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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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<title>10.3 Binary Search Edge Cases - Hello Algo</title>
<title>10.3 Binary Search Boundaries - Hello Algo</title>
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<div data-md-component="skip">
<a href="#103-binary-search-edge-cases" class="md-skip">
<a href="#103-binary-search-boundaries" class="md-skip">
Skip to content
</a>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
10.3 &nbsp; Binary Search Edge Cases
10.3 &nbsp; Binary Search Boundaries
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</div>
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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
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<span class="md-ellipsis">
Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
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4.4 Memory and Cache *
4.4 Random-Access Memory and Cache *
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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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<span class="md-ellipsis">
Chapter 6. Hashing
Chapter 6. Hash Table
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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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<span class="md-ellipsis">
10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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<span class="md-ellipsis">
10.3 Binary Search Edge Cases
10.3 Binary Search Boundaries
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<span class="md-ellipsis">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting Algorithms
11.1 Sorting Algorithm
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<span class="md-ellipsis">
12.4 Hanoi Tower Problem
12.4 Hanota Problem
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16.3 Terminology Table
16.3 Glossary
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<!-- Page content -->
<h1 id="103-binary-search-edge-cases">10.3 &nbsp; Binary Search Edge Cases<a class="headerlink" href="#103-binary-search-edge-cases" title="Permanent link">&para;</a></h1>
<h1 id="103-binary-search-boundaries">10.3 &nbsp; Binary Search Boundaries<a class="headerlink" href="#103-binary-search-boundaries" title="Permanent link">&para;</a></h1>
<h2 id="1031-finding-the-left-boundary">10.3.1 &nbsp; Finding the Left Boundary<a class="headerlink" href="#1031-finding-the-left-boundary" title="Permanent link">&para;</a></h2>
<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> that may contain duplicate elements, return the index of the leftmost element <code>target</code> in the array. If the array does not contain the element, return <span class="arithmatex">\(-1\)</span>.</p>
<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> that may contain duplicate elements, return the index of the leftmost occurrence of <code>target</code>. If the array does not contain <code>target</code>, return <span class="arithmatex">\(-1\)</span>.</p>
</div>
<p>Recall the method for finding the insertion point with binary search. After the search completes, <span class="arithmatex">\(i\)</span> points to the leftmost <code>target</code>, <strong>so finding the insertion point is essentially finding the index of the leftmost <code>target</code></strong>.</p>
<p>Consider implementing the left boundary search using the insertion point finding function. Note that the array may not contain <code>target</code>, which could result in the following two cases:</p>
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<p>So how do we find the rightmost <code>target</code>? The most direct approach is to modify the code and replace the pointer shrinking operation in the <code>nums[m] == target</code> case. The code is omitted here; interested readers can implement it themselves.</p>
<p>Below we introduce two more clever methods.</p>
<h3 id="1-reusing-left-boundary-search">1. &nbsp; Reusing Left Boundary Search<a class="headerlink" href="#1-reusing-left-boundary-search" title="Permanent link">&para;</a></h3>
<p>In fact, we can use the function for finding the leftmost element to find the rightmost element. The specific method is: <strong>Convert finding the rightmost <code>target</code> into finding the leftmost <code>target + 1</code></strong>.</p>
<p>As shown in Figure 10-7, after the search completes, pointer <span class="arithmatex">\(i\)</span> points to the leftmost <code>target + 1</code> (if it exists), while <span class="arithmatex">\(j\)</span> points to the rightmost <code>target</code>, <strong>so we can simply return <span class="arithmatex">\(j\)</span></strong>.</p>
<p>In fact, we can use the function for finding the leftmost <code>target</code> to find the rightmost <code>target</code>. The specific method is: <strong>convert finding the rightmost <code>target</code> into finding the leftmost <code>target + 1</code></strong>.</p>
<p>As shown in Figure 10-7, after the search completes, the pointer <span class="arithmatex">\(i\)</span> points to the leftmost <code>target + 1</code> (if it exists), while <span class="arithmatex">\(j\)</span> points to the rightmost <code>target</code>, <strong>so we can return <span class="arithmatex">\(j\)</span></strong>.</p>
<p><img alt="Converting right boundary search to left boundary search" class="animation-figure" src="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" /></p>
<p align="center"> Figure 10-7 &nbsp; Converting right boundary search to left boundary search </p>
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<p>We know that when the array does not contain <code>target</code>, <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> will eventually point to the first elements greater than and less than <code>target</code>, respectively.</p>
<p>Therefore, as shown in Figure 10-8, we can construct an element that does not exist in the array to find the left and right boundaries.</p>
<ul>
<li>Finding the leftmost <code>target</code>: Can be converted to finding <code>target - 0.5</code> and returning pointer <span class="arithmatex">\(i\)</span>.</li>
<li>Finding the rightmost <code>target</code>: Can be converted to finding <code>target + 0.5</code> and returning pointer <span class="arithmatex">\(j\)</span>.</li>
<li>Finding the leftmost <code>target</code>: This can be converted to finding <code>target - 0.5</code> and returning the pointer <span class="arithmatex">\(i\)</span>.</li>
<li>Finding the rightmost <code>target</code>: This can be converted to finding <code>target + 0.5</code> and returning the pointer <span class="arithmatex">\(j\)</span>.</li>
</ul>
<p><img alt="Converting boundary search to element search" class="animation-figure" src="../binary_search_edge.assets/binary_search_edge_by_element.png" /></p>
<p align="center"> Figure 10-8 &nbsp; Converting boundary search to element search </p>
<p>The code is omitted here, but the following two points are worth noting:</p>
<ul>
<li>Since the given array does not contain decimals, we don't need to worry about how to handle equal cases.</li>
<li>Since the given array does not contain decimal values, we do not need to worry about how to handle equality.</li>
<li>Because this method introduces decimals, the variable <code>target</code> in the function needs to be changed to a floating-point type (Python does not require this change).</li>
</ul>
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<a
href="../binary_search_insertion/"
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aria-label="Previous: 10.2 Binary Search Insertion"
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10.2 Binary Search Insertion
10.2 Binary Search Insertion Point
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