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Chapter 3. Data structures
|
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3.1 Classifying data structures
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</span>
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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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</span>
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3.4 Character encoding *
|
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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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|
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|
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4.2 Linked lists
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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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4.3 Lists
|
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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>
|
||
</li>
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4.4 Memory and cache *
|
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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>
|
||
</li>
|
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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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|
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</span>
|
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|
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|
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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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5.2 Queues
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|
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|
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|
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|
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5.3 Deques
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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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|
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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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|
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|
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|
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|
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|
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<span class="md-nav__icon md-icon"></span>
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Chapter 6. Hash tables
|
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|
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|
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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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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
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<span class="md-ellipsis">
|
||
|
||
|
||
6.1 Hash tables
|
||
|
||
|
||
</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_hashing/hash_collision/" class="md-nav__link">
|
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|
||
|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
6.2 Hash collisions
|
||
|
||
|
||
</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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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
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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|
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
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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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<li class="md-nav__item">
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<a href="../../chapter_hashing/exercises/" class="md-nav__link">
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<span class="md-ellipsis">
|
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|
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|
||
6.5 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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</ul>
|
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</nav>
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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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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chapter 7. Trees
|
||
|
||
|
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</span>
|
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|
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|
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</a>
|
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<span class="md-nav__icon md-icon"></span>
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Chapter 7. Trees
|
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|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
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|
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|
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|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.1 Binary 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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|
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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">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.2 Traversing a binary tree
|
||
|
||
|
||
</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_tree/array_representation_of_tree/" class="md-nav__link">
|
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|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.3 Array representation of a binary tree
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.4 Binary search trees
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.5 AVL trees *
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_tree/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.6 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
||
|
||
<li class="md-nav__item">
|
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<a href="../../chapter_tree/exercises/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
7.7 Exercises
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
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|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
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|
||
|
||
|
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|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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<div class="md-nav__link md-nav__container">
|
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<a href="../../chapter_heap/" class="md-nav__link ">
|
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|
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|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1"/></svg>
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chapter 8. Heaps
|
||
|
||
|
||
</span>
|
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|
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|
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|
||
</a>
|
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|
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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Chapter 8. Heaps
|
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|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_heap/heap/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
||
|
||
|
||
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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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.2 Building a heap
|
||
|
||
|
||
</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_heap/top_k/" class="md-nav__link">
|
||
|
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|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.3 The top-k problem
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
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<a href="../../chapter_heap/summary/" class="md-nav__link">
|
||
|
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|
||
|
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<span class="md-ellipsis">
|
||
|
||
|
||
8.4 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
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<a href="../../chapter_heap/exercises/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
8.5 Exercises
|
||
|
||
|
||
</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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|
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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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|
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|
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|
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|
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|
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|
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59zM6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43s.83.16 1.12.43l4.28-2.53zM12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68s1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25zm2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24z"/></svg>
|
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|
||
<span class="md-ellipsis">
|
||
|
||
|
||
Chapter 9. Graphs
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
|
||
|
||
<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
|
||
</div>
|
||
|
||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_11">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
|
||
|
||
Chapter 9. Graphs
|
||
|
||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.1 Graphs
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
9.2 Basic graph operations
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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" >
|
||
|
||
|
||
<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>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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--nested">
|
||
|
||
|
||
|
||
<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
|
||
|
||
|
||
<div class="md-nav__link md-nav__container">
|
||
<a href="../../chapter_sorting/" 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="false">
|
||
<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="../../chapter_sorting/sorting_algorithm/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.1 Sorting algorithms
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/selection_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.2 Selection sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.3 Bubble sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.4 Insertion sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.5 Quick sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.6 Merge sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/heap_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.7 Heap sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.8 Bucket sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/counting_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.9 Counting sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/radix_sort/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.10 Radix sort
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
|
||
|
||
11.11 Summary
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../chapter_sorting/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" >
|
||
|
||
|
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<div class="md-nav__link md-nav__container">
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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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12.2 Divide-and-conquer search
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12.3 Building a binary tree
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12.4 Tower of Hanoi
|
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12.5 Summary
|
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</a>
|
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</li>
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12.6 Exercises
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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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</a>
|
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13.2 Permutations problem
|
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</a>
|
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13.3 Subset sum 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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13.4 N-queens 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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13.5 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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13.6 Exercises
|
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|
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|
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|
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|
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Chapter 14. Dynamic programming
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14.1 Introduction to dynamic programming
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14.2 Properties of dynamic programming problems
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14.3 Solving dynamic programming problems
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14.4 0-1 knapsack problem
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14.5 Unbounded knapsack problem
|
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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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|
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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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14.7 Summary
|
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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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<h1 id="33-number-encoding">3.3 Number encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">¶</a></h1>
|
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<div class="admonition tip">
|
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<p class="admonition-title">Tip</p>
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<p>Sections marked with * in their titles are optional. If you are short on time or find them difficult, you can skip them initially and return after completing the required sections.</p>
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<h2 id="331-sign-magnitude-ones-complement-and-twos-complement">3.3.1 Sign-magnitude, one's complement, and two's complement<a class="headerlink" href="#331-sign-magnitude-ones-complement-and-twos-complement" title="Permanent link">¶</a></h2>
|
||
<p>The previous table showed that every integer type represents one more negative value than positive value. For example, <code>byte</code> ranges over <span class="arithmatex">\([-128, 127]\)</span>. Understanding this counterintuitive fact requires sign-magnitude, one's complement, and two's complement.</p>
|
||
<p>First, <strong>integers are stored in two's complement form</strong>. Before explaining why, let's define the three representations.</p>
|
||
<ul>
|
||
<li><strong>Sign-magnitude</strong>: The highest bit is the sign: <span class="arithmatex">\(0\)</span> for positive and <span class="arithmatex">\(1\)</span> for negative. The remaining bits represent the magnitude.</li>
|
||
<li><strong>One's complement</strong>: A positive number has the same representation as sign-magnitude. For a negative number, invert every bit except the sign bit of its sign-magnitude representation.</li>
|
||
<li><strong>Two's complement</strong>: A positive number again has the same representation as sign-magnitude. For a negative number, add <span class="arithmatex">\(1\)</span> to its one's complement.</li>
|
||
</ul>
|
||
<p>Figure 3-4 shows how to convert among these representations.</p>
|
||
<p><img alt="Converting between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></p>
|
||
<p align="center"> Figure 3-4 Converting between sign-magnitude, one's complement, and two's complement </p>
|
||
|
||
<p>Although <u>sign-magnitude</u> is intuitive, it has limitations. First, <strong>negative sign-magnitude values cannot be used directly in ordinary binary arithmetic</strong>. For example, directly adding <span class="arithmatex">\(1 + (-2)\)</span> in this representation gives <span class="arithmatex">\(-3\)</span>, which is clearly wrong.</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& 1 + (-2) \newline
|
||
& \rightarrow 0000 \; 0001 + 1000 \; 0010 \newline
|
||
& = 1000 \; 0011 \newline
|
||
& \rightarrow -3
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>To address this, computers introduced <u>one's complement</u>. Convert the operands to one's complement, calculate <span class="arithmatex">\(1 + (-2)\)</span>, and convert the result back to sign-magnitude to obtain the correct answer, <span class="arithmatex">\(-1\)</span>.</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& 1 + (-2) \newline
|
||
& \rightarrow 0000 \; 0001 \; \text{(sign-magnitude)} + 1000 \; 0010 \; \text{(sign-magnitude)} \newline
|
||
& = 0000 \; 0001 \; \text{(one's complement)} + 1111 \; 1101 \; \text{(one's complement)} \newline
|
||
& = 1111 \; 1110 \; \text{(one's complement)} \newline
|
||
& = 1000 \; 0001 \; \text{(sign-magnitude)} \newline
|
||
& \rightarrow -1
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>Second, <strong>sign-magnitude has two representations of zero, <span class="arithmatex">\(+0\)</span> and <span class="arithmatex">\(-0\)</span></strong>. Two binary encodings for zero can create ambiguity. Conditional checks may be wrong if they do not handle both forms, while special handling requires extra checks that can reduce efficiency.</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
+0 & \rightarrow 0000 \; 0000 \newline
|
||
-0 & \rightarrow 1000 \; 0000
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>One's complement also has positive and negative zero, so computers introduced <u>two's complement</u>. Consider converting negative zero through all three representations:</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
-0 \rightarrow \; & 1000 \; 0000 \; \text{(sign-magnitude)} \newline
|
||
= \; & 1111 \; 1111 \; \text{(one's complement)} \newline
|
||
= 1 \; & 0000 \; 0000 \; \text{(two's complement)} \newline
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>Adding <span class="arithmatex">\(1\)</span> to the one's complement of negative zero produces a carry. A <code>byte</code> has only 8 bits, so the <span class="arithmatex">\(1\)</span> carried into the ninth bit is discarded. <strong>Negative zero's two's complement is therefore <span class="arithmatex">\(0000 \; 0000\)</span>, the same as positive zero</strong>. Two's complement has only one zero, eliminating the ambiguity.</p>
|
||
<p>One question remains: if <code>byte</code> ranges over <span class="arithmatex">\([-128, 127]\)</span>, where does the extra negative value <span class="arithmatex">\(-128\)</span> come from? Every integer in <span class="arithmatex">\([-127, +127]\)</span> has sign-magnitude, one's complement, and two's complement representations, and its sign-magnitude and two's complement forms can be converted into each other.</p>
|
||
<p>However, <strong>the two's complement value <span class="arithmatex">\(1000 \; 0000\)</span> is an exception with no corresponding sign-magnitude form</strong>. Applying the conversion gives <span class="arithmatex">\(0000 \; 0000\)</span>, but that represents <span class="arithmatex">\(0\)</span>, whose two's complement should be itself. The special pattern <span class="arithmatex">\(1000 \; 0000\)</span> is defined to represent <span class="arithmatex">\(-128\)</span>. Indeed, two's complement addition of <span class="arithmatex">\((-1) + (-127)\)</span> gives <span class="arithmatex">\(-128\)</span>.</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
& (-127) + (-1) \newline
|
||
& \rightarrow 1111 \; 1111 \; \text{(sign-magnitude)} + 1000 \; 0001 \; \text{(sign-magnitude)} \newline
|
||
& = 1000 \; 0000 \; \text{(one's complement)} + 1111 \; 1110 \; \text{(one's complement)} \newline
|
||
& = 1000 \; 0001 \; \text{(two's complement)} + 1111 \; 1111 \; \text{(two's complement)} \newline
|
||
& = 1000 \; 0000 \; \text{(two's complement)} \newline
|
||
& \rightarrow -128
|
||
\end{aligned}
|
||
\]</div>
|
||
<p>All the calculations above use addition. This points to an important fact: <strong>computer arithmetic circuits are largely built around addition</strong>. Compared with operations such as multiplication, division, and subtraction, addition is simpler to implement in hardware, easier to parallelize, and faster.</p>
|
||
<p>This does not mean computers can only add. <strong>Combining addition with basic logical operations makes other arithmetic possible</strong>. For example, subtraction <span class="arithmatex">\(a - b\)</span> becomes addition <span class="arithmatex">\(a + (-b)\)</span>, while multiplication and division can be expressed as repeated additions or subtractions.</p>
|
||
<p>We can now summarize why computers use two's complement. It lets the same circuitry and operations handle both positive and negative addition, without dedicated subtraction circuitry or special handling for positive and negative zero. This simplifies hardware and improves efficiency.</p>
|
||
<p>Two's complement is an elegant design. Space limits our discussion here, but interested readers are encouraged to explore it further.</p>
|
||
<h2 id="332-floating-point-encoding">3.3.2 Floating-point encoding<a class="headerlink" href="#332-floating-point-encoding" title="Permanent link">¶</a></h2>
|
||
<p>Both <code>int</code> and <code>float</code> occupy 4 bytes, yet <code>float</code> has a much wider range. This seems surprising: since <code>float</code> also represents fractions, should its range not be smaller?</p>
|
||
<p><strong>The reason is that <code>float</code> uses a different representation</strong>. Write a 32-bit binary number as:</p>
|
||
<div class="arithmatex">\[
|
||
b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
|
||
\]</div>
|
||
<p>Under IEEE 754, a 32-bit <code>float</code> has three parts.</p>
|
||
<ul>
|
||
<li>Sign <span class="arithmatex">\(\mathrm{S}\)</span>: 1 bit, corresponding to <span class="arithmatex">\(b_{31}\)</span>.</li>
|
||
<li>Exponent <span class="arithmatex">\(\mathrm{E}\)</span>: 8 bits, corresponding to <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span>.</li>
|
||
<li>Fraction <span class="arithmatex">\(\mathrm{N}\)</span>: 23 bits, corresponding to <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span>.</li>
|
||
</ul>
|
||
<p>The value represented by the binary <code>float</code> is:</p>
|
||
<div class="arithmatex">\[
|
||
\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
|
||
\]</div>
|
||
<p>In decimal notation, this becomes:</p>
|
||
<div class="arithmatex">\[
|
||
\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
|
||
\]</div>
|
||
<p>The components have these ranges:</p>
|
||
<div class="arithmatex">\[
|
||
\begin{aligned}
|
||
\mathrm{S} \in & \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
|
||
(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
|
||
\end{aligned}
|
||
\]</div>
|
||
<p><img alt="Example of an IEEE 754 float calculation" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></p>
|
||
<p align="center"> Figure 3-5 Example of an IEEE 754 float calculation </p>
|
||
|
||
<p>In Figure 3-5, <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, and <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, giving:</p>
|
||
<div class="arithmatex">\[
|
||
\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
|
||
\]</div>
|
||
<p>We can now answer the original question: <strong>the exponent gives <code>float</code> a much wider range than <code>int</code></strong>. The largest positive <code>float</code> is <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>; changing the sign bit gives the most negative value.</p>
|
||
<p><strong>The wider range of <code>float</code> comes at the cost of precision</strong>. An <code>int</code> uses all 32 bits to represent evenly spaced integer values. Because <code>float</code> includes an exponent, the gap between adjacent representable values tends to grow as their magnitude increases.</p>
|
||
<p>As shown in Table 3-2, exponent values <span class="arithmatex">\(\mathrm{E} = 0\)</span> and <span class="arithmatex">\(\mathrm{E} = 255\)</span> have special meanings, <strong>representing zero, infinity, and <span class="arithmatex">\(\mathrm{NaN}\)</span>, among other values</strong>.</p>
|
||
<p align="center"> Table 3-2 Meanings of exponent values </p>
|
||
|
||
<div class="center-table">
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>Exponent E</th>
|
||
<th>Fraction <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
|
||
<th>Fraction <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
|
||
<th>Formula</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td><span class="arithmatex">\(0\)</span></td>
|
||
<td><span class="arithmatex">\(\pm 0\)</span></td>
|
||
<td>Subnormal number</td>
|
||
<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(1, 2, \dots, 254\)</span></td>
|
||
<td>Normal number</td>
|
||
<td>Normal number</td>
|
||
<td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(255\)</span></td>
|
||
<td><span class="arithmatex">\(\pm \infty\)</span></td>
|
||
<td><span class="arithmatex">\(\mathrm{NaN}\)</span></td>
|
||
<td></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
<p>Subnormal numbers substantially improve precision near zero. The smallest positive normal number is <span class="arithmatex">\(2^{-126}\)</span>, and the smallest positive subnormal number is <span class="arithmatex">\(2^{-126} \times 2^{-23}\)</span>.</p>
|
||
<p>Double-precision <code>double</code> uses a similar representation, which we will not detail here.</p>
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