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int *bc; + makeBC(text, &bc); + ...... +} +``` + +### `bc`表的构建 + +实际上,在前面的讨论中,已经涉及到了如何构造`bc`表的问题,这里做一个统一的总结。为了构造`bc`表,需要遍历模式串中的每一个字符,并且把每个字符`X`最后出现的位置保存在`bc[X]`中。为了简单起见,这里的`bc`表包含全部字符集,比如整个ASCII码字符集,这样便于判断某个字符是否出现在模式串当中,否则为了判断某个字符是否在模式串中出现过,还需要额外建立一个散列表或者位图,还是需要消耗同样的空间,因此这里的`bc`表兼具了给出移动位置以及散列表的作用。 + +在`bc`表的构建中采用`画家算法`(painter's algorithm),即从左至右遍历模式串,对于其中每一个出现的字符,都将其位置(或者秩)更新到其在`bc`表中对应的项中,这样,遍历结束时`bc`表保存的就是所有字符最后出现过的位置了。`bc`构造的算法如下: + +```c +void makeBC(char* const pattern, int* bc){ + bc = new int[256]; + for(int ix = 0; ix != 256; ++ix) + bc[ix] = -1; //initialize to -1 + for(int ix = 0; pattern[ix] != '\0'; ++ix) + bc[pattern[ix]] = ix; //painter's algorithm +} +``` + +使用`bc`策略时,最好可以达到`O(n/m)`的时间复杂度,对应了每次都在最右一个字符匹配失败,然后整体右移`m`个单位的情况,如下图所示: + +![bc_bestcase](bc_bestcase/png) + +这固然是一个非常好的结论,但是`bc`策略在最坏情况下却会达到`O(mn)`的时间复杂度,与蛮力策略相当,该情况如下图所示: + +![bc_worstcase](bc_worstcase.png) + +可以看到,在这种情况下,每次都需要进行`m - 1`次比对,才能在最左侧一次比对中失败,而该次失败只能让模式串右移一个单位。这种情况正与蛮力策略的最好情况相一致。一般地,单次匹配成功的概率越大,即字符集越小,就越接近于这种最坏的情况;单次匹配成功的概率越小,即字符集越大,就越接近与最好的情况。 diff --git a/thu_dsa/chp11/compare_backwards.png b/thu_dsa/chp11/compare_backwards.png new file mode 100644 index 0000000000000000000000000000000000000000..adc16b377be368d84a7cca73b01df18c35be794e GIT binary patch literal 45425 zcmeFZcT^PX_AP28DUAdXq)8&8B8udUfC4H(KyuEKnkHvZNup#$GN=eh2FWzR1{9Da zIW|p@*hHaeGWRQVpMB0gzd!DKRwaUT*Q@RQ2cvFG6bjytK!NgvDafX{<}oU)Wslsa~-DB{eX@oDgH zQlz}L)3IYz&d~pkk6AFggI^Nezbp0F-CzkrTmkR%Avkc@`>nri3^!#UP>DL&HnD$o z7;8iuHC@S|Q&YTML&sg?_Wp#t^i#4=7XrwnKRl48ihi_GLvZZ#A`Y0S-eFprLu_}B zaTR;HS^Ca;NKgnYVv@yCyj~slO51}61PSD&zm8}zGU9u8{~S~gCf>dHbI40ygVRv{ zIqqK#^#AJ6eV5hG*!7UI2C$;7bAFx_&L3wa_YfteM41L##VxoXg6@3Cb<;2_qIRi!_FaTO+)hX~Gc48n~os$HI9*Y<26W^Ijn{l|GW-03G%uu$> z4Srm9%8)Bf$n5bXNofEQ%IXq37~Z7(`ywUF||=kJo6H!y0>REn;tL4Y9d6gjk#W zTEu7i?JQw)#2$sfRK?>X^IZ}J5I!-o5{fCed&GjAZRL54z8n)Yu0JJcKPI-m{}o+g 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