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en/chapter_divide_and_conquer/binary_search_recur/index.html
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<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
<meta name="author" content="krahets">
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<span class="md-ellipsis">
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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<span class="md-nav__icon md-icon"></span>
Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
</label>
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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<span class="md-ellipsis">
4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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<span class="md-ellipsis">
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
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<span class="md-ellipsis">
5.3 Double-Ended Queue
5.3 Deque
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Chapter 6. Hashing
Chapter 6. Hash Table
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
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<span class="md-ellipsis">
7.3 Array Representation of Tree
7.3 Array Representation of Binary Trees
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<span class="md-ellipsis">
8.2 Building a Heap
8.2 Heap Construction Operation
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<span class="md-ellipsis">
8.3 Top-K Problem
8.3 Top-k Problem
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10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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11.1 Sorting Algorithms
11.1 Sorting Algorithm
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12.4 Hanoi Tower Problem
12.4 Hanota Problem
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<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
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<p>We have already learned that search algorithms are divided into two major categories.</p>
<ul>
<li><strong>Brute-force search</strong>: Implemented by traversing the data structure, with a time complexity of <span class="arithmatex">\(O(n)\)</span>.</li>
<li><strong>Adaptive search</strong>: Utilizes unique data organization forms or prior information, with time complexity reaching <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>.</li>
<li><strong>Adaptive search</strong>: Leverages specific data organization or prior information, with time complexity reaching <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>.</li>
</ul>
<p>In fact, <strong>search algorithms with time complexity of <span class="arithmatex">\(O(\log n)\)</span> are typically implemented based on the divide and conquer strategy</strong>, such as binary search and trees.</p>
<ul>
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<p>In previous sections, binary search was implemented based on iteration. Now we implement it based on divide and conquer (recursion).</p>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, where all elements are unique, find the element <code>target</code>.</p>
<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, where all elements are unique, find <code>target</code>.</p>
</div>
<p>From a divide and conquer perspective, we denote the subproblem corresponding to the search interval <span class="arithmatex">\([i, j]\)</span> as <span class="arithmatex">\(f(i, j)\)</span>.</p>
<p>Starting from the original problem <span class="arithmatex">\(f(0, n-1)\)</span>, perform binary search through the following steps.</p>
<ol>
<li>Calculate the midpoint <span class="arithmatex">\(m\)</span> of the search interval <span class="arithmatex">\([i, j]\)</span>, and use it to eliminate half of the search interval.</li>
<li>Recursively solve the subproblem reduced by half in size, which could be <span class="arithmatex">\(f(i, m-1)\)</span> or <span class="arithmatex">\(f(m+1, j)\)</span>.</li>
<li>Repeat steps <code>1.</code> and <code>2.</code> until <code>target</code> is found or the interval is empty and return.</li>
<li>Repeat steps <code>1.</code> and <code>2.</code> until <code>target</code> is found, or return when the interval is empty.</li>
</ol>
<p>Figure 12-4 shows the divide and conquer process of binary search for element <span class="arithmatex">\(6\)</span> in an array.</p>
<p><img alt="Divide and conquer process of binary search" class="animation-figure" src="../binary_search_recur.assets/binary_search_recur.png" /></p>