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<meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
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<title>11.11 Summary - Hello Algo</title>
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11.11 Summary
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3.1 Classifying data structures
|
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</a>
|
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3.2 Basic data types
|
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</a>
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3.3 Number encoding *
|
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3.4 Character encoding *
|
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3.5 Summary
|
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|
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|
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|
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|
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</a>
|
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</li>
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3.6 Exercises
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Chapter 4. Arrays and linked lists
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4.1 Arrays
|
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</span>
|
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|
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</a>
|
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</li>
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4.2 Linked lists
|
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|
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</span>
|
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|
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|
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|
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</a>
|
||
</li>
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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4.3 Lists
|
||
|
||
|
||
</span>
|
||
|
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|
||
|
||
</a>
|
||
</li>
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|
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4.4 Memory and cache *
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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<li class="md-nav__item">
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4.5 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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4.6 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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|
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Chapter 5. Stacks and queues
|
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Chapter 5. Stacks and queues
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5.1 Stacks
|
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|
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|
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</span>
|
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|
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|
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</a>
|
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5.2 Queues
|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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|
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5.3 Deques
|
||
|
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|
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</span>
|
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|
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|
||
|
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</a>
|
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</li>
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|
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|
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5.4 Summary
|
||
|
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|
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</span>
|
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|
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|
||
|
||
</a>
|
||
</li>
|
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5.5 Exercises
|
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|
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|
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|
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|
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|
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|
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Chapter 6. Hash tables
|
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Chapter 6. Hash tables
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6.1 Hash tables
|
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|
||
|
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</span>
|
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|
||
|
||
|
||
</a>
|
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</li>
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6.2 Hash collisions
|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
|
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</li>
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<li class="md-nav__item">
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|
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6.3 Hash algorithms
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
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|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
6.4 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
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|
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|
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|
||
6.5 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
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|
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|
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|
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</ul>
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<span class="md-ellipsis">
|
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|
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|
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Chapter 7. Trees
|
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|
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Chapter 7. Trees
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
||
7.1 Binary trees
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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|
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|
||
|
||
|
||
7.2 Traversing a binary tree
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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|
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|
||
|
||
|
||
7.3 Array representation of a binary tree
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
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|
||
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
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|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.4 Binary search trees
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.5 AVL trees *
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
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|
||
|
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|
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|
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|
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|
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<li class="md-nav__item">
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|
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|
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|
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|
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|
||
|
||
|
||
7.6 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
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|
||
|
||
|
||
7.7 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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Chapter 8. Heaps
|
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|
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|
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|
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Chapter 8. Heaps
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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|
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|
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|
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|
||
|
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|
||
8.1 Heaps
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
8.2 Building a heap
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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<li class="md-nav__item">
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
|
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|
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|
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|
||
|
||
|
||
8.3 The top-k problem
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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|
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|
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|
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|
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|
||
|
||
|
||
8.4 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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|
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|
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|
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|
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|
||
|
||
|
||
8.5 Exercises
|
||
|
||
|
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</span>
|
||
|
||
|
||
|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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Chapter 9. Graphs
|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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Chapter 9. Graphs
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<ul class="md-nav__list" data-md-scrollfix>
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|
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|
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|
||
9.1 Graphs
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
||
|
||
9.2 Basic graph operations
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.3 Graph traversal
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.4 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/exercises/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.5 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
|
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|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_searching/" class="md-nav__link ">
|
||
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4zM3 16v-2h6v2zm0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chapter 10. Searching
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_12">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
Chapter 10. Searching
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.1 Binary search
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.2 Finding an insertion point with binary search
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.3 Finding boundaries with binary search
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.4 Optimizing with hashing
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.5 Search algorithms revisited
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.6 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_searching/exercises/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
10.7 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--active md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../" class="md-nav__link ">
|
||
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2z"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chapter 11. Sorting
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
|
||
<label class="md-nav__title" for="__nav_13">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
Chapter 11. Sorting
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../sorting_algorithm/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.1 Sorting algorithms
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../selection_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.2 Selection sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../bubble_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.3 Bubble sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../insertion_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.4 Insertion sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../quick_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.5 Quick sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../merge_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.6 Merge sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../heap_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.7 Heap sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../bucket_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.8 Bucket sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../counting_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.9 Counting sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../radix_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.10 Radix sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--active">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__link md-nav__link--active" for="__toc">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.11 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
<a href="././" class="md-nav__link md-nav__link--active">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.11 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
|
||
|
||
<nav class="md-nav md-nav--secondary" aria-label="On this page">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__title" for="__toc">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
On this page
|
||
</label>
|
||
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#1-key-takeaways" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
<span class="md-typeset">
|
||
1. Key takeaways
|
||
</span>
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#2-q-a" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
<span class="md-typeset">
|
||
2. Q & A
|
||
</span>
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../exercises/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.12 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item md-nav__item--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
|
||
|
||
|
||
|
||
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Chapter 12. Divide and conquer
|
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Chapter 12. Divide and conquer
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12.1 Divide and conquer
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</a>
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12.2 Divide-and-conquer search
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</span>
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</a>
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12.3 Building a binary tree
|
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</span>
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</a>
|
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</li>
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12.4 Tower of Hanoi
|
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</span>
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</a>
|
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</li>
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12.5 Summary
|
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|
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|
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</span>
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|
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|
||
</a>
|
||
</li>
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12.6 Exercises
|
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|
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|
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|
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|
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|
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|
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Chapter 13. Backtracking
|
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Chapter 13. Backtracking
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13.1 Backtracking algorithms
|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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13.2 Permutations problem
|
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|
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|
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</span>
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|
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|
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|
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</a>
|
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</li>
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13.3 Subset sum problem
|
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|
||
|
||
</span>
|
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|
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|
||
|
||
</a>
|
||
</li>
|
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13.4 N-queens problem
|
||
|
||
|
||
</span>
|
||
|
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|
||
|
||
</a>
|
||
</li>
|
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|
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13.5 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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13.6 Exercises
|
||
|
||
|
||
</span>
|
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|
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|
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|
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Chapter 14. Dynamic programming
|
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Chapter 14. Dynamic programming
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14.1 Introduction to dynamic programming
|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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14.2 Properties of dynamic programming problems
|
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|
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|
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</span>
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|
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|
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|
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</a>
|
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14.3 Solving dynamic programming problems
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|
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|
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14.4 0-1 knapsack problem
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|
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</span>
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|
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|
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|
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14.5 Unbounded knapsack problem
|
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|
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|
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</span>
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|
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|
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|
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|
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14.6 Edit distance problem
|
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|
||
|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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14.7 Summary
|
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|
||
|
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</span>
|
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|
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|
||
|
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</a>
|
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</li>
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|
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<h1 id="1111-summary">11.11 Summary<a class="headerlink" href="#1111-summary" title="Permanent link">¶</a></h1>
|
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<h3 id="1-key-takeaways">1. Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">¶</a></h3>
|
||
<ul>
|
||
<li>Bubble sort sorts by swapping adjacent elements. Adding a flag to stop early improves its best-case time complexity to <span class="arithmatex">\(O(n)\)</span>.</li>
|
||
<li>Insertion sort inserts one element from the unsorted section into its correct position in the sorted section each round. Although its time complexity is <span class="arithmatex">\(O(n^2)\)</span>, its low operation count makes it popular for small datasets.</li>
|
||
<li>Quick sort sorts through partitioning. Repeatedly choosing the worst pivot can degrade its time complexity to <span class="arithmatex">\(O(n^2)\)</span>. Median or random pivot selection reduces this risk. Recursing on the shorter subarray reduces recursion depth and improves space complexity to <span class="arithmatex">\(O(\log n)\)</span>.</li>
|
||
<li>Merge sort has division and merging phases, exemplifying divide and conquer. Sorting an array requires an auxiliary array and <span class="arithmatex">\(O(n)\)</span> space, while sorting a linked list can be optimized to use <span class="arithmatex">\(O(1)\)</span> space.</li>
|
||
<li>Bucket sort has three steps: distribute data into buckets, sort within each bucket, and combine the results. It also uses divide and conquer and suits very large datasets. The key is distributing data evenly.</li>
|
||
<li>Counting sort is a special case of bucket sort that sorts by counting occurrences. It suits large datasets with a limited value range and requires data that can be converted to positive integers.</li>
|
||
<li>Radix sort sorts one digit at a time and requires data that can be represented as numbers with a fixed number of digits.</li>
|
||
<li>Ideally, a sorting algorithm would be efficient, stable, in-place, and adaptive. As with other data structures and algorithms, however, no single sorting algorithm meets all these requirements. In practice, choose one according to the data's properties.</li>
|
||
<li>Figure 11-19 compares common sorting algorithms in efficiency, stability, in-place operation, and adaptiveness.</li>
|
||
</ul>
|
||
<p><img alt="Comparison of sorting algorithms" class="animation-figure" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
|
||
<p align="center"> Figure 11-19 Comparison of sorting algorithms </p>
|
||
|
||
<h3 id="2-q-a">2. Q & A<a class="headerlink" href="#2-q-a" title="Permanent link">¶</a></h3>
|
||
<p><strong>Q</strong>: When is stability necessary in a sorting algorithm?</p>
|
||
<p>We often sort objects by one of their attributes. For example, students have names and heights, and we want a multilevel sort. First sort by name to obtain <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>, then sort by height. An unstable algorithm may produce <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>.</p>
|
||
<p>Students D and C have switched positions, breaking the ordering by name. This is not what we want.</p>
|
||
<p><strong>Q</strong>: Can the right-to-left and left-to-right scans in partitioning be reversed?</p>
|
||
<p>No. When the leftmost element is the pivot, the right-to-left scan must come first. This may seem counterintuitive, so let us examine why.</p>
|
||
<p>The final step of <code>partition()</code> swaps <code>nums[left]</code> and <code>nums[i]</code>. Afterward, every element to the left of the pivot must be <code><=</code> the pivot, <strong>so <code>nums[left] >= nums[i]</code> must hold before that swap</strong>. If we scan left to right first and fail to find a larger element, <strong>the loop stops at <code>i == j</code>, where it is possible that <code>nums[j] == nums[i] > nums[left]</code></strong>. The final swap would then move an element larger than the pivot to the far left, causing partitioning to fail.</p>
|
||
<p>For example, given <code>[0, 0, 0, 0, 1]</code>, scanning left to right first produces <code>[1, 0, 0, 0, 0]</code> after partitioning, which is incorrect.</p>
|
||
<p>Conversely, if we choose <code>nums[right]</code> as the pivot, we must scan left to right first.</p>
|
||
<p><strong>Q</strong>: In the recursion-depth optimization for quick sort, why does choosing the shorter subarray keep the depth at most <span class="arithmatex">\(\log n\)</span>?</p>
|
||
<p>Recursion depth is the number of recursive calls that have not yet returned. Each partition divides the array into two subarrays. After the optimization, the subarray chosen for recursion is at most half the original length. Even in the worst case, where it is always exactly half, the depth is <span class="arithmatex">\(\log n\)</span>.</p>
|
||
<p>In the original quick sort, we may repeatedly recurse on the larger subarray, with lengths <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> in the worst case, giving depth <span class="arithmatex">\(n\)</span>. The optimization prevents this.</p>
|
||
<p><strong>Q</strong>: If all array elements are equal, does quick sort take <span class="arithmatex">\(O(n^2)\)</span> time? How can we handle this degeneration?</p>
|
||
<p>Yes. In this case, consider partitioning into three sections: elements smaller than, equal to, and greater than the pivot. Recurse only on the smaller and greater sections. If all elements are equal, a single partition then completes the sort.</p>
|
||
<p><strong>Q</strong>: Why is bucket sort's worst-case time complexity <span class="arithmatex">\(O(n^2)\)</span>?</p>
|
||
<p>In the worst case, all elements fall into one bucket. If an <span class="arithmatex">\(O(n^2)\)</span> algorithm is used to sort that bucket, the overall time complexity is <span class="arithmatex">\(O(n^2)\)</span>.</p>
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