"
    1. \u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust\uff1b
    2. \u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 rust \uff0c\u5b89\u88c5 rust-analyzer\u3002
    "},{"location":"chapter_array_and_linkedlist/","title":"4. \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":""},{"location":"chapter_array_and_linkedlist/array/","title":"4.1. \u00a0 \u6570\u7ec4","text":"

    \u300c\u6570\u7ec4 Array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u5143\u7d20\u7684\u300c\u7d22\u5f15 Index\u300d\u3002

    Fig. \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f

    \u6570\u7ec4\u521d\u59cb\u5316\u3002\u901a\u5e38\u6709\u65e0\u521d\u59cb\u503c\u548c\u7ed9\u5b9a\u521d\u59cb\u503c\u4e24\u79cd\u65b9\u5f0f\uff0c\u6211\u4eec\u53ef\u6839\u636e\u9700\u6c42\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u3002\u5728\u672a\u7ed9\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u901a\u5e38\u4f1a\u88ab\u521d\u59cb\u5316\u4e3a\u9ed8\u8ba4\u503c \\(0\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
    array.cpp
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n
    array.py
    # \u521d\u59cb\u5316\u6570\u7ec4\narr: List[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: List[int] = [1, 3, 2, 5, 4]  \n
    array.go
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n
    array.js
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n
    array.ts
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n
    array.c
    int arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n
    array.cs
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n
    array.swift
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n
    array.zig
    // \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n
    array.dart
    /* \u521d\u59cb\u5316\u6570\u7ec4 */\nList<int> arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList<int> nums = [1, 3, 2, 5, 4];\n
    "},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1. \u00a0 \u6570\u7ec4\u4f18\u70b9","text":"

    \u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\u3002\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u6b64\u5143\u7d20\u3002

    Fig. \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 

    # \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n

    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u5143\u7d20\u7684\u7d22\u5f15\u8981\u4ece \\(0\\) \u5f00\u59cb\u7f16\u53f7\u5462\uff1f

    \u89c2\u5bdf\u4e0a\u56fe\uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002

    \u7136\u800c\uff0c\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u8868\u793a\u7684\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u7d22\u5f15\u4e3a \\(0\\) \u4e5f\u662f\u5408\u7406\u7684\u3002

    \u8bbf\u95ee\u5143\u7d20\u7684\u9ad8\u6548\u6027\u5e26\u6765\u4e86\u8bf8\u591a\u4fbf\u5229\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.cpp
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.py
    def random_access(nums: list[int]) -> int:\n\"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandom_index = random.randint(0, len(nums) - 1)\n# \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandom_num = nums[random_index]\nreturn random_num\n
    array.go
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandomIndex := rand.Intn(len(nums))\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandomNum = nums[randomIndex]\nreturn\n}\n
    array.js
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
    array.ts
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n
    array.c
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.cs
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(int[] nums) {\nRandom random = new();\n// \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = random.Next(nums.Length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.swift
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -> Int {\n// \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet randomIndex = nums.indices.randomElement()!\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet randomNum = nums[randomIndex]\nreturn randomNum\n}\n
    array.zig
    // \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n// \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\nvar randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nvar randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    array.dart
    /* \u968f\u673a\u8fd4\u56de\u4e00\u4e2a\u6570\u7ec4\u5143\u7d20 */\nint randomAccess(List nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = Random().nextInt(nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2. \u00a0 \u6570\u7ec4\u7f3a\u70b9","text":"

    \u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7531\u4e8e\u7cfb\u7edf\u65e0\u6cd5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u56e0\u6b64\u6570\u7ec4\u957f\u5ea6\u65e0\u6cd5\u6269\u5c55\u3002\u800c\u82e5\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u65b0\u5efa\u4e00\u4e2a\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u662f\u975e\u5e38\u8017\u65f6\u7684\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.cpp
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = new int[size + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] nums;\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.py
    def extend(nums: list[int], enlarge: int) -> list[int]:\n\"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n# \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres = [0] * (len(nums) + enlarge)\n# \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in range(len(nums)):\nres[i] = nums[i]\n# \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n
    array.go
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres := make([]int, len(nums)+enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i, num := range nums {\nres[i] = num\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
    array.js
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.ts
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u662f\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.c
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = (int *)malloc(sizeof(int) * (size + enlarge));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\nres[i] = nums[i];\n}\n// \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\nfor (int i = size; i < size + enlarge; i++) {\nres[i] = 0;\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.cs
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.Length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.swift
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -> [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = Array(repeating: 0, count: nums.count + enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in nums.indices {\nres[i] = nums[i]\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n
    array.zig
    // \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = try mem_allocator.alloc(i32, nums.len + enlarge);\n@memset(res, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(i32, res, nums);\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n
    array.dart
    /* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList extend(List nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nList<int> res = List.filled(nums.length + enlarge, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (var i = 0; i < nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n

    \u6570\u7ec4\u4e2d\u63d2\u5165\u6216\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\u4e0b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u653e\u4efb\u4f55\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4e0d\u5f97\u4e0d\u5c06\u6b64\u7d22\u5f15\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002

    Fig. \u6570\u7ec4\u63d2\u5165\u5143\u7d20

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.cpp
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.py
    def insert(nums: list[int], num: int, index: int) -> None:\n\"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n# \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in range(len(nums) - 1, index, -1):\nnums[i] = nums[i - 1]\n# \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n
    array.go
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i := len(nums) - 1; i > index; i-- {\nnums[i] = nums[i-1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
    array.js
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.ts
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.c
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.cs
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.Length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.swift
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: nums.count - 1, next: { $0 > index + 1 ? $0 - 1 : nil }) {\nnums[i] = nums[i - 1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num\n}\n
    array.zig
    // \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar i = nums.len - 1;\nwhile (i > index) : (i -= 1) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n
    array.dart
    /* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(List nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var i = nums.length - 1; i > index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = num;\n}\n

    \u5220\u9664\u5143\u7d20\u4e5f\u7c7b\u4f3c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5220\u9664\u5143\u7d20\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6211\u4eec\u65e0\u9700\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002

    Fig. \u6570\u7ec4\u5220\u9664\u5143\u7d20

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.cpp
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.py
    def remove(nums: list[int], index: int) -> None:\n\"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\"\"\"\n# \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in range(index, len(nums) - 1):\nnums[i] = nums[i + 1]\n
    array.go
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i := index; i < len(nums)-1; i++ {\nnums[i] = nums[i+1]\n}\n}\n
    array.js
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums, index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.ts
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.c
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.cs
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i < nums.Length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.swift
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\nlet count = nums.count\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in sequence(first: index, next: { $0 < count - 1 - 1 ? $0 + 1 : nil }) {\nnums[i] = nums[i + 1]\n}\n}\n
    array.zig
    // \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar i = index;\nwhile (i < nums.len - 1) : (i += 1) {\nnums[i] = nums[i + 1];\n}\n}\n
    array.dart
    /* \u5220\u9664\u7d22\u5f15 index \u5904\u5143\u7d20 */\nvoid remove(List nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var i = index; i < nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n

    \u603b\u7ed3\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\uff1a

    • \u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002
    • \u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002
    • \u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u6211\u4eec\u4e0d\u5173\u5fc3\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u540c\u65f6\u4e5f\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002
    "},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3. \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":"

    \u6570\u7ec4\u904d\u5386\u3002\u4ee5\u4e0b\u4ecb\u7ecd\u4e24\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int num : nums) {\ncount++;\n}\n}\n
    array.cpp
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
    array.py
    def traverse(nums: list[int]) -> None:\n\"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\ncount = 0\n# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor num in nums:\ncount += 1\n# \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\nfor i, num in enumerate(nums):\ncount += 1\n
    array.go
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\ncount := 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i := 0; i < len(nums); i++ {\ncount++\n}\ncount = 0\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor range nums {\ncount++\n}\n}\n
    array.js
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
    array.ts
    /* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let num of nums) {\ncount += 1;\n}\n}\n
    array.c
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < size; i++) {\ncount++;\n}\n}\n
    array.cs
    /* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i < nums.Length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nforeach (int num in nums) {\ncount++;\n}\n}\n
    array.swift
    /* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\nvar count = 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor _ in nums.indices {\ncount += 1\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor _ in nums {\ncount += 1\n}\n}\n
    array.zig
    // \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\nvar count: i32 = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nvar i: i32 = 0;\nwhile (i < nums.len) : (i += 1) {\ncount += 1;\n}\ncount = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (nums) |_| {\ncount += 1;\n}\n}\n
    array.dart
    /* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List nums) {\nvar count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (var i = 0; i < nums.length; i++) {\ncount++;\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (var num in nums) {\ncount++;\n}\n// \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\nnums.forEach((element) {\ncount++;\n});\n}\n

    \u6570\u7ec4\u67e5\u627e\u3002\u901a\u8fc7\u904d\u5386\u6570\u7ec4\uff0c\u67e5\u627e\u6570\u7ec4\u5185\u7684\u6307\u5b9a\u5143\u7d20\uff0c\u5e76\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array.java
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.cpp
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.py
    def find(nums: list[int], target: int) -> int:\n\"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\nfor i in range(len(nums)):\nif nums[i] == target:\nreturn i\nreturn -1\n
    array.go
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\nindex = -1\nfor i := 0; i < len(nums); i++ {\nif nums[i] == target {\nindex = i\nbreak\n}\n}\nreturn\n}\n
    array.js
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
    array.ts
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\nfor (let i = 0; i < nums.length; i++) {\nif (nums[i] === target) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    array.c
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i < size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.cs
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i < nums.Length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n
    array.swift
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -> Int {\nfor i in nums.indices {\nif nums[i] == target {\nreturn i\n}\n}\nreturn -1\n}\n
    array.zig
    // \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\nfor (nums, 0..) |num, i| {\nif (num == target) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    array.dart
    /* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List nums, int target) {\nfor (var i = 0; i < nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/array/#414","title":"4.1.4. \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"

    \u968f\u673a\u8bbf\u95ee\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u6837\u672c\u7684\u968f\u673a\u62bd\u53d6\u3002

    \u4e8c\u5206\u67e5\u627e\u3002\u4f8b\u5982\u524d\u6587\u67e5\u5b57\u5178\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u5178\u4e2d\u7684\u6240\u6709\u5b57\u6309\u7167\u62fc\u97f3\u987a\u5e8f\u5b58\u50a8\u5728\u6570\u7ec4\u4e2d\uff0c\u7136\u540e\u4f7f\u7528\u4e0e\u65e5\u5e38\u67e5\u7eb8\u8d28\u5b57\u5178\u76f8\u540c\u7684\u201c\u7ffb\u5f00\u4e2d\u95f4\uff0c\u6392\u9664\u4e00\u534a\u201d\u7684\u65b9\u5f0f\uff0c\u6765\u5b9e\u73b0\u4e00\u4e2a\u67e5\u7535\u5b50\u5b57\u5178\u7684\u7b97\u6cd5\u3002

    \u6df1\u5ea6\u5b66\u4e60\u3002\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2. \u00a0 \u94fe\u8868","text":"

    \u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u6392\u9664\u5df2\u88ab\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u7a7a\u95f2\u5185\u5b58\u7a7a\u95f4\u901a\u5e38\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6211\u4eec\u9700\u8981\u7533\u8bf7\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6570\u7ec4\u65f6\uff0c\u7a7a\u95f2\u5185\u5b58\u4e2d\u53ef\u80fd\u6ca1\u6709\u8fd9\u4e48\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u4e0e\u6570\u7ec4\u76f8\u6bd4\uff0c\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u5b83\u53ef\u4ee5\u88ab\u5b58\u50a8\u5728\u975e\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002

    \u300c\u94fe\u8868 Linked List\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u8fde\u63a5\uff0c\u4ece\u5f53\u524d\u8282\u70b9\u901a\u8fc7\u6307\u9488\u53ef\u4ee5\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002\u7531\u4e8e\u6307\u9488\u8bb0\u5f55\u4e86\u4e0b\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u56e0\u6b64\u65e0\u9700\u4fdd\u8bc1\u5185\u5b58\u5730\u5740\u7684\u8fde\u7eed\u6027\uff0c\u4ece\u800c\u53ef\u4ee5\u5c06\u5404\u4e2a\u8282\u70b9\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002

    \u94fe\u8868\u300c\u8282\u70b9 Node\u300d\u5305\u542b\u4e24\u9879\u6570\u636e\uff0c\u4e00\u662f\u8282\u70b9\u300c\u503c Value\u300d\uff0c\u4e8c\u662f\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u300c\u6307\u9488 Pointer\u300d\uff0c\u6216\u79f0\u300c\u5f15\u7528 Reference\u300d\u3002

    Fig. \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
    class ListNode:\n\"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                  # \u8282\u70b9\u503c\nself.next: Optional[ListNode] = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\nVal  int       // \u8282\u70b9\u503c\nNext *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\nreturn &ListNode{\nVal:  val,\nNext: nil,\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nconstructor(val, next) {\nthis.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\nthis.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nconstructor(val?: number, next?: ListNode | null) {\nthis.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nreturn node;\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode\n{\nint val;         // \u8282\u70b9\u503c\nListNode next;   // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\nListNode(int x) => val = x;  //\u6784\u9020\u51fd\u6570\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
    // \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\n}\n};\n}\n
    /* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n

    \u5c3e\u8282\u70b9\u6307\u5411\u4ec0\u4e48\uff1f

    \u6211\u4eec\u5c06\u94fe\u8868\u7684\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u300c\u5c3e\u8282\u70b9\u300d\uff0c\u5176\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5728 Java, C++, Python \u4e2d\u5206\u522b\u8bb0\u4e3a \\(\\text{null}\\) , \\(\\text{nullptr}\\) , \\(\\text{None}\\) \u3002\u5728\u4e0d\u5f15\u8d77\u6b67\u4e49\u7684\u524d\u63d0\u4e0b\uff0c\u672c\u4e66\u90fd\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u7a7a\u3002

    \u5982\u4f55\u79f0\u547c\u94fe\u8868\uff1f

    \u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u6574\u4f53\u5c31\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u4f8b\u5982\u6570\u7ec4 nums \uff0c\u5305\u542b\u5404\u4e2a\u5143\u7d20 nums[0] , nums[1] \u7b49\u7b49\u3002\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u4f8b\u5982\u5934\u8282\u70b9 head \u548c\u94fe\u8868 head \u5b9e\u9645\u4e0a\u662f\u540c\u4e49\u7684\u3002

    \u94fe\u8868\u521d\u59cb\u5316\u65b9\u6cd5\u3002\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u5f15\u7528\u6307\u5411\u5173\u7cfb\u3002\u5b8c\u6210\u540e\uff0c\u5373\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\uff08\u5373\u9996\u4e2a\u8282\u70b9\uff09\u51fa\u53d1\uff0c\u901a\u8fc7\u6307\u9488 next \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.cpp
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
    linked_list.py
    # \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
    linked_list.go
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n
    linked_list.js
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.ts
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.c
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0->next = n1;\nn1->next = n2;\nn2->next = n3;\nn3->next = n4;\n
    linked_list.cs
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    linked_list.swift
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n
    linked_list.zig
    // \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9 \nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = &n1;\nn1.next = &n2;\nn2.next = &n3;\nn3.next = &n4;\n
    linked_list.dart
    /* \u521d\u59cb\u5316\u94fe\u8868 1 -> 3 -> 2 -> 5 -> 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1. \u00a0 \u94fe\u8868\u4f18\u70b9","text":"

    \u94fe\u8868\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u7684\u64cd\u4f5c\u6548\u7387\u9ad8\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u60f3\u5728\u94fe\u8868\u4e2d\u95f4\u7684\u4e24\u4e2a\u8282\u70b9 A , B \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 P \uff0c\u6211\u4eec\u53ea\u9700\u8981\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u6307\u9488\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u64cd\u4f5c\u6548\u7387\u8981\u4f4e\u5f97\u591a\u3002

    Fig. \u94fe\u8868\u63d2\u5165\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.cpp
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
    linked_list.py
    def insert(n0: ListNode, P: ListNode) -> None:\n\"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\nn1 = n0.next\nP.next = n1\nn0.next = P\n
    linked_list.go
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\nn1 := n0.Next\nP.Next = n1\nn0.Next = P\n}\n
    linked_list.js
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.ts
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.c
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0->next;\nP->next = n1;\nn0->next = P;\n}\n
    linked_list.cs
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n
    linked_list.swift
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\nlet n1 = n0.next\nP.next = n1\nn0.next = P\n}\n
    linked_list.zig
    // \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\nvar n1 = n0.?.next;\nP.?.next = n1;\nn0.?.next = P;\n}\n
    linked_list.dart
    /* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n

    \u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u6307\u9488\u5373\u53ef\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u8282\u70b9 P \u4ecd\u7136\u6307\u5411 n1 \uff0c\u4f46\u5b9e\u9645\u4e0a P \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u6b64\u94fe\u8868\uff0c\u56e0\u4e3a\u904d\u5386\u6b64\u94fe\u8868\u65f6\u65e0\u6cd5\u8bbf\u95ee\u5230 P \u3002

    Fig. \u94fe\u8868\u5220\u9664\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.cpp
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\nif (n0->next == nullptr)\nreturn;\n// n0 -> P -> n1\nListNode *P = n0->next;\nListNode *n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\ndelete P;\n}\n
    linked_list.py
    def remove(n0: ListNode) -> None:\n\"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nif not n0.next:\nreturn\n# n0 -> P -> n1\nP = n0.next\nn1 = P.next\nn0.next = n1\n
    linked_list.go
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeNode(n0 *ListNode) {\nif n0.Next == nil {\nreturn\n}\n// n0 -> P -> n1\nP := n0.Next\nn1 := P.Next\nn0.Next = n1\n}\n
    linked_list.js
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\nif (!n0.next) return;\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.ts
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\nif (!n0.next) {\nreturn;\n}\n// n0 -> P -> n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.c
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeNode(ListNode *n0) {\nif (!n0->next)\nreturn;\n// n0 -> P -> n1\nListNode *P = n0->next;\nListNode *n1 = P->next;\nn0->next = n1;\n// \u91ca\u653e\u5185\u5b58\nfree(P);\n}\n
    linked_list.cs
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -> P -> n1\nListNode P = n0.next;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
    linked_list.swift
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\nif n0.next == nil {\nreturn\n}\n// n0 -> P -> n1\nlet P = n0.next\nlet n1 = P?.next\nn0.next = n1\nP?.next = nil\n}\n
    linked_list.zig
    // \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\nif (n0.?.next == null) return;\n// n0 -> P -> n1\nvar P = n0.?.next;\nvar n1 = P.?.next;\nn0.?.next = n1;\n}\n
    linked_list.dart
    /* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null) return;\n// n0 -> P -> n1\nListNode P = n0.next!;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#422","title":"4.2.2. \u00a0 \u94fe\u8868\u7f3a\u70b9","text":"

    \u94fe\u8868\u8bbf\u95ee\u8282\u70b9\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u8282\u6240\u8ff0\uff0c\u6570\u7ec4\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u4efb\u610f\u5143\u7d20\u3002\u7136\u800c\uff0c\u94fe\u8868\u65e0\u6cd5\u76f4\u63a5\u8bbf\u95ee\u4efb\u610f\u8282\u70b9\uff0c\u8fd9\u662f\u56e0\u4e3a\u7cfb\u7edf\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4f8b\u5982\uff0c\u82e5\u8981\u8bbf\u95ee\u94fe\u8868\u7d22\u5f15\u4e3a index\uff08\u5373\u7b2c index + 1 \u4e2a\uff09\u7684\u8282\u70b9\uff0c\u5219\u9700\u8981\u5411\u540e\u904d\u5386 index \u8f6e\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.cpp
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == nullptr)\nreturn nullptr;\nhead = head->next;\n}\nreturn head;\n}\n
    linked_list.py
    def access(head: ListNode, index: int) -> ListNode | None:\n\"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\nfor _ in range(index):\nif not head:\nreturn None\nhead = head.next\nreturn head\n
    linked_list.go
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\nfor i := 0; i < index; i++ {\nif head == nil {\nreturn nil\n}\nhead = head.Next\n}\nreturn head\n}\n
    linked_list.js
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.ts
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\nfor (let i = 0; i < index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.c
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nwhile (head && head->next && index) {\nhead = head->next;\nindex--;\n}\nreturn head;\n}\n
    linked_list.cs
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode head, int index) {\nfor (int i = 0; i < index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n
    linked_list.swift
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -> ListNode? {\nvar head: ListNode? = head\nfor _ in 0 ..< index {\nif head == nil {\nreturn nil\n}\nhead = head?.next\n}\nreturn head\n}\n
    linked_list.zig
    // \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\nvar head = node;\nvar i: i32 = 0;\nwhile (i < index) : (i += 1) {\nhead = head.?.next;\nif (head == null) return null;\n}\nreturn head;\n}\n
    linked_list.dart
    /* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\nfor (var i = 0; i < index; i++) {\nif (head == null) return null;\nhead = head.next;\n}\nreturn head;\n}\n

    \u94fe\u8868\u7684\u5185\u5b58\u5360\u7528\u8f83\u5927\u3002\u94fe\u8868\u4ee5\u8282\u70b9\u4e3a\u5355\u4f4d\uff0c\u6bcf\u4e2a\u8282\u70b9\u9664\u4e86\u4fdd\u5b58\u503c\u4e4b\u5916\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u8fd9\u610f\u5473\u7740\u5728\u76f8\u540c\u6570\u636e\u91cf\u7684\u60c5\u51b5\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    "},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3. \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u904d\u5386\u94fe\u8868\u67e5\u627e\u3002\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u94fe\u8868\u5185\u503c\u4e3a target \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linked_list.java
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.cpp
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head != nullptr) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.py
    def find(head: ListNode, target: int) -> int:\n\"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nindex = 0\nwhile head:\nif head.val == target:\nreturn index\nhead = head.next\nindex += 1\nreturn -1\n
    linked_list.go
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\nindex := 0\nfor head != nil {\nif head.Val == target {\nreturn index\n}\nhead = head.Next\nindex++\n}\nreturn -1\n}\n
    linked_list.js
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.ts
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.c
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head) {\nif (head->val == target)\nreturn index;\nhead = head->next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.cs
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    linked_list.swift
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -> Int {\nvar head: ListNode? = head\nvar index = 0\nwhile head != nil {\nif head?.val == target {\nreturn index\n}\nhead = head?.next\nindex += 1\n}\nreturn -1\n}\n
    linked_list.zig
    // \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\nvar head = node;\nvar index: i32 = 0;\nwhile (head != null) {\nif (head.?.val == target) return index;\nhead = head.?.next;\nindex += 1;\n}\nreturn -1;\n}\n
    linked_list.dart
    /* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target) {\nreturn index;\n}\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n
    "},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4. \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"

    \u5355\u5411\u94fe\u8868\u3002\u5373\u4e0a\u8ff0\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u6210\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002

    \u73af\u5f62\u94fe\u8868\u3002\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u5373\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002

    \u53cc\u5411\u94fe\u8868\u3002\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u8282\u70b9\uff09\u7684\u6307\u9488\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n
    class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                   # \u8282\u70b9\u503c\nself.next: Optional[ListNode] = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nself.prev: Optional[ListNode] = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\nVal  int             // \u8282\u70b9\u503c\nNext *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nPrev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\nreturn &DoublyListNode{\nVal:  val,\nNext: nil,\nPrev: nil,\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval;\nnext;\nprev;\nconstructor(val, next, prev) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nprev: ListNode | null;\nconstructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n}\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nstruct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n};\ntypedef struct ListNode ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node, *next;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode->val = val;\nnode->next = NULL;\nnode->prev = NULL;\nreturn node;\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(int x) => val = x;  // \u6784\u9020\u51fd\u6570\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nvar prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n
    // \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nprev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\nListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n

    Fig. \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b

    "},{"location":"chapter_array_and_linkedlist/list/","title":"4.3. \u00a0 \u5217\u8868","text":"

    \u6570\u7ec4\u957f\u5ea6\u4e0d\u53ef\u53d8\u5bfc\u81f4\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u5728\u8bb8\u591a\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u4e8b\u5148\u65e0\u6cd5\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u8fd9\u4f7f\u6570\u7ec4\u957f\u5ea6\u7684\u9009\u62e9\u53d8\u5f97\u56f0\u96be\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u9700\u8981\u5728\u6301\u7eed\u6dfb\u52a0\u6570\u636e\u65f6\u9891\u7e41\u6269\u5bb9\u6570\u7ec4\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002

    \u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u51fa\u73b0\u4e86\u4e00\u79cd\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u7ec4 Dynamic Array\u300d\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5373\u957f\u5ea6\u53ef\u53d8\u7684\u6570\u7ec4\uff0c\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5217\u8868 List\u300d\u3002\u5217\u8868\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\uff0c\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u52a8\u6001\u6269\u5bb9\u3002\u5728\u5217\u8868\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u6dfb\u52a0\u5143\u7d20\uff0c\u800c\u65e0\u9700\u62c5\u5fc3\u8d85\u8fc7\u5bb9\u91cf\u9650\u5236\u3002

    "},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1. \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u521d\u59cb\u5316\u5217\u8868\u3002\u901a\u5e38\u6211\u4eec\u4f1a\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<Integer> list1 = new ArrayList<>();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList<Integer> list = new ArrayList<>(Arrays.asList(numbers));\n
    list.cpp
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 list\n// \u65e0\u521d\u59cb\u503c\nvector<int> list1;\n// \u6709\u521d\u59cb\u503c\nvector<int> list = { 1, 3, 2, 5, 4 };\n
    list.py
    # \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nlist1: List[int] = []\n# \u6709\u521d\u59cb\u503c\nlist: List[int] = [1, 3, 2, 5, 4]\n
    list_test.go
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlist1 := []int\n// \u6709\u521d\u59cb\u503c\nlist := []int{1, 3, 2, 5, 4}\n
    list.js
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1 = [];\n// \u6709\u521d\u59cb\u503c\nconst list = [1, 3, 2, 5, 4];\n
    list.ts
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst list1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst list: number[] = [1, 3, 2, 5, 4];\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = new ();\n// \u6709\u521d\u59cb\u503c\nint[] numbers = new int[] { 1, 3, 2, 5, 4 };\nList<int> list = numbers.ToList();\n
    list.swift
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet list1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar list = [1, 3, 2, 5, 4]\n
    list.zig
    // \u521d\u59cb\u5316\u5217\u8868\nvar list = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list.deinit();\ntry list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 });\n
    list.dart
    /* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList<int> list1 = [];\n// \u6709\u521d\u59cb\u503c\nList<int> list = [1, 3, 2, 5, 4];\n

    \u8bbf\u95ee\u4e0e\u66f4\u65b0\u5143\u7d20\u3002\u7531\u4e8e\u5217\u8868\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.cpp
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.py
    # \u8bbf\u95ee\u5143\u7d20\nnum: int = list[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nlist[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list_test.go
    /* \u8bbf\u95ee\u5143\u7d20 */\nnum := list[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.js
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.ts
    /* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.swift
    /* \u8bbf\u95ee\u5143\u7d20 */\nlet num = list[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n
    list.zig
    // \u8bbf\u95ee\u5143\u7d20\nvar num = list.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n// \u66f4\u65b0\u5143\u7d20\nlist.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n
    list.dart
    /* \u8bbf\u95ee\u5143\u7d20 */\nint num = list[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nlist[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n

    \u5728\u5217\u8868\u4e2d\u6dfb\u52a0\u3001\u63d2\u5165\u3001\u5220\u9664\u5143\u7d20\u3002\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.cpp
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push_back(1);\nlist.push_back(3);\nlist.push_back(2);\nlist.push_back(5);\nlist.push_back(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(list.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.erase(list.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.py
    # \u6e05\u7a7a\u5217\u8868\nlist.clear()\n# \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n# \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nlist.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n# \u5220\u9664\u5143\u7d20\nlist.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list_test.go
    /* \u6e05\u7a7a\u5217\u8868 */\nlist = nil\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist = append(list, 1)\nlist = append(list, 3)\nlist = append(list, 2)\nlist = append(list, 5)\nlist = append(list, 4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist = append(list[:3], append([]int{6}, list[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist = append(list[:3], list[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.js
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
    list.ts
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.length = 0;\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.push(1);\nlist.push(3);\nlist.push(2);\nlist.push(5);\nlist.push(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.splice(3, 1);\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.Clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.Add(1);\nlist.Add(3);\nlist.Add(2);\nlist.Add(5);\nlist.Add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.Insert(3, 6);\n/* \u5220\u9664\u5143\u7d20 */\nlist.RemoveAt(3);\n
    list.swift
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.removeAll()\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.append(1)\nlist.append(3)\nlist.append(2)\nlist.append(5)\nlist.append(4)\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.zig
    // \u6e05\u7a7a\u5217\u8868\nlist.clearRetainingCapacity();\n// \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry list.append(1);\ntry list.append(3);\ntry list.append(2);\ntry list.append(5);\ntry list.append(4);\n// \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry list.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n// \u5220\u9664\u5143\u7d20\n_ = list.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n
    list.dart
    /* \u6e05\u7a7a\u5217\u8868 */\nlist.clear();\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nlist.add(1);\nlist.add(3);\nlist.add(2);\nlist.add(5);\nlist.add(4);\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nlist.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nlist.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n

    \u904d\u5386\u5217\u8868\u3002\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
    list.cpp
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.size(); i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n : list) {\ncount++;\n}\n
    list.py
    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(list)):\ncount += 1\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0\nfor n in list:\ncount += 1\n
    list_test.go
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i < len(list); i++ {\ncount++\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor range list {\ncount++\n}\n
    list.js
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
    list.ts
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const n of list) {\ncount++;\n}\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.Count(); i++)\n{\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int n in list)\n{\ncount++;\n}\n
    list.swift
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor _ in list.indices {\ncount += 1\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _ in list {\ncount += 1\n}\n
    list.zig
    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < list.items.len) : (i += 1) {\ncount += 1;\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (list.items) |_| {\ncount += 1;\n}\n
    list.dart
    /* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i < list.length; i++) {\ncount++;\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int n in list) {\ncount++;\n}\n

    \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\u3002\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 list1 \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u5217\u8868\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<Integer> list1 = new ArrayList<>(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.cpp
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector<int> list1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\nlist.insert(list.end(), list1.begin(), list1.end());\n
    list.py
    # \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nlist1: List[int] = [6, 8, 7, 10, 9]\nlist += list1  # \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list_test.go
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlist1 := []int{6, 8, 7, 10, 9}\nlist = append(list, list1...)  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.js
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1 = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.ts
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst list1: number[] = [6, 8, 7, 10, 9];\nlist.push(...list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = new() { 6, 8, 7, 10, 9 };\nlist.AddRange(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.swift
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet list1 = [6, 8, 7, 10, 9]\nlist.append(contentsOf: list1) // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.zig
    // \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar list1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer list1.deinit();\ntry list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 });\ntry list.insertSlice(list.items.len, list1.items); // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n
    list.dart
    /* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList<int> list1 = [6, 8, 7, 10, 9];\nlist.addAll(list1);  // \u5c06\u5217\u8868 list1 \u62fc\u63a5\u5230 list \u4e4b\u540e\n

    \u6392\u5e8f\u5217\u8868\u3002\u6392\u5e8f\u4e5f\u662f\u5e38\u7528\u7684\u65b9\u6cd5\u4e4b\u4e00\u3002\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u5bdf\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u53cc\u6307\u9488\u300d\u7b97\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart list.java
    /* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(list);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.cpp
    /* \u6392\u5e8f\u5217\u8868 */\nsort(list.begin(), list.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.py
    # \u6392\u5e8f\u5217\u8868\nlist.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list_test.go
    /* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(list)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.js
    /* \u6392\u5e8f\u5217\u8868 */  list.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.ts
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort((a, b) => a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n
    list.cs
    /* \u6392\u5e8f\u5217\u8868 */\nlist.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.swift
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    list.zig
    // \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32));\n
    list.dart
    /* \u6392\u5e8f\u5217\u8868 */\nlist.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n
    "},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2. \u00a0 \u5217\u8868\u5b9e\u73b0 *","text":"

    \u4e3a\u4e86\u5e2e\u52a9\u52a0\u6df1\u5bf9\u5217\u8868\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5728\u6b64\u63d0\u4f9b\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\u5b9e\u73b0\u3002\u9700\u8981\u5173\u6ce8\u4e09\u4e2a\u6838\u5fc3\u70b9\uff1a

    • \u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002
    • \u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf size\uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002
    • \u6269\u5bb9\u673a\u5236\uff1a\u63d2\u5165\u5143\u7d20\u65f6\u53ef\u80fd\u8d85\u51fa\u5217\u8868\u5bb9\u91cf\uff0c\u6b64\u65f6\u9700\u8981\u6269\u5bb9\u5217\u8868\u3002\u6269\u5bb9\u65b9\u6cd5\u662f\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002

    \u672c\u793a\u4f8b\u65e8\u5728\u5e2e\u52a9\u8bfb\u8005\u76f4\u89c2\u7406\u89e3\u5217\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002\u5b9e\u9645\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u5217\u8868\u5b9e\u73b0\u66f4\u52a0\u6807\u51c6\u548c\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u6709\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_list.java
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int capacity = 10; // \u5217\u8868\u5bb9\u91cf\nprivate int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[capacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\npublic int size() {\nreturn size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\nnums[size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = Arrays.copyOf(nums, capacity() * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\ncapacity = nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\nint size = size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[size];\nfor (int i = 0; i < size; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
    my_list.cpp
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate:\nint *nums;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint numsCapacity = 10; // \u5217\u8868\u5bb9\u91cf\nint numsSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\nnums = new int[numsCapacity];\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~MyList() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\nnums[size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size() - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index < 0 || index >= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < size() - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nint newCapacity = capacity() * extendRatio;\nint *tmp = nums;\nnums = new int[newCapacity];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i < size(); i++) {\nnums[i] = tmp[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] tmp;\nnumsCapacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> vec(size());\nfor (int i = 0; i < size(); i++) {\nvec[i] = nums[i];\n}\nreturn vec;\n}\n};\n
    my_list.py
    class MyList:\n\"\"\"\u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\nself.__nums: list[int] = [0] * self.__capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nself.__size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nself.__extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\nreturn self.__size\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\nreturn self.__capacity\ndef get(self, index: int) -> int:\n\"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nreturn self.__nums[index]\ndef set(self, num: int, index: int) -> None:\n\"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nself.__nums[index] = num\ndef add(self, num: int) -> None:\n\"\"\"\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size() == self.capacity():\nself.extend_capacity()\nself.__nums[self.__size] = num\nself.__size += 1\ndef insert(self, num: int, index: int) -> None:\n\"\"\"\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.__size == self.capacity():\nself.extend_capacity()\n# \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in range(self.__size - 1, index - 1, -1):\nself.__nums[j + 1] = self.__nums[j]\nself.__nums[index] = num\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size += 1\ndef remove(self, index: int) -> int:\n\"\"\"\u5220\u9664\u5143\u7d20\"\"\"\nif index < 0 or index >= self.__size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nnum = self.__nums[index]\n# \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in range(index, self.__size - 1):\nself.__nums[j] = self.__nums[j + 1]\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.__size -= 1\n# \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\ndef extend_capacity(self) -> None:\n\"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n# \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nself.__nums = self.__nums + [0] * self.capacity() * (self.__extend_ratio - 1)\n# \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.__capacity = len(self.__nums)\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\nreturn self.__nums[: self.__size]\n
    my_list.go
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\ntype myList struct {\nnumsCapacity int\nnums         []int\nnumsSize     int\nextendRatio  int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\nreturn &myList{\nnumsCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\nnums:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio:  2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\nreturn l.numsSize\n}\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\nreturn l.numsCapacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn l.nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nl.nums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\nl.nums[l.numsSize] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.numsSize == l.numsCapacity {\nl.extendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j := l.numsSize - 1; j >= index; j-- {\nl.nums[j+1] = l.nums[j]\n}\nl.nums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize++\n}\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\nif index < 0 || index >= l.numsSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnum := l.nums[index]\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j := index; j < l.numsSize-1; j++ {\nl.nums[j] = l.nums[j+1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.numsSize--\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nl.nums = append(l.nums, make([]int, l.numsCapacity*(l.extendRatio-1))...)\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nl.numsCapacity = len(l.nums)\n}\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nreturn l.nums[:l.numsSize]\n}\n
    my_list.js
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\n#nums = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n#capacity = 10; // \u5217\u8868\u5bb9\u91cf\n#size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n#extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#nums = new Array(this.#capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nsize() {\nreturn this.#size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\ncapacity() {\nreturn this.#capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nget(index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.#nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nset(index, num) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.#nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nadd(num) {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.#nums[this.#size] = num;\nthis.#size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\ninsert(index, num) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this.#size - 1; j >= index; j--) {\nthis.#nums[j + 1] = this.#nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#nums[index] = num;\nthis.#size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nremove(index) {\nif (index < 0 || index >= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.#nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this.#size - 1; j++) {\nthis.#nums[j] = this.#nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nextendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.#nums = this.#nums.concat(\nnew Array(this.capacity() * (this.#extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis.#capacity = this.#nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\ntoArray() {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
    my_list.ts
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate nums: Array<number>; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\nprivate _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.nums = new Array(this._capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic size(): number {\nreturn this._size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic capacity(): number {\nreturn this._capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic get(index: number): number {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic set(index: number, num: number): void {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic add(num: number): void {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this._size === this._capacity) this.extendCapacity();\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.nums[this._size] = num;\nthis._size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic insert(index: number, num: number): void {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this._size === this._capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this._size - 1; j >= index; j--) {\nthis.nums[j + 1] = this.nums[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.nums[index] = num;\nthis._size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic remove(index: number): number {\nif (index < 0 || index >= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j < this._size - 1; j++) {\nthis.nums[j] = this.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis._size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic extendCapacity(): void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.nums = this.nums.concat(\nnew Array(this.capacity() * (this.extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis._capacity = this.nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic toArray(): number[] {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst nums = new Array(size);\nfor (let i = 0; i < size; i++) {\nnums[i] = this.get(i);\n}\nreturn nums;\n}\n}\n
    my_list.c
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nstruct myList {\nint *nums;       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint capacity;    // \u5217\u8868\u5bb9\u91cf\nint size;        // \u5217\u8868\u5927\u5c0f\nint extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n};\ntypedef struct myList myList;\n/* \u6784\u9020\u51fd\u6570 */\nmyList *newMyList() {\nmyList *list = malloc(sizeof(myList));\nlist->capacity = 10;\nlist->nums = malloc(sizeof(int) * list->capacity);\nlist->size = 0;\nlist->extendRatio = 2;\nreturn list;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(myList *list) {\nfree(list->nums);\nfree(list);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(myList *list) {\nreturn list->size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(myList *list) {\nreturn list->capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(myList *list, int index) {\nassert(index >= 0 && index < list->size);\nreturn list->nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(myList *list, int index, int num) {\nassert(index >= 0 && index < list->size);\nlist->nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(myList *list, int num) {\nif (size(list) == capacity(list)) {\nextendCapacity(list); // \u6269\u5bb9\n}\nlist->nums[size(list)] = num;\nlist->size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(myList *list, int index, int num) {\nassert(index >= 0 && index < size(list));\nfor (int i = size(list); i > index; --i) {\nlist->nums[i] = list->nums[i - 1];\n}\nlist->nums[index] = num;\nlist->size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeNum(myList *list, int index) {\nassert(index >= 0 && index < size(list));\nint num = list->nums[index];\nfor (int i = index; i < size(list) - 1; i++) {\nlist->nums[i] = list->nums[i + 1];\n}\nlist->size--;\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(myList *list) {\n// \u5148\u5206\u914d\u7a7a\u95f4\nint newCapacity = capacity(list) * list->extendRatio;\nint *extend = (int *)malloc(sizeof(int) * newCapacity);\nint *temp = list->nums;\n// \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\nfor (int i = 0; i < size(list); i++)\nextend[i] = list->nums[i];\n// \u91ca\u653e\u65e7\u6570\u636e\nfree(temp);\n// \u66f4\u65b0\u65b0\u6570\u636e\nlist->nums = extend;\nlist->capacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(myList *list) {\nreturn list->nums;\n}\n
    my_list.cs
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate int[] nums;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int numsCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int numsSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\nnums = new int[numsCapacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int size() {\nreturn numsSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn numsCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nnums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\nnums[numsSize] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (numsSize == numsCapacity)\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = numsSize - 1; j >= index; j--) {\nnums[j + 1] = nums[j];\n}\nnums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index < 0 || index >= numsSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j < numsSize - 1; j++) {\nnums[j] = nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nnumsSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a numsCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nSystem.Array.Resize(ref nums, numsCapacity * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nnumsCapacity = nums.Length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] nums = new int[numsSize];\nfor (int i = 0; i < numsSize; i++) {\nnums[i] = get(i);\n}\nreturn nums;\n}\n}\n
    my_list.swift
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nprivate var nums: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\nprivate var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nnums = Array(repeating: 0, count: _capacity)\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nfunc size() -> Int {\n_size\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc capacity() -> Int {\n_capacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc get(index: Int) -> Int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn nums[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc set(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnums[index] = num\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc add(num: Int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\nnums[_size] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc insert(index: Int, num: Int) {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in sequence(first: _size - 1, next: { $0 >= index + 1 ? $0 - 1 : nil }) {\nnums[j + 1] = nums[j]\n}\nnums[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5220\u9664\u5143\u7d20 */\n@discardableResult\nfunc remove(index: Int) -> Int {\nif index < 0 || index >= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nlet num = nums[index]\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in index ..< (_size - 1) {\nnums[j] = nums[j + 1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size -= 1\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nnums = nums + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = nums.count\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nfunc toArray() -> [Int] {\nvar nums = Array(repeating: 0, count: _size)\nfor i in 0 ..< _size {\nnums[i] = get(index: i)\n}\nreturn nums\n}\n}\n
    my_list.zig
    // \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0\nfn MyList(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nnumsCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\nnumSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.nums = try self.mem_allocator.alloc(T, self.numsCapacity);\n@memset(self.nums, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\npub fn size(self: *Self) usize {\nreturn self.numSize;\n}\n// \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.numsCapacity;\n}\n// \u8bbf\u95ee\u5143\u7d20\npub fn get(self: *Self, index: usize) T {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn self.nums[index];\n}  // \u66f4\u65b0\u5143\u7d20\npub fn set(self: *Self, index: usize, num: T) void {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nself.nums[index] = num;\n}  // \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\npub fn add(self: *Self, num: T) !void {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\nself.nums[self.size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}  // \u4e2d\u95f4\u63d2\u5165\u5143\u7d20\npub fn insert(self: *Self, index: usize, num: T) !void {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar j = self.size() - 1;\nwhile (j >= index) : (j -= 1) {\nself.nums[j + 1] = self.nums[j];\n}\nself.nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}\n// \u5220\u9664\u5143\u7d20\npub fn remove(self: *Self, index: usize) T {\nif (index < 0 or index >= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nvar num = self.nums[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar j = index;\nwhile (j < self.size() - 1) : (j += 1) {\nself.nums[j] = self.nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n// \u5217\u8868\u6269\u5bb9\npub fn extendCapacity(self: *Self) !void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nvar newCapacity = self.capacity() * self.extendRatio;\nvar extend = try self.mem_allocator.alloc(T, newCapacity);\n@memset(extend, @as(T, 0));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(T, extend, self.nums);\nself.nums = extend;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.numsCapacity = newCapacity;\n}\n// \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar nums = try self.mem_allocator.alloc(T, self.size());\n@memset(nums, @as(T, 0));\nfor (nums, 0..) |*num, i| {\nnum.* = self.get(i);\n}\nreturn nums;\n}\n};\n}\n
    my_list.dart
    /* \u5217\u8868\u7c7b\u7b80\u6613\u5b9e\u73b0 */\nclass MyList {\nlate List<int> _nums; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint _capacity = 10; // \u5217\u8868\u5bb9\u91cf\nint _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\n_nums = List.filled(_capacity, 0);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5373\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() => _size;\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() => _capacity;\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nreturn _nums[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n_nums[index] = num;\n}\n/* \u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n_nums[_size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var j = _size - 1; j >= index; j--) {\n_nums[j + 1] = _nums[j];\n}\n_nums[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index >= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nint num = _nums[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var j = index; j < _size - 1; j++) {\n_nums[j] = _nums[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nfinal _newNums = List.filled(_capacity * _extendRatio, 0);\n// \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nList.copyRange(_newNums, 0, _nums);\n// \u66f4\u65b0 _nums \u7684\u5f15\u7528\n_nums = _newNums;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = _nums.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nList<int> toArray() {\nList<int> nums = [];\nfor (var i = 0; i < _size; i++) {\nnums.add(get(i));\n}\nreturn nums;\n}\n}\n
    "},{"location":"chapter_array_and_linkedlist/summary/","title":"4.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u65b9\u5f0f\u3002\u4e24\u8005\u7684\u4f18\u7f3a\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002
    • \u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002
    • \u94fe\u8868\u901a\u8fc7\u66f4\u6539\u6307\u9488\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u5faa\u73af\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\u3002
    • \u52a8\u6001\u6570\u7ec4\uff0c\u53c8\u79f0\u5217\u8868\uff0c\u662f\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u4e00\u79cd\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u6613\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002
    • \u4e0b\u8868\u603b\u7ed3\u5e76\u5bf9\u6bd4\u4e86\u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u5404\u9879\u7279\u6027\u4e0e\u64cd\u4f5c\u6548\u7387\u3002
    \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u79bb\u6563\u5185\u5b58\u7a7a\u95f4 \u6570\u636e\u7ed3\u6784\u957f\u5ea6 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u957f\u5ea6\u53ef\u53d8 \u5185\u5b58\u4f7f\u7528\u7387 \u5360\u7528\u5185\u5b58\u5c11\u3001\u7f13\u5b58\u5c40\u90e8\u6027\u597d \u5360\u7528\u5185\u5b58\u591a \u4f18\u52bf\u64cd\u4f5c \u968f\u673a\u8bbf\u95ee \u63d2\u5165\u3001\u5220\u9664 \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(N)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(N)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(N)\\) \\(O(1)\\)

    \u7f13\u5b58\u5c40\u90e8\u6027

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6570\u636e\u8bfb\u5199\u901f\u5ea6\u6392\u5e8f\u662f\u201c\u786c\u76d8 < \u5185\u5b58 < CPU \u7f13\u5b58\u201d\u3002\u5f53\u6211\u4eec\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u8ba1\u7b97\u673a\u53ea\u80fd\u6328\u4e2a\u5730\u7f13\u5b58\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u6837\u7684\u591a\u6b21\u201c\u642c\u8fd0\u201d\u964d\u4f4e\u4e86\u6574\u4f53\u6548\u7387\u3002

    "},{"location":"chapter_array_and_linkedlist/summary/#441-q-a","title":"4.4.1. \u00a0 Q & A","text":"

    \u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f

    \u6808\u5185\u5b58\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff0c\u800c\u5806\u5185\u5b58\u7531\u7a0b\u5e8f\u5458\u5728\u4ee3\u7801\u4e2d\u5206\u914d\uff08\u6ce8\u610f\uff0c\u8fd9\u91cc\u7684\u6808\u548c\u5806\u548c\u6570\u636e\u7ed3\u6784\u4e2d\u7684\u6808\u548c\u5806\u4e0d\u662f\u540c\u4e00\u6982\u5ff5\uff09\u3002

    1. \u6808\u4e0d\u7075\u6d3b\uff0c\u5206\u914d\u7684\u5185\u5b58\u5927\u5c0f\u4e0d\u53ef\u66f4\u6539\uff1b\u5806\u76f8\u5bf9\u7075\u6d3b\uff0c\u53ef\u4ee5\u52a8\u6001\u5206\u914d\u5185\u5b58\uff1b
    2. \u6808\u662f\u4e00\u5757\u6bd4\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5bb9\u6613\u51fa\u73b0\u5185\u5b58\u4e0d\u8db3\uff1b\u5806\u5185\u5b58\u5f88\u5927\uff0c\u4f46\u662f\u7531\u4e8e\u662f\u52a8\u6001\u5206\u914d\uff0c\u5bb9\u6613\u788e\u7247\u5316\uff0c\u7ba1\u7406\u5806\u5185\u5b58\u7684\u96be\u5ea6\u66f4\u5927\u3001\u6210\u672c\u66f4\u9ad8\uff1b
    3. \u8bbf\u95ee\u6808\u6bd4\u8bbf\u95ee\u5806\u66f4\u5feb\uff0c\u56e0\u4e3a\u6808\u5185\u5b58\u8f83\u5c0f\u3001\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u5806\u5e27\u5206\u6563\u5728\u5f88\u5927\u7684\u7a7a\u95f4\u5185\uff0c\u4f1a\u51fa\u73b0\u66f4\u591a\u7684\u7f13\u5b58\u672a\u547d\u4e2d\uff1b

    \u4e3a\u4ec0\u4e48\u6570\u7ec4\u4f1a\u5f3a\u8c03\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f

    \u94fe\u8868\u7531\u7ed3\u70b9\u7ec4\u6210\uff0c\u7ed3\u70b9\u4e4b\u95f4\u7531\u6307\u9488\u8fde\u63a5\uff0c\u5404\u4e2a\u7ed3\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int, double, string, object \u7b49\u3002

    \u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u90a3\u4e48\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd elementLength \u3002

    // \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\nelementAddr = firtstElementAddr + elementLength * elementIndex\n

    \u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a P.next \u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f

    \u4e0d\u4fee\u6539 P.next \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u7ed3\u70b9\u904d\u5386\u5230\u5c3e\u7ed3\u70b9\u5df2\u7ecf\u9047\u4e0d\u5230 P \u4e86\u3002\u8fd9\u610f\u5473\u7740\u7ed3\u70b9 P \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u7ed3\u70b9 P \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8fd9\u6761\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u4e86\u3002

    \u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java, Python, Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9 P \u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u6709\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f P.next \u7684\u503c\u3002\u5728 C, C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002

    \u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f

    \u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u7ed3\u70b9\u3001\u5c3e\u7ed3\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002

    \u56fe\u7247\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u7ed3\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u7ed3\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f

    \u6587\u4e2d\u53ea\u662f\u4e00\u4e2a\u793a\u610f\u56fe\uff0c\u53ea\u662f\u5b9a\u6027\u8868\u793a\u3002\u5b9a\u91cf\u7684\u8bdd\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u5206\u6790\uff1a

    • \u4e0d\u540c\u7c7b\u578b\u7684\u7ed3\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int, long, double, \u6216\u8005\u662f\u7c7b\u7684\u5b9e\u4f8b\u7b49\u7b49\u3002
    • \u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002

    \u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f

    \u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002

    \u201c\u5217\u8868\u7684\u51fa\u73b0\u5927\u5927\u63d0\u5347\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u526f\u4f5c\u7528\u662f\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f

    \u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u90fd\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002

    \u5728 Python \u4e2d\u521d\u59cb\u5316 n = [1, 2, 3] \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 m = [2, 1, 3] \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf n \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 m \u8fd8\u662f\u6570\u7ec4\u5417\uff1f

    \u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 n = [n1, n2, n3, n4, n5] \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9\u4e94\u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u662f\u88ab\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u7684\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5230\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002

    \u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\u7684\u662f\uff0c\u5728 Python \u4e2d\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u662f\u65e0\u9700\u8fde\u7eed\u7684\u3002

    "},{"location":"chapter_backtracking/","title":"12. \u00a0 \u56de\u6eaf","text":""},{"location":"chapter_backtracking/backtracking_algorithm/","title":"12.1. \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"

    \u300c\u56de\u6eaf\u7b97\u6cd5 Backtracking Algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002

    \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u4e0b\u9762\uff0c\u6211\u4eec\u5c06\u4ece\u524d\u5e8f\u904d\u5386\u5165\u624b\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002

    \u4f8b\u9898\u4e00

    \u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u524d\u5e8f\u904d\u5386\u8fd9\u9897\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u5230\u7ed3\u679c\u5217\u8868 res \u4e4b\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_i_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(root);\n}\npreOrder(root.left);\npreOrder(root.right);\n}\n
    preorder_traversal_i_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(root);\n}\npreOrder(root->left);\npreOrder(root->right);\n}\n
    preorder_traversal_i_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\nif root is None:\nreturn\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(root)\npre_order(root.left)\npre_order(root.right)\n
    preorder_traversal_i_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\nif root == nil {\nreturn\n}\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, root)\n}\npreOrderI(root.Left, res)\npreOrderI(root.Right, res)\n}\n
    preorder_traversal_i_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n
    preorder_traversal_i_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n
    preorder_traversal_i_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_i_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(root);\n}\npreOrder(root.left);\npreOrder(root.right);\n}\n
    preorder_traversal_i_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(root)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n
    preorder_traversal_i_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_i_compact.dart
    [class]{}-[func]{preOrder}\n

    Fig. \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1211","title":"12.1.1. \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"

    \u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002

    \u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 return \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002

    \u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u7b49\u4ef7\u4e8e\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002

    \u4f8b\u9898\u4e8c

    \u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 path \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 path \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 res \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0cres \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_ii_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList<>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n
    preorder_traversal_ii_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root->left);\npreOrder(root->right);\n// \u56de\u9000\npath.pop_back();\n}\n
    preorder_traversal_ii_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\nif root is None:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n
    preorder_traversal_ii_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\nif root == nil {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderII(root.Left, res, path)\npreOrderII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n
    preorder_traversal_ii_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_ii_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_ii_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_ii_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List<TreeNode>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n
    preorder_traversal_ii_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n
    preorder_traversal_ii_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_ii_compact.dart
    [class]{}-[func]{preOrder}\n

    \u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb path \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece path \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u662f\u4e92\u4e3a\u76f8\u53cd\u7684\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1212","title":"12.1.2. \u00a0 \u526a\u679d","text":"

    \u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002

    \u4f8b\u9898\u4e09

    \u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u8def\u5f84\u4e2d\u4e0d\u80fd\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002

    \u89e3\u9898\u601d\u8def\uff1a\u5728\u4f8b\u9898\u4e8c\u7684\u57fa\u7840\u4e0a\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff0c\u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u7ec8\u6b62\u7ee7\u7eed\u641c\u7d22\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_iii_compact.java
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList<>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n
    preorder_traversal_iii_compact.cpp
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n// \u526a\u679d\nif (root == nullptr || root->val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root->val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root->left);\npreOrder(root->right);\n// \u56de\u9000\npath.pop_back();\n}\n
    preorder_traversal_iii_compact.py
    def pre_order(root: TreeNode) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u526a\u679d\nif root is None or root.val == 3:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n
    preorder_traversal_iii_compact.go
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n// \u526a\u679d\nif root == nil || root.Val == 3 {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif int(root.Val) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderIII(root.Left, res, path)\npreOrderIII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n
    preorder_traversal_iii_compact.js
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_iii_compact.ts
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n
    preorder_traversal_iii_compact.c
    [class]{}-[func]{preOrder}\n
    preorder_traversal_iii_compact.cs
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List<TreeNode>(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n
    preorder_traversal_iii_compact.swift
    /* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n// \u526a\u679d\nguard let root = root, root.val != 3 else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n
    preorder_traversal_iii_compact.zig
    [class]{}-[func]{preOrder}\n
    preorder_traversal_iii_compact.dart
    [class]{}-[func]{preOrder}\n

    \u526a\u679d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5229\u7528\u7ea6\u675f\u6761\u4ef6\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002

    Fig. \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1213","title":"12.1.3. \u00a0 \u5e38\u7528\u672f\u8bed","text":"

    \u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\u3002

    \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3 Solution \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\u3002\u56de\u6eaf\u7b97\u6cd5\u7684\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a\u6216\u591a\u4e2a\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3 \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6240\u6709\u8def\u5f84\uff0c\u4e14\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9 \u72b6\u6001 State \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 path \u8282\u70b9\u5217\u8868 \u7ea6\u675f\u6761\u4ef6 Constraint \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u80fd\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9 \u5c1d\u8bd5 Attempt \u5c1d\u8bd5\u662f\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6839\u636e\u5f53\u524d\u72b6\u6001\u548c\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\u3002\u5c1d\u8bd5\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb path \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000 Backtracking \u56de\u9000\u6307\u5728\u641c\u7d22\u4e2d\u9047\u5230\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u6216\u65e0\u6cd5\u7ee7\u7eed\u641c\u7d22\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u7ed3\u70b9\u3001\u7ed3\u675f\u7ed3\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d Pruning \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u7ec8\u6b62\u7ee7\u7eed\u641c\u7d22

    Tip

    \u89e3\u3001\u72b6\u6001\u3001\u7ea6\u675f\u6761\u4ef6\u7b49\u672f\u8bed\u662f\u901a\u7528\u7684\uff0c\u9002\u7528\u4e8e\u56de\u6eaf\u7b97\u6cd5\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1214","title":"12.1.4. \u00a0 \u6846\u67b6\u4ee3\u7801","text":"

    \u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002\u4e3a\u63d0\u5347\u4ee3\u7801\u901a\u7528\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5c06\u56de\u6eaf\u7b97\u6cd5\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\u3002

    \u8bbe state \u4e3a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0cchoices \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff0c\u5219\u53ef\u5f97\u5230\u4ee5\u4e0b\u56de\u6eaf\u7b97\u6cd5\u7684\u6846\u67b6\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector<Choice *> &choices, vector<State *> &res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    def backtrack(state: State, choices: list[choice], res: list[state]) -> None:\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n# \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\nbacktrack(state, choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\nbacktrack(state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res, numRes);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < numChoices; i++) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, &choices[i])) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, &choices[i]);\nbacktrack(state, choices, numChoices, res, numRes);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, &choices[i]);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice> choices, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state: state, res: &res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &state, choice: choice)\nbacktrack(state: &state, choices: choices, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &state, choice: choice)\n}\n}\n}\n
    \n
    /* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List<Choice>, List<State> res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n

    \u4e0b\u9762\uff0c\u6211\u4eec\u5c1d\u8bd5\u57fa\u4e8e\u6b64\u6846\u67b6\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u5728\u4f8b\u9898\u4e09\u4e2d\uff0c\u72b6\u6001 state \u662f\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 choices \u662f\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c res \u662f\u8def\u5f84\u5217\u8868\uff0c\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart preorder_traversal_iii_template.java
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List<TreeNode> state) {\nreturn !state.isEmpty() && state.get(state.size() - 1).val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\nres.add(new ArrayList<>(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List<TreeNode> state, TreeNode choice) {\nreturn choice != null && choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\nstate.add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\nstate.remove(state.size() - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, Arrays.asList(choice.left, choice.right), res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.cpp
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector<TreeNode *> &state) {\nreturn !state.empty() && state.back()->val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {\nres.push_back(state);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector<TreeNode *> &state, TreeNode *choice) {\nreturn choice != nullptr && choice->val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector<TreeNode *> &state, TreeNode *choice) {\nstate.push_back(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector<TreeNode *> &state, TreeNode *choice) {\nstate.pop_back();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode *choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nvector<TreeNode *> nextChoices{choice->left, choice->right};\nbacktrack(state, nextChoices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.py
    def is_solution(state: list[TreeNode]) -> bool:\n\"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\nreturn state and state[-1].val == 7\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n\"\"\"\u8bb0\u5f55\u89e3\"\"\"\nres.append(list(state))\ndef is_valid(state: list[TreeNode], choice: TreeNode) -> bool:\n\"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\nreturn choice is not None and choice.val != 3\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\nstate.append(choice)\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u6062\u590d\u72b6\u6001\"\"\"\nstate.pop()\ndef backtrack(\nstate: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n
    preorder_traversal_iii_template.go
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\nreturn len(*state) != 0 && (*state)[len(*state)-1].Val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n*res = append(*res, *state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\nreturn choice != nil && choice.Val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = append(*state, choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = (*state)[:len(*state)-1]\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range *choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\ntemp := make([]*TreeNode, 0)\ntemp = append(temp, choice.Left, choice.Right)\nbacktrackIII(state, &temp, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n
    preorder_traversal_iii_template.js
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\nreturn state && state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\nreturn choice !== null && choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n
    preorder_traversal_iii_template.ts
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\nreturn state && state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\nreturn choice !== null && choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\nstate: TreeNode[],\nchoices: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n
    preorder_traversal_iii_template.c
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n
    preorder_traversal_iii_template.cs
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List<TreeNode> state) {\nreturn state.Count != 0 && state[^1].val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {\nres.Add(new List<TreeNode>(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List<TreeNode> state, TreeNode choice) {\nreturn choice != null && choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List<TreeNode> state, TreeNode choice) {\nstate.Add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List<TreeNode> state, TreeNode choice) {\nstate.RemoveAt(state.Count - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (TreeNode choice in choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, new List<TreeNode> { choice.left, choice.right }, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n
    preorder_traversal_iii_template.swift
    /* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -> Bool {\n!state.isEmpty && state.last!.val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\nres.append(state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -> Bool {\nchoice != nil && choice!.val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.append(choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.removeLast()\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\nrecordSolution(state: state, res: &res)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &state, choice: choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: [choice.left, choice.right].compactMap { $0 }, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &state, choice: choice)\n}\n}\n}\n
    preorder_traversal_iii_template.zig
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n
    preorder_traversal_iii_template.dart
    [class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n

    \u76f8\u8f83\u4e8e\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u5b9e\u73b0\u4ee3\u7801\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u5b9e\u73b0\u4ee3\u7801\u867d\u7136\u663e\u5f97\u5570\u55e6\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u6240\u6709\u56de\u6eaf\u95ee\u9898\u90fd\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u9700\u8981\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 state \u548c choices \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1215","title":"12.1.5. \u00a0 \u5178\u578b\u4f8b\u9898","text":"

    \u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002

    • \u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002
    • \u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002
    • \u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u4e2a\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u4e2a\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u4e2a\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002

    \u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002

    • \\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002
    • \u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002
    • \u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002

    \u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002

    • 0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002
    • \u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002
    • \u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002

    \u8bf7\u6ce8\u610f\uff0c\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u4e0d\u662f\u89e3\u51b3\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u7684\u6700\u4f18\u65b9\u6cd5\u30020-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff1b\u65c5\u884c\u5546\u662f\u4e00\u4e2a NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u51b3\u65b9\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\uff1b\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8f6e\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178 NP-Hard \u95ee\u9898\uff0c\u901a\u5e38\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002

    "},{"location":"chapter_backtracking/backtracking_algorithm/#1216","title":"12.1.6. \u00a0 \u4f18\u52bf\u4e0e\u5c40\u9650\u6027","text":"

    \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u52bf\u5728\u4e8e\u5b83\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002

    \u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u6700\u574f\u7684\u60c5\u51b5\u4e0b\uff0c\u56de\u6eaf\u7b97\u6cd5\u9700\u8981\u904d\u5386\u89e3\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\u89e3\u3002\u4f8b\u5982\uff0c\u6c42\u89e3 \\(n\\) \u7687\u540e\u95ee\u9898\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n!)\\) \u3002\u56de\u6eaf\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u53ef\u80fd\u8f83\u9ad8\u3002\u56e0\u4e3a\u5728\u6bcf\u4e00\u6b21\u9012\u5f52\u8c03\u7528\u65f6\uff0c\u90fd\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u9009\u62e9\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5bf9\u4e8e\u6df1\u5ea6\u5f88\u5927\u7684\u9012\u5f52\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u975e\u5e38\u5927\u3002

    \u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u6211\u4eec\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u8fdb\u884c\u6548\u7387\u4f18\u5316\uff1a

    • \u4e0a\u6587\u4ecb\u7ecd\u8fc7\u7684\u526a\u679d\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u3002\u5b83\u53ef\u4ee5\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002
    • \u53e6\u4e00\u4e2a\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u662f\u52a0\u5165\u300c\u542f\u53d1\u5f0f\u641c\u7d22 Heuristic Search\u300d\u7b56\u7565\uff0c\u5b83\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002
    "},{"location":"chapter_backtracking/n_queens_problem/","title":"12.4. \u00a0 N \u7687\u540e\u95ee\u9898","text":"

    Question

    \u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u4e4b\u5904\u5728\u540c\u4e00\u884c\u6216\u540c\u4e00\u5217\u6216\u540c\u4e00\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 choices \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 state \u3002

    Fig. 4 \u7687\u540e\u95ee\u9898\u7684\u89e3

    \u672c\u9898\u5171\u6709\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u548c\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf \\ \u548c\u526f\u5bf9\u89d2\u7ebf / \u4e24\u79cd\u3002

    Fig. n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6

    \u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u7b2c\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002\u8fd9\u610f\u5473\u7740\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002\u6b64\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86 \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u7bc7\u5e45\u9650\u5236\uff0c\u4e0b\u56fe\u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u4e00\u4e2a\u641c\u7d22\u5206\u652f\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u526a\u679d\u4e86\u3002

    Fig. \u9010\u884c\u653e\u7f6e\u7b56\u7565

    \u4e3a\u4e86\u5b9e\u73b0\u6839\u636e\u5217\u7ea6\u675f\u526a\u679d\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 cols \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 cols \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 cols \u7684\u72b6\u6001\u3002

    \u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a (row, col) \uff0c\u89c2\u5bdf\u77e9\u9635\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u76f8\u7b49\uff0c\u5373 row - col \u4e3a\u6052\u5b9a\u503c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u82e5\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 row1 - col1 == row2 - col2 \uff0c\u5219\u8fd9\u4e24\u4e2a\u683c\u5b50\u4e00\u5b9a\u5904\u5728\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002

    \u5229\u7528\u8be5\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4 diag1 \u6765\u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635 row - col \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\u56e0\u6b64\u5171\u6709 \\(2n - 1\\) \u6761\u4e3b\u5bf9\u89d2\u7ebf\u3002

    Fig. \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f

    \u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 row + col \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u540c\u6837\u7684\u65b9\u6cd5\uff0c\u501f\u52a9\u6570\u7ec4 diag2 \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002

    \u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u5199\u51fa \\(n\\) \u7687\u540e\u7684\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart n_queens.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<String>> state, List<List<List<String>>> res,\nboolean[] cols, boolean[] diags1, boolean[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList<List<String>> copyState = new ArrayList<>();\nfor (List<String> sRow : state) {\ncopyState.add(new ArrayList<>(sRow));\n}\nres.add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate.get(row).set(col, \"Q\");\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate.get(row).set(col, \"#\");\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<String>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList<List<String>> state = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<String> row = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\nrow.add(\"#\");\n}\nstate.add(row);\n}\nboolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nboolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nboolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nList<List<List<String>>> res = new ArrayList<>();\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,\nvector<bool> &diags1, vector<bool> &diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nvector<vector<vector<string>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvector<vector<string>> state(n, vector<string>(n, \"#\"));\nvector<bool> cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvector<bool> diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvector<bool> diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvector<vector<vector<string>>> res;\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.py
    def backtrack(\nrow: int,\nn: int,\nstate: list[list[str]],\nres: list[list[list[str]]],\ncols: list[bool],\ndiags1: list[bool],\ndiags2: list[bool],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n# \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n:\nres.append([list(row) for row in state])\nreturn\n# \u904d\u5386\u6240\u6709\u5217\nfor col in range(n):\n# \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 = row - col + n - 1\ndiag2 = row + col\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif not cols[col] and not diags1[diag1] and not diags2[diag2]:\n# \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = diags1[diag1] = diags2[diag2] = True\n# \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2)\n# \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = diags1[diag1] = diags2[diag2] = False\ndef n_queens(n: int) -> list[list[list[str]]]:\n\"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n# \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate = [[\"#\" for _ in range(n)] for _ in range(n)]\ncols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ndiags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\ndiags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nres = []\nbacktrack(0, n, state, res, cols, diags1, diags2)\nreturn res\n
    n_queens.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col < n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col < n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !(*cols)[col] && !(*diags1)[diag1] && !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\nfunc nQueens(n int) [][][]string {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate := make([][]string, n)\nfor i := 0; i < n; i++ {\nrow := make([]string, n)\nfor i := 0; i < n; i++ {\nrow[i] = \"#\"\n}\nstate[i] = row\n}\n// \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ncols := make([]bool, n)\ndiags1 := make([]bool, 2*n-1)\ndiags2 := make([]bool, 2*n-1)\nres := make([][][]string, 0)\nbacktrack(0, n, &state, &res, &cols, &diags1, &diags2)\nreturn res\n}\n
    n_queens.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) => row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () => Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst res = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\nrow: number,\nn: number,\nstate: string[][],\nres: string[][][],\ncols: boolean[],\ndiags1: boolean[],\ndiags2: boolean[]\n): void {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) => row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () => Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nconst res: string[][][] = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    n_queens.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List<List<string>> state, List<List<List<string>>> res,\nbool[] cols, bool[] diags1, bool[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList<List<string>> copyState = new List<List<string>>();\nforeach (List<string> sRow in state) {\ncopyState.Add(new List<string>(sRow));\n}\nres.Add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col < n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif (!cols[col] && !diags1[diag1] && !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList<List<List<string>>> nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList<List<string>> state = new List<List<string>>();\nfor (int i = 0; i < n; i++) {\nList<string> row = new List<string>();\nfor (int j = 0; j < n; j++) {\nrow.Add(\"#\");\n}\nstate.Add(row);\n}\nbool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nbool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nbool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nList<List<List<string>>> res = new List<List<List<string>>>();\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n
    n_queens.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col in 0 ..< n {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nlet diag1 = row - col + n - 1\nlet diag2 = row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u5b58\u5728\u7687\u540e\nif !cols[col] && !diags1[diag1] && !diags2[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = true\ndiags1[diag1] = true\ndiags2[diag2] = true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row: row + 1, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = false\ndiags1[diag1] = false\ndiags2[diag2] = false\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -> [[[String]]] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvar state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\nvar cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvar diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvar diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u6709\u7687\u540e\nvar res: [[[String]]] = []\nbacktrack(row: 0, n: n, state: &state, res: &res, cols: &cols, diags1: &diags1, diags2: &diags2)\nreturn res\n}\n
    n_queens.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    n_queens.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n
    "},{"location":"chapter_backtracking/n_queens_problem/#1241","title":"12.4.1. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n, n-1, \\cdots, 2, 1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u5730\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002

    state \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0ccols , diags1 , diags2 \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_backtracking/permutations_problem/","title":"12.2. \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"

    \u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u8fd9\u4e2a\u96c6\u5408\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002

    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u7ec4\u548c\u5176\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002

    \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1221","title":"12.2.1. \u00a0 \u65e0\u91cd\u590d\u7684\u60c5\u51b5","text":"

    Question

    \u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002

    \u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \u3001\u518d\u9009\u62e9 \\(3\\) \u3001\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002

    \u4ece\u56de\u6eaf\u7b97\u6cd5\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 choices \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 state \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64\u5728\u904d\u5386\u9009\u62e9\u65f6\uff0c\u5e94\u5f53\u6392\u9664\u5df2\u7ecf\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u4e2a\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 state \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002

    Fig. \u5168\u6392\u5217\u7684\u9012\u5f52\u6811

    \u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u4ee3\u7801\u884c\u6570\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u4ed6\u4eec\u5c55\u5f00\u5728 backtrack() \u51fd\u6570\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart permutations_i.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList<Integer>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList<List<Integer>> permutationsI(int[] nums) {\nList<List<Integer>> res = new ArrayList<List<Integer>>();\nbacktrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\nreturn res;\n}\n
    permutations_i.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nvector<vector<int>> permutationsI(vector<int> nums) {\nvector<int> state;\nvector<bool> selected(nums.size(), false);\nvector<vector<int>> res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n
    permutations_i.py
    def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif not selected[i]:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_i(nums: list[int]) -> list[list[int]]:\n\"\"\"\u5168\u6392\u5217 I\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n
    permutations_i.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i < len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackI(&state, &nums, &selected, &res)\nreturn res\n}\n
    permutations_i.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_i.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_i.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n
    permutations_i.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList<List<int>> permutationsI(int[] nums) {\nList<List<int>> res = new List<List<int>>();\nbacktrack(new List<int>(), nums, new bool[nums.Length], res);\nreturn res;\n}\n
    permutations_i.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: choices, selected: &selected, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -> [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &state, choices: nums, selected: &selected, res: &res)\nreturn res\n}\n
    permutations_i.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n
    permutations_i.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n

    \u9700\u8981\u91cd\u70b9\u5173\u6ce8\u7684\u662f\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 selected \uff0c\u5b83\u7684\u957f\u5ea6\u4e0e\u8f93\u5165\u6570\u7ec4\u957f\u5ea6\u76f8\u7b49\uff0c\u5176\u4e2d selected[i] \u8868\u793a choices[i] \u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u5229\u7528 selected \u907f\u514d\u67d0\u4e2a\u5143\u7d20\u88ab\u91cd\u590d\u9009\u62e9\uff0c\u4ece\u800c\u5b9e\u73b0\u526a\u679d\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1, 3 \u7684\u5206\u652f\u3002\u4ece\u672c\u8d28\u4e0a\u7406\u89e3\uff0c\u6b64\u526a\u679d\u64cd\u4f5c\u53ef\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u964d\u4f4e\u81f3 \\(O(n!)\\) \u3002

    Fig. \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b

    "},{"location":"chapter_backtracking/permutations_problem/#1222","title":"12.2.2. \u00a0 \u8003\u8651\u91cd\u590d\u7684\u60c5\u51b5","text":"

    Question

    \u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002

    \u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u7684\u5143\u7d20 \\(1\\) \uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a\u5143\u7d20\u8bb0\u4e3a \\(\\hat{1}\\) \u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u90fd\u662f\u91cd\u590d\u7684\u3002

    Fig. \u91cd\u590d\u6392\u5217

    \u90a3\u4e48\uff0c\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\uff0c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u662f\u6ca1\u6709\u5fc5\u8981\u7684\uff0c\u5e94\u5f53\u88ab\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u56e0\u4e3a\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u6389\u3002\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u9700\u8981\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002

    Fig. \u91cd\u590d\u6392\u5217\u526a\u679d

    \u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5b9e\u73b0\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u56e0\u6b64\uff0c\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 duplicated \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart permutations_ii.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<Integer> state, int[] choices, boolean[] selected, List<List<Integer>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList<Integer>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nSet<Integer> duplicated = new HashSet<Integer>();\nfor (int i = 0; i < choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList<List<Integer>> permutationsII(int[] nums) {\nList<List<Integer>> res = new ArrayList<List<Integer>>();\nbacktrack(new ArrayList<Integer>(), nums, new boolean[nums.length], res);\nreturn res;\n}\n
    permutations_ii.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector<int> &state, const vector<int> &choices, vector<bool> &selected, vector<vector<int>> &res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nunordered_set<int> duplicated;\nfor (int i = 0; i < choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && duplicated.find(choice) == duplicated.end()) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nvector<vector<int>> permutationsII(vector<int> nums) {\nvector<int> state;\nvector<bool> selected(nums.size(), false);\nvector<vector<int>> res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n
    permutations_ii.py
    def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated = set[int]()\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif not selected[i] and choice not in duplicated:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_ii(nums: list[int]) -> list[list[int]]:\n\"\"\"\u5168\u6392\u5217 II\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n
    permutations_ii.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated := make(map[int]struct{}, 0)\nfor i := 0; i < len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif _, ok := duplicated[choice]; !ok && !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n// \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nduplicated[choice] = struct{}{}\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackII(&state, &nums, &selected, &res)\nreturn res\n}\n
    permutations_ii.js
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_ii.ts
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) => {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n
    permutations_ii.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n
    permutations_ii.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List<int> state, int[] choices, bool[] selected, List<List<int>> res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nISet<int> duplicated = new HashSet<int>();\nfor (int i = 0; i < choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] && !duplicated.Contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList<List<int>> permutationsII(int[] nums) {\nList<List<int>> res = new List<List<int>>();\nbacktrack(new List<int>(), nums, new bool[nums.Length], res);\nreturn res;\n}\n
    permutations_ii.swift
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nvar duplicated: Set<Int> = []\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i], !duplicated.contains(choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &state, choices: choices, selected: &selected, res: &res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -> [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &state, choices: nums, selected: &selected, res: &res)\nreturn res\n}\n
    permutations_ii.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n
    permutations_ii.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n

    \u6ce8\u610f\uff0c\u867d\u7136 selected \u548c duplicated \u90fd\u8d77\u5230\u526a\u679d\u7684\u4f5c\u7528\uff0c\u4f46\u4ed6\u4eec\u526a\u6389\u7684\u662f\u4e0d\u540c\u7684\u5206\u652f\uff1a

    • \u526a\u679d\u6761\u4ef6\u4e00\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a selected \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u4f5c\u7528\u662f\u907f\u514d\u67d0\u4e2a\u5143\u7d20\u5728 state \u4e2d\u91cd\u590d\u51fa\u73b0\u3002
    • \u526a\u679d\u6761\u4ef6\u4e8c\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u5373\u6bcf\u4e2a\u5f00\u542f\u7684 backtrack \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a duplicated \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u904d\u5386\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u4ee5\u907f\u514d\u4ea7\u751f\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002

    Fig. \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4

    "},{"location":"chapter_backtracking/permutations_problem/#1223","title":"12.2.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\) \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002

    \u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002selected \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a duplicated \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u5168\u6392\u5217 I \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5168\u6392\u5217 II \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_backtracking/subset_sum_problem/","title":"12.3. \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 target \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e target \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002

    \u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u7531\u4e8e\u96c6\u5408\u4e2d\u7684\u6570\u5b57\u53ef\u4ee5\u88ab\u91cd\u590d\u9009\u53d6\uff0c\u56e0\u6b64\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u5b50\u96c6\u662f\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\u7684\uff0c\u4f8b\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002

    "},{"location":"chapter_backtracking/subset_sum_problem/#1231","title":"12.3.1. \u00a0 \u4ece\u5168\u6392\u5217\u5f15\u51fa\u89e3\u6cd5","text":"

    \u7c7b\u4f3c\u4e8e\u4e0a\u8282\u5168\u6392\u5217\u95ee\u9898\u7684\u89e3\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e target \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002

    \u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u5141\u8bb8\u91cd\u590d\u9009\u53d6\u540c\u4e00\u5143\u7d20\uff0c\u56e0\u6b64\u65e0\u9700\u501f\u52a9 selected \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_i_naive.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int total, int[] choices, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<Integer>> subsetSumINaive(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, int total, vector<int> &choices, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (size_t i = 0; i < choices.size(); i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector<vector<int>> subsetSumINaive(vector<int> &nums, int target) {\nvector<int> state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0;           // \u5b50\u96c6\u548c\nvector<vector<int>> res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.py
    def backtrack(\nstate: list[int],\ntarget: int,\ntotal: int,\nchoices: list[int],\nres: list[list[int]],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif total == target:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in range(len(choices)):\n# \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total + choices[i] > target:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i_naive(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal = 0  # \u5b50\u96c6\u548c\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res)\nreturn res\n
    subset_sum_i_naive.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == total {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i < len(*choices); i++ {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total+(*choices)[i] > target {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal := 0              // \u5b50\u96c6\u548c\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumINaive(total, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_i_naive.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<int> state, int target, int total, int[] choices, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i < choices.Length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] > target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList<List<int>> subsetSumINaive(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n
    subset_sum_i_naive.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n
    subset_sum_i_naive.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n

    \u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e24\u79cd\u4e0d\u540c\u7684\u60c5\u51b5\u3002

    Fig. \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d

    "},{"location":"chapter_backtracking/subset_sum_problem/#1232","title":"12.3.2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"

    \u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u56e0\u4e3a\uff1a

    • \u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 target \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002
    • \u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u662f\u5f88\u8017\u65f6\u7684\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002

    \u4e3a\u4e86\u8fbe\u5230\u6700\u4f73\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u4e0b\u56fe\uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u5177\u4f53\u6765\u770b\uff1a

    1. \u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) , \\(4\\) \uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\cdots]\\) \u3002
    2. \u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\cdots]\\) \u548c 1. \u4e2d\u63d0\u5230\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002
    3. \u540c\u7406\uff0c\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\cdots]\\) \u548c\u5b50\u96c6 \\([5, 4, \\cdots]\\) \u548c\u4e4b\u524d\u7684\u5b50\u96c6\u91cd\u590d\u3002

    Fig. \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6

    \u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\cdots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\cdots , x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\cdots \\leq i_m\\) \u3002\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u662f\u91cd\u590d\u5b50\u96c6\u3002

    \u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf start \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\uff0c\u4ece\u800c\u5b8c\u6210\u5b50\u96c6\u53bb\u91cd\u3002

    \u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4e24\u9879\u4f18\u5316\u3002\u9996\u5148\uff0c\u6211\u4eec\u5728\u5f00\u542f\u641c\u7d22\u524d\u5c06\u6570\u7ec4 nums \u6392\u5e8f\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 target \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u90fd\u4e00\u5b9a\u4f1a\u8d85\u8fc7 target \u3002\u5176\u6b21\uff0c\u6211\u4eec\u901a\u8fc7\u5728 target \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 target \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\uff0c\u7701\u53bb\u4e86\u5143\u7d20\u548c\u53d8\u91cf total \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_i.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<Integer>> subsetSumI(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector<vector<int>> subsetSumI(vector<int> &nums, int target) {\nvector<int> state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector<vector<int>> res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.py
    def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] < 0:\nbreak\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n
    subset_sum_i.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i := start; i < len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] < 0 {\nbreak\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumI(start, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_i.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i < choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList<List<int>> subsetSumI(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_i.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n
    subset_sum_i.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u5230\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002

    Fig. \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b

    "},{"location":"chapter_backtracking/subset_sum_problem/#1233","title":"12.3.3. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 target \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e target \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002

    \u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u4e5f\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002

    Fig. \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6

    \u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u5229\u7528\u8be5\u7279\u6027\uff0c\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002

    \u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf start \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u5373\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart subset_sum_ii.java
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<Integer> state, int target, int[] choices, int start, List<List<Integer>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList<>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<Integer>> subsetSumII(int[] nums, int target) {\nList<Integer> state = new ArrayList<>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<Integer>> res = new ArrayList<>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.cpp
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector<int> &state, int target, vector<int> &choices, int start, vector<vector<int>> &res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector<vector<int>> subsetSumII(vector<int> &nums, int target) {\nvector<int> state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector<vector<int>> res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.py
    def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n# \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] < 0:\nbreak\n# \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i > start and choices[i] == choices[i - 1]:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_ii(nums: list[int], target: int) -> list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n
    subset_sum_ii.go
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i := start; i < len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] < 0 {\nbreak\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i > start && (*choices)[i] == (*choices)[i-1] {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumII(start, target, &state, &nums, &res)\nreturn res\n}\n
    subset_sum_ii.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.cs
    /* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List<int> state, int target, int[] choices, int start, List<List<int>> res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List<int>(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i < choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] < 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i > start && choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList<List<int>> subsetSumII(int[] nums, int target) {\nList<int> state = new List<int>(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList<List<int>> res = new List<List<int>>(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n
    subset_sum_ii.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n
    subset_sum_ii.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n

    \u4e0b\u56fe\u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u5efa\u8bae\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002

    Fig. \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b

    "},{"location":"chapter_backtracking/summary/","title":"12.5. \u00a0 \u5c0f\u7ed3","text":"
    • \u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002
    • \u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u66f4\u9ad8\u6548\u7387\u6216\u66f4\u597d\u6548\u679c\u7684\u89e3\u6cd5\u3002
    • \u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u679d\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002
    • \u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002
    • \u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002
    • \u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002
    • \\(n\\) \u7687\u540e\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002
    • \u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002
    "},{"location":"chapter_computational_complexity/","title":"2. \u00a0 \u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1. \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1. \u00a0 \u7b97\u6cd5\u8bc4\u4ef7\u7ef4\u5ea6","text":"

    \u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u7b97\u6cd5\u8bbe\u8ba1\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\uff1a

    1. \u627e\u5230\u95ee\u9898\u89e3\u6cd5\u3002\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\uff0c\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002
    2. \u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\u3002\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002

    \u56e0\u6b64\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u6210\u4e3a\u4e3b\u8981\u7684\u8bc4\u4ef7\u7ef4\u5ea6\uff0c\u4e3b\u8981\u5305\u62ec\uff1a

    • \u65f6\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002
    • \u7a7a\u95f4\u6548\u7387\uff0c\u5373\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002

    \u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u638c\u63e1\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u65b9\u6cd5\u5219\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u4e86\u89e3\u8bc4\u4ef7\u6807\u51c6\uff0c\u6211\u4eec\u624d\u80fd\u8fdb\u884c\u7b97\u6cd5\u4e4b\u95f4\u7684\u5bf9\u6bd4\u5206\u6790\uff0c\u4ece\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2. \u00a0 \u6548\u7387\u8bc4\u4f30\u65b9\u6cd5","text":""},{"location":"chapter_computational_complexity/performance_evaluation/#_1","title":"\u5b9e\u9645\u6d4b\u8bd5","text":"

    \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 A \u548c\u7b97\u6cd5 B\uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6211\u4eec\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u5c31\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u5c40\u9650\u6027\u3002

    \u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u8868\u73b0\u3002\u4f8b\u5982\uff0c\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 A \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 B \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u6211\u4eec\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002

    \u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 A \u7684\u8fd0\u884c\u65f6\u95f4\u53ef\u80fd\u77ed\u4e8e\u7b97\u6cd5 B\uff1b\u800c\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u8fd9\u6837\u9700\u8981\u5360\u7528\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#_2","title":"\u7406\u8bba\u4f30\u7b97","text":"

    \u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u590d\u6742\u5ea6\u5206\u6790 Complexity Analysis\u300d\u6216\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 Asymptotic Complexity Analysis\u300d\u3002

    \u590d\u6742\u5ea6\u5206\u6790\u8bc4\u4f30\u7684\u662f\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\uff1a

    • \u201c\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u201d\u53ef\u5206\u4e3a\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u548c\u201c\u5360\u7528\u7a7a\u95f4\u201d\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5c06\u590d\u6742\u5ea6\u5206\u4e3a\u300c\u65f6\u95f4\u590d\u6742\u5ea6 Time Complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\uff1b
    • \u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u589e\u591a\u65f6\u201d\u8868\u793a\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u6709\u5173\uff0c\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff1b
    • \u201c\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u662f\u7b97\u6cd5\u65f6\u95f4\u4e0e\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u975e\u5177\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\uff1b

    \u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\u3002\u9996\u5148\uff0c\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u56e0\u6b64\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002\u5176\u6b21\uff0c\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002

    \u5982\u679c\u4f60\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u7684\u6982\u5ff5\u4ecd\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u9700\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u8be6\u7ec6\u4ecb\u7ecd\u3002

    "},{"location":"chapter_computational_complexity/performance_evaluation/#213","title":"2.1.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790\u91cd\u8981\u6027","text":"

    \u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u544a\u8bc9\u6211\u4eec\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5e76\u4f7f\u6211\u4eec\u80fd\u591f\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002

    \u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u7b2c\u4e00\u7ae0\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u6211\u4eec\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002\u56e0\u6b64\uff0c\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5efa\u8bae\u8bfb\u8005\u5148\u5bf9\u590d\u6742\u5ea6\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u5e76\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u6848\u4f8b\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/","title":"2.3. \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"

    \u300c\u7a7a\u95f4\u590d\u6742\u5ea6 Space Complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u4f7f\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#231","title":"2.3.1. \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"

    \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\uff1a

    • \u300c\u8f93\u5165\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\uff1b
    • \u300c\u6682\u5b58\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\uff1b
    • \u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\uff1b

    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7edf\u8ba1\u8303\u56f4\u662f\u300c\u6682\u5b58\u7a7a\u95f4\u300d+\u300c\u8f93\u51fa\u7a7a\u95f4\u300d\u3002

    \u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a

    • \u300c\u6682\u5b58\u6570\u636e\u300d\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002
    • \u300c\u6808\u5e27\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002
    • \u300c\u6307\u4ee4\u7a7a\u95f4\u300d\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002

    \u56e0\u6b64\uff0c\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u4e00\u822c\u7edf\u8ba1 \u6682\u5b58\u6570\u636e\u3001\u8f93\u51fa\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4 \u4e09\u90e8\u5206\u3002

    Fig. \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nfinal int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7ed3\u6784\u4f53 */\nstruct Node {\nint val;\nNode *next;\nNode(int x) : val(x), next(nullptr) {}\n};\n/* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    class Node:\n\"\"\"\u7c7b\"\"\"\ndef __init__(self, x: int):\nself.val: int = x                 # \u8282\u70b9\u503c\nself.next: Optional[Node] = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\uff08\u5f15\u7528\uff09\ndef function() -> int:\n\"\"\"\u51fd\u6570\"\"\"\n# do something...\nreturn 0\ndef algorithm(n) -> int:  # \u8f93\u5165\u6570\u636e\nA = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\nb = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnode = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn A + b + c      # \u8f93\u51fa\u6570\u636e\n
    /* \u7ed3\u6784\u4f53 */\ntype node struct {\nval  int\nnext *node\n}\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\nreturn &node{val: val}\n}\n/* \u51fd\u6570 */\nfunc function() int {\n// do something...\nreturn 0\n}\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\nconst a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nb := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnewNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nval;\nnext;\nconstructor(val) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc() {\n// do something\nreturn 0;\n}\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\nconst a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nval: number;\nnext: Node | null;\nconstructor(val?: number) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n// do something\nreturn 0;\n}\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\nconst a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u51fd\u6570 */\nint func() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\nconst int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nint c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node\n{\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function()\n{\n// do something...\nreturn 0;\n}\nint algorithm(int n)          // \u8f93\u5165\u6570\u636e\n{\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n
    /* \u7c7b */\nclass Node {\nvar val: Int\nvar next: Node?\ninit(x: Int) {\nval = x\n}\n}\n/* \u51fd\u6570 */\nfunc function() -> Int {\n// do something...\nreturn 0\n}\nfunc algorithm(n: Int) -> Int { // \u8f93\u5165\u6570\u636e\nlet a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nvar b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n
    \n
    /* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(this.val, [this.next]);\n}\n/* \u51fd\u6570 */\nint function() {\n// do something...\nreturn 0;\n}\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\nconst int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#232","title":"2.3.2. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"

    \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u662f\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u300c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002

    \u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\uff0c\u5206\u522b\u662f\u8f93\u5165\u6570\u636e\u7684\u6700\u5dee\u5206\u5e03\u548c\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u6700\u5dee\u65f6\u95f4\u70b9\u3002

    • \u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\u3002\u5f53 \\(n < 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n > 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 nums \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    • \u4ee5\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\u3002\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 nums \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff1b\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\nint[] nums = new int[n]; // O(n)\n}\n
    void algorithm(int n) {\nint a = 0;               // O(1)\nvector<int> b(10000);    // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
    def algorithm(n: int) -> None:\na = 0               # O(1)\nb = [0] * 10000     # O(1)\nif n > 10:\nnums = [0] * n  # O(n)\n
    func algorithm(n int) {\na := 0                      // O(1)\nb := make([]int, 10000)     // O(1)\nvar nums []int\nif n > 10 {\nnums := make([]int, n)  // O(n)\n}\nfmt.Println(a, b, nums)\n}\n
    function algorithm(n) {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
    function algorithm(n: number): void {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n > 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n
    void algorithm(int n) {\nint a = 0;               // O(1)\nint b[10000];            // O(1)\nif (n > 10)\nvector<int> nums(n); // O(n)\n}\n
    void algorithm(int n)\n{\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n > 10)\n{\nint[] nums = new int[n]; // O(n)\n}\n}\n
    func algorithm(n: Int) {\nlet a = 0 // O(1)\nlet b = Array(repeating: 0, count: 10000) // O(1)\nif n > 10 {\nlet nums = Array(repeating: 0, count: n) // O(n)\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 0;                            // O(1)\nList<int> b = List.filled(10000, 0);  // O(1)\nif (n > 10) {\nList<int> nums = List.filled(n, 0); // O(n)\n}\n}\n

    \u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u4f8b\u5982\uff0c\u51fd\u6570 loop() \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 function() \uff0c\u6bcf\u8f6e\u4e2d\u7684 function() \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002\u800c\u9012\u5f52\u51fd\u6570 recur() \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 recur() \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    def function() -> int:\n# do something\nreturn 0\ndef loop(n: int) -> None:\n\"\"\"\u5faa\u73af O(1)\"\"\"\nfor _ in range(n):\nfunction()\ndef recur(n: int) -> int:\n\"\"\"\u9012\u5f52 O(n)\"\"\"\nif n == 1: return\nreturn recur(n - 1)\n
    func function() int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\nfor i := 0; i < n; i++ {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\nif n == 1 {\nreturn\n}\nrecur(n - 1)\n}\n
    function constFunc() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
    function constFunc(): number {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n
    int func() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    int function()\n{\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nint recur(int n)\n{\nif (n == 1) return 1;\nreturn recur(n - 1);\n}\n
    @discardableResult\nfunc function() -> Int {\n// do something\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\nfor _ in 0 ..< n {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\nif n == 1 {\nreturn\n}\nrecur(n: n - 1)\n}\n
    \n
    int function() {\n// do something\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#233","title":"2.3.3. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u6709\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09

    \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n^2) < O(2^n) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} \\end{aligned} \\]

    Fig. \u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b

    Tip

    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u524d\u7f6e\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u770b\u4e0d\u61c2\u7684\u5730\u65b9\u65e0\u9700\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\uff0c\u73b0\u9636\u6bb5\u6211\u4eec\u5148\u4e13\u6ce8\u4e8e\u7406\u89e3\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"

    \u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u5373\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
    space_complexity.cpp
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nvector<int> nums(10000);\nListNode node(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
    space_complexity.py
    def constant(n: int) -> None:\n\"\"\"\u5e38\u6570\u9636\"\"\"\n# \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\na = 0\nnums = [0] * 10000\nnode = ListNode(0)\n# \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nc = 0\n# \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nfunction()\n
    space_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0\nb := 0\nnums := make([]int, 10000)\nListNode := newNode(0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nvar c int\nfor i := 0; i < n; i++ {\nc = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i := 0; i < n; i++ {\nfunction()\n}\nfmt.Println(a, b, nums, c, ListNode)\n}\n
    space_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
    space_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i < n; i++) {\nconstFunc();\n}\n}\n
    space_complexity.c
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nint nums[1000];\nListNode *node = newListNode(0);\nfree(node);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunc();\n}\n}\n
    space_complexity.cs
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nint a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i < n; i++) {\nfunction();\n}\n}\n
    space_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nlet a = 0\nvar b = 0\nlet nums = Array(repeating: 0, count: 10000)\nlet node = ListNode(x: 0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nlet c = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..< n {\nfunction()\n}\n}\n
    space_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a: i32 = 0;\nvar b: i32 = 0;\nvar nums = [_]i32{0}**10000;\nvar node = inc.ListNode(i32){.val = 0};\nvar i: i32 = 0;\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nwhile (i < n) : (i += 1) {\nvar c: i32 = 0;\n_ = c;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\ni = 0;\nwhile (i < n) : (i += 1) {\n_ = function();\n}\n_ = a;\n_ = b;\n_ = nums;\n_ = node;\n}\n
    space_complexity.dart
    /* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nList<int> nums = List.filled(10000, 0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i < n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i < n; i++) {\nfunction();\n}\n}\n
    "},{"location":"chapter_computational_complexity/space_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"

    \u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nnodes.add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<Integer, String> map = new HashMap<>();\nfor (int i = 0; i < n; i++) {\nmap.put(i, String.valueOf(i));\n}\n}\n
    space_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvector<int> nums(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvector<ListNode> nodes;\nfor (int i = 0; i < n; i++) {\nnodes.push_back(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nunordered_map<int, string> map;\nfor (int i = 0; i < n; i++) {\nmap[i] = to_string(i);\n}\n}\n
    space_complexity.py
    def linear(n: int) -> None:\n\"\"\"\u7ebf\u6027\u9636\"\"\"\n# \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nnums = [0] * n\n# \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nmapp = dict[int, str]()\nfor i in range(n):\nmapp[i] = str(i)\n
    space_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n_ = make([]int, n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes []*node\nfor i := 0; i < n; i++ {\nnodes = append(nodes, newNode(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nm := make(map[int]string, n)\nfor i := 0; i < n; i++ {\nm[i] = strconv.Itoa(i)\n}\n}\n
    space_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
    space_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes: ListNode[] = [];\nfor (let i = 0; i < n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i < n; i++) {\nmap.set(i, i.toString());\n}\n}\n
    space_complexity.c
    /* \u54c8\u5e0c\u8868 */\nstruct hashTable {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n};\ntypedef struct hashTable hashTable;\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint *nums = malloc(sizeof(int) * n);\nfree(nums);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nListNode **nodes = malloc(sizeof(ListNode *) * n);\nfor (int i = 0; i < n; i++) {\nnodes[i] = newListNode(i);\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i < n; i++) {\nfree(nodes[i]);\n}\nfree(nodes);\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nhashTable *h = NULL;\nfor (int i = 0; i < n; i++) {\nhashTable *tmp = malloc(sizeof(hashTable));\ntmp->key = i;\ntmp->val = i;\nHASH_ADD_INT(h, key, tmp);\n}\n// \u5185\u5b58\u91ca\u653e\nhashTable *curr, *tmp;\nHASH_ITER(hh, h, curr, tmp) {\nHASH_DEL(h, curr);\nfree(curr);\n}\n}\n
    space_complexity.cs
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = new();\nfor (int i = 0; i < n; i++) {\nnodes.Add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nDictionary<int, string> map = new();\nfor (int i = 0; i < n; i++) {\nmap.Add(i, i.ToString());\n}\n}\n
    space_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet nums = Array(repeating: 0, count: n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet nodes = (0 ..< n).map { ListNode(x: $0) }\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet map = Dictionary(uniqueKeysWithValues: (0 ..< n).map { ($0, \"\\($0)\") })\n}\n
    space_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvar nums = [_]i32{0}**n;\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ntry nodes.append(i);\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\ndefer map.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\nconst string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\ndefer std.heap.page_allocator.free(string);\ntry map.put(i, string);\n}\n_ = nums;\n}\n
    space_complexity.dart
    /* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nList<int> nums = List.filled(n, 0);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList<ListNode> nodes = [];\nfor (var i = 0; i < n; i++) {\nnodes.add(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap<int, String> map = HashMap();\nfor (var i = 0; i < n; i++) {\nmap.putIfAbsent(i, () => i.toString());\n}\n}\n

    \u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nSystem.out.println(\"\u9012\u5f52 n = \" + n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\ncout << \"\u9012\u5f52 n = \" << n << endl;\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.py
    def linear_recur(n: int) -> None:\n\"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nprint(\"\u9012\u5f52 n =\", n)\nif n == 1:\nreturn\nlinear_recur(n - 1)\n
    space_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\nfmt.Println(\"\u9012\u5f52 n =\", n)\nif n == 1 {\nreturn\n}\nspaceLinearRecur(n - 1)\n}\n
    space_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.c
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprintf(\"\u9012\u5f52 n = %d\\r\\n\", n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n
    space_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\nprint(\"\u9012\u5f52 n = \\(n)\")\nif n == 1 {\nreturn\n}\nlinearRecur(n: n - 1)\n}\n
    space_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\nstd.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n
    space_complexity.dart
    /* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprint('\u9012\u5f52 n = $n');\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n

    Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"

    \u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[][] numMatrix = new int[n][n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<Integer>> numList = new ArrayList<>();\nfor (int i = 0; i < n; i++) {\nList<Integer> tmp = new ArrayList<>();\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n
    space_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvector<vector<int>> numMatrix;\nfor (int i = 0; i < n; i++) {\nvector<int> tmp;\nfor (int j = 0; j < n; j++) {\ntmp.push_back(0);\n}\nnumMatrix.push_back(tmp);\n}\n}\n
    space_complexity.py
    def quadratic(n: int) -> None:\n\"\"\"\u5e73\u65b9\u9636\"\"\"\n# \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nnum_matrix = [[0] * n for _ in range(n)]\n
    space_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nnumMatrix := make([][]int, n)\nfor i := 0; i < n; i++ {\nnumMatrix[i] = make([]int, n)\n}\n}\n
    space_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
    space_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() => Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i < n; i++) {\nconst tmp = [];\nfor (let j = 0; j < n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n
    space_complexity.c
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nint **numMatrix = malloc(sizeof(int *) * n);\nfor (int i = 0; i < n; i++) {\nint *tmp = malloc(sizeof(int) * n);\nfor (int j = 0; j < n; j++) {\ntmp[j] = 0;\n}\nnumMatrix[i] = tmp;\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i < n; i++) {\nfree(numMatrix[i]);\n}\nfree(numMatrix);\n}\n
    space_complexity.cs
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[,] numMatrix = new int[n, n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = new();\nfor (int i = 0; i < n; i++) {\nList<int> tmp = new();\nfor (int j = 0; j < n; j++) {\ntmp.Add(0);\n}\nnumList.Add(tmp);\n}\n}\n
    space_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n
    space_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvar nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\nvar tmp = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer tmp.deinit();\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ntry tmp.append(0);\n}\ntry nodes.append(tmp);\n}\n}\n
    space_complexity.dart
    /* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numMatrix = List.generate(n, (_) => List.filled(n, 0));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList<List<int>> numList = [];\nfor (var i = 0; i < n; i++) {\nList<int> tmp = [];\nfor (int j = 0; j < n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n

    \u5728\u4ee5\u4e0b\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 algorithm() \uff0c\u5e76\u4e14\u6bcf\u4e2a\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n, n-1, n-2, ..., 2, 1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nint[] nums = new int[n];\nSystem.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\nvector<int> nums(n);\ncout << \"\u9012\u5f52 n = \" << n << \" \u4e2d\u7684 nums \u957f\u5ea6 = \" << nums.size() << endl;\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.py
    def quadratic_recur(n: int) -> int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 0:\nreturn 0\n# \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nnums = [0] * n\nreturn quadratic_recur(n - 1)\n
    space_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\nif n <= 0 {\nreturn 0\n}\nnums := make([]int, n)\nfmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\nreturn spaceQuadraticRecur(n - 1)\n}\n
    space_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\nif (n <= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.c
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0)\nreturn 0;\nint *nums = malloc(sizeof(int) * n);\nprintf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\nfree(nums);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nint[] nums = new int[n];\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -> Int {\nif n <= 0 {\nreturn 0\n}\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = Array(repeating: 0, count: n)\nprint(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\nreturn quadraticRecur(n: n - 1)\n}\n
    space_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\nif (n <= 0) return 0;\nvar nums = [_]i32{0}**n;\nstd.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\nreturn quadraticRecur(n - 1);\n}\n
    space_complexity.dart
    /* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n <= 0) return 0;\nList<int> nums = List.filled(n, 0);\nprint('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\nreturn quadraticRecur(n - 1);\n}\n

    Fig. \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"

    \u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart space_complexity.java
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\nif (n == 0)\nreturn null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn nullptr;\nTreeNode *root = new TreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.py
    def build_tree(n: int) -> TreeNode | None:\n\"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\nif n == 0:\nreturn None\nroot = TreeNode(0)\nroot.left = build_tree(n - 1)\nroot.right = build_tree(n - 1)\nreturn root\n
    space_complexity.go
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\nif n == 0 {\nreturn nil\n}\nroot := newTreeNode(0)\nroot.left = buildTree(n - 1)\nroot.right = buildTree(n - 1)\nreturn root\n}\n
    space_complexity.js
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.c
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn NULL;\nTreeNode *root = newTreeNode(0);\nroot->left = buildTree(n - 1);\nroot->right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n
    space_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -> TreeNode? {\nif n == 0 {\nreturn nil\n}\nlet root = TreeNode(x: 0)\nroot.left = buildTree(n: n - 1)\nroot.right = buildTree(n: n - 1)\nreturn root\n}\n
    space_complexity.zig
    // \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\nif (n == 0) return null;\nconst root = try mem_allocator.create(inc.TreeNode(i32));\nroot.init(0);\nroot.left = try buildTree(mem_allocator, n - 1);\nroot.right = try buildTree(mem_allocator, n - 1);\nreturn root;\n}\n
    space_complexity.dart
    /* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n

    Fig. \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/space_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"

    \u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u548c\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u7b49\u3002

    \u4f8b\u5982\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002

    \u518d\u4f8b\u5982\u201c\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\u201d\uff0c\u8f93\u5165\u4efb\u610f\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n) = O(\\log n)\\) \u3002

    "},{"location":"chapter_computational_complexity/space_complexity/#234","title":"2.3.4. \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"

    \u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u662f\u975e\u5e38\u56f0\u96be\u7684\u3002

    \u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002

    \u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002

    "},{"location":"chapter_computational_complexity/summary/","title":"2.4. \u00a0 \u5c0f\u7ed3","text":"

    \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30

    • \u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8bc4\u4ef7\u7b97\u6cd5\u6027\u80fd\u7684\u4e24\u4e2a\u5173\u952e\u7ef4\u5ea6\u3002
    • \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002
    • \u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u514b\u670d\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6

    • \u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002
    • \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5373\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002
    • \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002
    • \u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n \\log n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) , \\(O(n!)\\) \u7b49\u3002
    • \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002
    • \u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6

    • \u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002
    • \u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u8ba1\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6307\u4ee4\u7a7a\u95f4\u3001\u6570\u636e\u7a7a\u95f4\u3001\u6808\u5e27\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
    • \u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u95f4\u70b9\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002
    • \u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u5c0f\u5230\u5927\u6392\u5217\u6709 \\(O(1)\\) , \\(O(\\log n)\\) , \\(O(n)\\) , \\(O(n^2)\\) , \\(O(2^n)\\) \u7b49\u3002
    "},{"location":"chapter_computational_complexity/summary/#241-q-a","title":"2.4.1. \u00a0 Q & A","text":"

    \u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f

    \u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u88ab\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java, Python, C++, Go, C# \u7b49\uff09 \u90fd\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u4e00\u822c\u6765\u8bf4\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002

    \u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f

    \u51fd\u6570\uff08function\uff09\u53ef\u4ee5\u72ec\u7acb\u88ab\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002 \u65b9\u6cd5\uff08method\uff09\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u65b9\u6cd5\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u65b9\u6cd5\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002

    \u56e0\u6b64\uff0cC \u548c Go \u53ea\u6709\u51fd\u6570\uff0cJava \u548c C# \u53ea\u6709\u65b9\u6cd5\uff0c\u5728 C++, Python \u4e2d\u53d6\u51b3\u4e8e\u5b83\u662f\u5426\u5c5e\u4e8e\u4e00\u4e2a\u7c7b\u3002

    \u56fe\u7247\u201c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f

    \u4e0d\u662f\uff0c\u8be5\u56fe\u7247\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff08\u5373\u589e\u957f\u8d8b\u52bf\uff09\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002\u6bcf\u4e2a\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u6765\u628a\u6240\u6709\u66f2\u7ebf\u7684\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002 \u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\uff1b\u4f46\u76f8\u5bf9\u5730 \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u662f\u590d\u6742\u5ea6\u5360\u4e3b\u5bfc\u7684\u60c5\u51b5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/","title":"2.2. \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":""},{"location":"chapter_computational_complexity/time_complexity/#221","title":"2.2.1. \u00a0 \u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4","text":"

    \u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u60f3\u8981\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f

    1. \u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002
    2. \u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c + \u9700\u8981 1 ns\uff0c\u4e58\u6cd5\u64cd\u4f5c * \u9700\u8981 10 ns\uff0c\u6253\u5370\u64cd\u4f5c\u9700\u8981 5 ns \u7b49\u3002
    3. \u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002

    \u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e3a \\(6n + 12\\) ns \u3002

    \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nSystem.out.println(0);     // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\ncout << 0 << endl;         // 5 ns\n}\n}\n
    # \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int) -> None:\na = 2      # 1 ns\na = a + 1  # 1 ns\na = a * 2  # 10 ns\n# \u5faa\u73af n \u6b21\nfor _ in range(n):  # 1 ns\nprint(0)        # 5 ns\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\na := 2      // 1 ns\na = a + 1   // 1 ns\na = a * 2   // 10 ns\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {    // 1 ns\nfmt.Println(a)          // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\nvar a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\nvar a: number = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintf(\"%d\", 0);            // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n)\n{\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++)\n{  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nConsole.WriteLine(0);     // 5 ns\n}\n}\n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\nvar a = 2 // 1 ns\na = a + 1 // 1 ns\na = a * 2 // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // 1 ns\nprint(0) // 5 ns\n}\n}\n
    \n
    // \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprint(0); // 5 ns\n}\n}\n

    \u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#222","title":"2.2.2. \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"

    \u300c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u300d\u91c7\u53d6\u4e86\u4e00\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u5176\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002

    \u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u8f83\u4e3a\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 A , B , C \u3002

    • \u7b97\u6cd5 A \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
    • \u7b97\u6cd5 B \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u300c\u7ebf\u6027\u9636\u300d\u3002
    • \u7b97\u6cd5 C \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u4ecd\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c A \u76f8\u540c\uff0c\u4ecd\u4e3a\u300c\u5e38\u6570\u9636\u300d\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nSystem.out.println(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nSystem.out.println(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nSystem.out.println(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\ncout << 0 << endl;\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\ncout << 0 << endl;\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\ncout << 0 << endl;\n}\n}\n
    # \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int) -> None:\nprint(0)\n# \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int) -> None:\nfor _ in range(n):\nprint(0)\n# \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int) -> None:\nfor _ in range(1000000):\nprint(0)\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\nfmt.Println(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\nfor i := 0; i < n; i++ {\nfmt.Println(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\nfor i := 0; i < 1000000; i++ {\nfmt.Println(0)\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\nfor (let i = 0; i < n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\nfor (let i = 0; i < 1000000; i++) {\nconsole.log(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nprintf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i < n; i++) {\nprintf(\"%d\", 0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprintf(\"%d\", 0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n)\n{\nConsole.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n)\n{\nfor (int i = 0; i < n; i++)\n{\nConsole.WriteLine(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n)\n{\nfor (int i = 0; i < 1000000; i++)\n{\nConsole.WriteLine(0);\n}\n}\n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\nprint(0)\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\nfor _ in 0 ..< n {\nprint(0)\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\nfor _ in 0 ..< 1000000 {\nprint(0)\n}\n}\n
    \n
    // \u7b97\u6cd5 A \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\nprint(0);\n}\n// \u7b97\u6cd5 B \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\nfor (int i = 0; i < n; i++) {\nprint(0);\n}\n}\n// \u7b97\u6cd5 C \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\nfor (int i = 0; i < 1000000; i++) {\nprint(0);\n}\n}\n

    Fig. \u7b97\u6cd5 A, B, C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf

    \u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u4f18\u52bf\u548c\u5c40\u9650\u6027\u5462\uff1f

    \u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 B \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n > 1\\) \u65f6\u6bd4\u7b97\u6cd5 A \u6162\uff0c\u5728 \\(n > 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 C \u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u300c\u5e38\u6570\u9636\u300d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u300c\u7ebf\u6027\u9636\u300d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u6240\u8868\u8fbe\u7684\u542b\u4e49\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\u7684\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u7684\u6570\u91cf\u7684\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u7684\u7b80\u5316\u65b9\u6cd5\u5927\u5927\u964d\u4f4e\u4e86\u4f30\u7b97\u96be\u5ea6\u3002

    \u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 A \u548c C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 C \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 B \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 C \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#223","title":"2.2.3. \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"

    \u8bbe\u7b97\u6cd5\u7684\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0b\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a

    \\[ T(n) = 3 + 2n \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nSystem.out.println(0);    // +1\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\ncout << 0 << endl;    // +1\n}\n}\n
    def algorithm(n: int) -> None:\na = 1      # +1\na = a + 1  # +1\na = a * 2  # +1\n# \u5faa\u73af n \u6b21\nfor i in range(n):  # +1\nprint(0)        # +1\n
    func algorithm(n int) {\na := 1      // +1\na = a + 1   // +1\na = a * 2   // +1\n// \u5faa\u73af n \u6b21\nfor i := 0; i < n; i++ {   // +1\nfmt.Println(a)         // +1\n}\n}\n
    function algorithm(n) {\nvar a = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
    function algorithm(n: number): void{\nvar a: number = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i < n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintf(\"%d\", 0);            // +1\n}\n}  
    void algorithm(int n)\n{\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n{\nConsole.WriteLine(0);   // +1\n}\n}\n
    func algorithm(n: Int) {\nvar a = 1 // +1\na = a + 1 // +1\na = a * 2 // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..< n { // +1\nprint(0) // +1\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 1; // +1\na = a + 1; // +1\na = a * 2; // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i < n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprint(0); // +1\n}\n}\n

    \\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u53ef\u4ee5\u5f97\u51fa\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002

    \u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 Big-\\(O\\) Notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c Asymptotic Upper Bound\u300d\u3002

    \u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf\u51fd\u6570 \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u6765\u770b\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5b9a\u4e49\u3002

    \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c

    \u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n > n_0\\) \uff0c\u5747\u6709 $$ T(n) \\leq c \\cdot f(n) $$ \u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a $$ T(n) = O(f(n)) $$

    Fig. \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c

    \u4ece\u672c\u8d28\u4e0a\u8bb2\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#224","title":"2.2.4. \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"

    \u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u9700\u62c5\u5fc3\u3002\u56e0\u4e3a\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\uff0c\u6211\u4eec\u53ea\u9700\u8981\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u6570\u5b66\u610f\u4e49\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u3002

    \u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#_1","title":"\u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"

    \u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\uff1a

    1. \u5ffd\u7565\u4e0e \\(n\\) \u65e0\u5173\u7684\u64cd\u4f5c\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u662f \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\uff0c\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002
    2. \u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002
    3. \u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u4e0a\u8ff0 1. \u548c 2. \u6280\u5de7\u3002

    \u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u3001\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\u3002

    \\[ \\begin{aligned} T(n) & = 2n(n + 1) + (5n + 1) + 2 & \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline & = 2n^2 + 7n + 3 \\newline T(n) & = n^2 + n & \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]

    \u6700\u7ec8\uff0c\u4e24\u8005\u90fd\u80fd\u63a8\u51fa\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7ed3\u679c\uff0c\u5373 \\(O(n^2)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nSystem.out.println(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nSystem.out.println(0);\n}\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\ncout << 0 << endl;\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\ncout << 0 << endl;\n}\n}\n}\n
    def algorithm(n: int) -> None:\na = 1      # +0\uff08\u6280\u5de7 1\uff09\na = a + n  # +0\uff08\u6280\u5de7 1\uff09\n# +n\uff08\u6280\u5de7 2\uff09\nfor i in range(5 * n + 1):\nprint(0)\n# +n*n\uff08\u6280\u5de7 3\uff09\nfor i in range(2 * n):\nfor j in range(n + 1):\nprint(0)\n
    func algorithm(n int) {\na := 1     // +0\uff08\u6280\u5de7 1\uff09\na = a + n  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i := 0; i < 5 * n + 1; i++ {\nfmt.Println(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i := 0; i < 2 * n; i++ {\nfor j := 0; j < n + 1; j++ {\nfmt.Println(0)\n}\n}\n}\n
    function algorithm(n) {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
    function algorithm(n: number): void {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i < 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i < 2 * n; i++) {\nfor (let j = 0; j < n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n
    void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprintf(\"%d\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprintf(\"%d\", 0);\n}\n}\n}\n
    void algorithm(int n)\n{\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++)\n{\nConsole.WriteLine(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++)\n{\nfor (int j = 0; j < n + 1; j++)\n{\nConsole.WriteLine(0);\n}\n}\n}\n
    func algorithm(n: Int) {\nvar a = 1 // +0\uff08\u6280\u5de7 1\uff09\na = a + n // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor _ in 0 ..< (5 * n + 1) {\nprint(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor _ in 0 ..< (2 * n) {\nfor _ in 0 ..< (n + 1) {\nprint(0)\n}\n}\n}\n
    \n
    void algorithm(int n) {\nint a = 1; // +0\uff08\u6280\u5de7 1\uff09\na = a + n; // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i < 5 * n + 1; i++) {\nprint(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i < 2 * n; i++) {\nfor (int j = 0; j < n + 1; j++) {\nprint(0);\n}\n}\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#_2","title":"\u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\u7531\u591a\u9879\u5f0f \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u88ab\u5ffd\u7565\u3002

    \u4ee5\u4e0b\u8868\u683c\u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002

    \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#225","title":"2.2.5. \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5305\u62ec\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a

    \\[ \\begin{aligned} O(1) < O(\\log n) < O(n) < O(n \\log n) < O(n^2) < O(2^n) < O(n!) \\newline \\text{\u5e38\u6570\u9636} < \\text{\u5bf9\u6570\u9636} < \\text{\u7ebf\u6027\u9636} < \\text{\u7ebf\u6027\u5bf9\u6570\u9636} < \\text{\u5e73\u65b9\u9636} < \\text{\u6307\u6570\u9636} < \\text{\u9636\u4e58\u9636} \\end{aligned} \\]

    Fig. \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u89c1\u7c7b\u578b

    Tip

    \u90e8\u5206\u793a\u4f8b\u4ee3\u7801\u9700\u8981\u4e00\u4e9b\u9884\u5907\u77e5\u8bc6\uff0c\u5305\u62ec\u6570\u7ec4\u3001\u9012\u5f52\u7b97\u6cd5\u7b49\u3002\u5982\u679c\u9047\u5230\u4e0d\u7406\u89e3\u7684\u90e8\u5206\uff0c\u8bf7\u4e0d\u8981\u62c5\u5fc3\uff0c\u53ef\u4ee5\u5728\u5b66\u4e60\u5b8c\u540e\u9762\u7ae0\u8282\u540e\u518d\u56de\u987e\u3002\u73b0\u9636\u6bb5\uff0c\u8bf7\u5148\u4e13\u6ce8\u4e8e\u7406\u89e3\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u542b\u4e49\u548c\u63a8\u7b97\u65b9\u6cd5\u3002

    "},{"location":"chapter_computational_complexity/time_complexity/#o1","title":"\u5e38\u6570\u9636 \\(O(1)\\)","text":"

    \u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002

    \u5bf9\u4e8e\u4ee5\u4e0b\u7b97\u6cd5\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf size \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.py
    def constant(n: int) -> int:\n\"\"\"\u5e38\u6570\u9636\"\"\"\ncount = 0\nsize = 100000\nfor _ in range(size):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\ncount := 0\nsize := 100000\nfor i := 0; i < size; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e38\u6570\u9636 */\nfunction constant(n) {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i < size; i++) count++;\nreturn count;\n}\n
    time_complexity.c
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nint i = 0;\nfor (int i = 0; i < size; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i < size; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -> Int {\nvar count = 0\nlet size = 100_000\nfor _ in 0 ..< size {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n_ = n;\nvar count: i32 = 0;\nconst size: i32 = 100_000;\nvar i: i32 = 0;\nwhile(i<size) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (var i = 0; i < size; i++) {\ncount++;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on","title":"\u7ebf\u6027\u9636 \\(O(n)\\)","text":"

    \u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.py
    def linear(n: int) -> int:\n\"\"\"\u7ebf\u6027\u9636\"\"\"\ncount = 0\nfor _ in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\ncount := 0\nfor i := 0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u9636 */\nfunction linear(n) {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\nlet count = 0;\nfor (let i = 0; i < n; i++) count++;\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i < n; i++)\ncount++;\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -> Int {\nvar count = 0\nfor _ in 0 ..< n {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (var i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n

    \u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\u3002

    \u5982\u4f55\u786e\u5b9a\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \uff1f

    \u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u4f8b\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u76f4\u63a5\u5c06 \\(n\\) \u89c6\u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u4e0b\u9762\u904d\u5386\u6570\u7ec4\u7684\u793a\u4f8b\u4e2d\uff0c\u6570\u636e\u5927\u5c0f \\(n\\) \u4e3a\u6570\u7ec4\u7684\u957f\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector<int> &nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def array_traversal(nums: list[int]) -> int:\n\"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor num in nums:\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor range nums {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i < nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nforeach (int num in nums) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\nvar count: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (nums) |_| {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List<int> nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (var num in nums) {\ncount++;\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on2","title":"\u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"

    \u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u4e3a \\(O(n^2)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.py
    def quadratic(n: int) -> int:\n\"\"\"\u5e73\u65b9\u9636\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i in range(n):\nfor j in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i := 0; i < n; i++ {\nfor j := 0; j < n; j++ {\ncount++\n}\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -> Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0 ..< n {\nfor _ in 0 ..< n {\ncount += 1\n}\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < n) : (j += 1) {\ncount += 1;\n}\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n

    Fig. \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u3001\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u4ee5\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1, n-2, \\cdots, 2, 1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(\\frac{n}{2}\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    \\[ O((n - 1) \\frac{n}{2}) = O(n^2) \\] JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector<int> &nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.py
    def bubble_sort(nums: list[int]) -> int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\ncount = 0  # \u8ba1\u6570\u5668\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(len(nums) - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp: int = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\nreturn count\n
    time_complexity.go
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\ncount := 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp := nums[j]\nnums[j] = nums[j+1]\nnums[j+1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = n - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -> Int {\nvar count = 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0 ..< i {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\nvar count: i32 = 0;  // \u8ba1\u6570\u5668 \n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: i32 = @intCast(i32, nums.len ) - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List<int> nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (var i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (var j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#o2n","title":"\u6307\u6570\u9636 \\(O(2^n)\\)","text":"

    Note

    \u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002

    \u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u901a\u5e38\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002\u82e5\u4e00\u4e2a\u95ee\u9898\u4f7f\u7528\u300c\u66b4\u529b\u679a\u4e3e\u300d\u6c42\u89e3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u90a3\u4e48\u901a\u5e38\u9700\u8981\u4f7f\u7528\u300c\u52a8\u6001\u89c4\u5212\u300d\u6216\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u7b49\u65b9\u6cd5\u6765\u89e3\u51b3\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.py
    def exponential(n: int) -> int:\n\"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nbase = 1\n# cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in range(n):\nfor _ in range(base):\ncount += 1\nbase *= 2\n# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n
    time_complexity.go
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\ncount, base := 0, 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor i := 0; i < n; i++ {\nfor j := 0; j < base; j++ {\ncount++\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
    time_complexity.js
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\nlet count = 0,\nbase = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i < n; i++) {\nfor (let j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.c
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0;\nint bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, bas = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i < n; i++) {\nfor (int j = 0; j < bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -> Int {\nvar count = 0\nvar base = 1\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0 ..< n {\nfor _ in 0 ..< base {\ncount += 1\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n
    time_complexity.zig
    // \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\nvar count: i32 = 0;\nvar bas: i32 = 1;\nvar i: i32 = 0;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nwhile (i < n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j < bas) : (j += 1) {\ncount += 1;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n
    time_complexity.dart
    /* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// cell \u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (var i = 0; i < n; i++) {\nfor (var j = 0; j < base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n

    Fig. \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u4e0d\u65ad\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.cpp
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.py
    def exp_recur(n: int) -> int:\n\"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 1:\nreturn 1\nreturn exp_recur(n - 1) + exp_recur(n - 1) + 1\n
    time_complexity.go
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n-1) + expRecur(n-1) + 1\n}\n
    time_complexity.js
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.ts
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.c
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.cs
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.swift
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -> Int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n
    time_complexity.zig
    // \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    time_complexity.dart
    /* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#olog-n","title":"\u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"

    \u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u7684\u60c5\u51b5\u201d\u3002\u5bf9\u6570\u9636\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\uff0c\u65f6\u95f4\u589e\u957f\u7f13\u6162\uff0c\u662f\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002

    \u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u300c\u4e8c\u5206\u67e5\u627e\u300d\u548c\u300c\u5206\u6cbb\u7b97\u6cd5\u300d\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002

    \u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def logarithmic(n: float) -> int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nwhile n > 1:\nn = n / 2\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\ncount := 0\nfor n > 1 {\nn = n / 2\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\nlet count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -> Int {\nvar count = 0\nvar n = n\nwhile n > 1 {\nn = n / 2\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\nvar count: i32 = 0;\nvar n_var = n;\nwhile (n_var > 1)\n{\nn_var = n_var / 2;\ncount +=1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\nint count = 0;\nwhile (n > 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n

    Fig. \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    \u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u4e2a\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.cpp
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.py
    def log_recur(n: float) -> int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n <= 1:\nreturn 0\nreturn log_recur(n / 2) + 1\n
    time_complexity.go
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n/2) + 1\n}\n
    time_complexity.js
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.ts
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.c
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.cs
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.swift
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 0\n}\nreturn logRecur(n: n / 2) + 1\n}\n
    time_complexity.zig
    // \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    time_complexity.dart
    /* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\nif (n <= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n
    "},{"location":"chapter_computational_complexity/time_complexity/#on-log-n","title":"\u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"

    \u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002

    \u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.py
    def linear_log_recur(n: float) -> int:\n\"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\nif n <= 1:\nreturn 1\ncount: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\nfor _ in range(n):\ncount += 1\nreturn count\n
    time_complexity.go
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\nif n <= 1 {\nreturn 1\n}\ncount := linearLogRecur(n/2) +\nlinearLogRecur(n/2)\nfor i := 0.0; i < n; i++ {\ncount++\n}\nreturn count\n}\n
    time_complexity.js
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\nif (n <= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nfor (int i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -> Int {\nif n <= 1 {\nreturn 1\n}\nvar count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\nfor _ in stride(from: 0, to: n, by: 1) {\ncount += 1\n}\nreturn count\n}\n
    time_complexity.zig
    // \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\nif (n <= 1) return 1;\nvar count: i32 = linearLogRecur(n / 2) +\nlinearLogRecur(n / 2);\nvar i: f32 = 0;\nwhile (i < n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\nif (n <= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (var i = 0; i < n; i++) {\ncount++;\n}\nreturn count;\n}\n

    Fig. \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/time_complexity/#on_1","title":"\u9636\u4e58\u9636 \\(O(n!)\\)","text":"

    \u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u300c\u5168\u6392\u5217\u300d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a

    \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\cdots \\times 2 \\times 1 \\]

    \u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u7ec8\u6b62\u5206\u88c2\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart time_complexity.java
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.cpp
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.py
    def factorial_recur(n: int) -> int:\n\"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 0:\nreturn 1\ncount = 0\n# \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in range(n):\ncount += factorial_recur(n - 1)\nreturn count\n
    time_complexity.go
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\nif n == 0 {\nreturn 1\n}\ncount := 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor i := 0; i < n; i++ {\ncount += factorialRecur(n - 1)\n}\nreturn count\n}\n
    time_complexity.js
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.ts
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\nif (n == 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.c
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.cs
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.swift
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -> Int {\nif n == 0 {\nreturn 1\n}\nvar count = 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0 ..< n {\ncount += factorialRecur(n: n - 1)\n}\nreturn count\n}\n
    time_complexity.zig
    // \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\nif (n == 0) return 1;\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nwhile (i < n) : (i += 1) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n
    time_complexity.dart
    /* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (var i = 0; i < n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n

    Fig. \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6

    "},{"location":"chapter_computational_complexity/time_complexity/#226","title":"2.2.6. \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"

    \u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5176\u4e2d nums \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff1b\u7b97\u6cd5\u7684\u4efb\u52a1\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a

    • \u5f53 nums = [?, ?, ..., 1] \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\uff1b
    • \u5f53 nums = [1, ?, ?, ...] \uff0c\u5373\u5f53\u9996\u4e2a\u6570\u5b57\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u6b64\u65f6\u8fbe\u5230 \u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\)\uff1b

    \u201c\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\u201d\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002\u76f8\u5e94\u5730\uff0c\u201c\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\u201d\u7528 \\(\\Omega\\) \u8bb0\u53f7\u6765\u8868\u793a\uff0c\u4ee3\u8868\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart worst_best_time_complexity.java
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nInteger[] nums = new Integer[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nCollections.shuffle(Arrays.asList(nums));\n// Integer[] -> int[]\nint[] res = new int[n];\nfor (int i = 0; i < n; i++) {\nres[i] = nums[i];\n}\nreturn res;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.cpp
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector<int> randomNumbers(int n) {\nvector<int> nums(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\nunsigned seed = chrono::system_clock::now().time_since_epoch().count();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nshuffle(nums.begin(), nums.end(), default_random_engine(seed));\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector<int> &nums) {\nfor (int i = 0; i < nums.size(); i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.py
    def random_numbers(n: int) -> list[int]:\n\"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n# \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\nnums = [i for i in range(1, n + 1)]\n# \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrandom.shuffle(nums)\nreturn nums\ndef find_one(nums: list[int]) -> int:\n\"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\nfor i in range(len(nums)):\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1:\nreturn i\nreturn -1\n
    worst_best_time_complexity.go
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\nnums := make([]int, n)\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor i := 0; i < n; i++ {\nnums[i] = i + 1\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrand.Shuffle(len(nums), func(i, j int) {\nnums[i], nums[j] = nums[j], nums[i]\n})\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\nfor i := 0; i < len(nums); i++ {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
    worst_best_time_complexity.js
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    worst_best_time_complexity.ts
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i < n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\nfor (let i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n
    worst_best_time_complexity.c
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n// \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3an\uff0c\u5143\u7d20\u7c7b\u578b\u4e3aint\uff09\nint *nums = (int *)malloc(n * sizeof(int));\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = n - 1; i > 0; i--) {\nint j = rand() % (i + 1);\nint temp = nums[i];\nnums[i] = nums[j];\nnums[j] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\nfor (int i = 0; i < n; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.cs
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nint[] nums = new int[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = 0; i < nums.Length; i++) {\nvar index = new Random().Next(i, nums.Length);\nvar tmp = nums[i];\nvar ran = nums[index];\nnums[i] = ran;\nnums[index] = tmp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i < nums.Length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n
    worst_best_time_complexity.swift
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -> [Int] {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nvar nums = Array(1 ... n)\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle()\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -> Int {\nfor i in nums.indices {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n
    worst_best_time_complexity.zig
    // \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\npub fn randomNumbers(comptime n: usize) [n]i32 {\nvar nums: [n]i32 = undefined;\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (nums) |*num, i| {\nnum.* = @intCast(i32, i) + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nconst rand = std.crypto.random;\nrand.shuffle(i32, &nums);\nreturn nums;\n}\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\npub fn findOne(nums: []i32) i32 {\nfor (nums) |num, i| {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (num == 1) return @intCast(i32, i);\n}\nreturn -1;\n}\n
    worst_best_time_complexity.dart
    /* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList<int> randomNumbers(int n) {\nfinal nums = List.filled(n, 0);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (var i = 0; i < n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle();\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List<int> nums) {\nfor (var i = 0; i < nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1) return i;\n}\nreturn -1;\n}\n

    Tip

    \u5b9e\u9645\u5e94\u7528\u4e2d\u6211\u4eec\u5f88\u5c11\u4f7f\u7528\u300c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u300d\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u76f8\u53cd\uff0c\u300c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u201c\u6548\u7387\u5b89\u5168\u503c\u201d\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002

    \u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u6216\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u5728\u201c\u7279\u6b8a\u5206\u5e03\u7684\u6570\u636e\u201d\u4e2d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u56e0\u6b64\u5e76\u4e0d\u80fd\u6700\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u8f83\u4e4b\u4e0b\uff0c\u300c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u300d\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002

    \u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5219\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(\\frac{n}{2}\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(\\frac{n}{2}) = \\Theta(n)\\) \u3002

    \u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5c24\u5176\u662f\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u7b80\u4fbf\u5730\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002

    \u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f

    \u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u300c\u5e73\u5747\u590d\u6742\u5ea6\u300d\uff0c\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002

    "},{"location":"chapter_data_structure/","title":"3. \u00a0 \u6570\u636e\u7ed3\u6784","text":""},{"location":"chapter_data_structure/basic_data_types/","title":"3.2. \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"

    \u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002

    \u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\u3002\u5b83\u5305\u62ec\uff1a

    • \u6574\u6570\u7c7b\u578b byte , short , int , long \uff1b
    • \u6d6e\u70b9\u6570\u7c7b\u578b float , double \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\uff1b
    • \u5b57\u7b26\u7c7b\u578b char \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u3001\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\uff1b
    • \u5e03\u5c14\u7c7b\u578b bool \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\uff1b

    \u6240\u6709\u57fa\u672c\u6570\u636e\u7c7b\u578b\u90fd\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6211\u4eec\u5c06 \\(1\\) \u4e2a\u4e8c\u8fdb\u5236\u4f4d\u79f0\u4e3a \\(1\\) \u6bd4\u7279\uff0c\u5e76\u89c4\u5b9a \\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bits\uff09\u7ec4\u6210\u3002\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\uff0c\u4f8b\u5982\uff1a

    • \u6574\u6570\u7c7b\u578b byte \u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u4e0d\u540c\u7684\u6570\u5b57\uff1b
    • \u6574\u6570\u7c7b\u578b int \u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\uff1b

    \u4e0b\u8868\u5217\u4e3e\u4e86\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u9700\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002

    \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 byte 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) short 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) int 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) long 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 float 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0 f\\) double 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 char 2 bytes / 1 byte \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 bool 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\)

    Tip

    \u5b57\u7b26\u7684\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u300c\u5b57\u7b26\u7f16\u7801\u300d\u7ae0\u8282\u3002

    \u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002\u56e0\u6b64\uff0c\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u88ab\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002

    \u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u4e0e\u533a\u522b\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u5b83\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\uff0c\u800c\u975e\u201c\u6570\u636e\u201d\u3002\u5982\u679c\u60f3\u8981\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 int \u3001\u5c0f\u6570 float \u3001\u8fd8\u662f\u5b57\u7b26 char \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002

    \u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e0d\u540c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] booleans = new boolean[5];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool booleans[5];\n
    # Python \u7684 list \u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nlist = [0, 0.0, 'a', False]\n
    // \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar booleans = [5]bool{}\n
    /* JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61 */\nconst array = [0, 0.0, 'a', false];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst booleans: boolean[] = [];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool booleans[10];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] booleans = new bool[5];\n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet booleans = Array(repeating: Bool(), count: 5)\n
    \n
    /* \u4f7f\u7528\u591a\u79cd\u300c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u300d\u6765\u521d\u59cb\u5316\u300c\u6570\u7ec4\u300d */\nList<int> numbers = List.filled(5, 0);\nList<double> decimals = List.filled(5, 0.0);\nList<String> characters = List.filled(5, 'a');\nList<bool> booleans = List.filled(5, false);\n
    "},{"location":"chapter_data_structure/character_encoding/","title":"3.4. \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 char \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u300c\u5b57\u7b26\u96c6\u300d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002

    "},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1. \u00a0 ASCII \u5b57\u7b26\u96c6","text":"

    \u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5168\u79f0\u4e3a\u201c\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\u201d\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u5373\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u8fd9\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0-9 \u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002

    Fig. ASCII \u7801

    \u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002

    "},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2. \u00a0 GBK \u5b57\u7b26\u96c6","text":"

    \u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u4f8b\u5982\uff0c\u6c49\u5b57\u5927\u7ea6\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002

    \u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7684\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u4e4b\u540e\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\uff0c\u6269\u5c55\u5f97\u5230\u4e86\u300cGBK\u300d\u5b57\u7b26\u96c6\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002

    "},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3. \u00a0 Unicode \u5b57\u7b26\u96c6","text":"

    \u968f\u7740\u8ba1\u7b97\u673a\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u4e5f\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u7535\u8111\u5b89\u88c5\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002

    \u90a3\u4e2a\u65f6\u4ee3\u7684\u4eba\u4eec\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002

    \u300cUnicode\u300d\u7684\u5168\u79f0\u4e3a\u201c\u7edf\u4e00\u5b57\u7b26\u7f16\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3\u4e00\u767e\u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u5230\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002

    \u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u6b62 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u3001\u751a\u81f3\u662f\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002

    Unicode \u662f\u4e00\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u540c\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f

    \u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \uff0c\u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u51fa\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002

    Fig. Unicode \u7f16\u7801\u793a\u4f8b

    \u7136\u800c\uff0cASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700\u8981 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u5927\u5c0f\u7684 2 \u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002

    "},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4. \u00a0 UTF-8 \u7f16\u7801","text":"

    \u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u4e2a\u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700\u8981 1 \u4e2a\u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u4e2a\u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u4e2a\u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u4e2a\u5b57\u8282\u3002

    UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\uff1a

    • \u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \u3001\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002
    • \u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n > 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \u3001\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002

    \u4e0b\u56fe\u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u5c06\u6700\u9ad8 \\(n\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(1\\) \u6bd4\u8f83\u5bb9\u6613\u7406\u89e3\uff0c\u53ef\u4ee5\u5411\u7cfb\u7edf\u6307\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002\u90a3\u4e48\uff0c\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\uff0c\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u662f\u56e0\u4e3a\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\u7684\u6700\u9ad8\u4e00\u4f4d\u662f \\(0\\) \u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u7684\u5224\u65ad\u51fa\u5f02\u5e38\u3002

    Fig. UTF-8 \u7f16\u7801\u793a\u4f8b

    \u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec UTF-16 \u548c UTF-32 \u3002\u5b83\u4eec\u4e3a Unicode \u5b57\u7b26\u96c6\u63d0\u4f9b\u4e86\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6cd5\u3002

    • UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u4e2a\u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5f88\u591a\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u4e2a\u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u4e2a\u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002
    • UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u4e2a\u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u4f1a\u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u4e3b\u8981\u4f7f\u7528 ASCII \u5b57\u7b26\u7684\u6587\u672c\u3002

    \u4ece\u5b58\u50a8\u7a7a\u95f4\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u4e2a\u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u53ea\u9700\u8981 2 \u4e2a\u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u4e2a\u5b57\u8282\u3002\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u90fd\u4f18\u5148\u652f\u6301 UTF-8 \u3002

    "},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5. \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"

    \u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u5177\u4f53\u6765\u8bf4\uff1a

    • \u968f\u673a\u8bbf\u95ee: UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002
    • \u5b57\u7b26\u8ba1\u6570: \u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002
    • \u5b57\u7b26\u4e32\u64cd\u4f5c: \u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u90fd\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002

    \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u5230\u8bb8\u591a\u56e0\u7d20\uff1a

    • Java \u7684 String \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002
    • JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 JavaScript \u8bed\u8a00\u5728 1995 \u5e74\u88ab Netscape \u516c\u53f8\u9996\u6b21\u5f15\u5165\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u76f8\u5bf9\u65e9\u671f\u7684\u9636\u6bb5\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u591f\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002
    • C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff0c\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff0c\u90fd\u5e7f\u6cdb\u5730\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002

    \u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u56e0\u6b64\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u589e\u52a0\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c Debug \u96be\u5ea6\u3002

    \u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\uff1a

    • Python 3 \u4f7f\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u5bf9\u4e8e\u5168\u90e8\u662f ASCII \u5b57\u7b26\u7684\u5b57\u7b26\u4e32\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u5b57\u7b26\u4e32\u4e2d\u5305\u542b\u7684\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u5b57\u7b26\u4e32\u4e2d\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u90a3\u4e48\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u4e2a\u5b57\u8282\u3002
    • Go \u8bed\u8a00\u7684 string \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 rune \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002
    • Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 char \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002

    "},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1. \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"

    \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002

    "},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1. \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"

    \u300c\u903b\u8f91\u7ed3\u6784\u300d\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002

    \u903b\u8f91\u7ed3\u6784\u901a\u5e38\u5206\u4e3a\u300c\u7ebf\u6027\u300d\u548c\u300c\u975e\u7ebf\u6027\u300d\u4e24\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\uff0c\u4f8b\u5982\u7f51\u72b6\u6216\u6811\u72b6\u7ed3\u6784\u3002

    • \u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\uff1b
    • \u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u56fe\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff1b

    Fig. \u7ebf\u6027\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784

    "},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2. \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u79bb\u6563","text":"

    \u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u5185\u5b58\u548c\u786c\u76d8\u662f\u4e24\u79cd\u4e3b\u8981\u7684\u5b58\u50a8\u786c\u4ef6\u8bbe\u5907\u3002\u300c\u786c\u76d8\u300d\u4e3b\u8981\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5bb9\u91cf\u8f83\u5927\uff08\u901a\u5e38\u53ef\u8fbe\u5230 TB \u7ea7\u522b\uff09\u3001\u901f\u5ea6\u8f83\u6162\u3002\u300c\u5185\u5b58\u300d\u7528\u4e8e\u8fd0\u884c\u7a0b\u5e8f\u65f6\u6682\u5b58\u6570\u636e\uff0c\u901f\u5ea6\u8f83\u5feb\uff0c\u4f46\u5bb9\u91cf\u8f83\u5c0f\uff08\u901a\u5e38\u4e3a GB \u7ea7\u522b\uff09\u3002

    \u5728\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u5173\u6570\u636e\u90fd\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u4e0b\u56fe\u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8 1 byte \u7684\u6570\u636e\uff0c\u5728\u7b97\u6cd5\u8fd0\u884c\u65f6\uff0c\u6240\u6709\u6570\u636e\u90fd\u88ab\u5b58\u50a8\u5728\u8fd9\u4e9b\u5355\u5143\u683c\u4e2d\u3002

    \u7cfb\u7edf\u901a\u8fc7\u300c\u5185\u5b58\u5730\u5740 Memory Location\u300d\u6765\u8bbf\u95ee\u76ee\u6807\u5185\u5b58\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002

    Fig. \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740

    \u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5176\u4ed6\u7a0b\u5e8f\u65e0\u6cd5\u540c\u65f6\u4f7f\u7528\u3002\u56e0\u6b64\uff0c\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u8fd0\u884c\u7684\u7a0b\u5e8f\u5f88\u591a\u5e76\u4e14\u7f3a\u5c11\u5927\u91cf\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u79bb\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002

    \u300c\u7269\u7406\u7ed3\u6784\u300d\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u6570\u7ec4\u7684\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u94fe\u8868\u7684\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u540c\u65f6\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002

    Fig. \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8

    \u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002

    • \u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\uff1b
    • \u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\uff1b

    \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u300c\u9759\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u88ab\u79f0\u4e3a\u300c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u300d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002

    Tip

    \u5982\u82e5\u611f\u89c9\u7406\u89e3\u7269\u7406\u7ed3\u6784\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\u201c\u6570\u7ec4\u4e0e\u94fe\u8868\u201d\uff0c\u7136\u540e\u518d\u56de\u5934\u7406\u89e3\u7269\u7406\u7ed3\u6784\u7684\u542b\u4e49\u3002\u6570\u7ec4\u4e0e\u94fe\u8868\u662f\u5176\u4ed6\u6240\u6709\u6570\u636e\u7ed3\u6784\u7684\u57fa\u77f3\uff0c\u5efa\u8bae\u4f60\u6295\u5165\u66f4\u591a\u65f6\u95f4\u6df1\u5165\u4e86\u89e3\u8fd9\u4e24\u79cd\u57fa\u672c\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_data_structure/number_encoding/","title":"3.3. \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"

    Note

    \u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709\u7684 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002

    "},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1. \u00a0 \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801","text":"

    \u4ece\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\u3002\u4f8b\u5982\uff0cbyte \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u5230\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002\u5728\u5c55\u5f00\u5206\u6790\u4e4b\u524d\uff0c\u6211\u4eec\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\uff1a

    • \u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002
    • \u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002
    • \u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002

    Fig. \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362

    \u663e\u7136\uff0c\u300c\u539f\u7801\u300d\u6700\u4e3a\u76f4\u89c2\uff0c\u7136\u800c\u6570\u5b57\u5374\u662f\u4ee5\u300c\u8865\u7801\u300d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u539f\u7801\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002

    \u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002

    \\[ \\begin{aligned} & 1 + (-2) \\newline & = 0000 \\space 0001 + 1000 \\space 0010 \\newline & = 1000 \\space 0011 \\newline & = -3 \\end{aligned} \\]

    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801\u300d\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5e76\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u5316\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002

    \\[ \\begin{aligned} & 1 + (-2) \\newline & = 0000 \\space 0001 \\space \\text{(\u539f\u7801)} + 1000 \\space 0010 \\space \\text{(\u539f\u7801)} \\newline & = 0000 \\space 0001 \\space \\text{(\u53cd\u7801)} + 1111 \\space 1101 \\space \\text{(\u53cd\u7801)} \\newline & = 1111 \\space 1110 \\space \\text{(\u53cd\u7801)} \\newline & = 1000 \\space 0001 \\space \\text{(\u539f\u7801)} \\newline & = -1 \\end{aligned} \\]

    \u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u7740\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u800c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u53ef\u80fd\u4f1a\u5bfc\u81f4\u9519\u8bef\u7684\u5224\u65ad\u7ed3\u679c\u3002\u5982\u679c\u6211\u4eec\u60f3\u8981\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u5176\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002

    \\[ \\begin{aligned} +0 & = 0000 \\space 0000 \\newline -0 & = 1000 \\space 0000 \\end{aligned} \\]

    \u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\u3002\u4e3a\u6b64\uff0c\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801\u300d\u3002\u90a3\u4e48\uff0c\u8865\u7801\u6709\u4ec0\u4e48\u4f5c\u7528\u5462\uff1f\u6211\u4eec\u5148\u6765\u5206\u6790\u4e00\u4e0b\u8d1f\u96f6\u7684\u8865\u7801\u7684\u8ba1\u7b97\u8fc7\u7a0b\uff1a

    \\[ \\begin{aligned} -0 = \\space & 1000 \\space 0000 \\space \\text{(\u539f\u7801)} \\newline = \\space & 1111 \\space 1111 \\space \\text{(\u53cd\u7801)} \\newline = 1 \\space & 0000 \\space 0000 \\space \\text{(\u8865\u7801)} \\newline \\end{aligned} \\]

    \u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u800c\u7531\u4e8e byte \u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4ece\u800c\u5f97\u5230\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\space 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u4ece\u800c\u89e3\u51b3\u4e86\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\u3002

    \u8fd8\u5269\u4f59\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1abyte \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u662f\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u7684\u3002

    \u7136\u800c\uff0c\u8865\u7801 \\(1000 \\space 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\space 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\space 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002

    \\[ \\begin{aligned} & (-127) + (-1) \\newline & = 1111 \\space 1111 \\space \\text{(\u539f\u7801)} + 1000 \\space 0001 \\space \\text{(\u539f\u7801)} \\newline & = 1000 \\space 0000 \\space \\text{(\u53cd\u7801)} + 1111 \\space 1110 \\space \\text{(\u53cd\u7801)} \\newline & = 1000 \\space 0001 \\space \\text{(\u8865\u7801)} + 1111 \\space 1111 \\space \\text{(\u8865\u7801)} \\newline & = 1000 \\space 0000 \\space \\text{(\u8865\u7801)} \\newline & = -128 \\end{aligned} \\]

    \u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\uff0c\u4e0a\u8ff0\u7684\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u4ece\u800c\u63d0\u9ad8\u8fd0\u7b97\u901f\u5ea6\u3002

    \u7136\u800c\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002

    \u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u9700\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u5e76\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002

    \u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u7531\u4e8e\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\u3002\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5ea6\u4e86\u89e3\u3002

    "},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2. \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"

    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1aint \u548c float \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes\uff0c\u4f46\u4e3a\u4ec0\u4e48 float \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 float \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 float \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 float \u7531\u4ee5\u4e0b\u90e8\u5206\u6784\u6210\uff1a

    • \u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff1b
    • \u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff1b
    • \u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 24 bits \uff0c\u5176\u4e2d 23 \u4f4d\u663e\u5f0f\u5b58\u50a8\uff1b

    \u8bbe 32-bit \u4e8c\u8fdb\u5236\u6570\u7684\u7b2c \\(i\\) \u4f4d\u4e3a \\(b_i\\) \uff0c\u5219 float \u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u5b9a\u4e49\u4e3a\uff1a

    \\[ \\text { val } = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\]

    \u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a

    \\[ \\text { val }=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\]

    \u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a

    \\[ \\begin{aligned} \\mathrm{S} \\in & \\{ 0, 1\\} , \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = & (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\]

    Fig. IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u8868\u793a\u65b9\u5f0f

    \u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u6613\u5f97

    \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\]

    \u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1afloat \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e int \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0cfloat \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002

    \u5c3d\u7ba1\u6d6e\u70b9\u6570 float \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b int \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 float \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002

    \u8fdb\u4e00\u6b65\u5730\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002

    \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\)

    \u7279\u522b\u5730\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\uff0c\u8fd9\u662f\u56e0\u4e3a\uff1a

    • \u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\approx 1.18 \\times 10^{-38}\\) \uff1b
    • \u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23} \\approx 1.4 \\times 10^{-45}\\) \uff1b

    \u53cc\u7cbe\u5ea6 double \u4e5f\u91c7\u7528\u7c7b\u4f3c float \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u6b64\u5904\u4e0d\u518d\u8be6\u8ff0\u3002

    "},{"location":"chapter_data_structure/summary/","title":"3.5. \u00a0 \u5c0f\u7ed3","text":"

    \u6570\u636e\u7ed3\u6784\u5206\u7c7b

    • \u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002
    • \u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u548c\u975e\u7ebf\u6027\u7ed3\u6784\u3002
    • \u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002
    • \u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u79bb\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002

    \u6570\u636e\u7c7b\u578b\u4e0e\u7f16\u7801

    • \u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 byte, short, int, long \u3001\u6d6e\u70b9\u6570 float, double \u3001\u5b57\u7b26 char \u548c\u5e03\u5c14 boolean \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002
    • \u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u662f\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u7684\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002
    • \u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002
    • \u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002
    • ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002
    • UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java, C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002
    "},{"location":"chapter_data_structure/summary/#351-q-a","title":"3.5.1. \u00a0 Q & A","text":"

    \u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f

    \u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u62c9\u94fe\u6cd5\u201d\uff08\u540e\u7eed\u6563\u5217\u8868\u7ae0\u8282\u4f1a\u8bb2\uff09\u3002\u5728\u62c9\u94fe\u6cd5\u4e2d\uff0c\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5730\u5740\uff08\u6876\uff09\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff1b\u5f53\u8fd9\u4e2a\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\uff09\u6570\u636e\u7ed3\u6784\u3002

    char \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 bytes \u5417\uff1f

    \u8fd9\u4e2a\u4e0e\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u6709\u5173\u3002\u4f8b\u5982\uff0cJava, JS, TS, C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes \u3002

    "},{"location":"chapter_dynamic_programming/","title":"13. \u00a0 \u52a8\u6001\u89c4\u5212","text":""},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"13.2. \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"

    \u5728\u4e0a\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u66b4\u529b\u89e3\u6cd5\uff0c\u4ece\u9012\u5f52\u6811\u4e2d\u89c2\u5bdf\u5230\u6d77\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u4e86\u89e3\u5230\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u8bb0\u5f55\u89e3\u6765\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u3002

    \u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u6700\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u65b9\u6848\u95ee\u9898\uff0c\u4f8b\u5982\u5bfb\u627e\u6700\u77ed\u8def\u5f84\u3001\u6700\u5927\u5229\u6da6\u3001\u6700\u5c11\u65f6\u95f4\u7b49\u3002\u8fd9\u7c7b\u95ee\u9898\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002

    "},{"location":"chapter_dynamic_programming/dp_problem_features/#1321","title":"13.2.1. \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"

    \u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002

    \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7

    \u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\u8d77\u59cb\u70b9\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u82e5\u7b2c \\(1\\) , \\(2\\) , \\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\) , \\(10\\) , \\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002

    Fig. \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7

    \u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff0c\u5373\uff1a

    \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\]

    \u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002\u5bf9\u4e8e\u672c\u9898\uff0c\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) , \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002

    \u76f8\u8f83\u4e8e\u5206\u6cbb\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u89e3\u4e5f\u662f\u7531\u5176\u5b50\u95ee\u9898\u7684\u89e3\u6784\u6210\u7684\u3002\u4e0d\u540c\u7684\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u5b50\u95ee\u9898\u7684\u89e3\u4e0d\u4ec5\u63ed\u793a\u4e86\u95ee\u9898\u7684\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u800c\u4e14\u8fd8\u901a\u8fc7\u7279\u5b9a\u7684\u9012\u63a8\u5173\u7cfb\u94fe\u63a5\u8d77\u6765\uff0c\u5171\u540c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u5168\u5c40\u6700\u4f18\u89e3\u3002

    \u90a3\u4e48\uff0c\u4e0a\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u8981\u6c42\u89e3\u7684\u662f\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u662f\u4e00\u4e2a\u6bd4\u8f83\u5bbd\u6cdb\u7684\u6982\u5ff5\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002

    \u6839\u636e\u4ee5\u4e0a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) , \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u52a8\u6001\u89c4\u5212\u89e3\u9898\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart min_cost_climbing_stairs_dp.java
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector<int> &cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<int> dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.py
    def min_cost_climbing_stairs_dp(cost: list[int]) -> int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = cost[1], cost[2]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\nreturn dp[n]\n
    min_cost_climbing_stairs_dp.go
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.js
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.ts
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.c
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n
    min_cost_climbing_stairs_dp.swift
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.zig
    [class]{}-[func]{minCostClimbingStairsDP}\n
    min_cost_climbing_stairs_dp.dart
    [class]{}-[func]{minCostClimbingStairsDP}\n

    Fig. \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b

    \u8fd9\u9053\u9898\u540c\u6837\u4e5f\u53ef\u4ee5\u8fdb\u884c\u72b6\u6001\u538b\u7f29\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart min_cost_climbing_stairs_dp.java
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = Math.min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector<int> &cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.py
    def min_cost_climbing_stairs_dp_comp(cost: list[int]) -> int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\na, b = cost[1], cost[2]\nfor i in range(3, n + 1):\na, b = b, min(a, b) + cost[i]\nreturn b\n
    min_cost_climbing_stairs_dp.go
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.js
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.ts
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.c
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = Math.Min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n
    min_cost_climbing_stairs_dp.swift
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.zig
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    min_cost_climbing_stairs_dp.dart
    [class]{}-[func]{minCostClimbingStairsDPComp}\n
    "},{"location":"chapter_dynamic_programming/dp_problem_features/#1322","title":"13.2.2. \u00a0 \u65e0\u540e\u6548\u6027","text":"

    \u300c\u65e0\u540e\u6548\u6027\u300d\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u5f53\u524d\u72b6\u6001\u8fc7\u53bb\u6240\u7ecf\u5386\u8fc7\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002

    \u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u9700\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5373\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002

    \u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5411\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002

    \u5e26\u7ea6\u675f\u722c\u697c\u68af

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u4f8b\u5982\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002

    Fig. \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf

    \u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002

    \u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002

    \u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u3001\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u6765\u51b3\u5b9a\u4e0b\u4e00\u6b65\u8be5\u600e\u4e48\u8df3\uff1a

    • \u5f53 \\(j\\) \u7b49\u4e8e \\(1\\) \uff0c\u5373\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u8fd9\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff1b
    • \u5f53 \\(j\\) \u7b49\u4e8e \\(2\\) \uff0c\u5373\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u8fd9\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff1b

    \u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u7531\u6b64\uff0c\u6211\u4eec\u4fbf\u80fd\u63a8\u5bfc\u51fa\u4ee5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a

    \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\]

    Fig. \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb

    \u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_constraint_dp.java
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[][] dp = new int[n + 1][3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n
    climbing_stairs_constraint_dp.cpp
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<vector<int>> dp(n + 1, vector<int>(3, 0));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n
    climbing_stairs_constraint_dp.py
    def climbing_stairs_constraint_dp(n: int) -> int:\n\"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [[0] * 3 for _ in range(n + 1)]\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1], dp[1][2] = 1, 0\ndp[2][1], dp[2][2] = 0, 1\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i][1] = dp[i - 1][2]\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2]\nreturn dp[n][1] + dp[n][2]\n
    climbing_stairs_constraint_dp.go
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.js
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.ts
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.c
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.cs
    /* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn n;\n}\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[,] dp = new int[n + 1, 3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1, 1] = 1;\ndp[1, 2] = 0;\ndp[2, 1] = 0;\ndp[2, 2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i, 1] = dp[i - 1, 2];\ndp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n}\nreturn dp[n, 1] + dp[n, 2];\n}\n
    climbing_stairs_constraint_dp.swift
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.zig
    [class]{}-[func]{climbingStairsConstraintDP}\n
    climbing_stairs_constraint_dp.dart
    [class]{}-[func]{climbingStairsConstraintDP}\n

    \u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u6062\u590d\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u8bb8\u591a\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\uff0c\u4f8b\u5982\uff1a

    \u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u7ed9\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2, 3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4, 6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u4e8e\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\uff0c\u6216\u662f\u56e0\u4e3a\u8ba1\u7b97\u590d\u6742\u5ea6\u8fc7\u9ad8\u800c\u96be\u4ee5\u5e94\u7528\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u8457\u540d\u7684\u65c5\u884c\u5546\u95ee\u9898\uff09\u90fd\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u80fd\u591f\u63a5\u53d7\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"13.1. \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"

    \u300c\u52a8\u6001\u89c4\u5212 Dynamic Programming\u300d\u662f\u4e00\u79cd\u7528\u4e8e\u89e3\u51b3\u590d\u6742\u95ee\u9898\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u5b83\u628a\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u628a\u5b50\u95ee\u9898\u7684\u89e3\u5b58\u50a8\u8d77\u6765\u4ee5\u4f9b\u540e\u7eed\u4f7f\u7528\uff0c\u4ece\u800c\u907f\u514d\u4e86\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u5347\u4e86\u89e3\u9898\u6548\u7387\u3002

    \u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u4ece\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u7684\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u4e00\u6b65\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002

    \u722c\u697c\u68af

    \u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002

    Fig. \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf

    \u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_backtrack.java
    /* \u56de\u6eaf */\nvoid backtrack(List<Integer> choices, int state, int n, List<Integer> res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres.set(0, res.get(0) + 1);\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Integer choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList<Integer> choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList<Integer> res = new ArrayList<>();\nres.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res.get(0);\n}\n
    climbing_stairs_backtrack.cpp
    /* \u56de\u6eaf */\nvoid backtrack(vector<int> &choices, int state, int n, vector<int> &res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (auto &choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nvector<int> choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nvector<int> res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n
    climbing_stairs_backtrack.py
    def backtrack(choices: list[int], state: int, n: int, res: list[int]) -> int:\n\"\"\"\u56de\u6eaf\"\"\"\n# \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n:\nres[0] += 1\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state + choice > n:\nbreak\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res)\n# \u56de\u9000\ndef climbing_stairs_backtrack(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\nchoices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nstate = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nres = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res)\nreturn res[0]\n
    climbing_stairs_backtrack.go
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.js
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.ts
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.c
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.cs
    /* \u56de\u6eaf */\nvoid backtrack(List<int> choices, int state, int n, List<int> res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (int choice in choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice > n)\nbreak;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList<int> choices = new List<int> { 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList<int> res = new List<int> { 0 }; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n
    climbing_stairs_backtrack.swift
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.zig
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    climbing_stairs_backtrack.dart
    [class]{}-[func]{backtrack}\n[class]{}-[func]{climbingStairsBacktrack}\n
    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1311","title":"13.1.1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"

    \u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002

    \u5bf9\u4e8e\u672c\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u5c06\u95ee\u9898\u62c6\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec:

    \\[ dp[i-1] , dp[i-2] , \\cdots , dp[2] , dp[1] \\]

    \u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\uff0c\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u524d\u5f80\u7b2c \\(i\\) \u9636\u3002\u56e0\u6b64\uff0c\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\uff0c\u5373\uff1a

    \\[ dp[i] = dp[i-1] + dp[i-2] \\]

    Fig. \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb

    \u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u6210\u3002

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u6b64\u9012\u63a8\u516c\u5f0f\u5199\u51fa\u66b4\u529b\u641c\u7d22\u4ee3\u7801\uff1a\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373\u722c\u5230\u7b2c \\(1\\) , \\(2\\) \u9636\u5206\u522b\u6709 \\(1\\) , \\(2\\) \u79cd\u65b9\u6848\u3002

    \u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u4e0e\u56de\u6eaf\u89e3\u6cd5\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u6bd4\u56de\u6eaf\u7b97\u6cd5\u66f4\u52a0\u7b80\u6d01\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dfs.java
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.cpp
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.py
    def dfs(i: int) -> int:\n\"\"\"\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1) + dfs(i - 2)\nreturn count\ndef climbing_stairs_dfs(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\nreturn dfs(n)\n
    climbing_stairs_dfs.go
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.js
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.ts
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.cs
    /* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n
    climbing_stairs_dfs.swift
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n
    climbing_stairs_dfs.dart
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFS}\n

    \u4e0b\u56fe\u5c55\u793a\u4e86\u8be5\u65b9\u6cd5\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u7684\u8fd0\u884c\u65f6\u95f4\u589e\u957f\u5730\u975e\u5e38\u5feb\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002

    Fig. \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811

    \u5b9e\u9645\u4e0a\uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7531\u4e8e\u300c\u91cd\u53e0\u5b50\u95ee\u9898\u300d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982\uff0c\u95ee\u9898 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\u95ee\u9898 \\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \uff0c\u800c\u5b50\u95ee\u9898\u4e2d\u53c8\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\uff0c\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1312","title":"13.1.2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"

    \u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4 mem \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u8fd9\u6837\u505a\uff1a

    • \u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 mem[i] \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\uff1b
    • \u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece mem[i] \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\uff1b
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dfs_mem.java
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArrays.fill(mem, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.cpp
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector<int> &mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nvector<int> mem(n + 1, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.py
    def dfs(i: int, mem: list[int]) -> int:\n\"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1:\nreturn mem[i]\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1, mem) + dfs(i - 2, mem)\n# \u8bb0\u5f55 dp[i]\nmem[i] = count\nreturn count\ndef climbing_stairs_dfs_mem(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nmem = [-1] * (n + 1)\nreturn dfs(n, mem)\n
    climbing_stairs_dfs_mem.go
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.js
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.ts
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.cs
    /* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArray.Fill(mem, -1);\nreturn dfs(n, mem);\n}\n
    climbing_stairs_dfs_mem.swift
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n
    climbing_stairs_dfs_mem.dart
    [class]{}-[func]{dfs}\n[class]{}-[func]{climbingStairsDFSMem}\n

    \u89c2\u5bdf\u4e0b\u56fe\uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002\u5b9e\u9645\u4e0a\uff0c\u5982\u679c\u4e0d\u8003\u8651\u9012\u5f52\u5e26\u6765\u7684\u989d\u5916\u5f00\u9500\uff0c\u8bb0\u5fc6\u5316\u641c\u7d22\u89e3\u6cd5\u5df2\u7ecf\u51e0\u4e4e\u7b49\u540c\u4e8e\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u3002

    Fig. \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811

    "},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1313","title":"13.1.3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"

    \u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\uff1b\u6700\u7ec8\u901a\u8fc7\u56de\u6eaf\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u9010\u5c42\u6536\u96c6\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002

    \u6211\u4eec\u4e5f\u53ef\u4ee5\u76f4\u63a5\u201c\u4ece\u5e95\u81f3\u9876\u201d\u8fdb\u884c\u6c42\u89e3\uff0c\u5f97\u5230\u6807\u51c6\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002

    \u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u65e0\u9700\u4f7f\u7528\u9012\u5f52\uff0c\u800c\u53ef\u4ee5\u76f4\u63a5\u57fa\u4e8e\u9012\u63a8\u5b9e\u73b0\u3002\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 dp \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6570\u7ec4 dp \u8d77\u5230\u4e86\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 mem \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dp.java
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector<int> dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.py
    def climbing_stairs_dp(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\n# \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = 1, 2\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = dp[i - 1] + dp[i - 2]\nreturn dp[n]\n
    climbing_stairs_dp.go
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.js
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.ts
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.c
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u5217\u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i <= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n
    climbing_stairs_dp.swift
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.zig
    [class]{}-[func]{climbingStairsDP}\n
    climbing_stairs_dp.dart
    [class]{}-[func]{climbingStairsDP}\n

    \u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u67d0\u4e2a\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\u5bf9\u4e8e\u722c\u697c\u68af\u95ee\u9898\uff0c\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\u3002\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u5305\u62ec\uff1a

    • \u5c06 \\(dp\\) \u6570\u7ec4\u79f0\u4e3a\u300c\u72b6\u6001\u5217\u8868\u300d\uff0c\\(dp[i]\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u72b6\u6001\u7684\u89e3\uff1b
    • \u5c06\u6700\u7b80\u5355\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u5373\u7b2c \\(1\\) , \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\uff1b
    • \u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\uff1b

    Fig. \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b

    \u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u9700\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 dp \u6765\u5b58\u50a8\u6240\u6709\u72b6\u6001\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 dp \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart climbing_stairs_dp.java
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.cpp
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.py
    def climbing_stairs_dp_comp(n: int) -> int:\n\"\"\"\u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\na, b = 1, 2\nfor _ in range(3, n + 1):\na, b = b, a + b\nreturn b\n
    climbing_stairs_dp.go
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.js
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.ts
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.c
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.cs
    /* \u722c\u697c\u68af\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i <= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n
    climbing_stairs_dp.swift
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.zig
    [class]{}-[func]{climbingStairsDPComp}\n
    climbing_stairs_dp.dart
    [class]{}-[func]{climbingStairsDPComp}\n

    \u6211\u4eec\u5c06\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u79f0\u4e3a\u300c\u72b6\u6001\u538b\u7f29\u300d\u3002\u5728\u8bb8\u591a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u4e0d\u5fc5\u4fdd\u5b58\u6240\u6709\u7684\u5386\u53f2\u72b6\u6001\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u72b6\u6001\u538b\u7f29\uff0c\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002

    \u603b\u7684\u770b\u6765\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u7b97\u6cd5\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5404\u6709\u7279\u70b9\uff1a

    • \u5206\u6cbb\u7b97\u6cd5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u51e0\u4e2a\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u7136\u540e\u9012\u5f52\u89e3\u51b3\u5b50\u95ee\u9898\uff0c\u6700\u540e\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002\u4f8b\u5982\uff0c\u5f52\u5e76\u6392\u5e8f\u5c06\u957f\u6570\u7ec4\u4e0d\u65ad\u5212\u5206\u4e3a\u4e24\u4e2a\u77ed\u5b50\u6570\u7ec4\uff0c\u518d\u5c06\u6392\u5e8f\u597d\u7684\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u6392\u5e8f\u597d\u7684\u957f\u6570\u7ec4\u3002
    • \u52a8\u6001\u89c4\u5212\u4e5f\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u5f80\u5f80\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u539f\u95ee\u9898\u7684\u89e3\u4f9d\u8d56\u4e8e\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u5b50\u95ee\u9898\u7684\u89e3\u53c8\u4f9d\u8d56\u4e8e\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\u7684\u89e3\u3002\u56e0\u6b64\uff0c\u52a8\u6001\u89c4\u5212\u901a\u5e38\u4f1a\u5f15\u5165\u8bb0\u5fc6\u5316\uff0c\u4fdd\u5b58\u5df2\u7ecf\u89e3\u51b3\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002
    • \u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u540e\u7684\u5269\u4f59\u95ee\u9898\u770b\u4f5c\u4e3a\u4e00\u4e2a\u5b50\u95ee\u9898\u3002
    "},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"13.3. \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"

    \u80cc\u5305\u95ee\u9898\u662f\u5b66\u4e60\u52a8\u6001\u89c4\u5212\u7684\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u5165\u95e8\u9898\u76ee\uff0c\u5176\u6d89\u53ca\u5230\u201c\u9009\u62e9\u4e0e\u4e0d\u9009\u62e9\u201d\u548c\u201c\u9650\u5236\u6761\u4ef6\u4e0b\u7684\u6700\u4f18\u5316\u201d\u7b49\u95ee\u9898\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002

    \u80cc\u5305\u95ee\u9898\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u5b66\u4e60\u6700\u7b80\u5355\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002

    Question

    \u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\) \u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u73b0\u5728\u6709\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\uff0c\u8bf7\u6c42\u89e3\u5728\u4e0d\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002

    \u8bf7\u6ce8\u610f\uff0c\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u4f46\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002

    \u4e0b\u56fe\u7ed9\u51fa\u4e86\u4e00\u4e2a 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e\uff0c\u80cc\u5305\u5185\u7684\u6700\u5927\u4ef7\u503c\u4e3a \\(220\\) \u3002

    Fig. 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e

    \u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u4ecd\u7136\u5148\u4ece\u56de\u6eaf\u89d2\u5ea6\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\uff1b\u518d\u5f15\u5165\u8bb0\u5fc6\u5316\u5904\u7406\uff0c\u5f97\u5230\u8bb0\u5fc6\u5316\u641c\u7d22\u548c\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002

    "},{"location":"chapter_dynamic_programming/knapsack_problem/#1331","title":"13.3.1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"

    0-1 \u80cc\u5305\u95ee\u9898\u662f\u4e00\u9053\u5178\u578b\u7684\u201c\u9009\u6216\u4e0d\u9009\u201d\u7684\u95ee\u9898\uff0c0 \u4ee3\u8868\u4e0d\u9009\u30011 \u4ee3\u8868\u9009\u3002\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u770b\u4f5c\u662f\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u641c\u7d22\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\u3002\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\uff1a

    • \u72b6\u6001\u5305\u62ec\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u80cc\u5305\u5bb9\u91cf \\(c\\)\uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002
    • \u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u201c\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5bb9\u91cf\u4e3a \\(c\\) \u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, c]\\) \u3002

    \u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u5b50\u95ee\u9898\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u5206\u4e3a\u4e24\u79cd\uff1a

    • \u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \uff1b
    • \u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u8f6c\u79fb\u81f3 \\([i-1, c-wgt[i-1]]\\) \uff1b

    \u4e0a\u8ff0\u7684\u72b6\u6001\u8f6c\u79fb\u5411\u6211\u4eec\u5c55\u793a\u4e86\u672c\u9898\u7684\u300c\u6700\u4f18\u5b50\u7ed3\u6784\u300d\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u7684\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a

    \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\]

    \u4ee5\u4e0b\u662f\u66b4\u529b\u641c\u7d22\u7684\u5b9e\u73b0\u4ee3\u7801\uff0c\u5176\u4e2d\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\uff1a

    • \u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \uff1b\u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002
    • \u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u5df2\u5b8c\u6210 \\(n\\) \u8f6e\u51b3\u7b56\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\u4e3a\u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002
    • \u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart knapsack.java
    [class]{knapsack}-[func]{knapsackDFS}\n
    knapsack.cpp
    [class]{}-[func]{knapsackDFS}\n
    knapsack.py
    def knapsack_dfs(wgt, val, i, c):\n\"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n# \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 or c == 0:\nreturn 0\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] > c:\nreturn knapsack_dfs(wgt, val, i - 1, c)\n# \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno = knapsack_dfs(wgt, val, i - 1, c)\nyes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n# \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn max(no, yes)\n
    knapsack.go
    [class]{}-[func]{knapsackDFS}\n
    knapsack.js
    [class]{}-[func]{knapsackDFS}\n
    knapsack.ts
    [class]{}-[func]{knapsackDFS}\n
    knapsack.c
    [class]{}-[func]{knapsackDFS}\n
    knapsack.cs
    [class]{knapsack}-[func]{knapsackDFS}\n
    knapsack.swift
    [class]{}-[func]{knapsackDFS}\n
    knapsack.zig
    [class]{}-[func]{knapsackDFS}\n
    knapsack.dart
    [class]{}-[func]{knapsackDFS}\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002

    \u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u4e00\u4e9b\u300c\u91cd\u53e0\u5b50\u95ee\u9898\u300d\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u5f53\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002

    Fig. 0-1 \u80cc\u5305\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811

    "},{"location":"chapter_dynamic_programming/knapsack_problem/#1332","title":"13.3.2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"

    \u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u6c42\u89e3\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u8bb0\u5fc6\u5217\u8868 mem \u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2d mem[i][c] \u8868\u793a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5bb9\u91cf\u4e3a \\(c\\) \u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u5f53\u518d\u6b21\u9047\u5230\u76f8\u540c\u5b50\u95ee\u9898\u65f6\uff0c\u76f4\u63a5\u4ece mem \u4e2d\u83b7\u53d6\u8bb0\u5f55\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart knapsack.java
    [class]{knapsack}-[func]{knapsackDFSMem}\n
    knapsack.cpp
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.py
    def knapsack_dfs_mem(wgt, val, mem, i, c):\n\"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 or c == 0:\nreturn 0\n# \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][c] != -1:\nreturn mem[i][c]\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] > c:\nreturn knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n# \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\nyes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n# \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = max(no, yes)\nreturn mem[i][c]\n
    knapsack.go
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.js
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.ts
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.c
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.cs
    [class]{knapsack}-[func]{knapsackDFSMem}\n
    knapsack.swift
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.zig
    [class]{}-[func]{knapsackDFSMem}\n
    knapsack.dart
    [class]{}-[func]{knapsackDFSMem}\n

    \u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u6700\u591a\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002

    Fig. 0-1 \u80cc\u5305\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811

    "},{"location":"chapter_dynamic_programming/knapsack_problem/#1333","title":"13.3.3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"

    \u63a5\u4e0b\u6765\u5c31\u662f\u4f53\u529b\u6d3b\u4e86\uff0c\u6211\u4eec\u5c06\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u4ee3\u7801\u8bd1\u5199\u4e3a\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart knapsack.java
    [class]{knapsack}-[func]{knapsackDP}\n
    knapsack.cpp
    [class]{}-[func]{knapsackDP}\n
    knapsack.py
    def knapsack_dp(wgt, val, cap):\n\"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u5217\u8868\ndp = [[0] * (cap + 1) for _ in range(n + 1)]\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\nfor c in range(1, cap + 1):\nif wgt[i - 1] > c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c - wgt[i - 1]] + val[i - 1], dp[i - 1][c])\nreturn dp[n][cap]\n
    knapsack.go
    [class]{}-[func]{knapsackDP}\n
    knapsack.js
    [class]{}-[func]{knapsackDP}\n
    knapsack.ts
    [class]{}-[func]{knapsackDP}\n
    knapsack.c
    [class]{}-[func]{knapsackDP}\n
    knapsack.cs
    [class]{knapsack}-[func]{knapsackDP}\n
    knapsack.swift
    [class]{}-[func]{knapsackDP}\n
    knapsack.zig
    [class]{}-[func]{knapsackDP}\n
    knapsack.dart
    [class]{}-[func]{knapsackDP}\n

    \u89c2\u5bdf\u4e0b\u56fe\uff0c\u52a8\u6001\u89c4\u5212\u7684\u8fc7\u7a0b\u672c\u8d28\u4e0a\u5c31\u662f\u586b\u5145 \\(dp\\) \u5217\u8868\uff08\u77e9\u9635\uff09\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(n \\times cap)\\) \u3002

    <1><2><3><4><5><6><7><8><9><10><11><12><13><14>

    \u63a5\u4e0b\u6765\u8003\u8651\u72b6\u6001\u538b\u7f29\u3002\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684 \\(dp\\) \u77e9\u9635\u5360\u7528 \\(O(n \\times cap)\\) \u7a7a\u95f4\u3002\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u5c06\u4f4e\u81f3 \\(O(n)\\) \u3002\u4ee3\u7801\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002

    \u90a3\u4e48\uff0c\u6211\u4eec\u662f\u5426\u53ef\u4ee5\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u72b6\u6001\u538b\u7f29\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u5de6\u4e0a\u65b9\u6216\u6b63\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u904d\u5386\u5230\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\uff0c\u4e3a\u4e86\u907f\u514d\u5de6\u8fb9\u533a\u57df\u7684\u683c\u5b50\u88ab\u8986\u76d6\uff0c\u6211\u4eec\u5e94\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u8fd9\u6837\u65b9\u53ef\u5b9e\u73b0\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u3002

    \u4ee5\u4e0b\u52a8\u753b\u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i=1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i=2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u5efa\u8bae\u4f60\u601d\u8003\u4e00\u4e0b\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002

    <1><2><3><4><5><6>

    \u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6211\u4eec\u4ec5\u9700\u5c06 \\(dp\\) \u5217\u8868\u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u5c06\u5185\u5faa\u73af\u4fee\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart knapsack.java
    [class]{knapsack}-[func]{knapsackDPComp}\n
    knapsack.cpp
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.py
    def knapsack_dp_comp(wgt, val, cap):\n\"\"\"0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u5217\u8868\ndp = [0] * (cap + 1)\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\n# \u5012\u5e8f\u904d\u5386\nfor c in range(cap, 0, -1):\nif wgt[i - 1] > c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c - wgt[i - 1]] + val[i - 1], dp[c])\nreturn dp[cap]\n
    knapsack.go
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.js
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.ts
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.c
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.cs
    [class]{knapsack}-[func]{knapsackDPComp}\n
    knapsack.swift
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.zig
    [class]{}-[func]{knapsackDPComp}\n
    knapsack.dart
    [class]{}-[func]{knapsackDPComp}\n
    "},{"location":"chapter_graph/","title":"9. \u00a0 \u56fe","text":""},{"location":"chapter_graph/graph/","title":"9.1. \u00a0 \u56fe","text":"

    \u300c\u56fe Graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 Vertex\u300d\u548c\u300c\u8fb9 Edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002

    \\[ \\begin{aligned} V & = \\{ 1, 2, 3, 4, 5 \\} \\newline E & = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G & = \\{ V, E \\} \\newline \\end{aligned} \\]

    Fig. \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb

    \u90a3\u4e48\uff0c\u56fe\u4e0e\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb\u662f\u4ec0\u4e48\uff1f\u5982\u679c\u6211\u4eec\u628a\u300c\u9876\u70b9\u300d\u770b\u4f5c\u8282\u70b9\uff0c\u628a\u300c\u8fb9\u300d\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u6307\u9488\uff0c\u5219\u53ef\u5c06\u300c\u56fe\u300d\u770b\u4f5c\u662f\u4e00\u79cd\u4ece\u300c\u94fe\u8868\u300d\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u4ece\u800c\u66f4\u4e3a\u590d\u6742\u3002

    "},{"location":"chapter_graph/graph/#911","title":"9.1.1. \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b","text":"

    \u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe Undirected Graph\u300d\u548c\u300c\u6709\u5411\u56fe Directed Graph\u300d\u3002

    • \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\uff1b
    • \u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\uff1b

    Fig. \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe

    \u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe Connected Graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe Disconnected Graph\u300d\u3002

    • \u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\uff1b
    • \u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\uff1b

    Fig. \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe

    \u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u300c\u6709\u6743\u56fe Weighted Graph\u300d\u3002\u4f8b\u5982\uff0c\u5728\u738b\u8005\u8363\u8000\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002

    Fig. \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe

    "},{"location":"chapter_graph/graph/#912","title":"9.1.2. \u00a0 \u56fe\u5e38\u7528\u672f\u8bed","text":"
    • \u300c\u90bb\u63a5 Adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u4e0a\u56fe\u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002
    • \u300c\u8def\u5f84 Path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u4e0a\u56fe\u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002
    • \u300c\u5ea6 Degree\u300d\u8868\u793a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 In-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 Out-Degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002
    "},{"location":"chapter_graph/graph/#913","title":"9.1.3. \u00a0 \u56fe\u7684\u8868\u793a","text":"

    \u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u6cd5\u5305\u62ec\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002

    "},{"location":"chapter_graph/graph/#_1","title":"\u90bb\u63a5\u77e9\u9635","text":"

    \u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 Adjacency Matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\) \u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i][j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i][j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002

    Fig. \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a

    \u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002
    • \u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002
    • \u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) , \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002

    \u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002

    "},{"location":"chapter_graph/graph/#_2","title":"\u90bb\u63a5\u8868","text":"

    \u300c\u90bb\u63a5\u8868 Adjacency List\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u5373\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002

    Fig. \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a

    \u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002

    \u89c2\u5bdf\u4e0a\u56fe\u53ef\u53d1\u73b0\uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u300c\u94fe\u5730\u5740\u6cd5\u300d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u4e2d\u5e8f\u904d\u5386\u83b7\u53d6\u6709\u5e8f\u5e8f\u5217\uff1b\u6b64\u5916\uff0c\u8fd8\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u4f4e\u81f3 \\(O(1)\\) \u3002

    "},{"location":"chapter_graph/graph/#914","title":"9.1.4. \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"

    \u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u8bb8\u591a\u7cfb\u7edf\u90fd\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u5f85\u6c42\u89e3\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002

    \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2. \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"

    \u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u300c\u8fb9\u300d\u7684\u64cd\u4f5c\u548c\u5bf9\u300c\u9876\u70b9\u300d\u7684\u64cd\u4f5c\u3002\u5728\u300c\u90bb\u63a5\u77e9\u9635\u300d\u548c\u300c\u90bb\u63a5\u8868\u300d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002

    "},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"

    \u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u6709\uff1a

    • \u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002
    • \u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 vertices \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 adjMat \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_adjacency_matrix.java
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<Integer> vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<Integer>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new ArrayList<>();\nthis.adjMat = new ArrayList<>();\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (int[] e : edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<Integer> newRow = new ArrayList<>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.add(0);\n}\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<Integer> row : adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfBoundsException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<Integer> row : adjMat) {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat.get(i).set(j, 1);\nadjMat.get(j).set(i, 1);\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfBoundsException();\nadjMat.get(i).set(j, 0);\nadjMat.get(j).set(i, 0);\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nSystem.out.print(\"\u9876\u70b9\u5217\u8868 = \");\nSystem.out.println(vertices);\nSystem.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n
    graph_adjacency_matrix.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvector<int> vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvector<vector<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(const vector<int> &vertices, const vector<vector<int>> &edges) {\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const vector<int> &edge : edges) {\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() const {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.push_back(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nadjMat.emplace_back(vector<int>(n, 0));\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (vector<int> &row : adjMat) {\nrow.push_back(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.erase(vertices.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.erase(adjMat.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (vector<int> &row : adjMat) {\nrow.erase(row.begin() + index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid print() {\ncout << \"\u9876\u70b9\u5217\u8868 = \";\nprintVector(vertices);\ncout << \"\u90bb\u63a5\u77e9\u9635 =\" << endl;\nprintVectorMatrix(adjMat);\n}\n};\n
    graph_adjacency_matrix.py
    class GraphAdjMat:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n# \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices: list[int] = []\n# \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadj_mat: list[list[int]] = []\ndef __init__(self, vertices: list[int], edges: list[list[int]]) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.vertices: list[int] = []\nself.adj_mat: list[list[int]] = []\n# \u6dfb\u52a0\u9876\u70b9\nfor val in vertices:\nself.add_vertex(val)\n# \u6dfb\u52a0\u8fb9\n# \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges:\nself.add_edge(e[0], e[1])\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.vertices)\ndef add_vertex(self, val: int) -> None:\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nn = self.size()\n# \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.append(val)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnew_row = [0] * n\nself.adj_mat.append(new_row)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in self.adj_mat:\nrow.append(0)\ndef remove_vertex(self, index: int) -> None:\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif index >= self.size():\nraise IndexError()\n# \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in self.adj_mat:\nrow.pop(index)\ndef add_edge(self, i: int, j: int) -> None:\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\n# \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1\nself.adj_mat[j][i] = 1\ndef remove_edge(self, i: int, j: int) -> None:\n\"\"\"\u5220\u9664\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:\nraise IndexError()\nself.adj_mat[i][j] = 0\nself.adj_mat[j][i] = 0\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\nprint(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\nprint(\"\u90bb\u63a5\u77e9\u9635 =\")\nprint_matrix(self.adj_mat)\n
    graph_adjacency_matrix.go
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices []int\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat [][]int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n// \u6dfb\u52a0\u9876\u70b9\nn := len(vertices)\nadjMat := make([][]int, n)\nfor i := range adjMat {\nadjMat[i] = make([]int, n)\n}\n// \u521d\u59cb\u5316\u56fe\ng := &graphAdjMat{\nvertices: vertices,\nadjMat:   adjMat,\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor i := range edges {\ng.addEdge(edges[i][0], edges[i][1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\nreturn len(g.vertices)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\nn := g.size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\ng.vertices = append(g.vertices, val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnewRow := make([]int, n)\ng.adjMat = append(g.adjMat, newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i], 0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\nif index >= g.size() {\nreturn\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\ng.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\ng.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\ng.adjMat[i][j] = 1\ng.adjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= g.size() || j >= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\ng.adjMat[i][j] = 0\ng.adjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\nfmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\nfmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\nfor i := range g.adjMat {\nfmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n}\n}\n
    graph_adjacency_matrix.js
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices, edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val) {\nconst n = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow = [];\nfor (let j = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index) {\nif (index >= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint() {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n
    graph_adjacency_matrix.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices: number[], edges: number[][]) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val: number): void {\nconst n: number = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow: number[] = [];\nfor (let j: number = 0; j < n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index: number): void {\nif (index >= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= this.size() || j >= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint(): void {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n
    graph_adjacency_matrix.c
    [class]{graphAdjMat}-[func]{}\n
    graph_adjacency_matrix.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<int> vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new List<int>();\nthis.adjMat = new List<List<int>>();\n// \u6dfb\u52a0\u9876\u70b9\nforeach (int val in vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nforeach (int[] e in edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.Count;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.Add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = new List<int>(n);\nfor (int j = 0; j < n; j++) {\nnewRow.Add(0);\n}\nadjMat.Add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nforeach (List<int> row in adjMat) {\nrow.Add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index >= size())\nthrow new IndexOutOfRangeException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nforeach (List<int> row in adjMat) {\nrow.RemoveAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j)\nthrow new IndexOutOfRangeException();\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nConsole.Write(\"\u9876\u70b9\u5217\u8868 = \");\nPrintUtil.PrintList(vertices);\nConsole.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.PrintMatrix(adjMat);\n}\n}\n
    graph_adjacency_matrix.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nprivate var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nprivate var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(vertices: [Int], edges: [[Int]]) {\nself.vertices = []\nadjMat = []\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\naddVertex(val: val)\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges {\naddEdge(i: e[0], j: e[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc size() -> Int {\nvertices.count\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc addVertex(val: Int) {\nlet n = size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.append(val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nlet newRow = Array(repeating: 0, count: n)\nadjMat.append(newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i in adjMat.indices {\nadjMat[i].append(0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc removeVertex(index: Int) {\nif index >= size() {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i in adjMat.indices {\nadjMat[i].remove(at: index)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc addEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1\nadjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc removeEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i < 0 || j < 0 || i >= size() || j >= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\nadjMat[i][j] = 0\nadjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc print() {\nSwift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\nSwift.print(vertices)\nSwift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\nPrintUtil.printMatrix(matrix: adjMat)\n}\n}\n
    graph_adjacency_matrix.zig
    \n
    graph_adjacency_matrix.dart
    /* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList<int> vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList<List<int>> adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(List<int> vertices, List<List<int>> edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val in vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (List<int> e in edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList<int> newRow = List.filled(n, 0, growable: true);\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List<int> row in adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index >= size()) {\nthrow IndexError;\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List<int> row in adjMat) {\nrow.removeAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow IndexError;\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u6cbf\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i < 0 || j < 0 || i >= size() || j >= size() || i == j) {\nthrow IndexError;\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printAdjMat() {\nprint(\"\u9876\u70b9\u5217\u8868 = $vertices\");\nprint(\"\u90bb\u63a5\u77e9\u9635 = \");\nprintMatrix(adjMat);\n}\n}\n
    "},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2. \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"

    \u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\) \u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u6709\uff1a

    • \u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002
    • \u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002
    • \u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
    • \u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002
    \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u56fe\u7684\u4ee3\u7801\u793a\u4f8b\u3002\u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u6ce8\u610f\u5230\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 Vertex \u8282\u70b9\u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u6709\uff1a

    • \u5982\u679c\u6211\u4eec\u9009\u62e9\u901a\u8fc7\u9876\u70b9\u503c\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u503c\u91cd\u590d\u7684\u9876\u70b9\u5c06\u65e0\u6cd5\u88ab\u533a\u5206\u3002
    • \u5982\u679c\u7c7b\u4f3c\u90bb\u63a5\u77e9\u9635\u90a3\u6837\uff0c\u4f7f\u7528\u9876\u70b9\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\u3002\u90a3\u4e48\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u8981\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u5176\u4e2d \\(> i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u8fd9\u6837\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002
    • \u56e0\u6b64\u6211\u4eec\u8003\u8651\u5f15\u5165\u9876\u70b9\u7c7b Vertex \uff0c\u4f7f\u5f97\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684\u5bf9\u8c61\uff0c\u6b64\u65f6\u5220\u9664\u9876\u70b9\u65f6\u5c31\u65e0\u9700\u6539\u52a8\u5176\u4f59\u9876\u70b9\u4e86\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_adjacency_list.java
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new HashMap<>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (Vertex[] edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList.get(vet1).add(vet2);\nadjList.get(vet2).add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList.get(vet1).remove(vet2);\nadjList.get(vet2).remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.containsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.put(vet, new ArrayList<>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet))\nthrow new IllegalArgumentException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (List<Vertex> list : adjList.values()) {\nlist.remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nSystem.out.println(\"\u90bb\u63a5\u8868 =\");\nfor (Map.Entry<Vertex, List<Vertex>> pair : adjList.entrySet()) {\nList<Integer> tmp = new ArrayList<>();\nfor (Vertex vertex : pair.getValue())\ntmp.add(vertex.val);\nSystem.out.println(pair.getKey().val + \": \" + tmp + \",\");\n}\n}\n}\n
    graph_adjacency_list.cpp
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\npublic:\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nunordered_map<Vertex *, vector<Vertex *>> adjList;\n/* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\nvoid remove(vector<Vertex *> &vec, Vertex *vet) {\nfor (int i = 0; i < vec.size(); i++) {\nif (vec[i] == vet) {\nvec.erase(vec.begin() + i);\nbreak;\n}\n}\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(const vector<vector<Vertex *>> &edges) {\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const vector<Vertex *> &edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].push_back(vet2);\nadjList[vet2].push_back(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5220\u9664\u8fb9 vet1 - vet2\nremove(adjList[vet1], vet2);\nremove(adjList[vet2], vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex *vet) {\nif (adjList.count(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = vector<Vertex *>();\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex *vet) {\nif (!adjList.count(vet))\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.erase(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (auto &adj : adjList) {\nremove(adj.second, vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid print() {\ncout << \"\u90bb\u63a5\u8868 =\" << endl;\nfor (auto &adj : adjList) {\nconst auto &key = adj.first;\nconst auto &vec = adj.second;\ncout << key->val << \": \";\nprintVector(vetsToVals(vec));\n}\n}\n};\n
    graph_adjacency_list.py
    class GraphAdjList:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\ndef __init__(self, edges: list[list[Vertex]]) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nself.adj_list = dict[Vertex, Vertex]()\n# \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges:\nself.add_vertex(edge[0])\nself.add_vertex(edge[1])\nself.add_edge(edge[0], edge[1])\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.adj_list)\ndef add_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list[vet1].append(vet2)\nself.adj_list[vet2].append(vet1)\ndef remove_edge(self, vet1: Vertex, vet2: Vertex) -> None:\n\"\"\"\u5220\u9664\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list[vet1].remove(vet2)\nself.adj_list[vet2].remove(vet1)\ndef add_vertex(self, vet: Vertex) -> None:\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nif vet in self.adj_list:\nreturn\n# \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list[vet] = []\ndef remove_vertex(self, vet: Vertex) -> None:\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif vet not in self.adj_list:\nraise ValueError()\n# \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.pop(vet)\n# \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor vertex in self.adj_list:\nif vet in self.adj_list[vertex]:\nself.adj_list[vertex].remove(vet)\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\nprint(\"\u90bb\u63a5\u8868 =\")\nfor vertex in self.adj_list:\ntmp = [v.val for v in self.adj_list[vertex]]\nprint(f\"{vertex.val}: {tmp},\")\n
    graph_adjacency_list.go
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList map[Vertex][]Vertex\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\ng := &graphAdjList{\nadjList: make(map[Vertex][]Vertex),\n}\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor _, edge := range edges {\ng.addVertex(edge[0])\ng.addVertex(edge[1])\ng.addEdge(edge[0], edge[1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\nreturn len(g.adjList)\n}\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\ng.adjList[vet1] = append(g.adjList[vet1], vet2)\ng.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\ng.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\ng.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif ok {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ng.adjList[vet] = make([]Vertex, 0)\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif !ok {\npanic(\"error\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\ndelete(g.adjList, vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor v, list := range g.adjList {\ng.adjList[v] = DeleteSliceElms(list, vet)\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\nvar builder strings.Builder\nfmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\nfor k, v := range g.adjList {\nbuilder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\nfor _, vet := range v {\nbuilder.WriteString(strconv.Itoa(vet.Val) + \" \")\n}\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
    graph_adjacency_list.js
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet) {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet) {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint() {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
    graph_adjacency_list.ts
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList: Map<Vertex, Vertex[]>;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges: Vertex[][]) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet: Vertex): void {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet: Vertex): void {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (let set of this.adjList.values()) {\nconst index: number = set.indexOf(vet);\nif (index > -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint(): void {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList.entries()) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n
    graph_adjacency_list.c
    [class]{graphAdjList}-[func]{}\n
    graph_adjacency_list.cs
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic Dictionary<Vertex, List<Vertex>> adjList;\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new Dictionary<Vertex, List<Vertex>>();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nforeach (Vertex[] edge in edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.Count;\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].Add(vet2);\nadjList[vet2].Add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1].Remove(vet2);\nadjList[vet2].Remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.ContainsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.Add(vet, new List<Vertex>());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.ContainsKey(vet))\nthrow new InvalidOperationException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.Remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nforeach (List<Vertex> list in adjList.Values) {\nlist.Remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nConsole.WriteLine(\"\u90bb\u63a5\u8868 =\");\nforeach (KeyValuePair<Vertex, List<Vertex>> pair in adjList) {\nList<int> tmp = new List<int>();\nforeach (Vertex vertex in pair.Value)\ntmp.Add(vertex.val);\nConsole.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n}\n}\n}\n
    graph_adjacency_list.swift
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic private(set) var adjList: [Vertex: [Vertex]]\n/* \u6784\u9020\u65b9\u6cd5 */\npublic init(edges: [[Vertex]]) {\nadjList = [:]\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\naddVertex(vet: edge[0])\naddVertex(vet: edge[1])\naddEdge(vet1: edge[0], vet2: edge[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic func size() -> Int {\nadjList.count\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic func addEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]?.append(vet2)\nadjList[vet2]?.append(vet1)\n}\n/* \u5220\u9664\u8fb9 */\npublic func removeEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]?.removeAll(where: { $0 == vet2 })\nadjList[vet2]?.removeAll(where: { $0 == vet1 })\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic func addVertex(vet: Vertex) {\nif adjList[vet] != nil {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = []\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic func removeVertex(vet: Vertex) {\nif adjList[vet] == nil {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.removeValue(forKey: vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor key in adjList.keys {\nadjList[key]?.removeAll(where: { $0 == vet })\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic func print() {\nSwift.print(\"\u90bb\u63a5\u8868 =\")\nfor pair in adjList {\nvar tmp: [Int] = []\nfor vertex in pair.value {\ntmp.append(vertex.val)\n}\nSwift.print(\"\\(pair.key.val): \\(tmp),\")\n}\n}\n}\n
    graph_adjacency_list.zig
    [class]{GraphAdjList}-[func]{}\n
    graph_adjacency_list.dart
    /* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey: \u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap<Vertex, List<Vertex>> adjList = {};\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(List<List<Vertex>> edges) {\nfor (List<Vertex> edge in edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.length;\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]!.add(vet2);\nadjList[vet2]!.add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]!.remove(vet2);\nadjList[vet2]!.remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex vet) {\nif (adjList.containsKey(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = [];\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet)) {\nthrow ArgumentError;\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nadjList.forEach((key, value) {\nvalue.remove(vet);\n});\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid printAdjList() {\nprint(\"\u90bb\u63a5\u8868 =\");\nadjList.forEach((key, value) {\nList<int> tmp = [];\nfor (Vertex vertex in value) {\ntmp.add(vertex.val);\n}\nprint(\"${key.val}: $tmp,\");\n});\n}\n}\n
    "},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3. \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"

    \u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u4e0b\u8868\u4e3a\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6548\u7387\u5bf9\u6bd4\u3002

    \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\)

    \u89c2\u5bdf\u4e0a\u8868\uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002

    "},{"location":"chapter_graph/graph_traversal/","title":"9.3. \u00a0 \u56fe\u7684\u904d\u5386","text":"

    \u56fe\u4e0e\u6811\u7684\u5173\u7cfb

    \u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u5efa\u8bae\u4f60\u5728\u5b66\u4e60\u672c\u7ae0\u8282\u65f6\u878d\u4f1a\u8d2f\u901a\u4e24\u8005\u7684\u6982\u5ff5\u4e0e\u5b9e\u73b0\u65b9\u6cd5\u3002

    \u300c\u56fe\u300d\u548c\u300c\u6811\u300d\u90fd\u662f\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u90fd\u9700\u8981\u4f7f\u7528\u300c\u641c\u7d22\u7b97\u6cd5\u300d\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002

    \u4e0e\u6811\u7c7b\u4f3c\uff0c\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u4e5f\u53ef\u5206\u4e3a\u4e24\u79cd\uff0c\u5373\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 Breadth-First Traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u4e5f\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 Depth-First Search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS\u3002

    "},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1. \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\u5f00\u59cb\u8bbf\u95ee\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002

    Fig. \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386

    "},{"location":"chapter_graph/graph_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    BFS \u901a\u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002

    1. \u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 startVet \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\uff1b
    2. \u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\uff1b
    3. \u5faa\u73af\u6b65\u9aa4 2. \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6210\u540e\u7ed3\u675f\uff1b

    \u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_bfs.java
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>() {{ add(startVet); }};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new LinkedList<>() {{ offer(startVet); }};\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.isEmpty()) {\nVertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.cpp
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphBFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex *> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex *> visited = {startVet};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nqueue<Vertex *> que;\nque.push(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.empty()) {\nVertex *vet = que.front();\nque.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (auto adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.py
    def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n\"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]([start_vet])\n# \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nque = deque[Vertex]([start_vet])\n# \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile len(que) > 0:\nvet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adj_vet in graph.adj_list[vet]:\nif adj_vet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nque.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n
    graph_bfs.go
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\nvisited[startVet] = struct{}{}\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\nqueue := make([]Vertex, 0)\nqueue = append(queue, startVet)\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nfor len(queue) > 0 {\n// \u961f\u9996\u9876\u70b9\u51fa\u961f\nvet := queue[0]\nqueue = queue[1:]\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres = append(res, vet)\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nif !isExist {\nqueue = append(queue, adjVet)\nvisited[adjVet] = struct{}{}\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_bfs.js
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.ts
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.c
    [class]{}-[func]{graphBFS}\n
    graph_bfs.cs
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new List<Vertex>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet<Vertex> visited = new HashSet<Vertex>() { startVet };\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = new Queue<Vertex>();\nque.Enqueue(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.Count > 0) {\nVertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n
    graph_bfs.swift
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = [startVet]\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nvar que: [Vertex] = [startVet]\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.isEmpty {\nlet vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_bfs.zig
    [class]{}-[func]{graphBFS}\n
    graph_bfs.dart
    /* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList<Vertex> graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = {};\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue<Vertex> que = Queue();\nque.add(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.isNotEmpty) {\nVertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\nque.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n

    \u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u4ee5\u4e0b\u52a8\u753b\u56fe\u793a\u6765\u52a0\u6df1\u7406\u89e3\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f

    \u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u662f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u7684\u3002\u4ee5\u4e0a\u56fe\u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\) , \\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u3001\u9876\u70b9 \\(2\\) , \\(4\\) , \\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002

    "},{"location":"chapter_graph/graph_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \uff0c\u961f\u5217 que \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2. \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5177\u4f53\u5730\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002

    Fig. \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386

    "},{"location":"chapter_graph/graph_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    \u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934 + \u56de\u6eaf\u201d\u7684\u7b97\u6cd5\u5f62\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e BFS \u7c7b\u4f3c\uff0c\u5728 DFS \u4e2d\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 visited \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart graph_dfs.java
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new ArrayList<>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = new HashSet<>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.cpp
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &graph, unordered_set<Vertex *> &visited, vector<Vertex *> &res, Vertex *vet) {\nres.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex *adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector<Vertex *> graphDFS(GraphAdjList &graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector<Vertex *> res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set<Vertex *> visited;\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.py
    def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adj_list[vet]:\nif adjVet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n# \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet)\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]:\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]()\ndfs(graph, visited, res, start_vet)\nreturn res\n
    graph_dfs.go
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n// append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n*res = append(*res, vet)\nvisited[vet] = struct{}{}\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nif !isExist {\ndfs(g, visited, res, adjVet)\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\ndfs(g, visited, &res, startVet)\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n
    graph_dfs.js
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.ts
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\ngraph: GraphAdjList,\nvisited: Set<Vertex>,\nres: Vertex[],\nvet: Vertex\n): void {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set<Vertex> = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.c
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
    graph_dfs.cs
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, HashSet<Vertex> visited, List<Vertex> res, Vertex vet) {\nres.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9                             \n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = new List<Vertex>();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet<Vertex> visited = new HashSet<Vertex>();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n
    graph_dfs.swift
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set<Vertex>, res: inout [Vertex], vet: Vertex) {\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph: graph, visited: &visited, res: &res, vet: adjVet)\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set<Vertex> = []\ndfs(graph: graph, visited: &visited, res: &res, vet: startVet)\nreturn res\n}\n
    graph_dfs.zig
    [class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n
    graph_dfs.dart
    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\nGraphAdjList graph,\nSet<Vertex> visited,\nList<Vertex> res,\nVertex vet,\n) {\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList<Vertex> graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList<Vertex> res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet<Vertex> visited = {};\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5176\u4e2d\uff1a

    • \u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\uff1b
    • \u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u9012\u5f52\u65b9\u6cd5\u7684\u4f4d\u7f6e\uff1b

    \u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6a21\u62df\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    \u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f

    \u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002

    \u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u3001\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u4e0d\u540c\u7684\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002

    "},{"location":"chapter_graph/graph_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a \u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a \u5217\u8868 res \uff0c\u54c8\u5e0c\u8868 visited \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_graph/summary/","title":"9.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u88ab\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002
    • \u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002
    • \u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002
    • \u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002
    • \u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u6761\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002
    • \u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002
    • \u4ece\u7b97\u6cd5\u601d\u60f3\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002
    • \u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002
    • \u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002
    • \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002
    • \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002
    "},{"location":"chapter_graph/summary/#941-q-a","title":"9.4.1. \u00a0 Q & A","text":"

    \u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f

    \u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u8ba4\u4e3a\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002

    \u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f

    \u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002

    \u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f

    \u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u4f1a\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\u3001\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\u7b49\uff0c\u8fd9\u6837\u53ef\u4ee5\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002

    "},{"location":"chapter_hashing/","title":"6. \u00a0 \u6563\u5217\u8868","text":""},{"location":"chapter_hashing/hash_algorithm/","title":"6.3. \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"

    \u5728\u4e0a\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u4e86\u89e3\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\uff0c\u4ee5\u53ca\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\uff0c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5730\u5740\u6cd5\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u4f46\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002

    \u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u4f8b\u5982\u5bf9\u4e8e\u94fe\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5e73\u5747\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u597d\u7684\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u5168\u90e8\u952e\u503c\u90fd\u88ab\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002

    Fig. \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u4e0e\u6700\u5dee\u60c5\u51b5

    \u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u662f\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u7684\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a

    index = hash(key) % capacity\n

    \u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf capacity \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 hash() \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u51cf\u5c0f\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u9700\u8981\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 hash() \u7684\u8bbe\u8ba1\u4e0a\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1. \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"

    \u4e3a\u4e86\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5305\u542b\u4ee5\u4e0b\u7279\u70b9\uff1a

    • \u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002
    • \u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002
    • \u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5e73\u5747\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5e73\u5747\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002

    \u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\uff0c\u5305\u62ec\uff1a

    • \u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002
    • \u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u7684\u3002

    \u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u6ee1\u8db3\u66f4\u9ad8\u7684\u5b89\u5168\u6807\u51c6\uff0c\u4ee5\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u5305\u62ec\uff1a

    • \u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u5176\u56f0\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002
    • \u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002

    \u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 key \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 key % 100 \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\uff0c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 key \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 key \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2. \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"

    \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u590d\u6742\u4e14\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u4e00\u4e9b\u7b80\u5355\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4ee5\u5b57\u7b26\u4e32\u54c8\u5e0c\u4e3a\u4f8b\uff1a

    • \u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002
    • \u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e86\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002
    • \u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002
    • \u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart simple_hash.java
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (31 * hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash ^= (int) c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = ((hash << 4) ^ (hash >> 28) ^ (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n
    simple_hash.cpp
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (31 * hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\ncout<<(int)c<<endl;\nhash ^= (int)c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n
    simple_hash.py
    def add_hash(key: str) -> int:\n\"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash += ord(c)\nreturn hash % modulus\ndef mul_hash(key: str) -> int:\n\"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = 31 * hash + ord(c)\nreturn hash % modulus\ndef xor_hash(key: str) -> int:\n\"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash ^= ord(c)\nreturn hash % modulus\ndef rot_hash(key: str) -> int:\n\"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = (hash << 4) ^ (hash >> 28) ^ ord(c)\nreturn hash % modulus\n
    simple_hash.go
    /* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (31*hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\nhash := 0\nmodulus := 1000000007\nfor _, b := range []byte(key) {\nfmt.Println(int(b))\nhash ^= int(b)\nhash = (31*hash + int(b)) % modulus\n}\nreturn hash & modulus\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ int64(b)) % modulus\n}\nreturn int(hash)\n}\n
    simple_hash.js
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.ts
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.c
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.cs
    /* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (31 * hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash ^= c;\n}\nreturn hash & MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = ((hash << 4) ^ (hash >> 28) ^ c) % MODULUS;\n}\nreturn (int)hash;\n}\n
    simple_hash.swift
    /* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (31 * hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash ^= Int(scalar.value)\n}\n}\nreturn hash & MODULUS\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -> Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = ((hash << 4) ^ (hash >> 28) ^ Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n
    simple_hash.zig
    [class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n
    simple_hash.dart
    [class]{}-[func]{add_hash}\n[class]{}-[func]{mul_hash}\n[class]{}-[func]{xor_hash}\n[class]{}-[func]{rot_hash}\n

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002

    \u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u5f53\u6211\u4eec\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\u65f6\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4f1a\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002

    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u3002\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 key \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\) , \\(3\\) , \\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002

    \\[ \\begin{aligned} \\text{modulus} & = 9 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\cdots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\cdots \\} \\end{aligned} \\]

    \u5982\u679c\u8f93\u5165 key \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 modulus \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e key \u548c modulus \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002

    \\[ \\begin{aligned} \\text{modulus} & = 13 \\newline \\text{key} & = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\cdots \\} \\newline \\text{hash} & = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\cdots \\} \\end{aligned} \\]

    \u503c\u5f97\u5f3a\u8c03\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 key \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u662f\u53ef\u4ee5\u7684\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 key \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002

    \u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u5927\u4e00\u4e9b\uff0c\u4ee5\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002

    "},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3. \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"

    \u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002

    \u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5, SHA-1, SHA-2, SHA3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002

    \u76f4\u81f3\u76ee\u524d\uff0cMD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u88ab\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002

    MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256 / 512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4. \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"

    \u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 key \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 hash() \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\uff0c\u5305\u62ec\uff1a

    • \u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002
    • \u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u8bf7\u81ea\u884c\u5b66\u4e60\u3002
    • \u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002
    • \u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002

    Tip

    \u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart built_in_hash.java
    int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -727081396\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n
    built_in_hash.cpp
    int num = 3;\nsize_t hashNum = hash<int>()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbool bol = true;\nsize_t hashBol = hash<bool>()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndouble dec = 3.14159;\nsize_t hashDec = hash<double>()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash<string>()(str);\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n
    built_in_hash.py
    num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndec = 3.14159\nhash_dec = hash(dec) \n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 <ListNode object at 0x1058fd810> \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n
    built_in_hash.go
    \n
    built_in_hash.js
    \n
    built_in_hash.ts
    \n
    built_in_hash.c
    \n
    built_in_hash.cs
    int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\nobject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\nListNode obj = new ListNode(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n
    built_in_hash.swift
    let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n
    built_in_hash.zig
    \n
    built_in_hash.dart
    int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32 Hello \u7b97\u6cd5 \u7684\u54c8\u5e0c\u503c\u4e3a 468167534\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n

    \u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 key \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a key \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 value \u4e86\u3002

    \u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u4f8b\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\uff0c\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u9ed8\u8ba4\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002

    \u5411\u54c8\u5e0c\u51fd\u6570\u52a0\u76d0

    Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u56e0\u6b64\u5728\u4e0d\u540c\u7684 Python \u8fd0\u884c\u5b9e\u4f8b\u4e2d\uff0c\u540c\u4e00\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u4e0d\u540c\u7684\u3002\u6b64\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002

    "},{"location":"chapter_hashing/hash_collision/","title":"6.2. \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"

    \u4e0a\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6570\u7ec4\u7d22\u5f15\u3002

    \u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u65f6\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u6362\u4e00\u79cd\u601d\u8def\uff1a

    1. \u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u5b58\u5728\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002
    2. \u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002

    \u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u94fe\u5f0f\u5730\u5740\u548c\u5f00\u653e\u5bfb\u5740\u3002

    "},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1. \u00a0 \u94fe\u5f0f\u5730\u5740","text":"

    \u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 Separate Chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002

    Fig. \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868

    \u94fe\u5f0f\u5730\u5740\u4e0b\uff0c\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u5305\u62ec\uff1a

    • \u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 key \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6570\u7ec4\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 key \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002
    • \u6dfb\u52a0\u5143\u7d20\uff1a\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u5373\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002
    • \u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\uff0c\u5e76\u5c06\u5176\u5220\u9664\u3002

    \u8be5\u65b9\u6cd5\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\uff0c\u5305\u62ec\uff1a

    • \u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u7531\u4e8e\u94fe\u8868\u6216\u4e8c\u53c9\u6811\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\uff1b
    • \u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\uff1b

    \u4ee5\u4e0b\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\uff1a

    • \u4e3a\u4e86\u4f7f\u5f97\u4ee3\u7801\u5c3d\u91cf\u7b80\u77ed\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002
    • \u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5177\u4f53\u6765\u770b\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3 \\(2\\) \u500d\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map_chaining.java
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = new ArrayList<>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.add(new ArrayList<>());\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString get(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair : bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair : bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = new Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (Pair pair : bucket) {\nif (pair.key == key)\nbucket.remove(pair);\n}\nsize--;\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new ArrayList<>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.add(new ArrayList<>());\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List<Pair> bucket : bucketsTmp) {\nfor (Pair pair : bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (List<Pair> bucket : buckets) {\nList<String> res = new ArrayList<>();\nfor (Pair pair : bucket) {\nres.add(pair.key + \" -> \" + pair.val);\n}\nSystem.out.println(res);\n}\n}\n}\n
    hash_map_chaining.cpp
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nprivate:\nint size;                       // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;                // \u6269\u5bb9\u500d\u6570\nvector<vector<Pair *>> buckets; // \u6876\u6570\u7ec4\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3), extendRatio(2) {\nbuckets.resize(capacity);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double)size / (double)capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair *pair : buckets[index]) {\nif (pair->key == key) {\nreturn pair->val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nullptr\nreturn nullptr;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair *pair : buckets[index]) {\nif (pair->key == key) {\npair->val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].push_back(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nauto &bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (int i = 0; i < bucket.size(); i++) {\nif (bucket[i]->key == key) {\nPair *tmp = bucket[i];\nbucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\ndelete tmp;                       // \u91ca\u653e\u5185\u5b58\nsize--;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector<vector<Pair *>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets.clear();\nbuckets.resize(capacity);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (auto &bucket : bucketsTmp) {\nfor (Pair *pair : bucket) {\nput(pair->key, pair->val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (auto &bucket : buckets) {\ncout << \"[\";\nfor (Pair *pair : bucket) {\ncout << pair->key << \" -> \" << pair->val << \", \";\n}\ncout << \"]\\n\";\n}\n}\n};\n
    hash_map_chaining.py
    class HashMapChaining:\n\"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2 / 3  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -> float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket:\nif pair.key == key:\nreturn pair.val\n# \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\nreturn None\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() > self.load_thres:\nself.extend()\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket:\nif pair.key == key:\npair.val = val\nreturn\n# \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\npair = Pair(key, val)\nbucket.append(pair)\nself.size += 1\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor pair in bucket:\nif pair.key == key:\nbucket.remove(pair)\nself.size -= 1\nreturn\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [[] for _ in range(self.capacity)]\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in buckets:\nfor pair in bucket:\nself.put(pair.key, pair.val)\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor bucket in self.buckets:\nres = []\nfor pair in bucket:\nres.append(str(pair.key) + \" -> \" + pair.val)\nprint(res)\n
    hash_map_chaining.go
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\nsize        int      // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int      // \u6269\u5bb9\u500d\u6570\nbuckets     [][]pair // \u6876\u6570\u7ec4\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\nbuckets := make([][]pair, 4)\nfor i := 0; i < 4; i++ {\nbuckets[i] = make([]pair, 0)\n}\nreturn &hashMapChaining{\nsize:        0,\ncapacity:    4,\nloadThres:   2 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\nreturn float64(m.size / m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\nidx := m.hashFunc(key)\nbucket := m.buckets[idx]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor _, p := range bucket {\nif p.key == key {\nreturn p.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() > m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor _, p := range m.buckets[idx] {\nif p.key == key {\np.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\np := pair{\nkey: key,\nval: val,\n}\nm.buckets[idx] = append(m.buckets[idx], p)\nm.size += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor i, p := range m.buckets[idx] {\nif p.key == key {\n// \u5207\u7247\u5220\u9664\nm.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\nbreak\n}\n}\nm.size -= 1\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([][]pair, len(m.buckets))\nfor i := 0; i < len(m.buckets); i++ {\ntmpBuckets[i] = make([]pair, len(m.buckets[i]))\ncopy(tmpBuckets[i], m.buckets[i])\n}\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([][]pair, m.capacity)\nfor i := 0; i < m.capacity; i++ {\nm.buckets[i] = make([]pair, 0)\n}\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, bucket := range tmpBuckets {\nfor _, p := range bucket {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\nvar builder strings.Builder\nfor _, bucket := range m.buckets {\nbuilder.WriteString(\"[\")\nfor _, p := range bucket {\nbuilder.WriteString(strconv.Itoa(p.key) + \" -> \" + p.val + \" \")\n}\nbuilder.WriteString(\"]\")\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n
    hash_map_chaining.js
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.ts
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.c
    [class]{hashMapChaining}-[func]{}\n
    hash_map_chaining.cs
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = new List<List<Pair>>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.Add(new List<Pair>());\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate double loadFactor() {\nreturn (double)size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string get(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].Add(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nforeach (Pair pair in buckets[index].ToList()) { if (pair.key == key) {\nbuckets[index].Remove(pair);\n}\n}\nsize--;   }\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new List<List<Pair>>(capacity);\nfor (int i = 0; i < capacity; i++) {\nbuckets.Add(new List<Pair>());\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (List<Pair> bucket in bucketsTmp) {\nforeach (Pair pair in bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (List<Pair> bucket in buckets) {\nList<string> res = new List<string>();\nforeach (Pair pair in bucket) {\nres.Add(pair.key + \" -> \" + pair.val);\n}\nforeach (string kv in res) {\nConsole.WriteLine(kv);\n}\n}\n}\n}\n
    hash_map_chaining.swift
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [[Pair]] // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2 / 3\nextendRatio = 2\nbuckets = Array(repeating: [], count: capacity)\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -> Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -> Double {\nDouble(size / capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket {\nif pair.key == key {\nreturn pair.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() > loadThres {\nextend()\n}\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket {\nif pair.key == key {\npair.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nlet pair = Pair(key: key, val: val)\nbuckets[index].append(pair)\nsize += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (pairIndex, pair) in bucket.enumerated() {\nif pair.key == key {\nbuckets[index].remove(at: pairIndex)\n}\n}\nsize -= 1\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: [], count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in bucketsTmp {\nfor pair in bucket {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor bucket in buckets {\nlet res = bucket.map { \"\\($0.key) -> \\($0.val)\" }\nSwift.print(res)\n}\n}\n}\n
    hash_map_chaining.zig
    [class]{HashMapChaining}-[func]{}\n
    hash_map_chaining.dart
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nlate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nlate int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nlate double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nlate int extendRatio; // \u6269\u5bb9\u500d\u6570\nlate List<List<Pair>> buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2 / 3.0;\nextendRatio = 2;\nbuckets = List.generate(capacity, (_) => []);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair in bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair in bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList<Pair> bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nbucket.removeWhere((Pair pair) => pair.key == key);\nsize--;\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<List<Pair>> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = List.generate(capacity, (_) => []);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List<Pair> bucket in bucketsTmp) {\nfor (Pair pair in bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (List<Pair> bucket in buckets) {\nList<String> res = [];\nfor (Pair pair in bucket) {\nres.add(\"${pair.key} -> ${pair.val}\");\n}\nprint(res);\n}\n}\n}\n

    Tip

    \u4e3a\u4e86\u63d0\u9ad8\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u300cAVL \u6811\u300d\u6216\u300c\u7ea2\u9ed1\u6811\u300d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002

    "},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2. \u00a0 \u5f00\u653e\u5bfb\u5740","text":"

    \u300c\u5f00\u653e\u5bfb\u5740 Open Addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u3002

    "},{"location":"chapter_hashing/hash_collision/#_1","title":"\u7ebf\u6027\u63a2\u6d4b","text":"

    \u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u67e5\u627e\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5bf9\u5e94\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u65b9\u6cd5\u4e3a\uff1a

    • \u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6570\u7ec4\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u4f4d\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002
    • \u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de value \u5373\u53ef\uff1b\u6216\u8005\u82e5\u9047\u5230\u7a7a\u4f4d\uff0c\u8bf4\u660e\u76ee\u6807\u952e\u503c\u5bf9\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002

    Fig. \u7ebf\u6027\u63a2\u6d4b

    \u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5b58\u5728\u4ee5\u4e0b\u7f3a\u9677\uff1a

    • \u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u4f4d\uff0c\u67e5\u627e\u5176\u4ed6\u5143\u7d20\u65f6\uff0c\u8be5\u7a7a\u4f4d\u53ef\u80fd\u5bfc\u81f4\u7a0b\u5e8f\u8bef\u5224\u5143\u7d20\u4e0d\u5b58\u5728\u3002\u56e0\u6b64\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u6807\u8bb0\u5df2\u5220\u9664\u5143\u7d20\u3002
    • \u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u6570\u7ec4\u5185\u8fde\u7eed\u88ab\u5360\u7528\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8fd9\u4e00\u4f4d\u7f6e\u7684\u201c\u805a\u5806\u751f\u957f\u201d\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u964d\u4f4e\u3002

    \u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u4e3a\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u91cd\u70b9\u5305\u62ec\uff1a

    • \u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u56fa\u5b9a\u7684\u952e\u503c\u5bf9\u5b9e\u4f8b removed \u6765\u6807\u8bb0\u5df2\u5220\u9664\u5143\u7d20\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u4e00\u4e2a\u6876\u4e3a \\(\\text{None}\\) \u6216 removed \u65f6\uff0c\u8fd9\u4e2a\u6876\u90fd\u662f\u7a7a\u7684\uff0c\u53ef\u7528\u4e8e\u653e\u7f6e\u952e\u503c\u5bf9\u3002
    • \u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u662f\u53ef\u4ee5\u518d\u6b21\u88ab\u4f7f\u7528\u7684\u3002\u5f53\u63d2\u5165\u5143\u7d20\u65f6\uff0c\u82e5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u4e86\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7d22\u5f15\uff0c\u5219\u53ef\u5c06\u8be5\u5143\u7d20\u653e\u7f6e\u5230\u8be5\u7d22\u5f15\u3002
    • \u5728\u7ebf\u6027\u63a2\u6d4b\u65f6\uff0c\u6211\u4eec\u4ece\u5f53\u524d\u7d22\u5f15 index \u5411\u540e\u904d\u5386\uff1b\u800c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u9700\u8981\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map_open_addressing.java
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nprivate int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nprivate double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nprivate int extendRatio; // \u6269\u5bb9\u500d\u6570\nprivate Pair[] buckets; // \u6876\u6570\u7ec4\nprivate Pair removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new Pair[capacity];\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\npublic int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\npublic double loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (buckets[j] == null)\nreturn null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j].key == key && buckets[j] != removed)\nreturn buckets[j].val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == null || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j].key == key) {\nbuckets[j].val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j].key == key) {\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\npublic void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair pair : bucketsTmp) {\nif (pair != null && pair != removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair pair : buckets) {\nif (pair != null) {\nSystem.out.println(pair.key + \" -> \" + pair.val);\n} else {\nSystem.out.println(\"null\");\n}\n}\n}\n}\n
    hash_map_open_addressing.cpp
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate:\nint size;               // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;           // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres;       // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;        // \u6269\u5bb9\u500d\u6570\nvector<Pair *> buckets; // \u6876\u6570\u7ec4\nPair *removed;          // \u5220\u9664\u6807\u8bb0\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() {\n// \u6784\u9020\u65b9\u6cd5\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = vector<Pair *>(capacity, nullptr);\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn static_cast<double>(size) / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de nullptr\nif (buckets[j] == nullptr)\nreturn nullptr;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j]->key == key && buckets[j] != removed)\nreturn buckets[j]->val;\n}\nreturn nullptr;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres)\nextend();\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == nullptr || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j]->key == key) {\nbuckets[j]->val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == nullptr)\nreturn;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j]->key == key) {\ndelete buckets[j]; // \u91ca\u653e\u5185\u5b58\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector<Pair *> bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = vector<Pair *>(capacity, nullptr);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair *pair : bucketsTmp) {\nif (pair != nullptr && pair != removed) {\nput(pair->key, pair->val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (auto &pair : buckets) {\nif (pair != nullptr) {\ncout << pair->key << \" -> \" << pair->val << endl;\n} else {\ncout << \"nullptr\" << endl;\n}\n}\n}\n};\n
    hash_map_open_addressing.py
    class HashMapOpenAddressing:\n\"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2 / 3  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\nself.removed = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -> float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de None\nif self.buckets[j] is None:\nreturn None\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif self.buckets[j].key == key and self.buckets[j] != self.removed:\nreturn self.buckets[j].val\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() > self.load_thres:\nself.extend()\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif self.buckets[j] in [None, self.removed]:\nself.buckets[j] = Pair(key, val)\nself.size += 1\nreturn\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif self.buckets[j].key == key:\nself.buckets[j].val = val\nreturn\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in range(self.capacity):\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj = (index + i) % self.capacity\n# \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif self.buckets[j] is None:\nreturn\n# \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif self.buckets[j].key == key:\nself.buckets[j] = self.removed\nself.size -= 1\nreturn\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets_tmp = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [None] * self.capacity\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in buckets_tmp:\nif pair not in [None, self.removed]:\nself.put(pair.key, pair.val)\ndef print(self) -> None:\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\nelse:\nprint(\"None\")\n
    hash_map_open_addressing.go
    /* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\nsize        int     // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int     // \u6269\u5bb9\u500d\u6570\nbuckets     []pair  // \u6876\u6570\u7ec4\nremoved     pair    // \u5220\u9664\u6807\u8bb0\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\nbuckets := make([]pair, 4)\nreturn &hashMapOpenAddressing{\nsize:        0,\ncapacity:    4,\nloadThres:   2 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\nremoved: pair{\nkey: -1,\nval: \"-1\",\n},\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\nreturn float64(m.size) / float64(m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + 1) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif m.buckets[j] == (pair{}) {\nreturn \"\"\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif m.buckets[j].key == key && m.buckets[j] != m.removed {\nreturn m.buckets[j].val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() > m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + i) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\nm.buckets[j] = pair{\nkey: key,\nval: val,\n}\nm.size += 1\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif m.buckets[j].key == key {\nm.buckets[j].val = val\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i < m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + 1) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif m.buckets[j] == (pair{}) {\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif m.buckets[j].key == key {\nm.buckets[j] = m.removed\nm.size -= 1\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([]pair, len(m.buckets))\ncopy(tmpBuckets, m.buckets)\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([]pair, m.capacity)\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, p := range tmpBuckets {\nif p != (pair{}) && p != m.removed {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\nfor _, p := range m.buckets {\nif p != (pair{}) {\nfmt.Println(strconv.Itoa(p.key) + \" -> \" + p.val)\n} else {\nfmt.Println(\"nil\")\n}\n}\n}\n
    hash_map_open_addressing.js
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.ts
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.c
    [class]{hashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.cs
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nPair[] buckets; // \u6876\u6570\u7ec4\nPair removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new Pair[capacity];\nremoved = new Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate double loadFactor() {\nreturn (double)size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (buckets[j] == null)\nreturn null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[j].key == key && buckets[j] != removed)\nreturn buckets[j].val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (buckets[j] == null || buckets[j] == removed) {\nbuckets[j] = new Pair(key, val);\nsize += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (buckets[j].key == key) {\nbuckets[j].val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (buckets[j].key == key) {\nbuckets[j] = removed;\nsize -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (Pair pair in bucketsTmp) {\nif (pair != null && pair != removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (Pair pair in buckets) {\nif (pair != null) {\nConsole.WriteLine(pair.key + \" -> \" + pair.val);\n} else {\nConsole.WriteLine(\"null\");\n}\n}\n}\n}\n
    hash_map_open_addressing.swift
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [Pair?] // \u6876\u6570\u7ec4\nvar removed: Pair // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2 / 3\nextendRatio = 2\nbuckets = Array(repeating: nil, count: capacity)\nremoved = Pair(key: -1, val: \"-1\")\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -> Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -> Double {\nDouble(size / capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, to: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de nil\nif buckets[j] == nil {\nreturn nil\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif buckets[j]?.key == key, buckets[j] != removed {\nreturn buckets[j]?.val\n}\n}\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() > loadThres {\nextend()\n}\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, through: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif buckets[j] == nil || buckets[j] == removed {\nbuckets[j] = Pair(key: key, val: val)\nsize += 1\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif buckets[j]?.key == key {\nbuckets[j]?.val = val\nreturn\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i in stride(from: 0, to: capacity, by: 1) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nlet j = (index + i) % capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif buckets[j] == nil {\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif buckets[j]?.key == key {\nbuckets[j] = removed\nsize -= 1\nreturn\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: nil, count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in bucketsTmp {\nif let pair, pair != removed {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in buckets {\nif let pair {\nSwift.print(\"\\(pair.key) -> \\(pair.val)\")\n} else {\nSwift.print(\"null\")\n}\n}\n}\n}\n
    hash_map_open_addressing.zig
    [class]{HashMapOpenAddressing}-[func]{}\n
    hash_map_open_addressing.dart
    /* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nlate int _size; // \u952e\u503c\u5bf9\u6570\u91cf\nlate int _capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nlate double _loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nlate int _extendRatio; // \u6269\u5bb9\u500d\u6570\nlate List<Pair?> _buckets; // \u6876\u6570\u7ec4\nlate Pair _removed; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() {\n_size = 0;\n_capacity = 4;\n_loadThres = 2.0 / 3.0;\n_extendRatio = 2;\n_buckets = List.generate(_capacity, (index) => null);\n_removed = Pair(-1, \"-1\");\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % _capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn _size / _capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif (_buckets[j] == null) return null;\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (_buckets[j]!.key == key && _buckets[j] != _removed)\nreturn _buckets[j]!.val;\n}\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() > _loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif (_buckets[j] == null || _buckets[j] == _removed) {\n_buckets[j] = new Pair(key, val);\n_size += 1;\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif (_buckets[j]!.key == key) {\n_buckets[j]!.val = val;\nreturn;\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor (int i = 0; i < _capacity; i++) {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nint j = (index + i) % _capacity;\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (_buckets[j] == null) {\nreturn;\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif (_buckets[j]!.key == key) {\n_buckets[j] = _removed;\n_size -= 1;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList<Pair?> bucketsTmp = _buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n_capacity *= _extendRatio;\n_buckets = List.generate(_capacity, (index) => null);\n_size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair? pair in bucketsTmp) {\nif (pair != null && pair != _removed) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (Pair? pair in _buckets) {\nif (pair != null) {\nprint(\"${pair.key} -> ${pair.val}\");\n} else {\nprint(null);\n}\n}\n}\n}\n
    "},{"location":"chapter_hashing/hash_collision/#_2","title":"\u591a\u6b21\u54c8\u5e0c","text":"

    \u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u662f\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) , \\(f_2(x)\\) , \\(f_3(x)\\) , \\(\\cdots\\) \u8fdb\u884c\u63a2\u6d4b\u3002

    • \u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u4f4d\u540e\u63d2\u5165\u5143\u7d20\u3002
    • \u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u6216\u9047\u5230\u7a7a\u4f4d\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002

    \u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u589e\u52a0\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002

    \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9

    Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u88ab\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002

    Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002

    Golang \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\uff1b\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002

    "},{"location":"chapter_hashing/hash_map/","title":"6.1. \u00a0 \u54c8\u5e0c\u8868","text":"

    \u300c\u54c8\u5e0c\u8868 Hash Table\u300d\u901a\u8fc7\u5efa\u7acb\u952e key \u4e0e\u503c value \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a key \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684 value \u3002

    \u4ee5\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5b66\u751f\u7684\u6570\u636e\u5e93\u4e3a\u4f8b\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002

    Fig. \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a

    \u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\u3002\u82e5\u5c06\u5b66\u751f\u6570\u636e\u770b\u4f5c\u6570\u7ec4\uff08\u94fe\u8868\uff09\u5143\u7d20\uff0c\u5219\u6709\uff1a

    • \u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\uff1b
    • \u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
    • \u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b
    \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\)

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u5e38\u7528\u4e8e\u5bf9\u67e5\u627e\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u7684\u573a\u666f\u3002

    "},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1. \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"

    \u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map.java
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<Integer, String> map = new HashMap<>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   map.put(15937, \"\u5c0f\u5570\");   map.put(16750, \"\u5c0f\u7b97\");   map.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n
    hash_map.cpp
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map<int, string> map;\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n
    hash_map.py
    # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nmapp: Dict = {}\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = mapp[15937]\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmapp.pop(10583)\n
    hash_map.go
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nmapp := make(map[int]string)\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmapp[12836] = \"\u5c0f\u54c8\"\nmapp[15937] = \"\u5c0f\u5570\"\nmapp[16750] = \"\u5c0f\u7b97\"\nmapp[13276] = \"\u5c0f\u6cd5\"\nmapp[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := mapp[15937]\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(mapp, 10583)\n
    hash_map.js
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new ArrayHashMap();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n
    hash_map.ts
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map<number, string>();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -> Value');\nconsole.info(map);\n
    hash_map.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
    hash_map.cs
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary<int, String> map = new ();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.Add(12836, \"\u5c0f\u54c8\");\nmap.Add(15937, \"\u5c0f\u5570\");\nmap.Add(16750, \"\u5c0f\u7b97\");\nmap.Add(13276, \"\u5c0f\u6cd5\");\nmap.Add(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n
    hash_map.swift
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n
    hash_map.zig
    \n
    hash_map.dart
    /* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap<int, String> map = {};\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n

    \u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart hash_map.java
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (Map.Entry <Integer, String> kv: map.entrySet()) {\nSystem.out.println(kv.getKey() + \" -> \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\nSystem.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\nSystem.out.println(val);\n}\n
    hash_map.cpp
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor (auto kv: map) {\ncout << kv.first << \" -> \" << kv.second << endl;\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (auto key: map) {\ncout << key.first << endl;\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (auto val: map) {\ncout << val.second << endl;\n}\n
    hash_map.py
    # \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value in mapp.items():\nprint(key, \"->\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in mapp.keys():\nprint(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in mapp.values():\nprint(value)\n
    hash_map_test.go
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key->value\nfor key, value := range mapp {\nfmt.Println(key, \"->\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range mapp {\nfmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range mapp {\nfmt.Println(value)\n}\n
    hash_map.js
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
    hash_map.ts
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key->Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -> ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n
    hash_map.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n
    hash_map.cs
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nforeach (var kv in map)\n{\nConsole.WriteLine(kv.Key + \" -> \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys)\n{\nConsole.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (String val in map.Values)\n{\nConsole.WriteLine(val);\n}\n
    hash_map.swift
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nfor (key, value) in map {\nprint(\"\\(key) -> \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\nprint(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\nprint(value)\n}\n
    hash_map.zig
    \n
    hash_map.dart
    /* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key->Value\nmap.forEach((key, value) {\nprint('$key -> $value');\n});\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\nprint(key);\n});\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\nprint(value);\n});\n
    "},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2. \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"

    \u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 Bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 key \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 value \u3002

    \u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e key \u6765\u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 Hash Function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 key \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a key \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 key \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002

    \u8f93\u5165\u4e00\u4e2a key \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\uff0c\u901a\u8fc7\u54c8\u5e0c\u7b97\u6cd5 hash() \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\uff1b\u63a5\u4e0b\u6765\uff0c\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09capacity \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 key \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 index \u3002

    index = hash(key) % capacity\n

    \u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 index \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 value \u3002

    \u8bbe\u6570\u7ec4\u957f\u5ea6 capacity = 100 \u3001\u54c8\u5e0c\u7b97\u6cd5 hash(key) = key \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a key % 100 \u3002\u4e0b\u56fe\u4ee5 key \u5b66\u53f7\u548c value \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002

    Fig. \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406

    \u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 key \u548c value \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b Pair \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_hash_map.java
    /* \u952e\u503c\u5bf9 */\nclass Pair {\npublic int key;\npublic String val;\npublic Pair(int key, String val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Pair> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new ArrayList<>();\nfor (int i = 0; i < 100; i++) {\nbuckets.add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\nPair pair = buckets.get(index);\nif (pair == null)\nreturn null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\nPair pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets.set(index, pair);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets.set(index, null);\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Pair> pairSet() {\nList<Pair> pairSet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\npairSet.add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<Integer> keySet() {\nList<Integer> keySet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\nkeySet.add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<String> valueSet() {\nList<String> valueSet = new ArrayList<>();\nfor (Pair pair : buckets) {\nif (pair != null)\nvalueSet.add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair kv : pairSet()) {\nSystem.out.println(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
    array_hash_map.cpp
    /* \u952e\u503c\u5bf9 */\nstruct Pair {\npublic:\nint key;\nstring val;\nPair(int key, string val) {\nthis->key = key;\nthis->val = val;\n}\n};\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate:\nvector<Pair *> buckets;\npublic:\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = vector<Pair *>(100);\n}\n~ArrayHashMap() {\n// \u91ca\u653e\u5185\u5b58\nfor (const auto &bucket : buckets) {\ndelete bucket;\n}\nbuckets.clear();\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\nPair *pair = buckets[index];\nif (pair == nullptr)\nreturn nullptr;\nreturn pair->val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\nPair *pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\ndelete buckets[index];\nbuckets[index] = nullptr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvector<Pair *> pairSet() {\nvector<Pair *> pairSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\npairSet.push_back(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvector<int> keySet() {\nvector<int> keySet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nkeySet.push_back(pair->key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvector<string> valueSet() {\nvector<string> valueSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nvalueSet.push_back(pair->val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Pair *kv : pairSet()) {\ncout << kv->key << \" -> \" << kv->val << endl;\n}\n}\n};\n
    array_hash_map.py
    class Pair:\n\"\"\"\u952e\u503c\u5bf9\"\"\"\ndef __init__(self, key: int, val: str):\nself.key = key\nself.val = val\nclass ArrayHashMap:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets: list[Pair | None] = [None] * 100\ndef hash_func(self, key: int) -> int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nindex = key % 100\nreturn index\ndef get(self, key: int) -> str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\npair: Pair = self.buckets[index]\nif pair is None:\nreturn None\nreturn pair.val\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\npair = Pair(key, val)\nindex: int = self.hash_func(key)\nself.buckets[index] = pair\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\n# \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None\ndef entry_set(self) -> list[Pair]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\nresult: list[Pair] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair)\nreturn result\ndef key_set(self) -> list[int]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.key)\nreturn result\ndef value_set(self) -> list[str]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.val)\nreturn result\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"->\", pair.val)\n
    array_hash_map.go
    /* \u952e\u503c\u5bf9 */\ntype pair struct {\nkey int\nval string\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\nbuckets []*pair\n}\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets := make([]*pair, 100)\nreturn &arrayHashMap{buckets: buckets}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\nindex := key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\nindex := a.hashFunc(key)\npair := a.buckets[index]\nif pair == nil {\nreturn \"Not Found\"\n}\nreturn pair.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\npair := &pair{key: key, val: val}\nindex := a.hashFunc(key)\na.buckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\nindex := a.hashFunc(key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\na.buckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\nvar pairs []*pair\nfor _, pair := range a.buckets {\nif pair != nil {\npairs = append(pairs, pair)\n}\n}\nreturn pairs\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\nvar keys []int\nfor _, pair := range a.buckets {\nif pair != nil {\nkeys = append(keys, pair.key)\n}\n}\nreturn keys\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\nvar values []string\nfor _, pair := range a.buckets {\nif pair != nil {\nvalues = append(values, pair.val)\n}\n}\nreturn values\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\nfor _, pair := range a.buckets {\nif pair != nil {\nfmt.Println(pair.key, \"->\", pair.val)\n}\n}\n}\n
    array_hash_map.js
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\nconstructor(key, val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n#buckets;\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.#buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nlet index = this.#hashFunc(key);\nlet pair = this.#buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nset(key, val) {\nlet index = this.#hashFunc(key);\nthis.#buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\ndelete(key) {\nlet index = this.#hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.#buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nentries() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nkeys() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvalues() {\nlet arr = [];\nfor (let i = 0; i < this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nif (!pair) continue;\nconsole.info(`${pair.key} -> ${pair.val}`);\n}\n}\n}\n
    array_hash_map.ts
    /* \u952e\u503c\u5bf9 Number -> String */\nclass Pair {\npublic key: number;\npublic val: string;\nconstructor(key: number, val: string) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate readonly buckets: (Pair | null)[];\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic get(key: number): string | null {\nlet index = this.hashFunc(key);\nlet pair = this.buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic set(key: number, val: string) {\nlet index = this.hashFunc(key);\nthis.buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic delete(key: number) {\nlet index = this.hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic entries(): (Pair | null)[] {\nlet arr: (Pair | null)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic keys(): (number | undefined)[] {\nlet arr: (number | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic values(): (string | undefined)[] {\nlet arr: (string | undefined)[] = [];\nfor (let i = 0; i < this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]?.val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic print() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nif (!pair) continue;\nconsole.info(`${pair.key} -> ${pair.val}`);\n}\n}\n}\n
    array_hash_map.c
    /* \u952e\u503c\u5bf9 int->string */\nstruct pair {\nint key;\nchar *val;\n};\ntypedef struct pair pair;\n[class]{arrayHashMap}-[func]{}\n
    array_hash_map.cs
    /* \u952e\u503c\u5bf9 int->string */\nclass Pair {\npublic int key;\npublic string val;\npublic Pair(int key, string val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List<Pair?> buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new();\nfor (int i = 0; i < 100; i++) {\nbuckets.Add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string? get(int key) {\nint index = hashFunc(key);\nPair? pair = buckets[index];\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, string val) {\nPair pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List<Pair> pairSet() {\nList<Pair> pairSet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\npairSet.Add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List<int> keySet() {\nList<int> keySet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\nkeySet.Add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List<string> valueSet() {\nList<string> valueSet = new();\nforeach (Pair? pair in buckets) {\nif (pair != null)\nvalueSet.Add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nforeach (Pair kv in pairSet()) {\nConsole.WriteLine(kv.key + \" -> \" + kv.val);\n}\n}\n}\n
    array_hash_map.swift
    /* \u952e\u503c\u5bf9 */\nclass Pair {\nvar key: Int\nvar val: String\ninit(key: Int, val: String) {\nself.key = key\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate var buckets: [Pair?] = []\ninit() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nfor _ in 0 ..< 100 {\nbuckets.append(nil)\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate func hashFunc(key: Int) -> Int {\nlet index = key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -> String? {\nlet index = hashFunc(key: key)\nlet pair = buckets[index]\nreturn pair?.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\nlet pair = Pair(key: key, val: val)\nlet index = hashFunc(key: key)\nbuckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nfunc pairSet() -> [Pair] {\nvar pairSet: [Pair] = []\nfor pair in buckets {\nif let pair = pair {\npairSet.append(pair)\n}\n}\nreturn pairSet\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc keySet() -> [Int] {\nvar keySet: [Int] = []\nfor pair in buckets {\nif let pair = pair {\nkeySet.append(pair.key)\n}\n}\nreturn keySet\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc valueSet() -> [String] {\nvar valueSet: [String] = []\nfor pair in buckets {\nif let pair = pair {\nvalueSet.append(pair.val)\n}\n}\nreturn valueSet\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in pairSet() {\nSwift.print(\"\\(pair.key) -> \\(pair.val)\")\n}\n}\n}\n
    array_hash_map.zig
    // \u952e\u503c\u5bf9\nconst Pair = struct {\nkey: usize = undefined,\nval: []const u8 = undefined,\npub fn init(key: usize, val: []const u8) Pair {\nreturn Pair {\n.key = key,\n.val = val,\n};\n}\n};\n// \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\nreturn struct {\nbucket: ?std.ArrayList(?T) = null,\nmem_allocator: std.mem.Allocator = undefined,\nconst Self = @This();\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nself.mem_allocator = allocator;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\nself.bucket = std.ArrayList(?T).init(self.mem_allocator);\nvar i: i32 = 0;\nwhile (i < 100) : (i += 1) {\ntry self.bucket.?.append(null);\n}\n}\n// \u6790\u6784\u51fd\u6570\npub fn deinit(self: *Self) void {\nif (self.bucket != null) self.bucket.?.deinit();\n}\n// \u54c8\u5e0c\u51fd\u6570\nfn hashFunc(key: usize) usize {\nvar index = key % 100;\nreturn index;\n}\n// \u67e5\u8be2\u64cd\u4f5c\npub fn get(self: *Self, key: usize) []const u8 {\nvar index = hashFunc(key);\nvar pair = self.bucket.?.items[index];\nreturn pair.?.val;\n}\n// \u6dfb\u52a0\u64cd\u4f5c\npub fn put(self: *Self, key: usize, val: []const u8) !void {\nvar pair = Pair.init(key, val);\nvar index = hashFunc(key);\nself.bucket.?.items[index] = pair;\n}\n// \u5220\u9664\u64cd\u4f5c\npub fn remove(self: *Self, key: usize) !void {\nvar index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nself.bucket.?.items[index] = null;\n}       // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\npub fn pairSet(self: *Self) !std.ArrayList(T) {\nvar entry_set = std.ArrayList(T).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry entry_set.append(item.?);\n}\nreturn entry_set;\n}  // \u83b7\u53d6\u6240\u6709\u952e\npub fn keySet(self: *Self) !std.ArrayList(usize) {\nvar key_set = std.ArrayList(usize).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry key_set.append(item.?.key);\n}\nreturn key_set;\n}  // \u83b7\u53d6\u6240\u6709\u503c\npub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\nvar value_set = std.ArrayList([]const u8).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry value_set.append(item.?.val);\n}\nreturn value_set;\n}\n// \u6253\u5370\u54c8\u5e0c\u8868\npub fn print(self: *Self) !void {\nvar entry_set = try self.pairSet();\ndefer entry_set.deinit();\nfor (entry_set.items) |item| {\nstd.debug.print(\"{} -> {s}\\n\", .{item.key, item.val});\n}\n}\n};\n}\n
    array_hash_map.dart
    /* \u952e\u503c\u5bf9 */\nclass Pair {\nint key;\nString val;\nPair(this.key, this.val);\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u7b80\u6613\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nlate List<Pair?> _buckets;\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n_buckets = List.filled(100, null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint _hashFunc(int key) {\nfinal int index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nfinal int index = _hashFunc(key);\nfinal Pair? pair = _buckets[index];\nif (pair == null) {\nreturn null;\n}\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\nfinal Pair pair = Pair(key, val);\nfinal int index = _hashFunc(key);\n_buckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nfinal int index = _hashFunc(key);\n_buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nList<Pair> pairSet() {\nList<Pair> pairSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\npairSet.add(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nList<int> keySet() {\nList<int> keySet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nkeySet.add(pair.key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nList<String> values() {\nList<String> valueSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nvalueSet.add(pair.val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (final Pair kv in pairSet()) {\nprint(\"${kv.key} -> ${kv.val}\");\n}\n}\n}\n
    "},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3. \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"

    \u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u8f93\u5165\u7a7a\u95f4\uff08key \u8303\u56f4\uff09\u6620\u5c04\u5230\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\uff09\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002

    \u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 key \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a

    12386 % 100 = 36\n20386 % 100 = 36\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 Hash Collision\u300d\u3002

    Fig. \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b

    \u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a key \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 (136, A) \u548c (236, D) \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002

    Fig. \u54c8\u5e0c\u8868\u6269\u5bb9

    \u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\u3002\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf capacity \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u56e0\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002

    \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u300c\u8d1f\u8f7d\u56e0\u5b50 Load Factor\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u4e3a\u4e86\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u88ab\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u5bb9\u91cf\u6269\u5c55\u4e3a\u539f\u5148\u7684 \\(2\\) \u500d\u3002

    "},{"location":"chapter_hashing/summary/","title":"6.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u8f93\u5165 key \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 value \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002
    • \u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002
    • \u54c8\u5e0c\u51fd\u6570\u5c06 key \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 value \u3002
    • \u4e24\u4e2a\u4e0d\u540c\u7684 key \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002
    • \u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002
    • \u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002
    • \u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002
    • \u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002
    • \u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 HashMap \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 Dict \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002
    • \u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002
    • \u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002
    • \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5, SHA-1, SHA-2, SHA3 \u7b49\u3002MD5 \u5e38\u7528\u8bed\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002
    • \u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002
    "},{"location":"chapter_hashing/summary/#641-q-a","title":"6.4.1. \u00a0 Q & A","text":"

    \u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(O(n)\\) \uff1f

    \u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u7684\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002

    \u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86

    \u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002

    \u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u4ed6\u4eec\u66f4\u9ad8\u5462\uff1f

    \u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u7684\u5185\u5b58\u662f\u672a\u4f7f\u7528\u7684\uff0c

    \u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002

    \u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002

    \u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u5bf9\u4e8e\u6807\u8bb0\u5df2\u5220\u9664\u7684\u7a7a\u95f4\uff0c\u8fd9\u4e2a\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f

    \u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u662f\u53ef\u4ee5\u518d\u6b21\u88ab\u4f7f\u7528\u7684\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u4e86\u88ab\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u7684\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002

    \u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f

    \u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 key \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002

    \u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f

    \u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5165\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c key \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a key \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002

    "},{"location":"chapter_heap/","title":"8. \u00a0 \u5806","text":""},{"location":"chapter_heap/build_heap/","title":"8.2. \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"

    \u5982\u679c\u6211\u4eec\u60f3\u8981\u6839\u636e\u8f93\u5165\u5217\u8868\u751f\u6210\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u300c\u5efa\u5806\u300d\u3002

    "},{"location":"chapter_heap/build_heap/#821","title":"8.2.1. \u00a0 \u501f\u52a9\u5165\u5806\u65b9\u6cd5\u5b9e\u73b0","text":"

    \u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u501f\u52a9\u201c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\uff0c\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u5c06\u5217\u8868\u5143\u7d20\u4f9d\u6b21\u6dfb\u52a0\u5230\u5806\u4e2d\u3002

    \u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u5165\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u5728\u4f9d\u6b21\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5806\u7684\u5e73\u5747\u957f\u5ea6\u4e3a \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8be5\u65b9\u6cd5\u7684\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002

    "},{"location":"chapter_heap/build_heap/#822","title":"8.2.2. \u00a0 \u57fa\u4e8e\u5806\u5316\u64cd\u4f5c\u5b9e\u73b0","text":"

    \u6709\u8da3\u7684\u662f\uff0c\u5b58\u5728\u4e00\u79cd\u66f4\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u4ec5\u4e3a \\(O(n)\\) \u3002\u6211\u4eec\u5148\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u7136\u540e\u8fed\u4ee3\u5730\u5bf9\u5404\u4e2a\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002\u5f53\u7136\uff0c\u6211\u4eec\u4e0d\u9700\u8981\u5bf9\u53f6\u8282\u70b9\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff0c\u56e0\u4e3a\u5b83\u4eec\u6ca1\u6709\u5b50\u8282\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<Integer> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new ArrayList<>(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.cpp
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.py
    def __init__(self, nums: list[int]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n# \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nself.max_heap = nums\n# \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(self.parent(self.size() - 1), -1, -1):\nself.sift_down(i)\n
    my_heap.go
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nh := &maxHeap{data: nums}\nfor i := len(h.data) - 1; i >= 0; i-- {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nh.siftDown(i)\n}\nreturn h\n}\n
    my_heap.js
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.#maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.#parent(this.size() - 1); i >= 0; i--) {\nthis.#siftDown(i);\n}\n}\n
    my_heap.ts
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.parent(this.size() - 1); i >= 0; i--) {\nthis.siftDown(i);\n}\n}\n
    my_heap.c
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nmaxHeap *newMaxHeap(int nums[], int size) {\n// \u6240\u6709\u5143\u7d20\u5165\u5806\nmaxHeap *h = (maxHeap *)malloc(sizeof(maxHeap));\nh->size = size;\nmemcpy(h->data, nums, size * sizeof(int));\nfor (int i = size - 1; i >= 0; i--) {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nsiftDown(h, i);\n}\nreturn h;\n}\n
    my_heap.cs
    /* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new List<int>(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar size = parent(this.size() - 1);\nfor (int i = size; i >= 0; i--) {\nsiftDown(i);\n}\n}\n
    my_heap.swift
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\nsiftDown(i: i)\n}\n}\n
    my_heap.zig
    // \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\nif (self.max_heap != null) return;\nself.max_heap = std.ArrayList(T).init(allocator);\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\ntry self.max_heap.?.appendSlice(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar i: usize = parent(self.size() - 1) + 1;\nwhile (i > 0) : (i -= 1) {\ntry self.siftDown(i - 1);\n}\n}\n
    my_heap.dart
    /* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List<int> nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n_maxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = _parent(size() - 1); i >= 0; i--) {\n_siftDown(i);\n}\n}\n
    "},{"location":"chapter_heap/build_heap/#823","title":"8.2.3. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"

    \u4e3a\u4ec0\u4e48\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \uff1f\u6211\u4eec\u6765\u5c55\u5f00\u63a8\u7b97\u4e00\u4e0b\u3002

    • \u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\uff0c\u5728\u6392\u9664\u53f6\u8282\u70b9\u540e\uff0c\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1)/2\\) \uff0c\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b
    • \u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(O(\\log n)\\) \uff1b

    \u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u7279\u6027\u3002

    \u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u8be6\u7ec6\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u51cf\u5c0f\u8ba1\u7b97\u96be\u5ea6\uff0c\u6211\u4eec\u5047\u8bbe\u6811\u662f\u4e00\u4e2a\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002\u8bbe\u4e8c\u53c9\u6811\uff08\u5373\u5806\uff09\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \u3002\u4e0a\u6587\u63d0\u5230\uff0c\u8282\u70b9\u5806\u5316\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf

    \u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002

    \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{(h-1)}\\times1 \\]

    \u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230

    \\[ \\begin{aligned} T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \\cdots + 2^{h-1}\\times1 \\newline 2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \\cdots + 2^{h}\\times1 \\newline \\end{aligned} \\]

    \u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u4ee4\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97

    \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\cdots + 2^{h-1} + 2^h \\]

    \u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a

    \\[ \\begin{aligned} T(h) & = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline & = 2^{h+1} - h \\newline & = O(2^h) \\end{aligned} \\]

    \u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002

    "},{"location":"chapter_heap/heap/","title":"8.1. \u00a0 \u5806","text":"

    \u300c\u5806 Heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff1a

    • \u300c\u5927\u9876\u5806 Max Heap\u300d\uff0c\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\uff1b
    • \u300c\u5c0f\u9876\u5806 Min Heap\u300d\uff0c\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\uff1b

    Fig. \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806

    \u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002
    • \u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u300c\u5806\u9876\u300d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u300c\u5806\u5e95\u300d\u3002
    • \u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u5373\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002
    "},{"location":"chapter_heap/heap/#811","title":"8.1.1. \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"

    \u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 Priority Queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002

    \u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4f5c\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u300c\u4f18\u5148\u961f\u5217\u300d\u548c\u300c\u5806\u300d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u4f7f\u7528\u300c\u5806\u300d\u6765\u547d\u540d\u3002

    \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u4e0b\u8868\uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\)

    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002

    Tip

    \u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539 Comparator \u6765\u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart heap.java
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue<Integer> minHeap = new PriorityQueue<>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue<Integer> maxHeap = new PriorityQueue<>((a, b) -> b - a);\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = heap.poll();  // 5\npeek = heap.poll();  // 4\npeek = heap.poll();  // 3\npeek = heap.poll();  // 2\npeek = heap.poll();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4));\n
    heap.cpp
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue<int, vector<int>, greater<int>> minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue<int, vector<int>, less<int>> maxHeap;\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector<int> input{1, 3, 2, 5, 4};\npriority_queue<int, vector<int>, greater<int>> minHeap(input.begin(), input.end());\n
    heap.py
    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: List[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n
    heap.go
    // Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n// Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n// \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n*h = append(*h, x.(int))\n}\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n// \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\nlast := (*h)[len(*h)-1]\n*h = (*h)[:len(*h)-1]\nreturn last\n}\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\nreturn len(*h)\n}\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n// \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\nreturn (*h)[i].(int) > (*h)[j].(int)\n}\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n(*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\nreturn (*h)[0]\n}\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nmaxHeap := &intHeap{}\nheap.Init(maxHeap)\n/* \u5143\u7d20\u5165\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\nheap.Push(maxHeap, 1)\nheap.Push(maxHeap, 3)\nheap.Push(maxHeap, 2)\nheap.Push(maxHeap, 4)\nheap.Push(maxHeap, 5)\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\ntop := maxHeap.Top()\nfmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\nheap.Pop(maxHeap) // 5\nheap.Pop(maxHeap) // 4\nheap.Pop(maxHeap) // 3\nheap.Pop(maxHeap) // 2\nheap.Pop(maxHeap) // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nsize := len(*maxHeap)\nfmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(*maxHeap) == 0\nfmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n
    heap.js
    // JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.ts
    // TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.cs
    /* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue<int, int> minHeap = new PriorityQueue<int, int>();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue<int, int> maxHeap = new PriorityQueue<int, int>(Comparer<int>.Create((x, y) => y - x));\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue<int, int>(new List<(int, int)> { (1, 1), (3, 3), (2, 2), (5, 5), (4, 4), });\n
    heap.swift
    // Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    heap.zig
    \n
    heap.dart
    // Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n
    "},{"location":"chapter_heap/heap/#812","title":"8.1.2. \u00a0 \u5806\u7684\u5b9e\u73b0","text":"

    \u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002

    "},{"location":"chapter_heap/heap/#_1","title":"\u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"

    \u6211\u4eec\u5728\u4e8c\u53c9\u6811\u7ae0\u8282\u4e2d\u5b66\u4e60\u5230\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002

    \u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002

    \u5177\u4f53\u800c\u8a00\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u53d6\u6574\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002

    Fig. \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8

    \u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.cpp
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u53d6\u6574\n}\n
    my_heap.py
    def left(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -> int:\n\"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n
    my_heap.go
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n// \u5411\u4e0b\u6574\u9664\nreturn (i - 1) / 2\n}\n
    my_heap.js
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.ts
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.c
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(maxHeap *h, int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(maxHeap *h, int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(maxHeap *h, int i) {\nreturn (i - 1) / 2;\n}\n
    my_heap.cs
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.swift
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -> Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -> Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -> Int {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n
    my_heap.zig
    // \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\nreturn 2 * i + 1;\n}\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\nreturn 2 * i + 2;\n}\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n// return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\nreturn @divFloor(i - 1, 2);\n}\n
    my_heap.dart
    /* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\nreturn (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n
    "},{"location":"chapter_heap/heap/#_2","title":"\u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"

    \u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap.get(0);\n}\n
    my_heap.cpp
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
    my_heap.py
    def peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\nreturn self.max_heap[0]\n
    my_heap.go
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\nreturn h.data[0]\n}\n
    my_heap.js
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\nreturn this.#maxHeap[0];\n}\n
    my_heap.ts
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\nreturn this.maxHeap[0];\n}\n
    my_heap.c
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(maxHeap *h) {\nreturn h->data[0];\n}\n
    my_heap.cs
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n
    my_heap.swift
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nmaxHeap[0]\n}\n
    my_heap.zig
    // \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\nreturn self.max_heap.?.items[0];\n}
    my_heap.dart
    /* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn _maxHeap[0];\n}\n
    "},{"location":"chapter_heap/heap/#_3","title":"\u5143\u7d20\u5165\u5806","text":"

    \u7ed9\u5b9a\u5143\u7d20 val \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\u3002\u56e0\u6b64\uff0c\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 Heapify\u300d\u3002

    \u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002

    <1><2><3><4><5><6><7><8><9>

    \u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap.get(i) <= maxHeap.get(p))\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.cpp
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.push_back(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(maxHeap[i], maxHeap[p]);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.py
    def push(self, val: int):\n\"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n# \u6dfb\u52a0\u8282\u70b9\nself.max_heap.append(val)\n# \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1)\ndef sift_up(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\nwhile True:\n# \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np = self.parent(i)\n# \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 or self.max_heap[i] <= self.max_heap[p]:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, p)\n# \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n
    my_heap.go
    /* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n// \u6dfb\u52a0\u8282\u70b9\nh.data = append(h.data, val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nh.siftUp(len(h.data) - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\nfor true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np := h.parent(i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || h.data[i].(int) <= h.data[p].(int) {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
    my_heap.js
    /* \u5143\u7d20\u5165\u5806 */\npush(val) {\n// \u6dfb\u52a0\u8282\u70b9\nthis.#maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.#siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.#parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.#maxHeap[i] <= this.#maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.ts
    /* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n// \u6dfb\u52a0\u8282\u70b9\nthis.maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || this.maxHeap[i] <= this.maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.c
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(maxHeap *h, int val) {\n// \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\nif (h->size == MAX_SIZE) {\nprintf(\"heap is full!\");\nreturn;\n}\n// \u6dfb\u52a0\u8282\u70b9\nh->data[h->size] = val;\nh->size++;\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(h, h->size - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(maxHeap *h, int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(h, i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 || h->data[i] <= h->data[p]) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(h, i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.cs
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.Add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (p < 0 || maxHeap[i] <= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.swift
    /* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.append(val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(i: size() - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\nvar i = i\nwhile true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nlet p = parent(i: i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p < 0 || maxHeap[i] <= maxHeap[p] {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n
    my_heap.zig
    // \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n// \u6dfb\u52a0\u8282\u70b9\ntry self.max_heap.?.append(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\ntry self.siftUp(self.size() - 1);\n}  // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nvar p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u9700\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p < 0 or self.max_heap.?.items[i] <= self.max_heap.?.items[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n
    my_heap.dart
    /* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\n_maxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\n_siftUp(size() - 1);\n}\n[class]{MaxHeap}-[func]{siftUp}\n
    "},{"location":"chapter_heap/heap/#_4","title":"\u5806\u9876\u5143\u7d20\u51fa\u5806","text":"

    \u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u53d6\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\uff1a

    1. \u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u5373\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\uff1b
    2. \u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\uff1b
    3. \u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\uff1b

    \u987e\u540d\u601d\u4e49\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u4ece\u5e95\u81f3\u9876\u5806\u5316\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\uff1b\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u9700\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002

    <1><2><3><4><5><6><7><8><9><10>

    \u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart my_heap.java
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.remove(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap.get(l) > maxHeap.get(ma))\nma = l;\nif (r < size() && maxHeap.get(r) > maxHeap.get(ma))\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.cpp
    /* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n// \u5224\u7a7a\u5904\u7406\nif (empty()) {\nthrow out_of_range(\"\u5806\u4e3a\u7a7a\");\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap[0], maxHeap[size() - 1]);\n// \u5220\u9664\u8282\u70b9\nmaxHeap.pop_back();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\nswap(maxHeap[i], maxHeap[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.py
    def pop(self) -> int:\n\"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n# \u5224\u7a7a\u5904\u7406\nif self.is_empty():\nraise IndexError(\"\u5806\u4e3a\u7a7a\")\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1)\n# \u5220\u9664\u8282\u70b9\nval = self.max_heap.pop()\n# \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0)\n# \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\ndef sift_down(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl, r, ma = self.left(i), self.right(i), i\nif l < self.size() and self.max_heap[l] > self.max_heap[ma]:\nma = l\nif r < self.size() and self.max_heap[r] > self.max_heap[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, ma)\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n
    my_heap.go
    /* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n// \u5224\u7a7a\u5904\u7406\nif h.isEmpty() {\nfmt.Println(\"error\")\nreturn nil\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nh.swap(0, h.size()-1)\n// \u5220\u9664\u8282\u70b9\nval := h.data[len(h.data)-1]\nh.data = h.data[:len(h.data)-1]\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nh.siftDown(0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nl, r, max := h.left(i), h.right(i), i\nif l < h.size() && h.data[l].(int) > h.data[max].(int) {\nmax = l\n}\nif r < h.size() && h.data[r].(int) > h.data[max].(int) {\nmax = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif max == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, max)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max\n}\n}\n
    my_heap.js
    /* \u5143\u7d20\u51fa\u5806 */\npop() {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.#swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.#maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.#siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.#left(i),\nr = this.#right(i);\nlet ma = i;\nif (l < this.size() && this.#maxHeap[l] > this.#maxHeap[ma]) ma = l;\nif (r < this.size() && this.#maxHeap[r] > this.#maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.ts
    /* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new RangeError('Heap is empty.');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.left(i),\nr = this.right(i);\nlet ma = i;\nif (l < this.size() && this.maxHeap[l] > this.maxHeap[ma]) ma = l;\nif (r < this.size() && this.maxHeap[r] > this.maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.c
    /* \u5143\u7d20\u51fa\u5806 */\nint pop(maxHeap *h) {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty(h)) {\nprintf(\"heap is empty!\");\nreturn INT_MAX;\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(h, 0, size(h) - 1);\n// \u5220\u9664\u8282\u70b9\nint val = h->data[h->size - 1];\nh->size--;\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(h, 0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(maxHeap *h, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nint l = left(h, i);\nint r = right(h, i);\nint max = i;\nif (l < size(h) && h->data[l] > h->data[max]) {\nmax = l;\n}\nif (r < size(h) && h->data[r] > h->data[max]) {\nmax = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (max == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(h, i, max);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max;\n}\n}\n
    my_heap.cs
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfRangeException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.Last();\nmaxHeap.RemoveAt(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l < size() && maxHeap[l] > maxHeap[ma])\nma = l;\nif (r < size() && maxHeap[r] > maxHeap[ma])\nma = r;\n// \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.swift
    /* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -> Int {\n// \u5224\u7a7a\u5904\u7406\nif isEmpty() {\nfatalError(\"\u5806\u4e3a\u7a7a\")\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(i: 0, j: size() - 1)\n// \u5220\u9664\u8282\u70b9\nlet val = maxHeap.remove(at: size() - 1)\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(i: 0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = left(i: i)\nlet r = right(i: i)\nvar ma = i\nif l < size(), maxHeap[l] > maxHeap[ma] {\nma = l\n}\nif r < size(), maxHeap[r] > maxHeap[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n
    my_heap.zig
    // \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n// \u5224\u65ad\u5904\u7406\nif (self.isEmpty()) unreachable;\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\ntry self.swap(0, self.size() - 1);\n// \u5220\u9664\u8282\u70b9\nvar val = self.max_heap.?.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\ntry self.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n} // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nvar l = left(i);\nvar r = right(i);\nvar ma = i;\nif (l < self.size() and self.max_heap.?.items[l] > self.max_heap.?.items[ma]) ma = l;\nif (r < self.size() and self.max_heap.?.items[r] > self.max_heap.?.items[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n
    my_heap.dart
    /* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n_swap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = _maxHeap.removeLast();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\n_siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n[class]{MaxHeap}-[func]{siftDown}\n
    "},{"location":"chapter_heap/heap/#813","title":"8.1.3. \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"
    • \u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002
    • \u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u540e\u7eed\u7684\u5806\u6392\u5e8f\u7ae0\u8282\u3002
    • \u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002
    "},{"location":"chapter_heap/summary/","title":"8.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002
    • \u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002
    • \u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) \u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002
    • \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002
    • \u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002
    • \u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002
    • Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002
    "},{"location":"chapter_heap/summary/#841-q-a","title":"8.4.1. \u00a0 Q & A","text":"

    \u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f

    \u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u9732\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4e0d\u6070\u5f53\u7684\u4f7f\u7528\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u9732\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002

    "},{"location":"chapter_heap/top_k/","title":"8.3. \u00a0 Top-K \u95ee\u9898","text":"

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u65e0\u5e8f\u6570\u7ec4 nums \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002

    \u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002

    "},{"location":"chapter_heap/top_k/#831","title":"8.3.1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"

    \u6211\u4eec\u53ef\u4ee5\u8fdb\u884c \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\) , \\(2\\) , \\(\\cdots\\) , \\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002

    \u8be5\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002

    Fig. \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20

    Tip

    \u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\uff0c\u7b49\u4ef7\u4e8e\u300c\u9009\u62e9\u6392\u5e8f\u300d\u7b97\u6cd5\u3002

    "},{"location":"chapter_heap/top_k/#832","title":"8.3.2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"

    \u6211\u4eec\u53ef\u4ee5\u5bf9\u6570\u7ec4 nums \u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002

    \u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u8981\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002

    Fig. \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20

    "},{"location":"chapter_heap/top_k/#833","title":"8.3.3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\uff1b
    2. \u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\uff1b
    3. \u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\uff1b
    4. \u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1b
    <1><2><3><4><5><6><7><8><9>

    \u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002

    \u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart top_k.java
    /* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue<Integer> topKHeap(int[] nums, int k) {\nQueue<Integer> heap = new PriorityQueue<Integer>();\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i < k; i++) {\nheap.add(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i < nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] > heap.peek()) {\nheap.poll();\nheap.add(nums[i]);\n}\n}\nreturn heap;\n}\n
    top_k.cpp
    /* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue<int, vector<int>, greater<int>> topKHeap(vector<int> &nums, int k) {\npriority_queue<int, vector<int>, greater<int>> heap;\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i < k; i++) {\nheap.push(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i < nums.size(); i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] > heap.top()) {\nheap.pop();\nheap.push(nums[i]);\n}\n}\nreturn heap;\n}\n
    top_k.py
    def top_k_heap(nums: list[int], k: int) -> list[int]:\n\"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\nheap = []\n# \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor i in range(k):\nheapq.heappush(heap, nums[i])\n# \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor i in range(k, len(nums)):\n# \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif nums[i] > heap[0]:\nheapq.heappop(heap)\nheapq.heappush(heap, nums[i])\nreturn heap\n
    top_k.go
    [class]{maxHeap}-[func]{topKHeap}\n
    top_k.js
    [class]{}-[func]{topKHeap}\n
    top_k.ts
    [class]{}-[func]{topKHeap}\n
    top_k.c
    [class]{maxHeap}-[func]{topKHeap}\n
    top_k.cs
    [class]{top_k}-[func]{topKHeap}\n
    top_k.swift
    [class]{}-[func]{topKHeap}\n
    top_k.zig
    [class]{}-[func]{topKHeap}\n
    top_k.dart
    [class]{}-[func]{top_k_heap}\n
    "},{"location":"chapter_introduction/","title":"1. \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":""},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1. \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"

    \u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u4e8e\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002

    \u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\uff0c\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002

    \u4f8b\u4e00\uff1a\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002

    \u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u79ef\u6728\u7684\u5404\u79cd\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u800c\u7ec4\u88c5\u8bf4\u660e\u4e66\u4e0a\u7684\u4e00\u7cfb\u5217\u6b65\u9aa4\u5219\u662f\u7b97\u6cd5\u3002

    Fig. \u62fc\u88c5\u79ef\u6728

    \u4f8b\u4e8c\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u7684\u82f1\u6587\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u8fd9\u6837\u64cd\u4f5c\uff1a

    1. \u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff08\u5047\u8bbe\u4e3a \\(m\\) \uff09\uff1b
    2. \u7531\u4e8e\u5728\u82f1\u6587\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\uff1b
    3. \u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 1-2 \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002
    <1><2><3><4><5>

    \u67e5\u9605\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u300c\u4e8c\u5206\u67e5\u627e\u300d\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u300c\u6570\u7ec4\u300d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u662f\u300c\u4e8c\u5206\u67e5\u627e\u300d\u7b97\u6cd5\u3002

    \u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u6536\u94f6\u5458\u4ed8\u4e86 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u7ed9\u6211\u4eec\u627e \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u4ee5\u4e0b\u601d\u8003\uff1a

    1. \u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) , \\(5\\) , \\(10\\) , \\(20\\) \u5143\u3002
    2. \u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002
    3. \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002
    4. \u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002
    5. \u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002

    \u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u300c\u8d2a\u5fc3\u7b97\u6cd5\u300d\u3002

    Fig. \u8d27\u5e01\u627e\u96f6

    \u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002

    Tip

    \u9605\u8bfb\u81f3\u6b64\uff0c\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u90a3\u4e48\u592a\u597d\u4e86\uff01\u56e0\u4e3a\u8fd9\u6b63\u662f\u672c\u4e66\u5b58\u5728\u7684\u610f\u4e49\u3002\u63a5\u4e0b\u6765\uff0c\u8fd9\u672c\u4e66\u5c06\u4e00\u6b65\u6b65\u5f15\u5bfc\u4f60\u6df1\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002

    "},{"location":"chapter_introduction/summary/","title":"1.3. \u00a0 \u5c0f\u7ed3","text":"
    • \u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002
    • \u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002
    • \u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002
    • \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u5219\u662f\u53d1\u6325\u6570\u636e\u7ed3\u6784\u4f5c\u7528\u7684\u821e\u53f0\u3002
    • \u4e50\u9ad8\u79ef\u6728\u5bf9\u5e94\u4e8e\u6570\u636e\uff0c\u79ef\u6728\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002
    "},{"location":"chapter_introduction/what_is_dsa/","title":"1.2. \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1. \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"

    \u300c\u7b97\u6cd5 Algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\u3002\u7b97\u6cd5\u5177\u6709\u4ee5\u4e0b\u7279\u6027\uff1a

    • \u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002
    • \u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002
    • \u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002
    "},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2. \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"

    \u300c\u6570\u636e\u7ed3\u6784 Data Structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u4e3a\u4e86\u63d0\u9ad8\u6570\u636e\u5b58\u50a8\u548c\u64cd\u4f5c\u6027\u80fd\uff0c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u76ee\u6807\u5305\u62ec\uff1a

    • \u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u51cf\u5c11\uff0c\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002
    • \u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002
    • \u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u4f7f\u5f97\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002

    \u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\uff0c\u8fd9\u610f\u5473\u7740\u8981\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u4f18\u52bf\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4f8b\u5982\uff0c\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\uff1b\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002

    "},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3. \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"

    \u300c\u6570\u636e\u7ed3\u6784\u300d\u4e0e\u300c\u7b97\u6cd5\u300d\u9ad8\u5ea6\u76f8\u5173\u4e14\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\uff1a

    • \u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u7528\u4e8e\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002
    • \u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u901a\u8fc7\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002
    • \u7279\u5b9a\u7b97\u6cd5\u901a\u5e38\u6709\u5bf9\u5e94\u6700\u4f18\u7684\u6570\u636e\u7ed3\u6784\u3002\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u8fdb\u884c\u5b9e\u73b0\uff0c\u4f46\u6700\u7ec8\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\u3002

    Fig. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb

    \u7c7b\u6bd4\u300cLEGO \u4e50\u9ad8\u300d\u548c\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\uff0c\u5219\u5bf9\u5e94\u5173\u7cfb\u5982\u4e0b\u8868\u6240\u793a\u3002

    \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 LEGO \u4e50\u9ad8 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b

    \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002

    \u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0

    \u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u300c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u300d\u7b80\u79f0\u4e3a\u300c\u7b97\u6cd5\u300d\u3002\u4f8b\u5982\uff0c\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u5bdf\u4e86\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002

    "},{"location":"chapter_preface/","title":"0. \u00a0 \u524d\u8a00","text":""},{"location":"chapter_preface/about_the_book/","title":"0.1. \u00a0 \u5173\u4e8e\u672c\u4e66","text":"

    \u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002

    • \u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\uff1b
    • \u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Java, C++, Python, Go, JS, TS, C#, Swift, Zig \u7b49\u8bed\u8a00\uff1b
    • \u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\uff1b
    "},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1. \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"

    \u82e5\u60a8\u662f\u300c\u7b97\u6cd5\u521d\u5b66\u8005\u300d\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u8fd9\u672c\u4e66\u6b63\u662f\u4e3a\u60a8\u91cf\u8eab\u5b9a\u5236\uff01

    \u5982\u679c\u60a8\u662f\u300c\u7b97\u6cd5\u8001\u624b\u300d\uff0c\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u60a8\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u88ab\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002

    \u82e5\u60a8\u662f\u300c\u7b97\u6cd5\u5927\u795e\u300d\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u60a8\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002

    \u524d\u7f6e\u6761\u4ef6

    \u60a8\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002

    "},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2. \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"

    \u672c\u4e66\u4e3b\u8981\u5185\u5bb9\u5305\u62ec\uff1a

    • \u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u3001\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u4f30\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5305\u62ec\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002
    • \u6570\u636e\u7ed3\u6784\uff1a\u5e38\u89c1\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u5728\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u5f62\u5f0f\u3001\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6d89\u53ca\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u6563\u5217\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\uff0c\u5185\u5bb9\u5305\u62ec\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002
    • \u7b97\u6cd5\uff1a\u67e5\u627e\u7b97\u6cd5\u3001\u6392\u5e8f\u7b97\u6cd5\u3001\u641c\u7d22\u4e0e\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u5206\u6cbb\u7b97\u6cd5\u7b49\uff0c\u5185\u5bb9\u6db5\u76d6\u5b9a\u4e49\u3001\u5e94\u7528\u573a\u666f\u3001\u4f18\u7f3a\u70b9\u3001\u65f6\u7a7a\u6548\u7387\u3001\u5b9e\u73b0\u65b9\u6cd5\u3001\u793a\u4f8b\u9898\u76ee\u7b49\u3002

    Fig. Hello \u7b97\u6cd5\u5185\u5bb9\u7ed3\u6784

    "},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3. \u00a0 \u81f4\u8c22","text":"

    \u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a

    • \u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u60a8\u9f13\u52b1\u6211\u201c\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\u3002
    • \u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\u3002
    • \u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\u3002
    • \u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c LOGO\uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\u3002
    • \u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u6770\u51fa\u7684\u5f00\u6e90\u9879\u76ee Material-for-MkDocs \u3002

    \u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01

    \u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u8457\u4f5c\u3002

    \u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4e9b\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002

    "},{"location":"chapter_preface/suggestions/","title":"0.2. \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"

    Tip

    \u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u60a8\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002

    "},{"location":"chapter_preface/suggestions/#021","title":"0.2.1. \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"
    • \u6807\u9898\u540e\u6807\u6ce8 * \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002
    • \u6587\u7ae0\u4e2d\u7684\u91cd\u8981\u540d\u8bcd\u4f1a\u7528 \u300c \u300d \u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982 \u300c\u6570\u7ec4 Array\u300d \u3002\u8bf7\u52a1\u5fc5\u8bb0\u4f4f\u8fd9\u4e9b\u540d\u8bcd\uff0c\u5305\u62ec\u82f1\u6587\u7ffb\u8bd1\uff0c\u4ee5\u4fbf\u540e\u7eed\u9605\u8bfb\u6587\u732e\u65f6\u4f7f\u7528\u3002
    • \u52a0\u7c97\u7684\u6587\u5b57 \u8868\u793a\u91cd\u70b9\u5185\u5bb9\u6216\u603b\u7ed3\u6027\u8bed\u53e5\uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002
    • \u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 \u201c\u53cc\u5f15\u53f7\u201d \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002
    • \u6d89\u53ca\u5230\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002
    • \u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    \"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    // \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n// \u591a\u884c\n// \u6ce8\u91ca\n
    /* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n
    "},{"location":"chapter_preface/suggestions/#022","title":"0.2.2. \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"

    \u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u56e0\u6b64\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002

    \u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5982\u679c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u5efa\u8bae\u4ee5\u56fe\u4e3a\u4e3b\u7ebf\uff0c\u4ee5\u6587\u5b57\uff08\u901a\u5e38\u4f4d\u4e8e\u56fe\u50cf\u4e0a\u65b9\uff09\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002

    Fig. \u52a8\u753b\u56fe\u89e3\u793a\u4f8b

    "},{"location":"chapter_preface/suggestions/#023","title":"0.2.3. \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"

    \u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728GitHub \u4ed3\u5e93\uff0c\u6e90\u4ee3\u7801\u5305\u542b\u8be6\u7ec6\u6ce8\u91ca\uff0c\u5e76\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u76f4\u63a5\u8fd0\u884c\u3002

    \u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u5efa\u8bae\u4f60\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002\u5982\u679c\u65f6\u95f4\u5145\u88d5\uff0c\u5efa\u8bae\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u4e0e\u4ec5\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002

    Fig. \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b

    \u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002

    \u7b2c\u4e8c\u6b65\uff1a\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u3002\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\u3002

    git clone https://github.com/krahets/hello-algo.git\n

    \u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u70b9\u51fb\u201cDownload ZIP\u201d\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002

    Fig. \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801

    \u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u679c\u4ee3\u7801\u5757\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\uff0c\u5219\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 codes \u6587\u4ef6\u5939\u4e2d\u627e\u5230\u76f8\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002

    Fig. \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6

    "},{"location":"chapter_preface/suggestions/#024","title":"0.2.4. \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"

    \u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u201c\u60ef\u7740\u201d\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5176\u4ed6\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u5f97\u5230\u56de\u590d\u3002

    \u540c\u65f6\uff0c\u4e5f\u5e0c\u671b\u60a8\u80fd\u5728\u8bc4\u8bba\u533a\u591a\u82b1\u4e9b\u65f6\u95f4\u3002\u4e00\u65b9\u9762\uff0c\u60a8\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u8fd9\u5c06\u6709\u52a9\u4e8e\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5e0c\u671b\u60a8\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\u3001\u5206\u4eab\u60a8\u7684\u89c1\u89e3\uff0c\u8ba9\u5927\u5bb6\u5171\u540c\u5b66\u4e60\u548c\u8fdb\u6b65\u3002

    Fig. \u8bc4\u8bba\u533a\u793a\u4f8b

    "},{"location":"chapter_preface/suggestions/#025","title":"0.2.5. \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"

    \u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\uff1a

    1. \u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u5185\u5bb9\u3002
    2. \u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u5251\u6307 Offer\u548cLeetCode Hot 100\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3-5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002
    3. \u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002

    \u4f5c\u4e3a\u4e00\u672c\u5165\u95e8\u6559\u7a0b\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u7b2c\u4e00\u9636\u6bb5\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u7b2c\u4e8c\u548c\u7b2c\u4e09\u9636\u6bb5\u7684\u5b66\u4e60\u3002

    Fig. \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf

    "},{"location":"chapter_preface/summary/","title":"0.3. \u00a0 \u5c0f\u7ed3","text":"
    • \u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u60a8\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u5185\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002
    • \u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002
    • \u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u7c4d\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002
    • \u4e66\u5185\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002
    • \u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u6253\u4ee3\u7801\u3002
    • \u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002
    "},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"

    [1] Thomas H. Cormen, et al. Introduction to Algorithms (3rd Edition).

    [2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1st Edition).

    [3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.

    [4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.

    [5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.

    [6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.

    [7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.

    [8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6th Edition).

    [9] Aston Zhang, et al. Dive into Deep Learning.

    "},{"location":"chapter_searching/","title":"10. \u00a0 \u641c\u7d22","text":""},{"location":"chapter_searching/binary_search/","title":"10.1. \u00a0 \u4e8c\u5206\u67e5\u627e","text":"

    \u300c\u4e8c\u5206\u67e5\u627e Binary Search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u51cf\u5c11\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\uff0c\u6570\u7ec4\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 target \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002

    \u5bf9\u4e8e\u4e0a\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u5176\u4e2d\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u201c\u95ed\u533a\u95f4\u201d\uff0c\u5373\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002

    \u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u4e2a\u6b65\u9aa4\uff1a

    1. \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\space \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002
    2. \u5224\u65ad nums[m] \u548c target \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a
      1. \u5f53 nums[m] < target \u65f6\uff0c\u8bf4\u660e target \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \uff1b
      2. \u5f53 nums[m] > target \u65f6\uff0c\u8bf4\u660e target \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \uff1b
      3. \u5f53 nums[m] = target \u65f6\uff0c\u8bf4\u660e\u627e\u5230 target \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \uff1b

    \u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002

    <0><1><2><3><4><5><6><7>

    \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f int \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa int \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector<int> &nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.size() - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.py
    def binary_search(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n# \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j = 0, len(nums) - 1\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j:\n# \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u9700\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
    binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j := 0, len(nums)-1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nfor i <= j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] < target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
    binary_search.c
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.Length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i = 0\nvar j = nums.count - 1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile i <= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i: usize = 0;\nvar j: usize = nums.items.len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.dart
    /* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List<int> nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i > j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n

    \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u533a\u95f4\uff0c\u56e0\u6b64\u4e8c\u5206\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002\u6307\u9488 i , j \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002

    "},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1. \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"

    \u9664\u4e86\u4e0a\u8ff0\u7684\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j]\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002

    \u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search.java
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cpp
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(vector<int> &nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.size();\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.py
    def binary_search_lcro(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\"\"\"\n# \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j = 0, len(nums)\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\nelif nums[m] > target:\nj = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n
    binary_search.go
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j := 0, len(nums)\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nfor i < j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.js
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearchLCRO(nums, target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] < target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nelse return m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.ts
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n
    binary_search.c
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.cs
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.Length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] > target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.swift
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i = 0\nvar j = nums.count\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i < j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] > target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n
    binary_search.zig
    // \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i: usize = 0;\nvar j: usize = nums.items.len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i <= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] < target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums.items[m] > target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(T, m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n
    binary_search.dart
    /* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List<int> nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i < j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] > target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002

    \u5728\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u6cd5\u4e2d\uff0c\u7531\u4e8e\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u6307\u9488 \\(i\\) \u548c \\(j\\) \u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002

    Fig. \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49

    "},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2. \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"

    \u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\uff1a

    • \u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002
    • \u4e8c\u5206\u67e5\u627e\u65e0\u9700\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002

    \u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002
    • \u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
    • \u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700\u8981 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002
    "},{"location":"chapter_searching/binary_search_edge/","title":"10.2. \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":"

    \u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u9898\u76ee\u89c4\u5b9a\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\u3002\u5982\u679c\u76ee\u6807\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u591a\u6b21\u51fa\u73b0\uff0c\u4e0a\u8282\u4ecb\u7ecd\u7684\u65b9\u6cd5\u53ea\u80fd\u4fdd\u8bc1\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a\u76ee\u6807\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u7d22\u5f15\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11\u76ee\u6807\u5143\u7d20\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 nums \uff0c\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 target \u5728\u6570\u7ec4\u4e2d\u9996\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002

    "},{"location":"chapter_searching/binary_search_edge/#1021","title":"10.2.1. \u00a0 \u7ebf\u6027\u65b9\u6cd5","text":"

    \u4e3a\u4e86\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u8fb9\u7684 target \uff0c\u6211\u4eec\u53ef\u4ee5\u5206\u4e3a\u4e24\u6b65\uff1a

    1. \u8fdb\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5b9a\u4f4d\u5230\u4efb\u610f\u4e00\u4e2a target \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \uff1b
    2. \u4ee5\u7d22\u5f15 \\(k\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u627e\u5230\u6700\u5de6\u8fb9\u7684 target \u8fd4\u56de\u5373\u53ef\u3002

    Fig. \u7ebf\u6027\u67e5\u627e\u6700\u5de6\u8fb9\u7684\u5143\u7d20

    \u8fd9\u4e2a\u65b9\u6cd5\u867d\u7136\u6709\u6548\uff0c\u4f46\u7531\u4e8e\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5f53\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 target \u65f6\u6548\u7387\u8f83\u4f4e\u3002

    "},{"location":"chapter_searching/binary_search_edge/#1022","title":"10.2.2. \u00a0 \u4e8c\u5206\u65b9\u6cd5","text":"

    \u8003\u8651\u4ec5\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u89e3\u51b3\u8be5\u95ee\u9898\u3002\u6574\u4f53\u7b97\u6cd5\u6d41\u7a0b\u4e0d\u53d8\uff0c\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad target \u548c nums[m] \u5927\u5c0f\u5173\u7cfb\uff1a

    • \u5f53 nums[m] < target \u6216 nums[m] > target \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 target \uff0c\u56e0\u6b64\u91c7\u53d6\u4e0e\u4e0a\u8282\u4ee3\u7801\u76f8\u540c\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 target \u9760\u8fd1\u3002
    • \u5f53 nums[m] == target \u65f6\uff0c\u8bf4\u660e\u201c\u5c0f\u4e8e target \u7684\u5143\u7d20\u201d\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e target \u7684\u5143\u7d20\u9760\u8fd1\u3002

    \u4e8c\u5206\u67e5\u627e\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 target \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(i\\) \u5373\u53ef\u3002

    <1><2><3><4><5><6><7><8>

    \u6ce8\u610f\uff0c\u6570\u7ec4\u53ef\u80fd\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20 target \u3002\u56e0\u6b64\u5728\u51fd\u6570\u8fd4\u56de\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u5224\u65ad nums[i] \u4e0e target \u662f\u5426\u76f8\u7b49\uff0c\u4ee5\u53ca\u7d22\u5f15 \\(i\\) \u662f\u5426\u8d8a\u754c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_edge.java
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.length || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.cpp
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(vector<int> &nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.size() || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.py
    def binary_search_left_edge(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i <= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nj = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\nif i == len(nums) or nums[i] != target:\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i\n
    binary_search_edge.go
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i <= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] < target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n}\n}\nif i == len(nums) || nums[i] != target {\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\nreturn i\n}\n
    binary_search_edge.js
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchLeftEdge(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif (i == nums.length || nums[i] != target) {\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i;\n}\n
    binary_search_edge.ts
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchLeftEdge(nums: number[], target: number): number {\nlet i = 0, j = nums.length - 1;  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2);  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif (i == nums.length || nums[i] != target) {\nreturn -1;  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i;\n}\n
    binary_search_edge.c
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int *nums, int size, int target) {\nint i = 0, j = size - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == size || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.cs
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.Length || nums[i] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    binary_search_edge.swift
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nvar i = 0\nvar j = nums.count - 1\nwhile i <= j {\nlet m = i + (j - 1) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] > target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\nif i == nums.count || nums[i] != target {\nreturn -1 // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn i\n}\n
    binary_search_edge.zig
    [class]{}-[func]{binarySearchLeftEdge}\n
    binary_search_edge.dart
    /* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchLeftEdge(List<int> nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u95f4\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\nif (i == nums.length || nums[i] != target) return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn i;\n}\n
    "},{"location":"chapter_searching/binary_search_edge/#1023","title":"10.2.3. \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"

    \u7c7b\u4f3c\u5730\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u8fb9\u7684 target \u3002\u5f53 nums[m] == target \u65f6\uff0c\u8bf4\u660e\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c i = m + 1 \uff0c\u4f7f\u5f97\u6307\u9488 \\(i\\) \u5411\u5927\u4e8e target \u7684\u5143\u7d20\u9760\u8fd1\u3002

    \u5b8c\u6210\u4e8c\u5206\u540e\uff0c\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u6700\u53f3\u8fb9\u7684 target \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(j\\) \u5373\u53ef\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_edge.java
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.cpp
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(vector<int> &nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.py
    def binary_search_right_edge(nums: list[int], target: int) -> int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i <= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] > target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\ni = m + 1  # \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\nif j < 0 or nums[j] != target:\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j\n
    binary_search_edge.go
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchRightEdge(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i <= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] < target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] > target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n}\n}\nif j < 0 || nums[j] != target {\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\nreturn j\n}\n
    binary_search_edge.js
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchRightEdge(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif (j < 0 || nums[j] != target) {\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j;\n}\n
    binary_search_edge.ts
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunction binarySearchRightEdge(nums: number[], target: number): number {\nlet i = 0, j = nums.length - 1;  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nlet m = Math.floor((i + j) / 2);  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target) {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] > target) {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1;  // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif (j < 0 || nums[j] != target) {\nreturn -1;  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j;\n}\n
    binary_search_edge.c
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int *nums, int size, int target) {\nint i = 0, j = size - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.cs
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target)\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n
    binary_search_edge.swift
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -> Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nvar i = 0\nvar j = nums.count - 1\nwhile i <= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] < target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] > target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\ni = m + 1 // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\n}\nif j < 0 || nums[j] != target {\nreturn -1 // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\nreturn j\n}\n
    binary_search_edge.zig
    [class]{}-[func]{binarySearchRightEdge}\n
    binary_search_edge.dart
    /* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a\u5143\u7d20 */\nint binarySearchRightEdge(List<int> nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i <= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u95f4\u7d22\u5f15 m\nif (nums[m] < target)\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelse if (nums[m] > target)\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse\ni = m + 1; // \u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [m+1, j] \u4e2d\n}\nif (j < 0 || nums[j] != target) return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn j;\n}\n

    \u89c2\u5bdf\u4e0b\u56fe\uff0c\u641c\u7d22\u6700\u53f3\u8fb9\u5143\u7d20\u65f6\u6307\u9488 \\(j\\) \u7684\u4f5c\u7528\u4e0e\u641c\u7d22\u6700\u5de6\u8fb9\u5143\u7d20\u65f6\u6307\u9488 \\(i\\) \u7684\u4f5c\u7528\u4e00\u81f4\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u641c\u7d22\u6700\u5de6\u8fb9\u5143\u7d20\u548c\u6700\u53f3\u8fb9\u5143\u7d20\u7684\u5b9e\u73b0\u662f\u955c\u50cf\u5bf9\u79f0\u7684\u3002

    Fig. \u67e5\u627e\u6700\u5de6\u8fb9\u548c\u6700\u53f3\u8fb9\u5143\u7d20\u7684\u5bf9\u79f0\u6027

    Tip

    \u4ee5\u4e0a\u4ee3\u7801\u91c7\u53d6\u7684\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.3. \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"

    \u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002

    Question

    \u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 nums \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 target \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a target \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/#1031","title":"10.3.1. \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"

    \u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a target \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002

    Fig. \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart two_sum.java
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
    two_sum.cpp
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector<int> twoSumBruteForce(vector<int> &nums, int target) {\nint size = nums.size();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn {i, j};\n}\n}\nreturn {};\n}\n
    two_sum.py
    def two_sum_brute_force(nums: list[int], target: int) -> list[int]:\n\"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n# \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in range(len(nums) - 1):\nfor j in range(i + 1, len(nums)):\nif nums[i] + nums[j] == target:\nreturn [i, j]\nreturn []\n
    two_sum.go
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\nsize := len(nums)\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i := 0; i < size-1; i++ {\nfor j := i + 1; i < size; j++ {\nif nums[i]+nums[j] == target {\nreturn []int{i, j}\n}\n}\n}\nreturn nil\n}\n
    two_sum.js
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
    two_sum.ts
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (let i = 0; i < n; i++) {\nfor (let j = i + 1; j < n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n
    two_sum.c
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\nfor (int i = 0; i < numsSize; ++i) {\nfor (int j = i + 1; j < numsSize; ++j) {\nif (nums[i] + nums[j] == target) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = i, res[1] = j;\n*returnSize = 2;\nreturn res;\n}\n}\n}\n*returnSize = 0;\nreturn NULL;\n}\n
    two_sum.cs
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.Length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor (int i = 0; i < size - 1; i++) {\nfor (int j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n
    two_sum.swift
    /* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -> [Int] {\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nfor i in nums.indices.dropLast() {\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[i] + nums[j] == target {\nreturn [i, j]\n}\n}\n}\nreturn [0]\n}\n
    two_sum.zig
    // \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\nvar size: usize = nums.len;\nvar i: usize = 0;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n^2)\nwhile (i < size - 1) : (i += 1) {\nvar j = i + 1;\nwhile (j < size) : (j += 1) {\nif (nums[i] + nums[j] == target) {\nreturn [_]i32{@intCast(i32, i), @intCast(i32, j)};\n}\n}\n}\nreturn null;\n}\n
    two_sum.dart
    /* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList<int> twoSumBruteForce(List<int> nums, int target) {\nint size = nums.length;\nfor (var i = 0; i < size - 1; i++) {\nfor (var j = i + 1; j < size; j++) {\nif (nums[i] + nums[j] == target) return [i, j];\n}\n}\nreturn [0];\n}\n

    \u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002

    "},{"location":"chapter_searching/replace_linear_by_hashing/#1032","title":"10.3.2. \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"

    \u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\uff1a

    1. \u5224\u65ad\u6570\u5b57 target - nums[i] \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1b
    2. \u5c06\u952e\u503c\u5bf9 nums[i] \u548c\u7d22\u5f15 i \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\uff1b
    <1><2><3>

    \u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart two_sum.java
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nMap<Integer, Integer> dic = new HashMap<>();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn new int[] { dic.get(target - nums[i]), i };\n}\ndic.put(nums[i], i);\n}\nreturn new int[0];\n}\n
    two_sum.cpp
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector<int> twoSumHashTable(vector<int> &nums, int target) {\nint size = nums.size();\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nunordered_map<int, int> dic;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.find(target - nums[i]) != dic.end()) {\nreturn {dic[target - nums[i]], i};\n}\ndic.emplace(nums[i], i);\n}\nreturn {};\n}\n
    two_sum.py
    def two_sum_hash_table(nums: list[int], target: int) -> list[int]:\n\"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n# \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\ndic = {}\n# \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in range(len(nums)):\nif target - nums[i] in dic:\nreturn [dic[target - nums[i]], i]\ndic[nums[i]] = i\nreturn []\n
    two_sum.go
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nhashTable := map[int]int{}\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor idx, val := range nums {\nif preIdx, ok := hashTable[target-val]; ok {\nreturn []int{preIdx, idx}\n}\nhashTable[val] = idx\n}\nreturn nil\n}\n
    two_sum.js
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m = {};\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nif (m[target - nums[i]] !== undefined) {\nreturn [m[target-nums[i]], i];\n} else {\nm[nums[i]] = i;\n}\n}\nreturn [];\n}\n
    two_sum.ts
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nlet m: Map<number, number> = new Map();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (let i = 0; i < nums.length; i++) {\nlet index = m.get(target - nums[i]);\nif (index !== undefined) {\nreturn [index, i];\n} else {\nm.set(nums[i], i);\n}\n}\nreturn [];\n}\n
    two_sum.c
    /* \u54c8\u5e0c\u8868 */\nstruct hashTable {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n};\ntypedef struct hashTable hashTable;\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nhashTable *find(hashTable *h, int key) {\nhashTable *tmp;\nHASH_FIND_INT(h, &key, tmp);\nreturn tmp;\n}\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(hashTable *h, int key, int val) {\nhashTable *t = find(h, key);\nif (t == NULL) {\nhashTable *tmp = malloc(sizeof(hashTable));\ntmp->key = key, tmp->val = val;\nHASH_ADD_INT(h, key, tmp);\n} else {\nt->val = val;\n}\n}\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\nhashTable *hashtable = NULL;\nfor (int i = 0; i < numsSize; i++) {\nhashTable *t = find(hashtable, target - nums[i]);\nif (t != NULL) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = t->val, res[1] = i;\n*returnSize = 2;\nreturn res;\n}\ninsert(hashtable, nums[i], i);\n}\n*returnSize = 0;\nreturn NULL;\n}\n
    two_sum.cs
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.Length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nDictionary<int, int> dic = new();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor (int i = 0; i < size; i++) {\nif (dic.ContainsKey(target - nums[i])) {\nreturn new int[] { dic[target - nums[i]], i };\n}\ndic.Add(nums[i], i);\n}\nreturn new int[0];\n}\n
    two_sum.swift
    /* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -> [Int] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic: [Int: Int] = [:]\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nfor i in nums.indices {\nif let j = dic[target - nums[i]] {\nreturn [j, i]\n}\ndic[nums[i]] = i\n}\nreturn [0]\n}\n
    two_sum.zig
    // \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\nvar size: usize = nums.len;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)\nvar dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\ndefer dic.deinit();\nvar i: usize = 0;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nwhile (i < size) : (i += 1) {\nif (dic.contains(target - nums[i])) {\nreturn [_]i32{dic.get(target - nums[i]).?, @intCast(i32, i)};\n}\ntry dic.put(nums[i], @intCast(i32, i));\n}\nreturn null;\n}\n
    two_sum.dart
    /* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList<int> twoSumHashTable(List<int> nums, int target) {\nint size = nums.length;\nMap<int, int> dic = HashMap();\nfor (var i = 0; i < size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn [dic[target - nums[i]]!, i];\n}\ndic.putIfAbsent(nums[i], () => i);\n}\nreturn [0];\n}\n

    \u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u4f4e\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002

    \u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.4. \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"

    \u300c\u641c\u7d22\u7b97\u6cd5 Searching Algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002

    \u6839\u636e\u5b9e\u73b0\u601d\u8def\uff0c\u641c\u7d22\u7b97\u6cd5\u603b\u4f53\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a

    • \u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002
    • \u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002

    \u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1041","title":"10.4.1. \u00a0 \u66b4\u529b\u641c\u7d22","text":"

    \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    • \u300c\u7ebf\u6027\u641c\u7d22\u300d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002
    • \u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u300d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u662f\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\u4e3a\u6b62\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002

    \u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u9700\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002

    \u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1042","title":"10.4.2. \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"

    \u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    • \u300c\u4e8c\u5206\u67e5\u627e\u300d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002
    • \u300c\u54c8\u5e0c\u67e5\u627e\u300d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002
    • \u300c\u6811\u67e5\u627e\u300d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002

    \u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002

    \u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u652f\u3002

    Note

    \u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u5173\u6ce8\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002

    "},{"location":"chapter_searching/searching_algorithm_revisited/#1043","title":"10.4.3. \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"

    \u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u5728\u8be5\u6570\u636e\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u4e0b\u56fe\u6240\u793a\u3002

    Fig. \u591a\u79cd\u641c\u7d22\u7b56\u7565

    \u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u4e0b\u8868\u6240\u793a\u3002

    \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f

    \u9664\u4e86\u4ee5\u4e0a\u8868\u683c\u5185\u5bb9\uff0c\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002

    \u7ebf\u6027\u641c\u7d22

    • \u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u9700\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002
    • \u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002
    • \u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002

    \u4e8c\u5206\u67e5\u627e

    • \u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002
    • \u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002
    • \u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002

    \u54c8\u5e0c\u67e5\u627e

    • \u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002
    • \u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002
    • \u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002
    • \u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002

    \u6811\u67e5\u627e

    • \u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u79bb\u6563\u5b58\u50a8\u7684\u3002
    • \u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002
    • \u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
    • \u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002
    "},{"location":"chapter_searching/summary/","title":"10.5. \u00a0 \u5c0f\u7ed3","text":"
    • \u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u4e8e\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u5b9e\u73b0\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002
    • \u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u9700\u5bf9\u6570\u636e\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002
    • \u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002
    • \u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002
    • \u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u5408\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u9700\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002
    • \u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u4f4e\u81f3 \\(O(1)\\) \u3002
    "},{"location":"chapter_sorting/","title":"11. \u00a0 \u6392\u5e8f","text":""},{"location":"chapter_sorting/bubble_sort/","title":"11.3. \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"

    \u300c\u5192\u6ce1\u6392\u5e8f Bubble Sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002

    \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u4e0a\u8ff0\u8fc7\u7a0b\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 > \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u5b83\u4fe9\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002

    <1><2><3><4><5><6><7>

    "},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u4e3a\uff1a

    1. \u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c
    2. \u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
    3. \u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002
    4. \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u9700\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002

    Fig. \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\n}\n}\n}\n}\n
    bubble_sort.py
    def bubble_sort(nums: list[int]) -> None:\n\"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\n
    bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\n}\n}\n}\n}\n
    bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.c
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i < size - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < size - 1 - i; j++) {\nif (nums[j] > nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\n}\n}\n}\n}\n
    bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\n}\n}\n}\n}\n
    bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    bubble_sort.dart
    /* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2. \u00a0 \u6548\u7387\u4f18\u5316","text":"

    \u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d flag \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002

    \u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bubble_sort.java
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\nboolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.cpp
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.py
    def bubble_sort_with_flag(nums: list[int]) -> None:\n\"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\nflag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in range(i):\nif nums[j] > nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\nflag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\nif not flag:\nbreak  # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n
    bubble_sort.go
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i > 0; i-- {\nflag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j < i; j++ {\nif nums[j] > nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
    bubble_sort.js
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.ts
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i > 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (let j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.c
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i < size - 1; i++) {\nbool flag = false;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < size - 1 - i; j++) {\nif (nums[j] > nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\nflag = true;\n}\n}\nif (!flag)\nbreak;\n}\n}\n
    bubble_sort.cs
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.swift
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\nvar flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] > nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n
    bubble_sort.zig
    // \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i > 0) : (i -= 1) {\nvar flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j < i) : (j += 1) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;\n}\n}\nif (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    bubble_sort.dart
    /* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i > 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j < i; j++) {\nif (nums[j] > nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n
    "},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\) , \\(n - 2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \uff0c\u603b\u548c\u4e3a \\(\\frac{(n - 1) n}{2}\\) \u3002\u5728\u5f15\u5165 flag \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002
    "},{"location":"chapter_sorting/bucket_sort/","title":"11.8. \u00a0 \u6876\u6392\u5e8f","text":"

    \u524d\u8ff0\u7684\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u6c34\u5e73\u3002

    \u300c\u6876\u6392\u5e8f Bucket Sort\u300d\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002

    "},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\uff1b
    2. \u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u672c\u6587\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\uff1b
    3. \u6309\u7167\u6876\u7684\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\uff0c\u5408\u5e76\u7ed3\u679c\uff1b

    Fig. \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart bucket_sort.java
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length / 2;\nList<List<Float>> buckets = new ArrayList<>();\nfor (int i = 0; i < k; i++) {\nbuckets.add(new ArrayList<>());\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int) (num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets.get(i).add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List<Float> bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nCollections.sort(bucket);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List<Float> bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.cpp
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(vector<float> &nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.size() / 2;\nvector<vector<float>> buckets(k);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = num * k;\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].push_back(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (vector<float> &bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort(bucket.begin(), bucket.end());\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (vector<float> &bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.py
    def bucket_sort(nums: list[float]) -> None:\n\"\"\"\u6876\u6392\u5e8f\"\"\"\n# \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk = len(nums) // 2\nbuckets = [[] for _ in range(k)]\n# 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums:\n# \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni = int(num * k)\n# \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n# 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor bucket in buckets:\n# \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort()\n# 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni = 0\nfor bucket in buckets:\nfor num in bucket:\nnums[i] = num\ni += 1\n
    bucket_sort.go
    /* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk := len(nums) / 2\nbuckets := make([][]float64, k)\nfor i := 0; i < k; i++ {\nbuckets[i] = make([]float64, 0)\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor _, num := range nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni := int(num * float64(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i] = append(buckets[i], num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i := 0; i < k; i++ {\n// \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort.Float64s(buckets[i])\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni := 0\nfor _, bucket := range buckets {\nfor _, num := range bucket {\nnums[i] = num\ni++\n}\n}\n}\n
    bucket_sort.js
    /* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets = [];\nfor (let i = 0; i < k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) => a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.ts
    /* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets: number[][] = [];\nfor (let i = 0; i < k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) => a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n
    bucket_sort.c
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = size / 2;\nfloat **buckets = calloc(k, sizeof(float *));\nfor (int i = 0; i < k; i++) {\n// \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\nbuckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (int i = 0; i < size; i++) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint bucket_idx = nums[i] * k;\nint j = 0;\n// \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\nwhile (buckets[bucket_idx][j] > 0 && buckets[bucket_idx][j] < nums[i]) {\nj++;\n}\nfloat temp = nums[i];\nwhile (j < ARRAY_SIZE && buckets[bucket_idx][j] > 0) {\nswap(&temp, &buckets[bucket_idx][j]);\nj++;\n}\nbuckets[bucket_idx][j] = temp;\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (int i = 0; i < k; i++) {\nqsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nfor (int i = 0, j = 0; j < k; j++) {\nfor (int l = 0; l < ARRAY_SIZE; l++) {\nif (buckets[j][l] > 0) {\nnums[i++] = buckets[j][l];\n}\n}\n}\n// \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\nfor (int i = 0; i < k; i++) {\nfree(buckets[i]);\n}\nfree(buckets);\n}\n
    bucket_sort.cs
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.Length / 2;\nList<List<float>> buckets = new List<List<float>>();\nfor (int i = 0; i < k; i++) {\nbuckets.Add(new List<float>());\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nforeach (float num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int) (num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].Add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nforeach (List<float> bucket in buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.Sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint j = 0;\nforeach (List<float> bucket in buckets) {\nforeach (float num in bucket) {\nnums[j++] = num;\n}\n}\n}\n
    bucket_sort.swift
    /* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nlet k = nums.count / 2\nvar buckets = (0 ..< k).map { _ in [Double]() }\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nlet i = Int(num * Double(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i in buckets.indices {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbuckets[i].sort()\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nvar i = nums.startIndex\nfor bucket in buckets {\nfor num in bucket {\nnums[i] = num\nnums.formIndex(after: &i)\n}\n}\n}\n
    bucket_sort.zig
    [class]{}-[func]{bucketSort}\n
    bucket_sort.dart
    /* \u6876\u6392\u5e8f */\nvoid bucketSort(List<double> nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length ~/ 2;\nList<List<double>> buckets = List.generate(k, (index) => []);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (double num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4 [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (num * k).toInt();\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List<double> bucket in buckets) {\nbucket.sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List<double> bucket in buckets) {\nfor (double num in bucket) {\nnums[i++] = num;\n}\n}\n}\n

    \u6876\u6392\u5e8f\u7684\u9002\u7528\u573a\u666f\u662f\u4ec0\u4e48\uff1f

    \u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002

    "},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386 \\(n\\) \u4e2a\u6876\uff0c\u82b1\u8d39 \\(O(k)\\) \u65f6\u95f4\u3002
    • \u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u574f\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002
    "},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3. \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"

    \u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4efd\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002

    \u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u4e2a\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u4e2a\u9012\u5f52\u6811\uff0c\u4f7f\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002

    Fig. \u9012\u5f52\u5212\u5206\u6876

    \u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002

    Fig. \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876

    "},{"location":"chapter_sorting/counting_sort/","title":"11.9. \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"

    \u300c\u8ba1\u6570\u6392\u5e8f Counting Sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002

    "},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1. \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"

    \u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 nums \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\u3002\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u6570\u7ec4\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 counter \uff1b
    2. \u501f\u52a9 counter \u7edf\u8ba1 nums \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d counter[num] \u5bf9\u5e94\u6570\u5b57 num \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 nums\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a num\uff09\uff0c\u6bcf\u8f6e\u5c06 counter[num] \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002
    3. \u7531\u4e8e counter \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u88ab\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 counter \uff0c\u6839\u636e\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5c06\u5b83\u4eec\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 nums \u5373\u53ef\u3002

    Fig. \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector<int> &nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.py
    def counting_sort_naive(nums: list[int]) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = 0\nfor num in nums:\nm = max(m, num)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\ni = 0\nfor num in range(m + 1):\nfor _ in range(counter[num]):\nnums[i] = num\ni += 1\n
    counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nfor i, num := 0, 0; num < m+1; num++ {\nfor j := 0; j < counter[num]; j++ {\nnums[i] = num\ni++\n}\n}\n}\n
    counting_sort.js
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num < m + 1; num++) {\nfor (let j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.ts
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array<number>(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num < m + 1; num++) {\nfor (let j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.c
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i < size; i++) {\nif (nums[i] > m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = malloc(sizeof(int) * m);\nfor (int i = 0; i < size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.cs
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n
    counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nvar i = 0\nfor num in stride(from: 0, to: m + 1, by: 1) {\nfor _ in stride(from: 0, to: counter[num], by: 1) {\nnums[i] = num\ni += 1\n}\n}\n}\n
    counting_sort.zig
    [class]{}-[func]{countingSortNaive}\n
    counting_sort.dart
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List<int> nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num in nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nList<int> counter = List.filled(m + 1, 0);\nfor (int num in nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num < m + 1; num++) {\nfor (int j = 0; j < counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n

    \u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb

    \u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 counter \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u662f\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002

    "},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2. \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"

    \u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u53d1\u73b0\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 3. \u5c31\u5931\u6548\u4e86\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u8981\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002

    \u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 counter \u7684\u300c\u524d\u7f00\u548c\u300d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 i \u5904\u7684\u524d\u7f00\u548c prefix[i] \u7b49\u4e8e\u6570\u7ec4\u524d i \u4e2a\u5143\u7d20\u4e4b\u548c\uff0c\u5373

    \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\]

    \u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0cprefix[num] - 1 \u4ee3\u8868\u5143\u7d20 num \u5728\u7ed3\u679c\u6570\u7ec4 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 nums \u7684\u6bcf\u4e2a\u5143\u7d20 num \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\uff1a

    1. \u5c06 num \u586b\u5165\u6570\u7ec4 res \u7684\u7d22\u5f15 prefix[num] - 1 \u5904\uff1b
    2. \u4ee4\u524d\u7f00\u548c prefix[num] \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\uff1b

    \u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 res \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 res \u8986\u76d6\u539f\u6570\u7ec4 nums \u5373\u53ef\u3002

    <1><2><3><4><5><6><7><8>

    \u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart counting_sort.java
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.cpp
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector<int> &nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector<int> counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.size();\nvector<int> res(n);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums = res;\n}\n
    counting_sort.py
    def counting_sort(nums: list[int]) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = max(nums)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n# \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in range(m):\ncounter[i + 1] += counter[i]\n# 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n# \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn = len(nums)\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nnum = nums[i]\nres[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n# \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\n
    counting_sort.go
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num > m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i := 0; i < m; i++ {\ncounter[i+1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn := len(nums)\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nnum := nums[i]\n// \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\nres[counter[num]-1] = num\n// \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\ncounter[num]--\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\ncopy(nums, res)\n}\n
    counting_sort.js
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res = new Array(n);\nfor (let i = n - 1; i >= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.ts
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array<number>(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res: number[] = new Array<number>(n);\nfor (let i = n - 1; i >= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.c
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i < size; i++) {\nif (nums[i] > m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = malloc(sizeof(int) * m);\nfor (int i = 0; i < size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint *res = malloc(sizeof(int) * size);\nfor (int i = size - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nmemcpy(nums, res, size * sizeof(int));\n}\n
    counting_sort.cs
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.Length;\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n
    counting_sort.swift
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in stride(from: 0, to: m, by: 1) {\ncounter[i + 1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nvar res = Array(repeating: 0, count: nums.count)\nfor i in stride(from: nums.count - 1, through: 0, by: -1) {\nlet num = nums[i]\nres[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in stride(from: 0, to: nums.count, by: 1) {\nnums[i] = res[i]\n}\n}\n
    counting_sort.zig
    [class]{}-[func]{countingSort}\n
    counting_sort.dart
    /* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List<int> nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num in nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nList<int> counter = List.filled(m + 1, 0);\nfor (int num in nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i < m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nList<int> res = List.filled(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums.setAll(0, res);\n}\n
    "},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 nums \u548c\u904d\u5386 counter \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 res \u548c counter \u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 res \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 nums \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 nums \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002
    "},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4. \u00a0 \u5c40\u9650\u6027","text":"

    \u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u5de5\u4f5c\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002

    \u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u8981\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u88ab\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u5373\u53ef\u3002

    \u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002

    "},{"location":"chapter_sorting/heap_sort/","title":"11.7. \u00a0 \u5806\u6392\u5e8f","text":"

    Tip

    \u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u300c\u5806\u300d\u7ae0\u8282\u3002

    \u300c\u5806\u6392\u5e8f Heap Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\uff1a

    1. \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
    2. \u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002

    \u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002

    "},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002
    2. \u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002
    3. \u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002
    4. \u5faa\u73af\u6267\u884c\u7b2c 2. \u548c 3. \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002

    \u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c 2. \u548c 3. \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002

    <1><2><3><4><5><6><7><8><9><10><11><12>

    \u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u5806\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316\uff08Sift Down\uff09\u7684\u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 Sift Down \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart heap_sort.java
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length / 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.cpp
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector<int> &nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector<int> &nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.size() / 2 - 1; i >= 0; --i) {\nsiftDown(nums, nums.size(), i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.size() - 1; i > 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.py
    def sift_down(nums: list[int], n: int, i: int):\n\"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl = 2 * i + 1\nr = 2 * i + 2\nma = i\nif l < n and nums[l] > nums[ma]:\nma = l\nif r < n and nums[r] > nums[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nnums[i], nums[ma] = nums[ma], nums[i]\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\ndef heap_sort(nums: list[int]):\n\"\"\"\u5806\u6392\u5e8f\"\"\"\n# \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(len(nums) // 2 - 1, -1, -1):\nsift_down(nums, len(nums), i)\n# \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in range(len(nums) - 1, 0, -1):\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums[0], nums[i] = nums[i], nums[0]\n# \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsift_down(nums, i, 0)\n
    heap_sort.go
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl := 2*i + 1\nr := 2*i + 2\nma := i\nif l < n && (*nums)[l] > (*nums)[ma] {\nma = l\n}\nif r < n && (*nums)[r] > (*nums)[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i := len(*nums)/2 - 1; i >= 0; i-- {\nsiftDown(nums, len(*nums), i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i := len(*nums) - 1; i > 0; i-- {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0)\n}\n}\n
    heap_sort.js
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l < n && nums[l] > nums[ma]) {\nma = l;\n}\nif (r < n && nums[r] > nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.ts
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l < n && nums[l] > nums[ma]) {\nma = l;\n}\nif (r < n && nums[r] > nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.c
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\nwhile (1) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = n / 2 - 1; i >= 0; --i) {\nsiftDown(nums, n, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = n - 1; i > 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.cs
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma])\nma = l;\nif (r < n && nums[r] > nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(nums[ma], nums[i]) = (nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.Length / 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.Length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.Length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(nums[i], nums[0]) = (nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    heap_sort.swift
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1\nlet r = 2 * i + 2\nvar ma = i\nif l < n, nums[l] > nums[ma] {\nma = l\n}\nif r < n, nums[r] > nums[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nnums.swapAt(i, ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\nsiftDown(nums: &nums, n: nums.count, i: i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums.swapAt(0, i)\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums: &nums, n: i, i: 0)\n}\n}\n
    heap_sort.zig
    [class]{}-[func]{siftDown}\n[class]{}-[func]{heapSort}\n
    heap_sort.dart
    /* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List<int> nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l < n && nums[l] > nums[ma]) ma = l;\nif (r < n && nums[r] > nums[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u9700\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List<int> nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length ~/ 2 - 1; i >= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i > 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u5373\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n
    "},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002
    "},{"location":"chapter_sorting/insertion_sort/","title":"11.4. \u00a0 \u63d2\u5165\u6392\u5e8f","text":"

    \u300c\u63d2\u5165\u6392\u5e8f Insertion Sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002

    \u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002

    \u56de\u5fc6\u6570\u7ec4\u7684\u5143\u7d20\u63d2\u5165\u64cd\u4f5c\uff0c\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a base \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 base \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u518d\u5c06 base \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002

    Fig. \u5355\u6b21\u63d2\u5165\u64cd\u4f5c

    "},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002
    2. \u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    3. \u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    4. \u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a base \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002

    Fig. \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart insertion_sort.java
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.cpp
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector<int> &nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.size(); i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.py
    def insertion_sort(nums: list[int]) -> None:\n\"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n# \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\nfor i in range(1, len(nums)):\nbase = nums[i]\nj = i - 1\n# \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0 and nums[j] > base:\nnums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\nnums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n
    insertion_sort.go
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := 1; i < len(nums); i++ {\nbase := nums[i]\nj := i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nfor j >= 0 && nums[j] > base {\nnums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--\n}\nnums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.js
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i < nums.length; i++) {\nlet base = nums[i],\nj = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.ts
    /* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i < nums.length; i++) {\nconst base = nums[i];\nlet j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.c
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < size; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\n// \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nnums[j + 1] = nums[j];\nj--;\n}\n// \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\nnums[j + 1] = base;\n}\n}\n
    insertion_sort.cs
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.Length; i++) {\nint bas = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > bas) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.swift
    /* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor i in stride(from: 1, to: nums.count, by: 1) {\nlet base = nums[i]\nvar j = i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j >= 0, nums[j] > base {\nnums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\n}\nnums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.zig
    // \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nvar i: usize = 1;\nwhile (i < nums.len) : (i += 1) {\nvar base = nums[i];\nvar j: usize = i;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 1 and nums[j - 1] > base) : (j -= 1) {\nnums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n}\nnums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    insertion_sort.dart
    /* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List<int> nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i < nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j >= 0 && nums[j] > base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n
    "},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\) , \\(n-2\\) , \\(\\cdots\\) , \\(2\\) , \\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\(\\frac{(n - 1) n}{2}\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002
    "},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3. \u00a0 \u63d2\u5165\u6392\u5e8f\u4f18\u52bf","text":"

    \u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u6bd4\u5feb\u901f\u6392\u5e8f\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002

    \u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u4f5c\u7528\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u56e0\u7d20\u3002

    \u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u90fd\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002

    \u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\u3002\u8fd9\u662f\u56e0\u4e3a\uff1a

    • \u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002
    • \u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002
    • \u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002
    "},{"location":"chapter_sorting/merge_sort/","title":"11.6. \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"

    \u300c\u5f52\u5e76\u6392\u5e8f Merge Sort\u300d\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u5b9e\u73b0\u6392\u5e8f\uff0c\u5305\u542b\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u4e24\u4e2a\u9636\u6bb5\uff1a

    1. \u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\uff1b
    2. \u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\uff1b

    Fig. \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5

    "},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u957f\u5ea6\u4e3a 1 \uff1b

    1. \u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 mid \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [left, mid] \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 [mid + 1, right] \uff09\uff1b
    2. \u9012\u5f52\u6267\u884c\u6b65\u9aa4 1. \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5212\u5206\uff1b

    \u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002

    <1><2><3><4><5><6><7><8><9><10>

    \u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u7684\u9012\u5f52\u987a\u5e8f\u4e0e\u4e8c\u53c9\u6811\u7684\u540e\u5e8f\u904d\u5386\u76f8\u540c\uff0c\u5177\u4f53\u6765\u770b\uff1a

    • \u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002
    • \u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002
    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart merge_sort.java
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = Arrays.copyOfRange(nums, left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn;                      // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.cpp
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(vector<int> &nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvector<int> tmp(nums.begin() + left, nums.begin() + right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector<int> &nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.py
    def merge(nums: list[int], left: int, mid: int, right: int) -> None:\n\"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n# \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n# \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\n# \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\ntmp = list(nums[left : right + 1])\n# \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleft_start = 0\nleft_end = mid - left\n# \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nright_start = mid + 1 - left\nright_end = right - left\n# i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni = left_start\nj = right_start\n# \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in range(left, right + 1):\n# \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > left_end:\nnums[k] = tmp[j]\nj += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelif j > right_end or tmp[i] <= tmp[j]:\nnums[k] = tmp[i]\ni += 1\n# \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse:\nnums[k] = tmp[j]\nj += 1\ndef merge_sort(nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif left >= right:\nreturn  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n# \u5212\u5206\u9636\u6bb5\nmid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n# \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n
    merge_sort.go
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums []int, left, mid, right int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4 \u501f\u52a9 copy \u6a21\u5757\ntmp := make([]int, right-left+1)\nfor i := left; i <= right; i++ {\ntmp[i-left] = nums[i]\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nleftStart, leftEnd := left-left, mid-left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nrightStart, rightEnd := mid+1-left, right-left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\ni, j := leftStart, rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k := left; k <= right; k++ {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj++\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni++\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j]\nj++\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right {\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nmid := (left + right) / 2\nmergeSort(nums, left, mid)\nmergeSort(nums, mid+1, right)\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n}\n
    merge_sort.js
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums, left, mid, right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left,\nleftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left,\nrightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart,\nj = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\nif (i > leftEnd) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nnums[k] = tmp[i++];\n} else {\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.ts
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = nums.slice(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left,\nleftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left,\nrightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nlet i = leftStart,\nj = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (let k = left; k <= right; k++) {\nif (i > leftEnd) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if (j > rightEnd || tmp[i] <= tmp[j]) {\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j++];\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.c
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int *nums, int left, int mid, int right) {\nint index;\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint tmp[right + 1 - left];\nfor (index = left; index < right + 1; index++) {\ntmp[index - left] = nums[index];\n}\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.cs
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nint[] tmp = nums[left..(right + 1)];\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n
    merge_sort.swift
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nlet tmp = Array(nums[left ..< (right + 1)])\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet leftStart = left - left\nlet leftEnd = mid - left\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nlet rightStart = mid + 1 - left\nlet rightEnd = right - left\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart\nvar j = rightStart\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor k in left ... right {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif i > leftEnd {\nnums[k] = tmp[j]\nj += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if j > rightEnd || tmp[i] <= tmp[j] {\nnums[k] = tmp[i]\ni += 1\n}\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse {\nnums[k] = tmp[j]\nj += 1\n}\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left >= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums: &nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums: &nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums: &nums, left: left, mid: mid, right: right)\n}\n
    merge_sort.zig
    // \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\ndefer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar tmp = try mem_allocator.alloc(i32, right + 1 - left);\nstd.mem.copy(i32, tmp, nums[left..right+1]);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nvar leftStart = left - left;\nvar leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nvar rightStart = mid + 1 - left;\nvar rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart;\nvar j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nvar k = left;\nwhile (k <= right) : (k += 1) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd) {\nnums[k] = tmp[j];\nj += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if  (j > rightEnd or tmp[i] <= tmp[j]) {\nnums[k] = tmp[i];\ni += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j];\nj += 1;\n}\n}\n}\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nvar mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\ntry mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\ntry mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\ntry merge(nums, left, mid, right);\n}\n
    merge_sort.dart
    /* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nvoid merge(List<int> nums, int left, int mid, int right) {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nList<int> tmp = nums.sublist(left, right + 1);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint leftStart = left - left, leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15\nint rightStart = mid + 1 - left, rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nint i = leftStart, j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nfor (int k = left; k <= right; k++) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i > leftEnd)\nnums[k] = tmp[j++];\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 <= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\nelse if (j > rightEnd || tmp[i] <= tmp[j])\nnums[k] = tmp[i++];\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 > \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nelse\nnums[k] = tmp[j++];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List<int> nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left >= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n

    \u5408\u5e76\u65b9\u6cd5 merge() \u4ee3\u7801\u4e2d\u7684\u96be\u70b9\u5305\u62ec\uff1a

    • \u5728\u9605\u8bfb\u4ee3\u7801\u65f6\uff0c\u9700\u8981\u7279\u522b\u6ce8\u610f\u5404\u4e2a\u53d8\u91cf\u7684\u542b\u4e49\u3002nums \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a [left, right] \uff0c\u4f46\u7531\u4e8e tmp \u4ec5\u590d\u5236\u4e86 nums \u8be5\u533a\u95f4\u7684\u5143\u7d20\uff0c\u56e0\u6b64 tmp \u5bf9\u5e94\u533a\u95f4\u4e3a [0, right - left] \u3002
    • \u5728\u6bd4\u8f83 tmp[i] \u548c tmp[j] \u7684\u5927\u5c0f\u65f6\uff0c\u8fd8\u9700\u8003\u8651\u5b50\u6570\u7ec4\u904d\u5386\u5b8c\u6210\u540e\u7684\u7d22\u5f15\u8d8a\u754c\u95ee\u9898\uff0c\u5373 i > leftEnd \u548c j > rightEnd \u7684\u60c5\u51b5\u3002\u7d22\u5f15\u8d8a\u754c\u7684\u4f18\u5148\u7ea7\u662f\u6700\u9ad8\u7684\uff0c\u5982\u679c\u5de6\u5b50\u6570\u7ec4\u5df2\u7ecf\u88ab\u5408\u5e76\u5b8c\u4e86\uff0c\u90a3\u4e48\u4e0d\u9700\u8981\u7ee7\u7eed\u6bd4\u8f83\uff0c\u76f4\u63a5\u5408\u5e76\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u5373\u53ef\u3002
    "},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002
    "},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3. \u00a0 \u94fe\u8868\u6392\u5e8f *","text":"

    \u5f52\u5e76\u6392\u5e8f\u5728\u6392\u5e8f\u94fe\u8868\u65f6\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u7531\u4e8e\u94fe\u8868\u4ec5\u9700\u6539\u53d8\u6307\u9488\u5c31\u53ef\u5b9e\u73b0\u8282\u70b9\u7684\u589e\u5220\u64cd\u4f5c\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u9700\u521b\u5efa\u8f85\u52a9\u94fe\u8868\u3002
    • \u901a\u8fc7\u4f7f\u7528\u201c\u8fed\u4ee3\u5212\u5206\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u5212\u5206\u201d\uff0c\u53ef\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\uff1b

    \u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002

    "},{"location":"chapter_sorting/quick_sort/","title":"11.5. \u00a0 \u5feb\u901f\u6392\u5e8f","text":"

    \u300c\u5feb\u901f\u6392\u5e8f Quick Sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u601d\u60f3\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002

    \u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u4e3a\uff1a

    1. \u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 i \u548c j \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\uff1b
    2. \u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 i\uff08j\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\uff1b
    3. \u5faa\u73af\u6267\u884c\u6b65\u9aa4 2. \uff0c\u76f4\u5230 i \u548c j \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\uff1b

    \u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002

    <1><2><3><4><5><6><7><8><9>

    \u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u601d\u60f3

    \u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cpp
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector<int> &nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector<int> &nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.py
    def partition(self, nums: list[int], left: int, right: int) -> int:\n\"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
    quick_sort.go
    /* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.js
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.ts
    /* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.c
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i < j && nums[i] <= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n
    quick_sort.cs
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.swift
    /* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\nlet tmp = nums[i]\nnums[i] = nums[j]\nnums[j] = tmp\n}\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left\nvar j = right\nwhile i < j {\nwhile i < j, nums[j] >= nums[left] {\nj -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i < j, nums[i] <= nums[left] {\ni += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums: &nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.zig
    // \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\nvar tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.dart
    /* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List<int> nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List<int> nums, int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"
    1. \u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u300c\u54e8\u5175\u5212\u5206\u300d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\uff1b
    2. \u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u300c\u54e8\u5175\u5212\u5206\u300d\uff1b
    3. \u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\uff1b

    Fig. \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector<int> &nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.py
    def quick_sort(self, nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right:\nreturn\n# \u54e8\u5175\u5212\u5206\npivot = self.partition(nums, left, right)\n# \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nself.quick_sort(nums, left, pivot - 1)\nself.quick_sort(nums, pivot + 1, right)\n
    quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\npivot := q.partition(nums, left, right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nq.quickSort(nums, left, pivot-1)\nq.quickSort(nums, pivot+1, right)\n}\n
    quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
    quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i < j && nums[i] <= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left >= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums: &nums, left: left, right: pivot - 1)\nquickSort(nums: &nums, left: pivot + 1, right: right)\n}\n
    quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nvar pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    quick_sort.dart
    /* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List<int> nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left >= right) return;\n// \u54e8\u5175\u5212\u5206\nint pivot = _partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\) \u3001\u81ea\u9002\u5e94\u6392\u5e8f \uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \u5c42\uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3001\u539f\u5730\u6392\u5e8f \uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002
    "},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3. \u00a0 \u5feb\u6392\u4e3a\u4ec0\u4e48\u5feb\uff1f","text":"

    \u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u300c\u5f52\u5e76\u6392\u5e8f\u300d\u548c\u300c\u5806\u6392\u5e8f\u300d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u539f\u56e0\u5982\u4e0b\uff1a

    • \u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002
    • \u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u300c\u5806\u6392\u5e8f\u300d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002
    • \u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u4f4e\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u300c\u63d2\u5165\u6392\u5e8f\u300d\u6bd4\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002
    "},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4. \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"

    \u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\) \u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u300c\u5192\u6ce1\u6392\u5e8f\u300d\u3002

    \u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002

    \u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cpp
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector<int> &nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector<int> &nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.py
    def median_three(self, nums: list[int], left: int, mid: int, right: int) -> int:\n\"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n# \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n# \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) ^ (nums[left] < nums[right]):\nreturn left\nelif (nums[mid] < nums[left]) ^ (nums[mid] < nums[right]):\nreturn mid\nreturn right\ndef partition(self, nums: list[int], left: int, right: int) -> int:\n\"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed = self.median_three(nums, left, (left + right) // 2, right)\n# \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n# \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i < j:\nwhile i < j and nums[j] >= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i < j and nums[i] <= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n
    quick_sort.go
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n}\nreturn right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nmed := q.medianThree(nums, left, (left+right)/2, right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i < j {\nfor i < j && nums[j] >= nums[left] {\nj-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i < j && nums[i] <= nums[left] {\ni++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n//\u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n//\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.js
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right])) return left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse return right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.ts
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\nnums: number[],\nleft: number,\nmid: number,\nright: number\n): number {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (Number(nums[left] < nums[mid]) ^ Number(nums[left] < nums[right])) {\nreturn left;\n} else if (\nNumber(nums[mid] < nums[left]) ^ Number(nums[mid] < nums[right])\n) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) {\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i < j && nums[i] <= nums[left]) {\ni++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nint partitionMedian(int nums[], int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.cs
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.swift
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -> Int {\nif (nums[left] < nums[mid]) != (nums[left] < nums[right]) {\nreturn left\n} else if (nums[mid] < nums[left]) != (nums[mid] < nums[right]) {\nreturn mid\n} else {\nreturn right\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -> Int {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums: &nums, i: left, j: med)\nreturn partition(nums: &nums, left: left, right: right)\n}\n
    quick_sort.zig
    // \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) != (nums[left] < nums[right])) {\nreturn left;\n} else if ((nums[mid] < nums[left]) != (nums[mid] < nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nvar med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i < j) {\nwhile (i < j and nums[j] >= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j and nums[i] <= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    quick_sort.dart
    /* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List<int> nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] < nums[mid]) ^ (nums[left] < nums[right]))\nreturn left;\nelse if ((nums[mid] < nums[left]) ^ (nums[mid] < nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List<int> nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = _medianThree(nums, left, (left + right) ~/ 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n_swap(nums, left, med);\n// \u4ee5 nums[left] \u4f5c\u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i < j) {\nwhile (i < j && nums[j] >= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i < j && nums[i] <= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n
    "},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5. \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"

    \u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u5012\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u7531\u4e8e\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002

    \u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(\\frac{n}{2}\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart quick_sort.java
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.cpp
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector<int> &nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.py
    def quick_sort(self, nums: list[int], left: int, right: int) -> None:\n\"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right:\n# \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot = self.partition(nums, left, right)\n# \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot - left < right - pivot:\nself.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\nelse:\nself.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n
    quick_sort.go
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nfor left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot := q.partition(nums, left, right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif pivot-left < right-pivot {\nq.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nq.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.js
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.ts
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.c
    /* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.cs
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.swift
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\nvar left = left\nvar right = right\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left < right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = partition(nums: &nums, left: left, right: right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left) < (right - pivot) {\nquickSortTailCall(nums: &nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums: &nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.zig
    // \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\nvar left = left_;\nvar right = right_;\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nvar pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    quick_sort.dart
    /* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List<int> nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left < right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = _partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u6392\nif (pivot - left < right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n
    "},{"location":"chapter_sorting/radix_sort/","title":"11.10. \u00a0 \u57fa\u6570\u6392\u5e8f","text":"

    \u4e0a\u4e00\u8282\u6211\u4eec\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002

    \u300c\u57fa\u6570\u6392\u5e8f Radix Sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002

    "},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1. \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"

    \u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \uff1b
    2. \u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u300c\u8ba1\u6570\u6392\u5e8f\u300d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff1b
    3. \u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 2. \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\uff1b

    Fig. \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b

    \u4e0b\u9762\u6765\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a

    \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\]

    \u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\space d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002

    \u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart radix_sort.java
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint[] counter = new int[10];\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = Integer.MIN_VALUE;\nfor (int num : nums)\nif (num > m)\nm = num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n
    radix_sort.cpp
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector<int> &nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvector<int> counter(10, 0);\nint n = nums.size();\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvector<int> res(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector<int> &nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = *max_element(nums.begin(), nums.end());\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n
    radix_sort.py
    def digit(num: int, exp: int) -> int:\n\"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n# \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num // exp) % 10\ndef counting_sort_digit(nums: list[int], exp: int) -> None:\n\"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n# \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\ncounter = [0] * 10\nn = len(nums)\n# \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in range(n):\nd = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n# \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in range(1, 10):\ncounter[i] += counter[i - 1]\n# \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nd = digit(nums[i], exp)\nj = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n# \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\ndef radix_sort(nums: list[int]) -> None:\n\"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n# \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nm = max(nums)\n# \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nexp = 1\nwhile exp <= m:\n# \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n# k = 1 -> exp = 1\n# k = 2 -> exp = 10\n# \u5373 exp = 10^(k-1)\ncounting_sort_digit(nums, exp)\nexp *= 10\n
    radix_sort.go
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\ncounter := make([]int, 10)\nn := len(nums)\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i := 0; i < n; i++ {\nd := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i := 1; i < 10; i++ {\ncounter[i] += counter[i-1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres := make([]int, n)\nfor i := n - 1; i >= 0; i-- {\nd := digit(nums[i], exp)\nj := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i := 0; i < n; i++ {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nmax := math.MinInt\nfor _, num := range nums {\nif num > max {\nmax = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp := 1; max >= exp; exp *= 10 {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp)\n}\n}\n
    radix_sort.js
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i < n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i >= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num > m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.ts
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i < n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i >= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num > m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.c
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint *counter = (int *)malloc((sizeof(int) * 10));\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < size; i++) {\n// \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\nint d = digit(nums[i], exp);\n// \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\ncounter[d]++;\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint *res = (int *)malloc(sizeof(int) * size);\nfor (int i = size - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < size; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint max = INT32_MIN;\nfor (size_t i = 0; i < size - 1; i++) {\nif (nums[i] > max) {\nmax = nums[i];\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; max >= exp; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, size, exp);\n}\n
    radix_sort.cs
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nint[] counter = new int[10];\nint n = nums.Length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = int.MinValue;\nforeach (int num in nums) {\nif (num > m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n
    radix_sort.swift
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -> Int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n(num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvar counter = Array(repeating: 0, count: 10)\nlet n = nums.count\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in nums.indices {\nlet d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in 1 ..< 10 {\ncounter[i] += counter[i - 1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = Array(repeating: 0, count: n)\nfor i in stride(from: n - 1, through: 0, by: -1) {\nlet d = digit(num: nums[i], exp: exp)\nlet j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in nums.indices {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m = Int.min\nfor num in nums {\nif num > m {\nm = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp in sequence(first: 1, next: { m >= ($0 * 10) ? $0 * 10 : nil }) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums: &nums, exp: exp)\n}\n}\n
    radix_sort.zig
    // \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn @mod(@divFloor(num, exp), 10);\n}\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n// defer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar counter = try mem_allocator.alloc(usize, 10);\n@memset(counter, 0);\nvar n = nums.len;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (nums) |num| {\nvar d = @bitCast(u32, digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nvar i: usize = 1;\nwhile (i < 10) : (i += 1) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = try mem_allocator.alloc(i32, n);\ni = n - 1;\nwhile (i >= 0) : (i -= 1) {\nvar d = @bitCast(u32, digit(nums[i], exp));\nvar j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\nif (i == 0) break;\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\ni = 0;\nwhile (i < n) : (i += 1) {\nnums[i] = res[i];\n}\n}\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m: i32 = std.math.minInt(i32);\nfor (nums) |num| {\nif (num > m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nvar exp: i32 = 1;\nwhile (exp <= m) : (exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ntry countingSortDigit(nums, exp);    }\n}
    radix_sort.dart
    /* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num ~/ exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List<int> nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\nList<int> counter = List<int>.filled(10, 0);\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i < n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i < 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nList<int> res = List<int>.filled(n, 0);\nfor (int i = n - 1; i >= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i < n; i++) nums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List<int> nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n// dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\nint m = -1 << 63; for (int num in nums) if (num > m) m = num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp <= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -> exp = 1\n// k = 2 -> exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n

    \u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f

    \u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a < b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a > b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u6211\u4eec\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002

    "},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"

    \u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002

    • \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\) \uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\) \u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\) \u3001\u975e\u539f\u5730\u6392\u5e8f \uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 res \u548c counter \u3002
    • \u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\u3002
    "},{"location":"chapter_sorting/selection_sort/","title":"11.2. \u00a0 \u9009\u62e9\u6392\u5e8f","text":"

    \u300c\u9009\u62e9\u6392\u5e8f Selection Sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u76f4\u63a5\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002

    \u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u4e0b\uff1a

    1. \u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002
    2. \u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    3. \u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    4. \u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002
    5. \u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u9700\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002
    <1><2><3><4><5><6><7><8><9><10><11>

    \u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart selection_sort.java
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    selection_sort.cpp
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector<int> &nums) {\nint n = nums.size();\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nswap(nums[i], nums[k]);\n}\n}\n
    selection_sort.py
    def selection_sort(nums: list[int]):\n\"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in range(n - 1):\n# \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk = i\nfor j in range(i + 1, n):\nif nums[j] < nums[k]:\nk = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n# \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n
    selection_sort.go
    /* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\nn := len(nums)\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i := 0; i < n-1; i++ {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk := i\nfor j := i + 1; j < n; j++ {\nif nums[j] < nums[k] {\n// \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\nk = j\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n}\n}\n
    selection_sort.js
    /* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n
    selection_sort.ts
    /* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n
    selection_sort.c
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j;  // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    selection_sort.cs
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\nint n = nums.Length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n(nums[k], nums[i]) = (nums[i], nums[k]);\n}\n}\n
    selection_sort.swift
    /* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in nums.indices.dropLast() {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nvar k = i\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[j] < nums[k] {\nk = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums.swapAt(i, k)\n}\n}\n
    selection_sort.zig
    [class]{}-[func]{selectionSort}\n
    selection_sort.dart
    /* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List<int> nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i < n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j < n; j++) {\nif (nums[j] < nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n
    "},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1. \u00a0 \u7b97\u6cd5\u7279\u6027","text":"
    • \u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\) , \\(n - 1\\) , \\(\\cdots\\) , \\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002
    • \u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) , \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002
    • \u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5143\u7d20\u65f6\uff0c\u6709\u53ef\u80fd\u5c06 nums[i] \u4ea4\u6362\u81f3\u5176\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002

    Fig. \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b

    "},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1. \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"

    \u300c\u6392\u5e8f\u7b97\u6cd5 Sorting Algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u6709\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002

    \u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\uff1b\u987a\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002

    Fig. \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b

    "},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1. \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"

    \u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u5373\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u964d\u4f4e\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002

    \u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u9700\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002

    \u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002\u7a33\u5b9a\u6392\u5e8f\u662f\u4f18\u826f\u7279\u6027\uff0c\u4e5f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002

    \u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1, 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\u3002

    # \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n('A', 19)\n('B', 18)\n('C', 21)\n('D', 19)\n('E', 23)\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n('B', 18)\n('D', 19)\n('A', 19)\n('C', 21)\n('E', 23)\n

    \u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u3001\u6700\u5dee\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002

    \u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002

    \u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u4e8e\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(<\\) , \\(=\\) , \\(>\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002

    "},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2. \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"

    \u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002

    \u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002

    "},{"location":"chapter_sorting/summary/","title":"11.11. \u00a0 \u5c0f\u7ed3","text":"
    • \u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002
    • \u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5b83\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002
    • \u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002
    • \u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002
    • \u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002
    • \u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002
    • \u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002

    Fig. \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4

    • \u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002
    "},{"location":"chapter_sorting/summary/#11111-q-a","title":"11.11.1. \u00a0 Q & A","text":"

    \u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u987b\u7684\uff1f

    \u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u5728\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u4e0a\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f/

    \u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 (A, 180) (B, 185) (C, 170) (D, 170) \uff1b\u63a5\u4e0b\u6765\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u6211\u4eec\u53ef\u80fd\u5f97\u5230 (D, 170) (C, 170) (A, 180) (B, 185) \u3002

    \u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002

    \u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f

    \u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002

    \u54e8\u5175\u5212\u5206 partition() \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 nums[left] \u548c nums[i] \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <= \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d nums[left] >= nums[i] \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 i == j \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd nums[j] == nums[i] > nums[left]\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002

    \u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 [0, 0, 0, 0, 1] \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a [1, 0, 0, 0, 0] \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002

    \u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 nums[right] \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002

    \u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(log n\\) \uff1f

    \u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u7684\u4e00\u534a\u957f\u5ea6\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(log n\\) \u3002

    \u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n, n - 1, n - 2, ..., 2, 1\\) \uff0c\u4ece\u800c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u7684\u51fa\u73b0\u3002

    \u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f

    \u662f\u7684\u3002\u8fd9\u79cd\u60c5\u51b5\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002

    \u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f

    \u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002

    "},{"location":"chapter_stack_and_queue/","title":"5. \u00a0 \u6808\u4e0e\u961f\u5217","text":""},{"location":"chapter_stack_and_queue/deque/","title":"5.3. \u00a0 \u53cc\u5411\u961f\u5217","text":"

    \u5bf9\u4e8e\u961f\u5217\uff0c\u6211\u4eec\u4ec5\u80fd\u5728\u5934\u90e8\u5220\u9664\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u7136\u800c\uff0c\u300c\u53cc\u5411\u961f\u5217 Deque\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002

    Fig. \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c

    "},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1. \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"

    \u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\)

    \u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart deque.java
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque<Integer> deque = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n
    deque.cpp
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque<int> deque;\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n
    deque.py
    # \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: Deque[int] = collections.deque()\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n
    deque_test.go
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n
    deque.js
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
    deque.ts
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n
    deque.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n
    deque.cs
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList<int> deque = new LinkedList<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n
    deque.swift
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n
    deque.zig
    \n
    deque.dart
    /* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue<int> deque = Queue<int>();\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n
    "},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2. \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"

    \u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002

    "},{"location":"chapter_stack_and_queue/deque/#_1","title":"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002

    \u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u300c\u53cc\u5411\u94fe\u8868\u300d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002

    \u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002

    LinkedListDequepushLast()pushFirst()popLast()popFirst()

    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_deque.java
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(int val) {\nthis.val = val;\nprev = next = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, boolean isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate Integer pop(boolean isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty())\nreturn null;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic Integer popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic Integer popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic Integer peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic Integer peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_deque.cpp
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\nint val;              // \u8282\u70b9\u503c\nDoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\nDoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\nDoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n}\n};\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate:\nDoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nLinkedListDeque() : front(nullptr), rear(nullptr) {\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~LinkedListDeque() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nDoublyListNode *pre, *cur = front;\nwhile (cur != nullptr) {\npre = cur;\ncur = cur->next;\ndelete pre;\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nDoublyListNode *node = new DoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront->prev = node;\nnode->next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear->next = node;\nnode->prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de -1\nif (isEmpty())\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front->val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nDoublyListNode *fNext = front->next;\nif (fNext != nullptr) {\nfNext->prev = nullptr;\nfront->next = nullptr;\ndelete front;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear->val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nDoublyListNode *rPrev = rear->prev;\nif (rPrev != nullptr) {\nrPrev->next = nullptr;\nrear->prev = nullptr;\ndelete rear;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nreturn isEmpty() ? -1 : front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nreturn isEmpty() ? -1 : rear->val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\nDoublyListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_deque.py
    class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\ndef __init__(self, val: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.val: int = val\nself.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nself.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nclass LinkedListDeque:\n\"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself.rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself.__size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.size() == 0\ndef push(self, num: int, is_front: bool) -> None:\n\"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\nnode = ListNode(num)\n# \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif self.is_empty():\nself.front = self.rear = node\n# \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelif is_front:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.prev = node\nnode.next = self.front\nself.front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.next = node\nnode.prev = self.rear\nself.rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself.__size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\ndef push_first(self, num: int) -> None:\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nself.push(num, True)\ndef push_last(self, num: int) -> None:\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nself.push(num, False)\ndef pop(self, is_front: bool) -> int:\n\"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n# \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\nif self.is_empty():\nreturn None\n# \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front:\nval: int = self.front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n# \u5220\u9664\u5934\u8282\u70b9\nfnext: ListNode | None = self.front.next\nif fnext != None:\nfnext.prev = None\nself.front.next = None\nself.front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse:\nval: int = self.rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n# \u5220\u9664\u5c3e\u8282\u70b9\nrprev: ListNode | None = self.rear.prev\nif rprev != None:\nrprev.next = None\nself.rear.prev = None\nself.rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself.__size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\ndef pop_first(self) -> int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nreturn self.pop(True)\ndef pop_last(self) -> int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nreturn self.pop(False)\ndef peek_first(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nreturn None if self.is_empty() else self.front.val\ndef peek_last(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nreturn None if self.is_empty() else self.rear.val\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\nnode = self.front\nres = [0] * self.size()\nfor i in range(self.size()):\nres[i] = node.val\nnode = node.next\nreturn res\n
    linkedlist_deque.go
    /* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\nreturn &linkedListDeque{\ndata: list.New(),\n}\n}\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\ns.data.PushFront(value)\n}\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\ns.data.PushBack(value)\n}\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_deque.js
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval; // \u8282\u70b9\u503c\nconstructor(val) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n#front; // \u5934\u8282\u70b9 front\n#rear; // \u5c3e\u8282\u70b9 rear\n#queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\nthis.#queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.#rear.next = node;\nnode.prev = this.#rear;\nthis.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.#front.prev = node;\nnode.next = this.#front;\nthis.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp = this.#rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.#rear.prev = null;\n}\nthis.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp = this.#front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.#front.next = null;\n}\nthis.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nreturn this.#queSize === 0 ? null : this.#rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nreturn this.#queSize === 0 ? null : this.#front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint() {\nconst arr = [];\nlet temp = this.#front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n
    linkedlist_deque.ts
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval: number; // \u8282\u70b9\u503c\nconstructor(val: number) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate front: ListNode; // \u5934\u8282\u70b9 front\nprivate rear: ListNode; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.front = null;\nthis.rear = null;\nthis.queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.rear.next = node;\nnode.prev = this.rear;\nthis.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.front.prev = node;\nnode.next = this.front;\nthis.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp: ListNode = this.rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.rear.prev = null;\n}\nthis.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp: ListNode = this.front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.front.next = null;\n}\nthis.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nreturn this.queSize === 0 ? null : this.rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nreturn this.queSize === 0 ? null : this.front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint(): void {\nconst arr: number[] = [];\nlet temp: ListNode = this.front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n
    linkedlist_deque.c
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct doublyListNode {\nint val;                     // \u8282\u70b9\u503c\nstruct doublyListNode *next; // \u540e\u7ee7\u8282\u70b9\nstruct doublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n};\ntypedef struct doublyListNode doublyListNode;\n/* \u6784\u9020\u51fd\u6570 */\ndoublyListNode *newDoublyListNode(int num) {\ndoublyListNode *new = (doublyListNode *)malloc(sizeof(doublyListNode));\nnew->val = num;\nnew->next = NULL;\nnew->prev = NULL;\nreturn new;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(doublyListNode *node) {\nfree(node);\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct linkedListDeque {\ndoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n};\ntypedef struct linkedListDeque linkedListDeque;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListDeque *newLinkedListDeque() {\nlinkedListDeque *deque = (linkedListDeque *)malloc(sizeof(linkedListDeque));\ndeque->front = NULL;\ndeque->rear = NULL;\ndeque->queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(linkedListDeque *deque) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i < deque->queSize && deque->front != NULL; i++) {\ndoublyListNode *tmp = deque->front;\ndeque->front = deque->front->next;\nfree(tmp);\n}\n// \u91ca\u653e deque \u7ed3\u6784\u4f53\nfree(deque);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(linkedListDeque *deque) {\nreturn deque->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(linkedListDeque *deque) {\nreturn (size(deque) == 0);\n}\n/* \u5165\u961f */\nvoid push(linkedListDeque *deque, int num, bool isFront) {\ndoublyListNode *node = newDoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411node\nif (empty(deque)) {\ndeque->front = deque->rear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\ndeque->front->prev = node;\nnode->next = deque->front;\ndeque->front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u5bf9\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\ndeque->rear->next = node;\nnode->prev = deque->rear;\ndeque->rear = node;\n}\ndeque->queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(linkedListDeque *deque, int num) {\npush(deque, num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(linkedListDeque *deque, int num) {\npush(deque, num, false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(linkedListDeque *deque) {\nassert(size(deque) && deque->front);\nreturn deque->front->val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(linkedListDeque *deque) {\nassert(size(deque) && deque->rear);\nreturn deque->rear->val;\n}\n/* \u51fa\u961f */\nint pop(linkedListDeque *deque, bool isFront) {\nif (empty(deque))\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\ndoublyListNode *fNext = deque->front->next;\nif (fNext) {\nfNext->prev = NULL;\ndeque->front->next = NULL;\ndelDoublyListNode(deque->front);\n}\ndeque->front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\ndoublyListNode *rPrev = deque->rear->prev;\nif (rPrev) {\nrPrev->next = NULL;\ndeque->rear->prev = NULL;\ndelDoublyListNode(deque->rear);\n}\ndeque->rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\ndeque->queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(linkedListDeque *deque) {\nreturn pop(deque, true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(linkedListDeque *deque) {\nreturn pop(deque, false);\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(linkedListDeque *deque) {\nint arr[deque->queSize];\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\ndoublyListNode *node;\nfor (i = 0, node = deque->front; i < deque->queSize; i++) {\narr[i] = node->val;\nnode = node->next;\n}\nprintArray(arr, deque->queSize);\n}\n
    linkedlist_deque.cs
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\npublic int val;       // \u8282\u70b9\u503c\npublic ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\npublic ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\npublic ListNode(int val) {\nthis.val = val;\nprev = null;\nnext = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\nprivate int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, bool isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (isEmpty()) {\nfront = node;\nrear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate int? pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty()) {\nreturn null;\n}\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int? popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int? popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int? peekFirst() {\nreturn isEmpty() ? null : front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int? peekLast() {\nreturn isEmpty() ? null : rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_deque.swift
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nvar prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\ninit(val: Int) {\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate var front: ListNode? // \u5934\u8282\u70b9 front\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9 rear\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ninit() {\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate func push(num: Int, isFront: Bool) {\nlet node = ListNode(val: num)\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif isEmpty() {\nfront = node\nrear = node\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if isFront {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront?.prev = node\nnode.next = front\nfront = node // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear?.next = node\nnode.prev = rear\nrear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\npush(num: num, isFront: true)\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\npush(num: num, isFront: false)\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate func pop(isFront: Bool) -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nlet val: Int\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif isFront {\nval = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet fNext = front?.next\nif fNext != nil {\nfNext?.prev = nil\nfront?.next = nil\n}\nfront = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet rPrev = rear?.prev\nif rPrev != nil {\nrPrev?.next = nil\nrear?.prev = nil\n}\nrear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\npop(isFront: true)\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\npop(isFront: false)\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int? {\nisEmpty() ? nil : front?.val\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int? {\nisEmpty() ? nil : rear?.val\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_deque.zig
    // \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = undefined,     // \u8282\u70b9\u503c\nnext: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nprev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\n// Initialize a list node with specific value\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\nrear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u5165\u961f\u64cd\u4f5c\npub fn push(self: *Self, num: T, is_front: bool) !void {\nvar node = try self.mem_allocator.create(ListNode(T));\nnode.init(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front, rear \u90fd\u6307\u5411 node\nif (self.isEmpty()) {\nself.front = node;\nself.rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n} else if (is_front) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.?.prev = node;\nnode.next = self.front;\nself.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.?.next = node;\nnode.prev = self.rear;\nself.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n} // \u961f\u9996\u5165\u961f\npub fn pushFirst(self: *Self, num: T) !void {\ntry self.push(num, true);\n} // \u961f\u5c3e\u5165\u961f\npub fn pushLast(self: *Self, num: T) !void {\ntry self.push(num, false);\n} // \u51fa\u961f\u64cd\u4f5c\npub fn pop(self: *Self, is_front: bool) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nvar val: T = undefined;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (is_front) {\nval = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nvar fNext = self.front.?.next;\nif (fNext != null) {\nfNext.?.prev = null;\nself.front.?.next = null;\n}\nself.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nvar rPrev = self.rear.?.prev;\nif (rPrev != null) {\nrPrev.?.next = null;\nself.rear.?.prev = null;\n}\nself.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n} // \u961f\u9996\u51fa\u961f\npub fn popFirst(self: *Self) T {\nreturn self.pop(true);\n} // \u961f\u5c3e\u51fa\u961f\npub fn popLast(self: *Self) T {\nreturn self.pop(false);\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peekFirst(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\npub fn peekLast(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.rear.?.val;\n}\n// \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_deque.dart
    /* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\nListNode(this.val, {this.next, this.prev});\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\nlate ListNode? _front; // \u5934\u8282\u70b9 _front\nlate ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nLinkedListDeque() {\nthis._front = null;\nthis._rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\nint size() {\nreturn this._queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nfinal ListNode node = ListNode(num);\nif (isEmpty()) {\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front\uff0c_rear \u90fd\u6307\u5411 node\n_front = _rear = node;\n} else if (isFront) {\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n_front!.prev = node;\nnode.next = _front;\n_front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n_rear!.next = node;\nnode.prev = _rear;\n_rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint? pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty()) {\nreturn null;\n}\nfinal int val;\nif (isFront) {\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nval = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode? fNext = _front!.next;\nif (fNext != null) {\nfNext.prev = null;\n_front!.next = null;\n}\n_front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nval = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode? rPrev = _rear!.prev;\nif (rPrev != null) {\nrPrev.next = null;\n_rear!.prev = null;\n}\n_rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint? popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint? popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint? peekFirst() {\nreturn _front?.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint? peekLast() {\nreturn _rear?.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList<int> toArray() {\nListNode? node = _front;\nfinal List<int> res = [];\nfor (int i = 0; i < _queSize; i++) {\nres.add(node!.val);\nnode = node.next;\n}\nreturn res;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\u3002

    ArrayDequepushLast()pushFirst()popLast()popFirst()

    \u4ee5\u4e0b\u662f\u5177\u4f53\u5b9e\u73b0\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_deque.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate:\nvector<int> nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nnums.resize(capacity);\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn nums.size();\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (queSize == capacity()) {\ncout << \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> res(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n};\n
    array_deque.py
    class ArrayDeque:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self, capacity: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__nums: list[int] = [0] * capacity\nself.__front: int = 0\nself.__size: int = 0\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__size == 0\ndef index(self, i: int) -> int:\n\"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + self.capacity()) % self.capacity()\ndef push_first(self, num: int) -> None:\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself.__front = self.index(self.__front - 1)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself.__nums[self.__front] = num\nself.__size += 1\ndef push_last(self, num: int) -> None:\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear = self.index(self.__front + self.__size)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop_first(self) -> int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nnum = self.peek_first()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself.__front = self.index(self.__front + 1)\nself.__size -= 1\nreturn num\ndef pop_last(self) -> int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nnum = self.peek_last()\nself.__size -= 1\nreturn num\ndef peek_first(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\nreturn self.__nums[self.__front]\ndef peek_last(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n# \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast = self.index(self.__front + self.__size - 1)\nreturn self.__nums[last]\ndef to_array(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n# \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres = []\nfor i in range(self.__size):\nres.append(self.__nums[self.index(self.__front + i)])\nreturn res\n
    array_deque.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\nreturn &arrayDeque{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + q.queCapacity) % q.queCapacity\n}\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nq.front = q.index(q.front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[q.front] = num\nq.queSize++\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear := q.index(q.front + q.queSize)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[rear] = num\nq.queSize++\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\nnum := q.peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nq.front = q.index(q.front + 1)\nq.queSize--\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\nnum := q.peekLast()\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\nif q.isEmpty() {\nreturn nil\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast := q.index(q.front + q.queSize - 1)\nreturn q.nums[last]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres := make([]int, q.queSize)\nfor i, j := 0, q.front; i < q.queSize; i++ {\nres[i] = q.nums[q.index(j)]\nj++\n}\nreturn res\n}\n
    array_deque.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\nthis.#front = 0;\nthis.#queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.#front = this.index(this.#front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.#nums[this.#front] = num;\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear = this.index(this.#front + this.#queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst() {\nconst num = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.#front = this.index(this.#front + 1);\nthis.#queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast() {\nconst num = this.peekLast();\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.#nums[this.#front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.#front + this.#queSize - 1);\nreturn this.#nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res = [];\nfor (let i = 0, j = this.#front; i < this.#queSize; i++, j++) {\nres[i] = this.#nums[this.index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = 0;\nthis.queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i: number): number {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.front = this.index(this.front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.nums[this.front] = num;\nthis.queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear: number = this.index(this.front + this.queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst(): number {\nconst num: number = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.front = this.index(this.front + 1);\nthis.queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast(): number {\nconst num: number = this.peekLast();\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.nums[this.front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.front + this.queSize - 1);\nreturn this.nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res: number[] = [];\nfor (let i = 0, j = this.front; i < this.queSize; i++, j++) {\nres[i] = this.nums[this.index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.c
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct arrayDeque {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n};\ntypedef struct arrayDeque arrayDeque;\n/* \u6784\u9020\u51fd\u6570 */\narrayDeque *newArrayDeque(int capacity) {\narrayDeque *deque = (arrayDeque *)malloc(sizeof(arrayDeque));\n// \u521d\u59cb\u5316\u6570\u7ec4\ndeque->queCapacity = capacity;\ndeque->nums = (int *)malloc(sizeof(int) * deque->queCapacity);\ndeque->front = deque->queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(arrayDeque *deque) {\nfree(deque->nums);\ndeque->queCapacity = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(arrayDeque *deque) {\nreturn deque->queCapacity;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(arrayDeque *deque) {\nreturn deque->queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(arrayDeque *deque) {\nreturn deque->queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(arrayDeque *deque, int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn ((i + capacity(deque)) % capacity(deque));\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(arrayDeque *deque, int num) {\nif (deque->queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\ndeque->front = dequeIndex(deque, deque->front - 1);\n// \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\ndeque->nums[deque->front] = num;\ndeque->queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(arrayDeque *deque, int num) {\nif (deque->queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = dequeIndex(deque, deque->front + deque->queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque->nums[rear] = num;\ndeque->queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(arrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nreturn deque->nums[deque->front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(arrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nint last = dequeIndex(deque, deque->front + deque->queSize - 1);\nreturn deque->nums[last];\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(arrayDeque *deque) {\nint num = peekFirst(deque);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\ndeque->front = dequeIndex(deque, deque->front + 1);\ndeque->queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(arrayDeque *deque) {\nint num = peekLast(deque);\ndeque->queSize--;\nreturn num;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printArrayDeque(arrayDeque *deque) {\nint arr[deque->queSize];\n// \u62f7\u8d1d\nfor (int i = 0, j = deque->front; i < deque->queSize; i++, j++) {\narr[i] = deque->nums[j % deque->queCapacity];\n}\nprintArray(arr, deque->queSize);\n}\n
    array_deque.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate readonly int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty()) {\nthrow new InvalidOperationException();\n}\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty()) {\nthrow new InvalidOperationException();\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n
    array_deque.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(capacity: Int) {\nnums = Array(repeating: 0, count: capacity)\nfront = 0\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate func index(i: Int) -> Int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n(i + capacity()) % capacity()\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(i: front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num\nqueSize += 1\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = index(i: front + size())\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -> Int {\nlet num = peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(i: front + 1)\nqueSize -= 1\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -> Int {\nlet num = peekLast()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -> Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = index(i: front + size() - 1)\nreturn nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: size())\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[index(i: j)]\n}\nreturn res\n}\n}\n
    array_deque.zig
    [class]{ArrayDeque}-[func]{}\n
    array_deque.dart
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nlate List<int> _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nthis._nums = List.filled(capacity, 0);\nthis._front = this._queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n_front = index(_front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n_nums[_front] = num;\n_queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(_front + _queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = num;\n_queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n_front = index(_front + 1);\n_queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(_front + _queSize - 1);\nreturn _nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList<int> toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nList<int> res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i < _queSize; i++, j++) {\nres[i] = _nums[index(j)];\n}\nreturn res;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3. \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"

    \u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002

    \u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c push \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 pop \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u5373\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002

    "},{"location":"chapter_stack_and_queue/queue/","title":"5.2. \u00a0 \u961f\u5217","text":"

    \u300c\u961f\u5217 Queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\uff08First In, First Out\uff09\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u7684\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002

    \u6211\u4eec\u628a\u961f\u5217\u7684\u5934\u90e8\u79f0\u4e3a\u300c\u961f\u9996\u300d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u300c\u961f\u5c3e\u300d\uff0c\u628a\u5c06\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u5165\u961f\u300d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u300c\u51fa\u961f\u300d\u3002

    Fig. \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219

    "},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1. \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"

    \u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002

    \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\)

    \u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart queue.java
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<Integer> queue = new LinkedList<>();\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n
    queue.cpp
    /* \u521d\u59cb\u5316\u961f\u5217 */\nqueue<int> queue;\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n
    queue.py
    # \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u770b\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u5efa\u8bae\nque: Deque[int] = collections.deque()\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n
    queue_test.go
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n
    queue.js
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
    queue.ts
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n
    queue.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n
    queue.cs
    /* \u521d\u59cb\u5316\u961f\u5217 */\nQueue<int> queue = new();\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count() == 0;\n
    queue.swift
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n
    queue.zig
    \n
    queue.dart
    /* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue<int> queue = Queue();\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n
    "},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2. \u00a0 \u961f\u5217\u5b9e\u73b0","text":"

    \u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u56e0\u6b64\uff0c\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u53ef\u4ee5\u7528\u6765\u5b9e\u73b0\u961f\u5217\u3002

    "},{"location":"chapter_stack_and_queue/queue/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u5bf9\u4e8e\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u300c\u5934\u8282\u70b9\u300d\u548c\u300c\u5c3e\u8282\u70b9\u300d\u5206\u522b\u89c6\u4e3a\u961f\u9996\u548c\u961f\u5c3e\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u800c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002

    LinkedListQueuepush()pop()

    \u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_queue.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new IndexOutOfBoundsException();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate:\nListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;\npublic:\nLinkedListQueue() {\nfront = nullptr;\nrear = nullptr;\nqueSize = 0;\n}\n~LinkedListQueue() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(front);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode *node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == nullptr) {\nfront = node;\nrear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear->next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nListNode *tmp = front;\nfront = front->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (size() == 0)\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn front->val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode *node = front;\nvector<int> res(size());\nfor (int i = 0; i < res.size(); i++) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_queue.py
    class LinkedListQueue:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself.__rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self.__front\ndef push(self, num: int) -> None:\n\"\"\"\u5165\u961f\"\"\"\n# \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nnode = ListNode(num)\n# \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif self.__front is None:\nself.__front = node\nself.__rear = node\n# \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse:\nself.__rear.next = node\nself.__rear = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u961f\"\"\"\nnum = self.peek()\n# \u5220\u9664\u5934\u8282\u70b9\nself.__front = self.__front.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.size() == 0:\nprint(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn False\nreturn self.__front.val\ndef to_list(self) -> list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nqueue = []\ntemp = self.__front\nwhile temp:\nqueue.append(temp.val)\ntemp = temp.next\nreturn queue\n
    linkedlist_queue.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\nreturn &linkedListQueue{\ndata: list.New(),\n}\n}\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\ns.data.PushBack(value)\n}\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_queue.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n#front; // \u5934\u8282\u70b9 #front\n#rear; // \u5c3e\u8282\u70b9 #rear\n#queSize = 0;\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.#front) {\nthis.#front = node;\nthis.#rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.#rear.next = node;\nthis.#rear = node;\n}\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nthis.#front = this.#front.next;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#front;\nconst res = new Array(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate front: ListNode | null; // \u5934\u8282\u70b9 front\nprivate rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number = 0;\nconstructor() {\nthis.front = null;\nthis.rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.front) {\nthis.front = node;\nthis.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.rear!.next = node;\nthis.rear = node;\n}\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\nif (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n// \u5220\u9664\u5934\u8282\u70b9\nthis.front = this.front.next;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.front;\nconst res = new Array<number>(this.size);\nfor (let i = 0; i < res.length; i++) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.c
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct linkedListQueue {\nListNode *front, *rear;\nint queSize;\n};\ntypedef struct linkedListQueue linkedListQueue;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListQueue *newLinkedListQueue() {\nlinkedListQueue *queue = (linkedListQueue *)malloc(sizeof(linkedListQueue));\nqueue->front = NULL;\nqueue->rear = NULL;\nqueue->queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(linkedListQueue *queue) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i < queue->queSize && queue->front != NULL; i++) {\nListNode *tmp = queue->front;\nqueue->front = queue->front->next;\nfree(tmp);\n}\n// \u91ca\u653e queue \u7ed3\u6784\u4f53\nfree(queue);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(linkedListQueue *queue) {\nreturn queue->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(linkedListQueue *queue) {\nreturn (size(queue) == 0);\n}\n/* \u5165\u961f */\nvoid push(linkedListQueue *queue, int num) {\n// \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\nListNode *node = newListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (queue->front == NULL) {\nqueue->front = node;\nqueue->rear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nqueue->rear->next = node;\nqueue->rear = node;\n}\nqueue->queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(linkedListQueue *queue) {\nassert(size(queue) && queue->front);\nreturn queue->front->val;\n}\n/* \u51fa\u961f */\nvoid pop(linkedListQueue *queue) {\nint num = peek(queue);\nListNode *tmp = queue->front;\nqueue->front = queue->front->next;\nfree(tmp);\nqueue->queSize--;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(linkedListQueue *queue) {\nint arr[queue->queSize];\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\nListNode *node;\nfor (i = 0, node = queue->front; i < queue->queSize && queue->front != queue->rear; i++) {\narr[i] = node->val;\nnode = node->next;\n}\nprintArray(arr, queue->queSize);\n}\n
    linkedlist_queue.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else if (rear != null) {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (size() == 0 || front == null)\nthrow new Exception();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nif (front == null)\nreturn Array.Empty<int>();\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i < res.Length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_queue.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate var front: ListNode? // \u5934\u8282\u70b9\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9\nprivate var _size = 0\ninit() {}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nlet node = ListNode(x: num)\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif front == nil {\nfront = node\nrear = node\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear?.next = node\nrear = node\n}\n_size += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn front!.val\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_queue.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\nrear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u5165\u961f\npub fn push(self: *Self, num: T) !void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (self.front == null) {\nself.front = node;\nself.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nself.rear.?.next = node;\nself.rear = node;\n}\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nself.front = self.front.?.next;\nself.que_size -= 1;\nreturn num;\n} // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_queue.dart
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nListNode? _front; // \u5934\u8282\u70b9 _front\nListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u961f\u5217\u957f\u5ea6\nLinkedListQueue() {\n_front = null;\n_rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nfinal node = ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (_front == null) {\n_front = node;\n_rear = node;\n} else {\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n_rear!.next = node;\n_rear = node;\n}\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nfinal int num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\n_front = _front!.next;\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (_queSize == 0) {\nthrow Exception('\u961f\u5217\u4e3a\u7a7a');\n}\nreturn _front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList<int> toArray() {\nListNode? node = _front;\nfinal List<int> queue = [];\nwhile (node != null) {\nqueue.add(node.val);\nnode = node.next;\n}\nreturn queue;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/queue/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u7531\u4e8e\u6570\u7ec4\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002

    \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf front \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf queSize \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 rear = front + queSize \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 rear \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002

    \u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a [front, rear - 1]\uff0c\u8fdb\u800c\uff1a

    • \u5bf9\u4e8e\u5165\u961f\u64cd\u4f5c\uff0c\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 rear \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 queSize \u589e\u52a0 1 \uff1b
    • \u5bf9\u4e8e\u51fa\u961f\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06 front \u589e\u52a0 1 \uff0c\u5e76\u5c06 queSize \u51cf\u5c11 1 \uff1b

    \u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002

    ArrayQueuepush()pop()

    \u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0cfront \u548c rear \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u300c\u73af\u5f62\u6570\u7ec4\u300d\u3002

    \u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 front \u6216 rear \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_queue.java
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % capacity()];\n}\nreturn res;\n}\n}\n
    array_queue.cpp
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate:\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u961f\u5217\u957f\u5ea6\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\npublic:\nArrayQueue(int capacity) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = new int[capacity];\nqueCapacity = capacity;\nfront = queSize = 0;\n}\n~ArrayQueue() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (queSize == queCapacity) {\ncout << \"\u961f\u5217\u5df2\u6ee1\" << endl;\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % queCapacity;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (empty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector<int> toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector<int> arr(queSize);\nfor (int i = 0, j = front; i < queSize; i++, j++) {\narr[i] = nums[j % queCapacity];\n}\nreturn arr;\n}\n};\n
    array_queue.py
    class ArrayQueue:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self, size: int) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nself.__front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nself.__size: int = 0  # \u961f\u5217\u957f\u5ea6\ndef capacity(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self.__nums)\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__size == 0\ndef push(self, num: int) -> None:\n\"\"\"\u5165\u961f\"\"\"\nif self.__size == self.capacity():\nraise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8F\nrear: int = (self.__front + self.__size) % self.capacity()\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.__nums[rear] = num\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u961f\"\"\"\nnum: int = self.peek()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.__front = (self.__front + 1) % self.capacity()\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn self.__nums[self.__front]\ndef to_list(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nres = [0] * self.size()\nj: int = self.__front\nfor i in range(self.size()):\nres[i] = self.__nums[(j % self.capacity())]\nj += 1\nreturn res\n
    array_queue.go
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\nreturn &arrayQueue{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n// \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\nif q.queSize == q.queCapacity {\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear := (q.front + q.queSize) % q.queCapacity\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nq.nums[rear] = num\nq.queSize++\n}\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\nnum := q.peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nq.front = (q.front + 1) % q.queCapacity\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\nrear := (q.front + q.queSize)\nif rear >= q.queCapacity {\nrear %= q.queCapacity\nreturn append(q.nums[q.front:], q.nums[:rear]...)\n}\nreturn q.nums[q.front:rear]\n}\n
    array_queue.js
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize = 0; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#queSize == 0;\n}\n/* \u5165\u961f */\npush(num) {\nif (this.size == this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.#front + this.size) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.#front = (this.#front + 1) % this.capacity;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.empty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#nums[this.#front];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.#front; i < this.size; i++, j++) {\narr[i] = this.#nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
    array_queue.ts
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = this.queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.queSize == 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\nif (this.size == this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.front + this.queSize) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.front = (this.front + 1) % this.capacity;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.empty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.nums[this.front];\n}\n/* \u8fd4\u56de Array */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.front; i < this.size; i++, j++) {\narr[i] = this.nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n
    array_queue.c
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct arrayQueue {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n};\ntypedef struct arrayQueue arrayQueue;\n/* \u6784\u9020\u51fd\u6570 */\narrayQueue *newArrayQueue(int capacity) {\narrayQueue *queue = (arrayQueue *)malloc(sizeof(arrayQueue));\n// \u521d\u59cb\u5316\u6570\u7ec4\nqueue->queCapacity = capacity;\nqueue->nums = (int *)malloc(sizeof(int) * queue->queCapacity);\nqueue->front = queue->queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(arrayQueue *queue) {\nfree(queue->nums);\nqueue->queCapacity = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(arrayQueue *queue) {\nreturn queue->queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(arrayQueue *queue) {\nreturn queue->queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(arrayQueue *queue) {\nreturn queue->queSize == 0;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(arrayQueue *queue) {\nassert(size(queue) != 0);\nreturn queue->nums[queue->front];\n}\n/* \u5165\u961f */\nvoid push(arrayQueue *queue, int num) {\nif (size(queue) == capacity(queue)) {\nprintf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (queue->front + queue->queSize) % queue->queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nqueue->nums[rear] = num;\nqueue->queSize++;\n}\n/* \u51fa\u961f */\nvoid pop(arrayQueue *queue) {\nint num = peek(queue);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nqueue->front = (queue->front + 1) % queue->queCapacity;\nqueue->queSize--;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printArrayQueue(arrayQueue *queue) {\nint arr[queue->queSize];\n// \u62f7\u8d1d\nfor (int i = 0, j = queue->front; i < queue->queSize; i++, j++) {\narr[i] = queue->nums[j % queue->queCapacity];\n}\nprintArray(arr, queue->queSize);\n}\n
    array_queue.cs
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nConsole.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new Exception();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i < queSize; i++, j++) {\nres[i] = nums[j % this.capacity()];\n}\nreturn res;\n}\n}\n
    array_queue.swift
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize = 0 // \u961f\u5217\u957f\u5ea6\ninit(capacity: Int) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = Array(repeating: 0, count: capacity)\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -> Int {\nnums.count\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nqueSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nqueSize == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\nif size() == capacity() {\nprint(\"\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (front + queSize) % capacity()\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfunc toArray() -> [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: queSize)\nfor (i, j) in sequence(first: (0, front), next: { $0 < self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[j % capacity()]\n}\nreturn res\n}\n}\n
    array_queue.zig
    // \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \ncap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\nfront: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.cap = cap;\nself.nums = try self.mem_allocator.alloc(T, self.cap);\n@memset(self.nums, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.cap;\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.queSize;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.queSize == 0;\n}\n// \u5165\u961f\npub fn push(self: *Self, num: T) !void {\nif (self.size() == self.capacity()) {\nstd.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nvar rear = (self.front + self.queSize) % self.capacity();\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nself.nums[rear] = num;\nself.queSize += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.capacity();\nself.queSize -= 1;\nreturn num;\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.nums[self.front];\n} // \u8fd4\u56de\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nvar j: usize = self.front;\nwhile (i < self.size()) : ({ i += 1; j += 1; }) {\nres[i] = self.nums[j % self.capacity()];\n}\nreturn res;\n}\n};\n}\n
    array_queue.dart
    /* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nlate List<int> _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u961f\u5217\u957f\u5ea6\nArrayQueue(int capacity) {\n_nums = List.filled(capacity, 0);\n_front = _queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capaCity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (_queSize == capaCity()) {\nthrow Exception(\"\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (_front + _queSize) % capaCity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = num;\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n_front = (_front + 1) % capaCity();\n_queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8fd4\u56de Array */\nList<int> toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nfinal List<int> res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i < _queSize; i++, j++) {\nres[i] = _nums[j % capaCity()];\n}\nreturn res;\n}\n}\n

    \u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff0c\u5373\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002

    \u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002

    "},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3. \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"
    • \u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u4f9d\u6b21\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002
    • \u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\u3002\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002
    "},{"location":"chapter_stack_and_queue/stack/","title":"5.1. \u00a0 \u6808","text":"

    \u300c\u6808 Stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\uff08First In, Last Out\uff09\u539f\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002

    \u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u9700\u8981\u62ff\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u53d6\u51fa\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u6570\u636e\u7ed3\u6784\u3002

    \u5728\u6808\u4e2d\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u300c\u6808\u9876\u300d\uff0c\u5e95\u90e8\u79f0\u4e3a\u300c\u6808\u5e95\u300d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u505a\u300c\u5165\u6808\u300d\uff0c\u800c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u505a\u300c\u51fa\u6808\u300d\u3002

    Fig. \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219

    "},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1. \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"

    \u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u4e0b\u8868\u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 push() , pop() , peek() \u547d\u540d\u4e3a\u4f8b\u3002

    \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\)

    \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u300c\u6570\u7ec4\u300d\u6216\u300c\u94fe\u8868\u300d\u89c6\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u901a\u8fc7\u201c\u8111\u8865\u201d\u6765\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart stack.java
    /* \u521d\u59cb\u5316\u6808 */\nStack<Integer> stack = new Stack<>();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n
    stack.cpp
    /* \u521d\u59cb\u5316\u6808 */\nstack<int> stack;\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n
    stack.py
    # \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: List[int] = []\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n
    stack_test.go
    /* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n
    stack.js
    /* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
    stack.ts
    /* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n
    stack.c
    // C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n
    stack.cs
    /* \u521d\u59cb\u5316\u6808 */\nStack<int> stack = new ();\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count()==0;\n
    stack.swift
    /* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n
    stack.zig
    \n
    stack.dart
    /* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList<int> stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n
    "},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2. \u00a0 \u6808\u7684\u5b9e\u73b0","text":"

    \u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002

    \u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u88ab\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002

    "},{"location":"chapter_stack_and_queue/stack/#_1","title":"\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"

    \u4f7f\u7528\u94fe\u8868\u6765\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002

    \u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002

    LinkedListStackpush()pop()

    \u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart linkedlist_stack.java
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0; // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0)\nthrow new IndexOutOfBoundsException();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.cpp
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate:\nListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize;        // \u6808\u7684\u957f\u5ea6\npublic:\nLinkedListStack() {\nstackTop = nullptr;\nstkSize = 0;\n}\n~LinkedListStack() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(stackTop);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nListNode *node = new ListNode(num);\nnode->next = stackTop;\nstackTop = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint num = top();\nListNode *tmp = stackTop;\nstackTop = stackTop->next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nstkSize--;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (size() == 0)\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stackTop->val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nvector<int> toVector() {\nListNode *node = stackTop;\nvector<int> res(size());\nfor (int i = res.size() - 1; i >= 0; i--) {\nres[i] = node->val;\nnode = node->next;\n}\nreturn res;\n}\n};\n
    linkedlist_stack.py
    class LinkedListStack:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__peek: ListNode | None = None\nself.__size: int = 0\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn self.__size\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self.__peek\ndef push(self, val: int) -> None:\n\"\"\"\u5165\u6808\"\"\"\nnode = ListNode(val)\nnode.next = self.__peek\nself.__peek = node\nself.__size += 1\ndef pop(self) -> int:\n\"\"\"\u51fa\u6808\"\"\"\nnum: int = self.peek()\nself.__peek = self.__peek.next\nself.__size -= 1\nreturn num\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n# \u5224\u7a7a\u5904\u7406\nif not self.__peek:\nreturn None\nreturn self.__peek.val\ndef to_list(self) -> list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\narr = []\nnode = self.__peek\nwhile node:\narr.append(node.val)\nnode = node.next\narr.reverse()\nreturn arr\n
    linkedlist_stack.go
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\nreturn &linkedListStack{\ndata: list.New(),\n}\n}\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\ns.data.PushBack(value)\n}\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\nreturn s.data\n}\n
    linkedlist_stack.js
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n#stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n#stkSize = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.#stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num) {\nconst node = new ListNode(num);\nnode.next = this.#stackPeek;\nthis.#stackPeek = node;\nthis.#stkSize++;\n}\n/* \u51fa\u6808 */\npop() {\nconst num = this.peek();\nthis.#stackPeek = this.#stackPeek.next;\nthis.#stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek() {\nif (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#stackPeek;\nconst res = new Array(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.ts
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size == 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nconst node = new ListNode(num);\nnode.next = this.stackPeek;\nthis.stackPeek = node;\nthis.stkSize++;\n}\n/* \u51fa\u6808 */\npop(): number {\nconst num = this.peek();\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nthis.stackPeek = this.stackPeek.next;\nthis.stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek(): number {\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.stackPeek;\nconst res = new Array<number>(this.size);\nfor (let i = res.length - 1; i >= 0; i--) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.c
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nstruct linkedListStack {\nListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint size;      // \u6808\u7684\u957f\u5ea6\n};\ntypedef struct linkedListStack linkedListStack;\n/* \u6784\u9020\u51fd\u6570 */\nlinkedListStack *newLinkedListStack() {\nlinkedListStack *s = malloc(sizeof(linkedListStack));\ns->top = NULL;\ns->size = 0;\nreturn s;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(linkedListStack *s) {\nwhile (s->top) {\nListNode *n = s->top->next;\nfree(s->top);\ns->top = n;\n}\nfree(s);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(linkedListStack *s) {\nassert(s);\nreturn s->size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(linkedListStack *s) {\nassert(s);\nreturn size(s) == 0;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(linkedListStack *s) {\nassert(s);\nassert(size(s) != 0);\nreturn s->top->val;\n}\n/* \u5165\u6808 */\nvoid push(linkedListStack *s, int num) {\nassert(s);\nListNode *node = (ListNode *)malloc(sizeof(ListNode));\nnode->next = s->top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\nnode->val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\ns->top = node;       // \u66f4\u65b0\u6808\u9876\ns->size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n/* \u51fa\u6808 */\nint pop(linkedListStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nassert(s);\nint val = peek(s);\nListNode *tmp = s->top;\ns->top = s->top->next;\n// \u91ca\u653e\u5185\u5b58\nfree(tmp);\ns->size--;\nreturn val;\n}\n
    linkedlist_stack.cs
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (stackPeek == null)\nthrow new Exception();\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (size() == 0 || stackPeek == null)\nthrow new Exception();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nif (stackPeek == null)\nreturn Array.Empty<int>();\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.Length - 1; i >= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n
    linkedlist_stack.swift
    /* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate var _size = 0 // \u6808\u7684\u957f\u5ea6\ninit() {}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\n_size\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nsize() == 0\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nlet node = ListNode(x: num)\nnode.next = _peek\n_peek = node\n_size += 1\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nlet num = peek()\n_peek = _peek?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn _peek!.val\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nvar node = _peek\nvar res = Array(repeating: 0, count: _size)\nfor i in sequence(first: res.count - 1, next: { $0 >= 0 + 1 ? $0 - 1 : nil }) {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n
    linkedlist_stack.zig
    // \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.stack_top = null;\nself.stk_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stk_size;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack_top.?.val;\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\nnode.next = self.stack_top;\nself.stack_top = node;\nself.stk_size += 1;\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.peek();\nself.stack_top = self.stack_top.?.next;\nself.stk_size -= 1;\nreturn num;\n} // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.stack_top;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i < res.len) : (i += 1) {\nres[res.len - i - 1] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n
    linkedlist_stack.dart
    /* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint _stkSize = 0; // \u6808\u7684\u957f\u5ea6\nLinkedListStack() {\n_stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stkSize == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nfinal ListNode node = ListNode(num);\nnode.next = _stackPeek;\n_stackPeek = node;\n_stkSize++;\n}\n/* \u51fa\u6808 */\nint pop() {\nfinal int num = peek();\n_stackPeek = _stackPeek!.next;\n_stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (_stackPeek == null) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stackPeek!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\nList<int> toList() {\nListNode? node = _stackPeek;\nList<int> list = [];\nwhile (node != null) {\nlist.add(node.val);\nnode = node.next;\n}\nlist = list.reversed.toList();\nreturn list;\n}\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#_2","title":"\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"

    \u5728\u57fa\u4e8e\u300c\u6570\u7ec4\u300d\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5728\u8fd9\u6837\u7684\u8bbe\u8ba1\u4e0b\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5c31\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002

    ArrayStackpush()pop()

    \u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u9700\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart array_stack.java
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate ArrayList<Integer> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new ArrayList<>();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.remove(size() - 1);\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.get(size() - 1);\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic Object[] toArray() {\nreturn stack.toArray();\n}\n}\n
    array_stack.cpp
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate:\nvector<int> stack;\npublic:\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool empty() {\nreturn stack.empty();\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nstack.push_back(num);\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint oldTop = top();\nstack.pop_back();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (empty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stack.back();\n}\n/* \u8fd4\u56de Vector */\nvector<int> toVector() {\nreturn stack;\n}\n};\n
    array_stack.py
    class ArrayStack:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self) -> None:\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.__stack: list[int] = []\ndef size(self) -> int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn len(self.__stack)\ndef is_empty(self) -> bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.__stack == []\ndef push(self, item: int) -> None:\n\"\"\"\u5165\u6808\"\"\"\nself.__stack.append(item)\ndef pop(self) -> int:\n\"\"\"\u51fa\u6808\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self.__stack.pop()\ndef peek(self) -> int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self.__stack[-1]\ndef to_list(self) -> list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nreturn self.__stack\n
    array_stack.go
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\ndata []int // \u6570\u636e\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\nreturn &arrayStack{\n// \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\ndata: make([]int, 0, 16),\n}\n}\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\nreturn len(s.data)\n}\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\nreturn s.size() == 0\n}\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n// \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\ns.data = append(s.data, v)\n}\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\nval := s.peek()\ns.data = s.data[:len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\nval := s.data[len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\nreturn s.data\n}\n
    array_stack.js
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n#stack;\nconstructor() {\nthis.#stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty() {\nreturn this.#stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nthis.#stack.push(num);\n}\n/* \u51fa\u6808 */\npop() {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop() {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack[this.#stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.#stack;\n}\n}\n
    array_stack.ts
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate stack: number[];\nconstructor() {\nthis.stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nempty(): boolean {\nreturn this.stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nthis.stack.push(num);\n}\n/* \u51fa\u6808 */\npop(): number | undefined {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop(): number | undefined {\nif (this.empty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack[this.stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.stack;\n}\n}\n
    array_stack.c
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct arrayStack {\nint *data;\nint size;\n};\ntypedef struct arrayStack arrayStack;\n/* \u6784\u9020\u51fd\u6570 */\narrayStack *newArrayStack() {\narrayStack *s = malloc(sizeof(arrayStack));\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\ns->data = malloc(sizeof(int) * MAX_SIZE);\ns->size = 0;\nreturn s;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(arrayStack *s) {\nreturn s->size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(arrayStack *s) {\nreturn s->size == 0;\n}\n/* \u5165\u6808 */\nvoid push(arrayStack *s, int num) {\nif (s->size == MAX_SIZE) {\nprintf(\"stack is full.\\n\");\nreturn;\n}\ns->data[s->size] = num;\ns->size++;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(arrayStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nreturn s->data[s->size - 1];\n}\n/* \u51fa\u6808 */\nint pop(arrayStack *s) {\nif (s->size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nint val = peek(s);\ns->size--;\nreturn val;\n}\n
    array_stack.cs
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate List<int> stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.Count();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.Add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new Exception();\nvar val = peek();\nstack.RemoveAt(size() - 1);\nreturn val;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new Exception();\nreturn stack[size() - 1];\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nreturn stack.ToArray();\n}\n}\n
    array_stack.swift
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate var stack: [Int]\ninit() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = []\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -> Int {\nstack.count\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -> Bool {\nstack.isEmpty\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nstack.append(num)\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.removeLast()\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -> Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.last!\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -> [Int] {\nstack\n}\n}\n
    array_stack.zig
    // \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack: ?std.ArrayList(T) = null,     // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) void {\nif (self.stack == null) {\nself.stack = std.ArrayList(T).init(allocator);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.stack == null) return;\nself.stack.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stack.?.items.len;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack.?.items[self.size() - 1];\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\ntry self.stack.?.append(num);\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.stack.?.pop();\nreturn num;\n} // \u8fd4\u56de ArrayList\npub fn toList(self: *Self) std.ArrayList(T) {\nreturn self.stack.?;\n}\n};\n}\n
    array_stack.dart
    /* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nlate List<int> _stack;\nArrayStack() {\n_stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stack.isEmpty;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\n_stack.add(num);\n}\n/* \u51fa\u6808 */\nint pop() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.removeLast();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.last;\n}\n/* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList<int> toArray() => _stack;\n}\n
    "},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3. \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":""},{"location":"chapter_stack_and_queue/stack/#_3","title":"\u652f\u6301\u64cd\u4f5c","text":"

    \u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002

    "},{"location":"chapter_stack_and_queue/stack/#_4","title":"\u65f6\u95f4\u6548\u7387","text":"

    \u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u662f\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002

    \u5728\u94fe\u8868\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002

    \u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff08\u5982 int , double \uff09\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a

    • \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\uff1b
    • \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\uff1b
    "},{"location":"chapter_stack_and_queue/stack/#_5","title":"\u7a7a\u95f4\u6548\u7387","text":"

    \u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u8fc7\u5b9e\u9645\u9700\u6c42\u3002\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002

    \u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002

    \u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002

    "},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4. \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"
    • \u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5c06\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u300c\u540e\u9000\u300d\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u9875\u9762\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002
    • \u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002
    "},{"location":"chapter_stack_and_queue/summary/","title":"5.4. \u00a0 \u5c0f\u7ed3","text":"
    • \u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002
    • \u4ece\u65f6\u95f4\u6548\u7387\u89d2\u5ea6\u770b\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u964d\u4f4e\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002
    • \u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002
    • \u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002
    • \u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002
    "},{"location":"chapter_stack_and_queue/summary/#541-q-a","title":"5.4.1. \u00a0 Q & A","text":"

    \u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f

    \u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u53cc\u5411\u961f\u5217\u7ae0\u8282\u6709\u63d0\u5230\u3002

    \u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f

    \u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0cJava \u548c Python \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 C \u548c C++ \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002

    \u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f

    \u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u8005\u662f\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002

    "},{"location":"chapter_tree/","title":"7. \u00a0 \u6811","text":""},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3. \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"

    \u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 TreeNode \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u5728\u4e0a\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u5728\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002

    \u90a3\u4e48\uff0c\u80fd\u5426\u7528\u300c\u6570\u7ec4\u300d\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002

    "},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1. \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"

    \u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u7f16\u53f7\uff08\u4ece \\(0\\) \u5f00\u59cb\uff09\uff0c\u6b64\u65f6\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u7d22\u5f15\u3002

    \u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    \u6620\u5c04\u516c\u5f0f\u7684\u4f5c\u7528\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u5982\u679c\u6211\u4eec\u5c06\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u90a3\u4e48\u5bf9\u4e8e\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5176\u5b50\u8282\u70b9\u3002

    "},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2. \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"

    \u7136\u800c\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u53ea\u662f\u4e00\u4e2a\u7279\u4f8b\u3002\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\uff0c\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \uff0c\u800c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \u3002\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002\u663e\u7136\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0a\u8ff0\u7684\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002

    Fig. \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027

    \u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\)\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nvector<int> tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n
    # \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u76f4\u63a5\u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u76f4\u63a5\u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n
    \n
    /* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList<int?> tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n

    Fig. \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    "},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3. \u00a0 \u4f18\u52bf\u4e0e\u5c40\u9650\u6027","text":"

    \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u5b58\u5728\u4ee5\u4e0b\u4f18\u70b9\uff1a

    • \u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\uff1b
    • \u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\uff1b
    • \u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\uff1b

    \u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5177\u6709\u4e00\u4e9b\u5c40\u9650\u6027\uff1a

    • \u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002
    • \u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\uff1b
    • \u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002

    \u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u8fd9\u610f\u5473\u7740\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002\u56e0\u6b64\uff0c\u5728\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \u3002

    Fig. \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a

    "},{"location":"chapter_tree/avl_tree/","title":"7.5. \u00a0 AVL \u6811 *","text":"

    \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u4e86\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\)\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002

    Fig. AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316

    \u518d\u4f8b\u5982\uff0c\u5728\u4ee5\u4e0b\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002

    Fig. AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316

    G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u5176 1962 \u5e74\u53d1\u8868\u7684\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002

    "},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1. \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"

    \u300cAVL \u6811\u300d\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u3002

    "},{"location":"chapter_tree/avl_tree/#_1","title":"\u8282\u70b9\u9ad8\u5ea6","text":"

    \u5728\u64cd\u4f5c AVL \u6811\u65f6\uff0c\u6211\u4eec\u9700\u8981\u83b7\u53d6\u8282\u70b9\u7684\u9ad8\u5ea6\uff0c\u56e0\u6b64\u9700\u8981\u4e3a AVL \u6811\u7684\u8282\u70b9\u7c7b\u6dfb\u52a0 height \u53d8\u91cf\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\npublic int val;        // \u8282\u70b9\u503c\npublic int height;     // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode left;  // \u5de6\u5b50\u8282\u70b9\npublic TreeNode right; // \u53f3\u5b50\u8282\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\nint val{};          // \u8282\u70b9\u503c\nint height = 0;     // \u8282\u70b9\u9ad8\u5ea6\nTreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\nTreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\nTreeNode() = default;\nexplicit TreeNode(int x) : val(x){}\n};\n
    class TreeNode:\n\"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                    # \u8282\u70b9\u503c\nself.height: int = 0                   # \u8282\u70b9\u9ad8\u5ea6\nself.left: Optional[TreeNode] = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nself.right: Optional[TreeNode] = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n
    /* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal    int       // \u8282\u70b9\u503c\nHeight int       // \u8282\u70b9\u9ad8\u5ea6\nLeft   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nRight  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval; // \u8282\u70b9\u503c\nheight; //\u8282\u70b9\u9ad8\u5ea6\nleft; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val, left, right, height) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;            // \u8282\u70b9\u503c\nheight: number;         // \u8282\u70b9\u9ad8\u5ea6\nleft: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height; this.left = left === undefined ? null : left; this.right = right === undefined ? null : right; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;\nint height;\nstruct TreeNode *left;\nstruct TreeNode *right;\n};\ntypedef struct TreeNode TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode->val = val;\nnode->height = 0;\nnode->left = NULL;\nnode->right = NULL;\nreturn node;\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\npublic int val;          // \u8282\u70b9\u503c\npublic int height;       // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode? left;   // \u5de6\u5b50\u8282\u70b9\npublic TreeNode? right;  // \u53f3\u5b50\u8282\u70b9\npublic TreeNode(int x) { val = x; }\n}\n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar height: Int // \u8282\u70b9\u9ad8\u5ea6\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\ninit(x: Int) {\nval = x\nheight = 0\n}\n}\n
    \n
    /* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nint height;      // \u8282\u70b9\u9ad8\u5ea6\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\nTreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n

    \u300c\u8282\u70b9\u9ad8\u5ea6\u300d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a 0 \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a -1 \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n
    avl_tree.cpp
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == nullptr ? -1 : node->height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode->height = max(height(node->left), height(node->right)) + 1;\n}\n
    avl_tree.py
    def height(self, node: TreeNode | None) -> int:\n\"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node is not None:\nreturn node.height\nreturn -1\ndef __update_height(self, node: TreeNode | None):\n\"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = max([self.height(node.left), self.height(node.right)]) + 1\n
    avl_tree.go
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node != nil {\nreturn node.Height\n}\nreturn -1\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\nlh := t.height(node.Left)\nrh := t.height(node.Right)\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif lh > rh {\nnode.Height = lh + 1\n} else {\nnode.Height = rh + 1\n}\n}\n
    avl_tree.js
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
    avl_tree.ts
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n
    avl_tree.c
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif (node != NULL) {\nreturn node->height;\n}\nreturn -1;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\nint lh = height(node->left);\nint rh = height(node->right);\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif (lh > rh) {\nnode->height = lh + 1;\n} else {\nnode->height = rh + 1;\n}\n}\n
    avl_tree.cs
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.Max(height(node.left), height(node.right)) + 1;\n}\n
    avl_tree.swift
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -> Int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nnode == nil ? -1 : node!.height\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n
    avl_tree.zig
    // \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n_ = self;\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn if (node == null) -1 else node.?.height;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.?.height = std.math.max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n
    avl_tree.dart
    /* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode!.height = max(height(node.left), height(node.right)) + 1;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_2","title":"\u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"

    \u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 Balance Factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a 0 \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
    avl_tree.cpp
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == nullptr)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
    avl_tree.py
    def balance_factor(self, node: TreeNode | None) -> int:\n\"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n# \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node is None:\nreturn 0\n# \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.left) - self.height(node.right)\n
    avl_tree.go
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node == nil {\nreturn 0\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn t.height(node.Left) - t.height(node.Right)\n}\n
    avl_tree.js
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
    avl_tree.ts
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n
    avl_tree.c
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == NULL) {\nreturn 0;\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node->left) - height(node->right);\n}\n
    avl_tree.cs
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n
    avl_tree.swift
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -> Int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nguard let node = node else { return 0 }\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node: node.left) - height(node: node.right)\n}\n
    avl_tree.zig
    // \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.?.left) - self.height(node.?.right);\n}\n
    avl_tree.dart
    /* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n

    Note

    \u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002

    "},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2. \u00a0 AVL \u6811\u65cb\u8f6c","text":"

    AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u300c\u65cb\u8f6c Rotation\u300d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u6811\u7684\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5c5e\u6027\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u6811\u300d\u3002

    \u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(> 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u300c\u5931\u8861\u8282\u70b9\u300d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u6211\u4eec\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002

    "},{"location":"chapter_tree/avl_tree/#_3","title":"\u53f3\u65cb","text":"

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a node \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a child \uff0c\u6267\u884c\u300c\u53f3\u65cb\u300d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u5df2\u7ecf\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u7279\u6027\u3002

    <1><2><3><4>

    \u6b64\u5916\uff0c\u5982\u679c\u8282\u70b9 child \u672c\u8eab\u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u53f3\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u5de6\u5b50\u8282\u70b9\u3002

    Fig. \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c

    \u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\nTreeNode child = node.left;\nTreeNode grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cpp
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child = node->left;\nTreeNode *grandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.py
    def __right_rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\nchild = node.left\ngrand_child = child.right\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node\nnode.left = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n
    avl_tree.go
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\nchild := node.Left\ngrandChild := child.Right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.Right = node\nnode.Left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.js
    /* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.ts
    /* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.c
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node->left;\ngrandChild = child->right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild->right = node;\nnode->left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cs
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\nTreeNode? child = node.left;\nTreeNode? grandChild = child?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.swift
    /* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.left\nlet grandChild = child?.right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild?.right = node\nnode?.left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.zig
    // \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.left;\nvar grandChild = child.?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.?.right = node;\nnode.?.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.dart
    /* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\nTreeNode? child = node!.left;\nTreeNode? grandChild = child!.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_4","title":"\u5de6\u65cb","text":"

    \u76f8\u5e94\u7684\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u300c\u5de6\u65cb\u300d\u64cd\u4f5c\u3002

    Fig. \u5de6\u65cb\u64cd\u4f5c

    \u540c\u7406\uff0c\u82e5\u8282\u70b9 child \u672c\u8eab\u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a grandChild \uff09\uff0c\u5219\u9700\u8981\u5728\u300c\u5de6\u65cb\u300d\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 grandChild \u4f5c\u4e3a node \u7684\u53f3\u5b50\u8282\u70b9\u3002

    Fig. \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c

    \u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u677e\u5730\u4ece\u53f3\u65cb\u7684\u4ee3\u7801\u63a8\u5bfc\u51fa\u5de6\u65cb\u7684\u4ee3\u7801\u3002\u5177\u4f53\u5730\uff0c\u53ea\u9700\u5c06\u300c\u53f3\u65cb\u300d\u4ee3\u7801\u4e2d\u7684\u628a\u6240\u6709\u7684 left \u66ff\u6362\u4e3a right \uff0c\u5c06\u6240\u6709\u7684 right \u66ff\u6362\u4e3a left \uff0c\u5373\u53ef\u5f97\u5230\u300c\u5de6\u65cb\u300d\u4ee3\u7801\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\nTreeNode child = node.right;\nTreeNode grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cpp
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child = node->right;\nTreeNode *grandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.py
    def __left_rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\nchild = node.right\ngrand_child = child.left\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node\nnode.right = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\nself.__update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n
    avl_tree.go
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\nchild := node.Right\ngrandChild := child.Left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.Left = node\nnode.Right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.js
    /* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.ts
    /* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.c
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node->right;\ngrandChild = child->left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild->left = node;\nnode->right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.cs
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\nTreeNode? child = node.right;\nTreeNode? grandChild = child?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.swift
    /* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -> TreeNode? {\nlet child = node?.right\nlet grandChild = child?.left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild?.left = node\nnode?.right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n
    avl_tree.zig
    // \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.right;\nvar grandChild = child.?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.?.left = node;\nnode.?.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    avl_tree.dart
    /* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\nTreeNode? child = node!.right;\nTreeNode? grandChild = child!.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_5","title":"\u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"

    \u5bf9\u4e8e\u4e0b\u56fe\u4e2d\u7684\u5931\u8861\u8282\u70b9 3\uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5de6\u65cb\u540e\u53f3\u65cb\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u5de6\u65cb\u300d\uff0c\u518d\u5bf9 node \u6267\u884c\u300c\u53f3\u65cb\u300d\u3002

    Fig. \u5148\u5de6\u65cb\u540e\u53f3\u65cb

    "},{"location":"chapter_tree/avl_tree/#_6","title":"\u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"

    \u540c\u7406\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u53f3\u65cb\u540e\u5de6\u65cb\uff0c\u5373\u5148\u5bf9 child \u6267\u884c\u300c\u53f3\u65cb\u300d\uff0c\u7136\u540e\u5bf9 node \u6267\u884c\u300c\u5de6\u65cb\u300d\u3002

    Fig. \u5148\u53f3\u65cb\u540e\u5de6\u65cb

    "},{"location":"chapter_tree/avl_tree/#_7","title":"\u65cb\u8f6c\u7684\u9009\u62e9","text":"

    \u4e0b\u56fe\u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u540e\u5de6\u3001\u5148\u5de6\u540e\u53f3\u7684\u65cb\u8f6c\u64cd\u4f5c\u3002

    Fig. AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5

    \u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u4e0a\u56fe\u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002

    \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(>0\\) \uff08\u5373\u5de6\u504f\u6811\uff09 \\(<0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(<0\\) \uff08\u5373\u53f3\u504f\u6811\uff09 \\(>0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb

    \u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.cpp
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint _balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (_balanceFactor > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (_balanceFactor < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.py
    def __rotate(self, node: TreeNode | None) -> TreeNode | None:\n\"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n# \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nbalance_factor = self.balance_factor(node)\n# \u5de6\u504f\u6811\nif balance_factor > 1:\nif self.balance_factor(node.left) >= 0:\n# \u53f3\u65cb\nreturn self.__right_rotate(node)\nelse:\n# \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = self.__left_rotate(node.left)\nreturn self.__right_rotate(node)\n# \u53f3\u504f\u6811\nelif balance_factor < -1:\nif self.balance_factor(node.right) <= 0:\n# \u5de6\u65cb\nreturn self.__left_rotate(node)\nelse:\n# \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = self.__right_rotate(node.right)\nreturn self.__left_rotate(node)\n# \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n
    avl_tree.go
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n// Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\nbf := t.balanceFactor(node)\n// \u5de6\u504f\u6811\nif bf > 1 {\nif t.balanceFactor(node.Left) >= 0 {\n// \u53f3\u65cb\nreturn t.rightRotate(node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.Left = t.leftRotate(node.Left)\nreturn t.rightRotate(node)\n}\n}\n// \u53f3\u504f\u6811\nif bf < -1 {\nif t.balanceFactor(node.Right) <= 0 {\n// \u5de6\u65cb\nreturn t.leftRotate(node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.Right = t.rightRotate(node.Right)\nreturn t.leftRotate(node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
    avl_tree.js
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.#rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.#leftRotate(node.left);\nreturn this.#rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.#leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.#rightRotate(node.right);\nreturn this.#leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.ts
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor > 1) {\nif (this.balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn this.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.leftRotate(node.left);\nreturn this.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor < -1) {\nif (this.balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn this.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.rightRotate(node.right);\nreturn this.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.c
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint bf = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (bf > 1) {\nif (balanceFactor(node->left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode->left = leftRotate(node->left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (bf < -1) {\nif (balanceFactor(node->right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode->right = rightRotate(node->right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.cs
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactorInt = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactorInt > 1) {\nif (balanceFactor(node.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node?.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactorInt < -1) {\nif (balanceFactor(node.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node?.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.swift
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -> TreeNode? {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balanceFactor = balanceFactor(node: node)\n// \u5de6\u504f\u6811\nif balanceFactor > 1 {\nif self.balanceFactor(node: node?.left) >= 0 {\n// \u53f3\u65cb\nreturn rightRotate(node: node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode?.left = leftRotate(node: node?.left)\nreturn rightRotate(node: node)\n}\n}\n// \u53f3\u504f\u6811\nif balanceFactor < -1 {\nif self.balanceFactor(node: node?.right) <= 0 {\n// \u5de6\u65cb\nreturn leftRotate(node: node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode?.right = rightRotate(node: node?.right)\nreturn leftRotate(node: node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n
    avl_tree.zig
    // \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nvar balance_factor = self.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balance_factor > 1) {\nif (self.balanceFactor(node.?.left) >= 0) {\n// \u53f3\u65cb\nreturn self.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.?.left = self.leftRotate(node.?.left);\nreturn self.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balance_factor < -1) {\nif (self.balanceFactor(node.?.right) <= 0) {\n// \u5de6\u65cb\nreturn self.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.?.right = self.rightRotate(node.?.right);\nreturn self.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    avl_tree.dart
    /* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint factor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (factor > 1) {\nif (balanceFactor(node!.left) >= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (factor < -1) {\nif (balanceFactor(node!.right) <= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u9700\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3. \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#_8","title":"\u63d2\u5165\u8282\u70b9","text":"

    \u300cAVL \u6811\u300d\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u300c\u4e8c\u53c9\u641c\u7d22\u6811\u300d\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\nif (node == null)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cpp
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node->val)\nnode->left = insertHelper(node->left, val);\nelse if (val > node->val)\nnode->right = insertHelper(node->right, val);\nelse\nreturn node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.py
    def insert(self, val) -> None:\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\nself.root = self.__insert_helper(self.root, val)\ndef __insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:\n\"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn TreeNode(val)\n# 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif val < node.val:\nnode.left = self.__insert_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__insert_helper(node.right, val)\nelse:\n# \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
    avl_tree.go
    /* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\nt.root = t.insertHelper(t.root, val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn NewTreeNode(val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val < node.Val {\nnode.Left = t.insertHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.insertHelper(node.Right, val)\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.js
    /* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\nthis.root = this.#insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val) node.left = this.#insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = this.#insertHelper(node.right, val);\nelse return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.ts
    /* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\nthis.root = this.insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val) {\nnode.left = this.insertHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.insertHelper(node.right, val);\n} else {\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.c
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(aVLTree *tree, int val) {\ntree->root = insertHelper(tree->root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == NULL) {\nreturn newTreeNode(val);\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node->val) {\nnode->left = insertHelper(node->left, val);\n} else if (val > node->val) {\nnode->right = insertHelper(node->right, val);\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cs
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.swift
    /* \u63d2\u5165\u8282\u70b9 */\n@discardableResult\nfunc insert(val: Int) {\nroot = insertHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn TreeNode(x: val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val < node!.val {\nnode?.left = insertHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = insertHelper(node: node?.right, val: val)\n} else {\nreturn node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.zig
    // \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\nself.root = (try self.insertHelper(self.root, val)).?;\n}\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) {\nvar tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\ntmp_node.init(val);\nreturn tmp_node;\n}\n// 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif (val < node.?.val) {\nnode.?.left = try self.insertHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = try self.insertHelper(node.?.right, val);\n} else {\nreturn node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nself.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.dart
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\nif (node == null) return TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val < node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val > node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_9","title":"\u5220\u9664\u8282\u70b9","text":"

    \u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u5730\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart avl_tree.java
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\nif (node == null)\nreturn null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode child = node.left != null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cpp
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn nullptr;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val)\nnode->left = removeHelper(node->left, val);\nelse if (val > node->val)\nnode->right = removeHelper(node->right, val);\nelse {\nif (node->left == nullptr || node->right == nullptr) {\nTreeNode *child = node->left != nullptr ? node->left : node->right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == nullptr) {\ndelete node;\nreturn nullptr;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\ndelete node;\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node->right;\nwhile (temp->left != nullptr) {\ntemp = temp->left;\n}\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.py
    def remove(self, val: int) -> None:\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\nself.root = self.__remove_helper(self.root, val)\ndef __remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:\n\"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn None\n# 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif val < node.val:\nnode.left = self.__remove_helper(node.left, val)\nelif val > node.val:\nnode.right = self.__remove_helper(node.right, val)\nelse:\nif node.left is None or node.right is None:\nchild = node.left or node.right\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child is None:\nreturn None\n# \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse:\nnode = child\nelse:\n# \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp = node.right\nwhile temp.left is not None:\ntemp = temp.left\nnode.right = self.__remove_helper(node.right, temp.val)\nnode.val = temp.val\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.__update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.__rotate(node)\n
    avl_tree.go
    /* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\nt.root = t.removeHelper(t.root, val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node.Val {\nnode.Left = t.removeHelper(node.Left, val)\n} else if val > node.Val {\nnode.Right = t.removeHelper(node.Right, val)\n} else {\nif node.Left == nil || node.Right == nil {\nchild := node.Left\nif node.Right != nil {\nchild = node.Right\n}\nif child == nil {\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nreturn nil\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp := node.Right\nfor temp.Left != nil {\ntemp = temp.Left\n}\nnode.Right = t.removeHelper(node.Right, temp.Val)\nnode.Val = temp.Val\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.js
    /* \u5220\u9664\u8282\u70b9 */\nremove(val) {\nthis.root = this.#removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) node.left = this.#removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = this.#removeHelper(node.right, val);\nelse {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) return null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse node = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.#removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.ts
    /* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\nthis.root = this.removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val) {\nnode.left = this.removeHelper(node.left, val);\n} else if (val > node.val) {\nnode.right = this.removeHelper(node.right, val);\n} else {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) {\nreturn null;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.c
    /* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeNode(aVLTree *tree, int val) {\nTreeNode *root = removeHelper(tree->root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nTreeNode *child, *grandChild;\nif (node == NULL) {\nreturn NULL;\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node->val) {\nnode->left = removeHelper(node->left, val);\n} else if (val > node->val) {\nnode->right = removeHelper(node->right, val);\n} else {\nif (node->left == NULL || node->right == NULL) {\nchild = node->left;\nif (node->right != NULL) {\nchild = node->right;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == NULL) {\nreturn NULL;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node->right;\nwhile (temp->left != NULL) {\ntemp = temp->left;\n}\nint tempVal = temp->val;\nnode->right = removeHelper(node->right, temp->val);\nnode->val = tempVal;\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.cs
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left != null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.swift
    /* \u5220\u9664\u8282\u70b9 */\n@discardableResult\nfunc remove(val: Int) {\nroot = removeHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -> TreeNode? {\nvar node = node\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val < node!.val {\nnode?.left = removeHelper(node: node?.left, val: val)\n} else if val > node!.val {\nnode?.right = removeHelper(node: node?.right, val: val)\n} else {\nif node?.left == nil || node?.right == nil {\nlet child = node?.left != nil ? node?.left : node?.right\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node?.right\nwhile temp?.left != nil {\ntemp = temp?.left\n}\nnode?.right = removeHelper(node: node?.right, val: temp!.val)\nnode?.val = temp!.val\n}\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n
    avl_tree.zig
    // \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\nself.root = self.removeHelper(self.root, val).?;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) return null;\n// 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif (val < node.?.val) {\nnode.?.left = self.removeHelper(node.?.left, val);\n} else if (val > node.?.val) {\nnode.?.right = self.removeHelper(node.?.right, val);\n} else {\nif (node.?.left == null or node.?.right == null) {\nvar child = if (node.?.left != null) node.?.left else node.?.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null) {\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n} else {\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node.?.right;\nwhile (temp.?.left != null) {\ntemp = temp.?.left;\n}\nnode.?.right = self.removeHelper(node.?.right, temp.?.val);\nnode.?.val = temp.?.val;\n}\n}\nself.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    avl_tree.dart
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val < node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val > node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left ?? node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp!.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n
    "},{"location":"chapter_tree/avl_tree/#_10","title":"\u67e5\u627e\u8282\u70b9","text":"

    AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002

    "},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4. \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"
    • \u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\uff1b
    • \u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\uff1b

    \u4e3a\u4ec0\u4e48\u7ea2\u9ed1\u6811\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\uff1f

    \u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u56e0\u6b64\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5728\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4e0a\u7684\u5e73\u5747\u6548\u7387\u9ad8\u4e8e AVL \u6811\u3002

    "},{"location":"chapter_tree/binary_search_tree/","title":"7.4. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"

    \u300c\u4e8c\u53c9\u641c\u7d22\u6811 Binary Search Tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a

    1. \u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(<\\) \u6839\u8282\u70b9\u7684\u503c \\(<\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\uff1b
    2. \u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 1. \uff1b

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811

    "},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_search_tree/#_1","title":"\u67e5\u627e\u8282\u70b9","text":"

    \u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c num \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 cur \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 root \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c cur.val \u548c num \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb

    • \u82e5 cur.val < num \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.right \uff1b
    • \u82e5 cur.val > num \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c cur = cur.left \uff1b
    • \u82e5 cur.val = num \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\uff1b
    <1><2><3><4>

    \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\nTreeNode cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.cpp
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\nTreeNode *cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur->val > num)\ncur = cur->left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.py
    def search(self, num: int) -> TreeNode | None:\n\"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\ncur: TreeNode | None = self.root\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur is not None:\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelif cur.val > num:\ncur = cur.left\n# \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse:\nbreak\nreturn cur\n
    binary_search_tree.go
    /* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\nnode := bst.root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor node != nil {\nif node.Val < num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nnode = node.Right\n} else if node.Val > num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nnode = node.Left\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn node\n}\n
    binary_search_tree.js
    /* \u67e5\u627e\u8282\u70b9 */\nfunction search(num) {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.ts
    /* \u67e5\u627e\u8282\u70b9 */\nfunction search(num: number): TreeNode | null {\nlet cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val < num) {\ncur = cur.right; // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else if (cur.val > num) {\ncur = cur.left; // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\nbreak; // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.c
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(binarySearchTree *bst, int num) {\nTreeNode *cur = bst->root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\nif (cur->val < num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else if (cur->val > num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.cs
    /* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int num) {\nTreeNode? cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur =\ncur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val > num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.swift
    /* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -> TreeNode? {\nvar cur = root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if cur!.val > num {\ncur = cur?.left\n}\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur\n}\n
    binary_search_tree.zig
    // \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\nvar cur = self.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else if (cur.?.val > num) {\ncur = cur.?.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n} else {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{search}\n
    "},{"location":"chapter_tree/binary_search_tree/#_2","title":"\u63d2\u5165\u8282\u70b9","text":"

    \u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 num \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u5206\u4e3a\u4e24\u6b65\uff1a

    1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c num \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\uff1b
    2. \u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 num \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\uff1b

    \u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new TreeNode(num);\nif (pre.val < num)\npre.right = node;\nelse\npre.left = node;\n}\n
    binary_search_tree.cpp
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr)\nreturn;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur->left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = new TreeNode(num);\nif (pre->val < num)\npre->right = node;\nelse\npre->left = node;\n}\n
    binary_search_tree.py
    def insert(self, num: int) -> None:\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.root is None:\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self.root, None\nwhile cur is not None:\n# \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur.val == num:\nreturn\npre = cur\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u63d2\u5165\u8282\u70b9\nnode = TreeNode(num)\nif pre.val < num:\npre.right = node\nelse:\npre.left = node\n
    binary_search_tree.go
    /* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nreturn\n}\npre = cur\nif cur.Val < num {\ncur = cur.Right\n} else {\ncur = cur.Left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nnode := NewTreeNode(num)\nif pre.Val < num {\npre.Right = node\n} else {\npre.Left = node\n}\n}\n
    binary_search_tree.js
    /* \u63d2\u5165\u8282\u70b9 */\nfunction insert(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return;\nlet cur = root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = new TreeNode(num);\nif (pre.val < num) pre.right = node;\nelse pre.left = node;\n}\n
    binary_search_tree.ts
    /* \u63d2\u5165\u8282\u70b9 */\nfunction insert(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\nif (cur.val === num) {\nreturn; // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = new TreeNode(num);\nif (pre!.val < num) {\npre!.right = node;\n} else {\npre!.left = node;\n}\n}\n
    binary_search_tree.c
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(binarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (bst->root == NULL)\nreturn;\nTreeNode *cur = bst->root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur->val == num) {\nreturn;\n}\npre = cur;\nif (cur->val < num) {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = newTreeNode(num);\nif (pre->val < num) {\npre->right = node;\n} else {\npre->left = node;\n}\n}\n
    binary_search_tree.cs
    /* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new TreeNode(num);\nif (pre != null) {\nif (pre.val < num)\npre.right = node;\nelse\npre.left = node;\n}\n}\n
    binary_search_tree.swift
    /* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur!.val == num {\nreturn\n}\npre = cur\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = TreeNode(x: num)\nif pre!.val < num {\npre?.right = node\n} else {\npre?.left = node\n}\n}\n
    binary_search_tree.zig
    // \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.?.val == num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nvar node = try self.mem_allocator.create(inc.TreeNode(T));\nnode.init(num);\nif (pre.?.val < num) {\npre.?.right = node;\n} else {\npre.?.left = node;\n}\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{insert}\n

    \u4e3a\u4e86\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u5229\u7528\u8f85\u52a9\u8282\u70b9 pre \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\uff0c\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002

    \u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    "},{"location":"chapter_tree/binary_search_tree/#_3","title":"\u5220\u9664\u8282\u70b9","text":"

    \u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u5728\u5220\u9664\u64cd\u4f5c\u540e\u7ef4\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 < \u6839\u8282\u70b9 < \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u3002\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u5728\u4e8c\u53c9\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u83b7\u53d6\u5f85\u5220\u9664\u8282\u70b9\u3002\u63a5\u4e0b\u6765\uff0c\u6839\u636e\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5220\u9664\u64cd\u4f5c\u9700\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\uff1a

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u5f85\u5220\u9664\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0\uff09

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002

    Fig. \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1\uff09

    \u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6 \\(<\\) \u6839 \\(<\\) \u53f3\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u6216\u8005\u79f0\u4e3a\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u4e3a\uff1a

    1. \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a tmp \uff1b
    2. \u5c06 tmp \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 tmp \uff1b
    <1><2><3><4>

    \u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_search_tree.java
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.cpp
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr)\nreturn;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur->val < num)\ncur = cur->right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur->left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == nullptr)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur->left == nullptr || cur->right == nullptr) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur->left != nullptr ? cur->left : cur->right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre->left == cur)\npre->left = child;\nelse\npre->right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n// \u91ca\u653e\u5185\u5b58\ndelete cur;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur->right;\nwhile (tmp->left != nullptr) {\ntmp = tmp->left;\n}\nint tmpVal = tmp->val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp->val);\n// \u7528 tmp \u8986\u76d6 cur\ncur->val = tmpVal;\n}\n}\n
    binary_search_tree.py
    def remove(self, num: int) -> None:\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.root is None:\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self.root, None\nwhile cur is not None:\n# \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur.val == num:\nbreak\npre = cur\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val < num:\ncur = cur.right\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur is None:\nreturn\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur.left is None or cur.right is None:\n# \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nchild = cur.left or cur.right\n# \u5220\u9664\u8282\u70b9 cur\nif cur != self.root:\nif pre.left == cur:\npre.left = child\nelse:\npre.right = child\nelse:\n# \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nself.root = child\n# \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse:\n# \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp: TreeNode = cur.right\nwhile tmp.left is not None:\ntmp = tmp.left\n# \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.val)\n# \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val\n
    binary_search_tree.go
    /* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nbreak\n}\npre = cur\nif cur.Val < num {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\ncur = cur.Right\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\ncur = cur.Left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\nif cur.Left == nil || cur.Right == nil {\nvar child *TreeNode = nil\n// \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\nif cur.Left != nil {\nchild = cur.Left\n} else {\nchild = cur.Right\n}\n// \u5220\u9664\u8282\u70b9 cur\nif cur != bst.root {\nif pre.Left == cur {\npre.Left = child\n} else {\npre.Right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nbst.root = child\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp := cur.Right\nfor tmp.Left != nil {\ntmp = tmp.Left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nbst.remove(tmp.Val)\n// \u7528 tmp \u8986\u76d6 cur\ncur.Val = tmp.Val\n}\n}\n
    binary_search_tree.js
    /* \u5220\u9664\u8282\u70b9 */\nfunction remove(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) return;\nlet cur = root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num) cur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left === cur) pre.left = child;\nelse pre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp = cur.right;\nwhile (tmp.left !== null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.ts
    /* \u5220\u9664\u8282\u70b9 */\nfunction remove(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root === null) {\nreturn;\n}\nlet cur = root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) {\nbreak;\n}\npre = cur;\nif (cur.val < num) {\ncur = cur.right as TreeNode; // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n} else {\ncur = cur.left as TreeNode; // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) {\nreturn;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre!.left === cur) {\npre!.left = child;\n} else {\npre!.right = child;\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp = cur.right;\nwhile (tmp.left !== null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp!.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.c
    /* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeNode(binarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (bst->root == NULL)\nreturn;\nTreeNode *cur = bst->root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur->val == num)\nbreak;\npre = cur;\nif (cur->val < num) {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur->right;\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur->left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == NULL)\nreturn;\n// \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\nif (cur->left == NULL || cur->right == NULL) {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur->left != NULL ? cur->left : cur->right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre->left == cur) {\npre->left = child;\n} else {\npre->right = child;\n}\n} else {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur->right;\nwhile (tmp->left != NULL) {\ntmp = tmp->left;\n}\nint tmpVal = tmp->val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremoveNode(bst, tmp->val);\n// \u7528 tmp \u8986\u76d6 cur\ncur->val = tmpVal;\n}\n}\n
    binary_search_tree.cs
    /* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val < num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null || pre == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode? child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode? tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n
    binary_search_tree.swift
    /* \u5220\u9664\u8282\u70b9 */\n@discardableResult\nfunc remove(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur!.val == num {\nbreak\n}\npre = cur\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val < num {\ncur = cur?.right\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur?.left == nil || cur?.right == nil {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur?.left != nil ? cur?.left : cur?.right\n// \u5220\u9664\u8282\u70b9 cur\nif cur !== root {\nif pre?.left === cur {\npre?.left = child\n} else {\npre?.right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur?.right\nwhile tmp?.left != nil {\ntmp = tmp?.left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(num: tmp!.val)\n// \u7528 tmp \u8986\u76d6 cur\ncur?.val = tmp!.val\n}\n}\n
    binary_search_tree.zig
    // \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.?.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val < num) {\ncur = cur.?.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.?.left == null or cur.?.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nvar child = if (cur.?.left != null) cur.?.left else cur.?.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre.?.left == cur) {\npre.?.left = child;\n} else {\npre.?.right = child;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur.?.right;\nwhile (tmp.?.left != null) {\ntmp = tmp.?.left;\n}\nvar tmp_val = tmp.?.val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.?.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.?.val = tmp_val;\n}\n}\n
    binary_search_tree.dart
    [class]{BinarySearchTree}-[func]{remove}\n
    "},{"location":"chapter_tree/binary_search_tree/#_4","title":"\u6392\u5e8f","text":"

    \u6211\u4eec\u77e5\u9053\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(<\\) \u6839\u8282\u70b9 \\(<\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002

    \u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u9700\u989d\u5916\u6392\u5e8f\uff0c\u975e\u5e38\u9ad8\u6548\u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217

    "},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"

    \u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002

    \u89c2\u5bdf\u53ef\u77e5\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u8868\u73b0\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u7684\u6570\u636e\u9002\u7528\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002

    \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\)

    \u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002

    \u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5e73\u8861\u4e0e\u9000\u5316

    "},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3. \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"
    • \u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002
    • \u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002
    • \u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002
    "},{"location":"chapter_tree/binary_tree/","title":"7.1. \u00a0 \u4e8c\u53c9\u6811","text":"

    \u300c\u4e8c\u53c9\u6811 Binary Tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u7740\u7956\u5148\u4e0e\u540e\u4ee3\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u7740\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u300c\u503c\u300d\u548c\u4e24\u4e2a\u300c\u6307\u9488\u300d\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart 
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;          // \u8282\u70b9\u503c\nTreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n
    class TreeNode:\n\"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                   # \u8282\u70b9\u503c\nself.left: Optional[TreeNode] = None  # \u5de6\u5b50\u8282\u70b9\u6307\u9488\nself.right: Optional[TreeNode] = None # \u53f3\u5b50\u8282\u70b9\u6307\u9488\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal   int\nLeft  *TreeNode\nRight *TreeNode\n}\n/* \u8282\u70b9\u521d\u59cb\u5316\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\nreturn &TreeNode{\nLeft:  nil,\nRight: nil,\nVal:   v,\n}\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nfunction TreeNode(val, left, right) {\nthis.val = (val === undefined ? 0 : val); // \u8282\u70b9\u503c\nthis.left = (left === undefined ? null : left); // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nthis.right = (right === undefined ? null : right); // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;\nleft: TreeNode | null;\nright: TreeNode | null;\nconstructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nthis.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n}\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;                // \u8282\u70b9\u503c\nint height;             // \u8282\u70b9\u9ad8\u5ea6\nstruct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nstruct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n};\ntypedef struct TreeNode TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode->val = val;\nnode->height = 0;\nnode->left = NULL;\nnode->right = NULL;\nreturn node;\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;          // \u8282\u70b9\u503c\nTreeNode? left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode? right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) { val = x; }\n}\n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u6307\u9488\ninit(x: Int) {\nval = x\n}\n}\n
    \n
    /* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(this.val, [this.left, this.right]);\n}\n

    \u8282\u70b9\u7684\u4e24\u4e2a\u6307\u9488\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9\u300d\uff0c\u540c\u65f6\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811\u300d\u3002

    \u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u4f8b\u5982\uff0c\u5728\u4ee5\u4e0b\u793a\u4f8b\u4e2d\uff0c\u82e5\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002

    Fig. \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811

    "},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1. \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"

    \u4e8c\u53c9\u6811\u6d89\u53ca\u7684\u672f\u8bed\u8f83\u591a\uff0c\u5efa\u8bae\u5c3d\u91cf\u7406\u89e3\u5e76\u8bb0\u4f4f\u3002

    • \u300c\u6839\u8282\u70b9 Root Node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\uff1b
    • \u300c\u53f6\u8282\u70b9 Leaf Node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \uff1b
    • \u8282\u70b9\u7684\u300c\u5c42 Level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \uff1b
    • \u8282\u70b9\u7684\u300c\u5ea6 Degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u8303\u56f4\u662f 0, 1, 2 \uff1b
    • \u300c\u8fb9 Edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u6307\u9488\uff1b
    • \u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b
    • \u8282\u70b9\u7684\u300c\u6df1\u5ea6 Depth\u300d \uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b
    • \u8282\u70b9\u7684\u300c\u9ad8\u5ea6 Height\u300d\uff1a\u4ece\u6700\u8fdc\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\uff1b

    Fig. \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed

    \u9ad8\u5ea6\u4e0e\u6df1\u5ea6\u7684\u5b9a\u4e49

    \u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u300c\u9ad8\u5ea6\u300d\u548c\u300c\u6df1\u5ea6\u300d\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002

    "},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2. \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":"

    \u521d\u59cb\u5316\u4e8c\u53c9\u6811\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree.java
    // \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.cpp
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
    binary_tree.py
    # \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
    binary_tree.go
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n
    binary_tree.js
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.ts
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.c
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1->left = n2;\nn1->right = n3;\nn2->left = n4;\nn2->right = n5;\n
    binary_tree.cs
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n
    binary_tree.swift
    // \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n
    binary_tree.zig
    \n
    binary_tree.dart
    /* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u5f15\u7528\u6307\u5411\uff08\u5373\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n

    \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u3002

    Fig. \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree.java
    TreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.cpp
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
    binary_tree.py
    # \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
    binary_tree.go
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n
    binary_tree.js
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.ts
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.c
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1->left = P;\nP->left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1->left = n2;\n
    binary_tree.cs
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n
    binary_tree.swift
    let P = TreeNode(x: 0)\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n
    binary_tree.zig
    \n
    binary_tree.dart
    /* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -> n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n

    Note

    \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002

    "},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3. \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#_1","title":"\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 Perfect Binary Tree\u300d\u9664\u4e86\u6700\u5e95\u5c42\u5916\uff0c\u5176\u4f59\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002

    Tip

    \u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\uff0c\u8bf7\u6ce8\u610f\u533a\u5206\u3002

    Fig. \u5b8c\u7f8e\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_2","title":"\u5b8c\u5168\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u5168\u4e8c\u53c9\u6811 Complete Binary Tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002

    Fig. \u5b8c\u5168\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_3","title":"\u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"

    \u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 Full Binary Tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002

    Fig. \u5b8c\u6ee1\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#_4","title":"\u5e73\u8861\u4e8c\u53c9\u6811","text":"

    \u300c\u5e73\u8861\u4e8c\u53c9\u6811 Balanced Binary Tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002

    Fig. \u5e73\u8861\u4e8c\u53c9\u6811

    "},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4. \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"

    \u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u300d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u300c\u94fe\u8868\u300d\u3002

    • \u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\uff1b
    • \u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \uff1b

    Fig. \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u4e0e\u6700\u5dee\u7ed3\u6784

    \u5982\u4e0b\u8868\u6240\u793a\uff0c\u5728\u6700\u4f73\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u6216\u6781\u5c0f\u503c\u3002

    \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \u65f6\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u6811\u7684\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u65f6\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2. \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"

    \u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002

    \u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002

    "},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1. \u00a0 \u5c42\u5e8f\u904d\u5386","text":"

    \u300c\u5c42\u5e8f\u904d\u5386 Level-Order Traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002

    \u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 Breadth-First Traversal\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u641c\u7d22\u65b9\u5f0f\u3002

    Fig. \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386

    "},{"location":"chapter_tree/binary_tree_traversal/#_1","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"

    \u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u300c\u961f\u5217\u300d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002

    JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree_bfs.java
    /* \u5c42\u5e8f\u904d\u5386 */\nList<Integer> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode> queue = new LinkedList<>() {{ add(root); }};\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<Integer> list = new ArrayList<>();\nwhile (!queue.isEmpty()) {\nTreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\nlist.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.cpp
    /* \u5c42\u5e8f\u904d\u5386 */\nvector<int> levelOrder(TreeNode *root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue<TreeNode *> queue;\nqueue.push(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvector<int> vec;\nwhile (!queue.empty()) {\nTreeNode *node = queue.front();\nqueue.pop();              // \u961f\u5217\u51fa\u961f\nvec.push_back(node->val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node->left != nullptr)\nqueue.push(node->left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node->right != nullptr)\nqueue.push(node->right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn vec;\n}\n
    binary_tree_bfs.py
    def level_order(root: TreeNode | None) -> list[int]:\n\"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n# \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue: deque[TreeNode] = deque()\nqueue.append(root)\n# \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nres = []\nwhile queue:\nnode: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\nres.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\nif node.left is not None:\nqueue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif node.right is not None:\nqueue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\nreturn res\n
    binary_tree_bfs.go
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []int {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue := list.New()\nqueue.PushBack(root)\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nnums := make([]int, 0)\nfor queue.Len() > 0 {\n// \u961f\u5217\u51fa\u961f\nnode := queue.Remove(queue.Front()).(*TreeNode)\n// \u4fdd\u5b58\u8282\u70b9\u503c\nnums = append(nums, node.Val)\nif node.Left != nil {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Left)\n}\nif node.Right != nil {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Right)\n}\n}\nreturn nums\n}\n
    binary_tree_bfs.js
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list = [];\nwhile (queue.length) {\nlet node = queue.shift(); // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.ts
    /* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list: number[] = [];\nwhile (queue.length) {\nlet node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) {\nqueue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right) {\nqueue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list;\n}\n
    binary_tree_bfs.c
    /* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n/* \u8f85\u52a9\u961f\u5217 */\nint front, rear;\nint index, *arr;\nTreeNode *node;\nTreeNode **queue;\n/* \u8f85\u52a9\u961f\u5217 */\nqueue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_NODE_SIZE);\n// \u961f\u5217\u6307\u9488\nfront = 0, rear = 0;\n// \u52a0\u5165\u6839\u8282\u70b9\nqueue[rear++] = root;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n/* \u8f85\u52a9\u6570\u7ec4 */\narr = (int *)malloc(sizeof(int) * MAX_NODE_SIZE);\n// \u6570\u7ec4\u6307\u9488\nindex = 0;\nwhile (front < rear) {\n// \u961f\u5217\u51fa\u961f\nnode = queue[front++];\n// \u4fdd\u5b58\u8282\u70b9\u503c\narr[index++] = node->val;\nif (node->left != NULL) {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node->left;\n}\nif (node->right != NULL) {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node->right;\n}\n}\n// \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n*size = index;\narr = realloc(arr, sizeof(int) * (*size));\n// \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\nfree(queue);\nreturn arr;\n}\n
    binary_tree_bfs.cs
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode> queue = new();\nqueue.Enqueue(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> list = new();\nwhile (queue.Count != 0) {\nTreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\nlist.Add(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n
    binary_tree_bfs.swift
    /* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -> [Int] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nvar queue: [TreeNode] = [root]\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list: [Int] = []\nwhile !queue.isEmpty {\nlet node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\nlist.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\nif let left = node.left {\nqueue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif let right = node.right {\nqueue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list\n}\n
    binary_tree_bfs.zig
    // \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst L = std.TailQueue(*inc.TreeNode(T));\nvar queue = L{};\nvar root_node = try mem_allocator.create(L.Node);\nroot_node.data = root;\nqueue.append(root_node); // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list = std.ArrayList(T).init(std.heap.page_allocator);\nwhile (queue.len > 0) {\nvar queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\nvar node = queue_node.data;\ntry list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.left.?;\nqueue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.right.?;\nqueue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}        }\nreturn list;\n}\n
    binary_tree_bfs.dart
    /* \u5c42\u5e8f\u904d\u5386 */\nList<int> levelOrder(TreeNode? root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue<TreeNode?> queue = Queue();\nqueue.add(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList<int> res = [];\nwhile (queue.isNotEmpty) {\nTreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\nres.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn res;\n}\n
    "},{"location":"chapter_tree/binary_tree_traversal/#_2","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\(\\frac{n + 1}{2}\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002

    "},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2. \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"

    \u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 Depth-First Traversal\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002

    \u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5de6\u4fa7\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u793a\u610f\u56fe\uff0c\u53f3\u4e0a\u65b9\u662f\u5bf9\u5e94\u7684\u9012\u5f52\u5b9e\u73b0\u4ee3\u7801\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u4e2a\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002

    Fig. \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386

    \u4f4d\u7f6e \u542b\u4e49 \u6b64\u5904\u8bbf\u95ee\u8282\u70b9\u65f6\u5bf9\u5e94 \u6a59\u8272\u5706\u5708\u5904 \u521a\u8fdb\u5165\u6b64\u8282\u70b9\uff0c\u5373\u5c06\u8bbf\u95ee\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u6811 \u524d\u5e8f\u904d\u5386 Pre-Order Traversal \u84dd\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\uff0c\u5373\u5c06\u8bbf\u95ee\u53f3\u5b50\u6811 \u4e2d\u5e8f\u904d\u5386 In-Order Traversal \u7d2b\u8272\u5706\u5708\u5904 \u5df2\u8bbf\u95ee\u5b8c\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff0c\u5373\u5c06\u8fd4\u56de \u540e\u5e8f\u904d\u5386 Post-Order Traversal"},{"location":"chapter_tree/binary_tree_traversal/#_3","title":"\u7b97\u6cd5\u5b9e\u73b0","text":"JavaC++PythonGoJavaScriptTypeScriptCC#SwiftZigDart binary_tree_dfs.java
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.add(root.val);\n}\n
    binary_tree_dfs.cpp
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nvec.push_back(root->val);\npreOrder(root->left);\npreOrder(root->right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left);\nvec.push_back(root->val);\ninOrder(root->right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root->left);\npostOrder(root->right);\nvec.push_back(root->val);\n}\n
    binary_tree_dfs.py
    def pre_order(root: TreeNode | None) -> None:\n\"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nres.append(root.val)\npre_order(root=root.left)\npre_order(root=root.right)\ndef in_order(root: TreeNode | None) -> None:\n\"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\nin_order(root=root.left)\nres.append(root.val)\nin_order(root=root.right)\ndef post_order(root: TreeNode | None) -> None:\n\"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npost_order(root=root.left)\npost_order(root=root.right)\nres.append(root.val)\n
    binary_tree_dfs.go
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nnums = append(nums, node.Val)\npreOrder(node.Left)\npreOrder(node.Right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.Left)\nnums = append(nums, node.Val)\ninOrder(node.Right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(node.Left)\npostOrder(node.Right)\nnums = append(nums, node.Val)\n}\n
    binary_tree_dfs.js
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
    binary_tree_dfs.ts
    /* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n
    binary_tree_dfs.c
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\narr[(*size)++] = root->val;\npreOrder(root->left, size);\npreOrder(root->right, size);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root->left, size);\narr[(*size)++] = root->val;\ninOrder(root->right, size);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root->left, size);\npostOrder(root->right, size);\narr[(*size)++] = root->val;\n}\n
    binary_tree_dfs.cs
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.Add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.Add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.Add(root.val);\n}\n
    binary_tree_dfs.swift
    /* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.append(root.val)\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(root: root.left)\nlist.append(root.val)\ninOrder(root: root.right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(root: root.left)\npostOrder(root: root.right)\nlist.append(root.val)\n}\n
    binary_tree_dfs.zig
    // \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\ntry list.append(root.?.val);\ntry preOrder(T, root.?.left);\ntry preOrder(T, root.?.right);\n}\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ntry inOrder(T, root.?.left);\ntry list.append(root.?.val);\ntry inOrder(T, root.?.right);\n}\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\ntry postOrder(T, root.?.left);\ntry postOrder(T, root.?.right);\ntry list.append(root.?.val);\n}\n
    binary_tree_dfs.dart
    /* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -> \u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811\nlist.add(node.val);\npreOrder(node.left);\npreOrder(node.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u6839\u8282\u70b9 -> \u53f3\u5b50\u6811\ninOrder(node.left);\nlist.add(node.val);\ninOrder(node.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -> \u53f3\u5b50\u6811 -> \u6839\u8282\u70b9\npostOrder(node.left);\npostOrder(node.right);\nlist.add(node.val);\n}\n

    Note

    \u6211\u4eec\u4e5f\u53ef\u4ee5\u4ec5\u57fa\u4e8e\u5faa\u73af\u5b9e\u73b0\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\uff0c\u6709\u5174\u8da3\u7684\u540c\u5b66\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002

    \u9012\u5f52\u8fc7\u7a0b\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u76f8\u53cd\u7684\u90e8\u5206\u3002\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\uff1b\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u8be5\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u4e3a\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\u3002

    <1><2><3><4><5><6><7><8><9><10><11>

    "},{"location":"chapter_tree/binary_tree_traversal/#_4","title":"\u590d\u6742\u5ea6\u5206\u6790","text":"

    \u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002

    \u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002

    "},{"location":"chapter_tree/summary/","title":"7.6. \u00a0 \u5c0f\u7ed3","text":"
    • \u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002
    • \u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002
    • \u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002
    • \u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002
    • \u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002
    • \u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002
    • \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u201d\u7684\u5206\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002
    • \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u5934\u7ee7\u7eed\u201d\u7684\u56de\u6eaf\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002
    • \u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002
    • AVL \u6811\uff0c\u4e5f\u79f0\u4e3a\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\uff0c\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002
    • AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002
    "},{"location":"chapter_tree/summary/#761-q-a","title":"7.6.1. \u00a0 Q & A","text":"

    \u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f

    \u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u8d70\u8fc7\u8fb9\u7684\u6570\u91cf\u201d\u3002

    \u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u90fd\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f

    \u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e3a\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002

    \u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f

    DFS \u7684\u524d\u3001\u4e2d\u3001\u540e\u5e8f\u904d\u5386\u548c\u8bbf\u95ee\u6570\u7ec4\u7684\u987a\u5e8f\u7c7b\u4f3c\uff0c\u662f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u65b9\u6cd5\uff0c\u5229\u7528\u8fd9\u4e09\u79cd\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u7ed3\u70b9\u5927\u5c0f\u6ee1\u8db3 \u5de6\u5b50\u7ed3\u70b9\u503c < \u6839\u7ed3\u70b9\u503c < \u53f3\u5b50\u7ed3\u70b9\u503c \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167 \u5de6->\u6839->\u53f3 \u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002

    \u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 node , child , grand_child \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 node \u7684\u7236\u8282\u70b9\u548c node \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f

    \u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c right_rotate(root) \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 return child \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002

    \u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 private \u548c public \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 height() \u51fd\u6570\u548c updateHeight() \u51fd\u6570\u5206\u522b\u653e\u5728 public \u548c private \u4e2d\u5462\uff1f

    \u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a private \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 updateHeight() \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c height() \u662f\u8bbf\u95ee\u7ed3\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e vector.size() \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 public \u4ee5\u4fbf\u4f7f\u7528\u3002

    \u8bf7\u95ee\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u4e2a\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f

    \u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u662f build_tree() \uff0c\u5df2\u5728\u6e90\u4ee3\u7801\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u7528\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002

    "}]} \ No newline at end of file diff --git a/sitemap.xml b/sitemap.xml index 2a01ee4dd..80dc6ea7f 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -2,412 +2,417 @@ https://www.hello-algo.com/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_appendix/contribution/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_appendix/installation/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_array_and_linkedlist/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_array_and_linkedlist/array/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_array_and_linkedlist/linked_list/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_array_and_linkedlist/list/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_array_and_linkedlist/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/backtracking_algorithm/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/n_queens_problem/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/permutations_problem/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/subset_sum_problem/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_backtracking/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_computational_complexity/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_computational_complexity/performance_evaluation/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_computational_complexity/space_complexity/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_computational_complexity/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_computational_complexity/time_complexity/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/basic_data_types/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/character_encoding/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/classification_of_data_structure/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/number_encoding/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_data_structure/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_dynamic_programming/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_dynamic_programming/dp_problem_features/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_dynamic_programming/intro_to_dynamic_programming/ - 2023-07-03 + 2023-07-05 + daily + + + https://www.hello-algo.com/chapter_dynamic_programming/knapsack_problem/ + 2023-07-05 daily https://www.hello-algo.com/chapter_graph/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_graph/graph/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_graph/graph_operations/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_graph/graph_traversal/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_graph/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_hashing/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_hashing/hash_algorithm/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_hashing/hash_collision/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_hashing/hash_map/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_hashing/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_heap/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_heap/build_heap/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_heap/heap/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_heap/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_heap/top_k/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_introduction/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_introduction/algorithms_are_everywhere/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_introduction/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_introduction/what_is_dsa/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_preface/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_preface/about_the_book/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_preface/suggestions/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_preface/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_reference/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/binary_search/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/binary_search_edge/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/replace_linear_by_hashing/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/searching_algorithm_revisited/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_searching/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/bubble_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/bucket_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/counting_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/heap_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/insertion_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/merge_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/quick_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/radix_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/selection_sort/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/sorting_algorithm/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_sorting/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_stack_and_queue/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_stack_and_queue/deque/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_stack_and_queue/queue/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_stack_and_queue/stack/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_stack_and_queue/summary/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/array_representation_of_tree/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/avl_tree/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/binary_search_tree/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/binary_tree/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/binary_tree_traversal/ - 2023-07-03 + 2023-07-05 daily https://www.hello-algo.com/chapter_tree/summary/ - 2023-07-03 + 2023-07-05 daily \ No newline at end of file diff --git a/sitemap.xml.gz b/sitemap.xml.gz index 11476bc7b68e3555b032fe179eddb104b3730646..da1aa235026bebbfb0481233090026c9df2f02d0 100644 GIT binary patch literal 823 zcmV-71IYXziwFpNm8E0?|8r?{Wo=<_E_iKh0NtBQj~gKjfbaV&qTG{7x2pP3Hhb$Y zXwS$4&I}5#V94X&7be+kBDIHJD%>10;2`t?wy_zqemz+;J76U3-OK#Lay6f!lU`{z zyv%?8_`diuf89K-b2Nl$RvJ5cnU5)jZ(Vmh9+#vOS_riCap*{nr9$0R*sYdd=bNWl z`P2?BrRlRaUT$hHbnRwDWBdZl;Fr?dZr#-e_w9j_^=k9Hdj7areO|0Sl_po&T{Ag? z8*m$OzfqvgdEMPfE+E~Fq-xDmty%UTLT}^`--SdqfVpc!|%jCBRUcx z`raTT@_^CiH0?DE=_&Um;1R_zIkLC!O-eajwFAslGkDU8DRj~Gt&bL*M1cn|x&099 zk7J<#Wfa_3gtZMMZP~1Vf&_w6%rfPOoKSlpEib}!4oIBXi@13E`omkc&WgTZklt!J z>Avt=##-dE?U^nE2Op`<4s0>g5`+yAtZjaZf5WT7TMQ|-2zBa--59h>(Tn8&iv`&^ z2vmY)7NLEi4kH6XR-D3@ko}UIvOzBST|KA=!+@B>)-nQ(t$VLG#`x(N6uq;drTc6Ad*k^4N6ILFSY;-gEAa+e-PJy>78C&a4^a1p?Rgr*L&qOhv zk0Umo+CL0|gAP9cA4f<+lTZXrLx+f4kjR1&dY_X>BZ}Vp9Sda2!+_L&YV*$9zMU1+ z;=5eqN=pb)HlNbGQ})`Gg}(|oG7D4#7nO(@OntFuv@0^vpIPL;@BFvYdm3EK@}TQo zL&aqb3|t-gn3~s$D`TJg-r!Va!Yg<6QVADc>9eZ^KEySt<3p<*VVqupvO%^O2SygR z&sy&I;6_mT#0QtqpyZ{$*C&&+^cyvYz#-K)s`X;?K4V5!tP1zci22R&ROdy^!0J*& zY#;FQr-zb)MkI9UcI}G^{lPFHeLS6D#mo+t$$g9o`=5ET=|Pw_zW`CSQQ=uS004m> Blo$X2 literal 812 zcmV+{1JnE;iwFo0jiO`%|8r?{Wo=<_E_iKh0M(mIliMH+fbabkncNenyN5m|o4xfH zv}Z;J#b&}Q2=e&%1t*)wOnd01gM5fVVVDn)WC`Q-`^lQw0V8Se-sYc|tN9F_^h&$o zZT{=$kHy#d`{vU+M?;uqrLm*8`Iu7p-gU?0aY;I%g+NOmhmPb}D%4$t-D>%5zWFpO zSMA_Zny$6+a#MSuYd0er;}>8Czm(o~>#jCsdD1kB1K5>k27T}aH z+U-yoO^F}yB;<#EDGHu*&Xbz5wGi9Zl^*0S#qv`g4SYZ!E21`yI*pbOzZ3h6=tzX< zdxMO~14f(E^sHe>Pq{Avk0^%8k-dFyQp(|~9bl%K!IMr*p^L6>eYD^t3OsBoLHhmMKT%gxUjXc@d^_K;pz+#Kqg!KR&W`R`dmf^j6DB z_l4gw)*_c}&vY3$_(*kjV2hcSAZ&9M`nH8|l+9)QhJSf@dX7WMon#P>sR#g(7)}81R=MRhjV0T|JY*g;)CQYJm@NP3rj2YDXBSH;8PIEyjV7#qG0}J3hD(ls@sn zB{V2`>F@Q