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krahets
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width,initial-scale=1">
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<meta name="description" content="Data Structures and Algorithms Crash Course with Animated Illustrations and Off-the-Shelf Code">
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<meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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<meta name="author" content="krahets">
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<span class="md-ellipsis">
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Chapter 1. Encounter With Algorithms
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Chapter 1. Encounter with Algorithms
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<span class="md-nav__icon md-icon"></span>
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Chapter 1. Encounter With Algorithms
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Chapter 1. Encounter with Algorithms
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</label>
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<span class="md-ellipsis">
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Chapter 4. Array and Linked List
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Chapter 4. Arrays and Linked Lists
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<span class="md-nav__icon md-icon"></span>
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Chapter 4. Array and Linked List
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Chapter 4. Arrays and Linked Lists
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</label>
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<span class="md-ellipsis">
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4.4 Memory and Cache *
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4.4 Random-Access Memory and Cache *
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<span class="md-ellipsis">
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Chapter 5. Stack and Queue
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Chapter 5. Stacks and Queues
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<span class="md-nav__icon md-icon"></span>
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Chapter 5. Stack and Queue
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Chapter 5. Stacks and Queues
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</label>
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<span class="md-ellipsis">
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5.3 Double-Ended Queue
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5.3 Deque
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<span class="md-ellipsis">
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Chapter 6. Hashing
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Chapter 6. Hash Table
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<span class="md-nav__icon md-icon"></span>
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Chapter 6. Hashing
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Chapter 6. Hash Table
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</label>
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<span class="md-ellipsis">
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7.3 Array Representation of Tree
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7.3 Array Representation of Binary Trees
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<span class="md-ellipsis">
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8.2 Building a Heap
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8.2 Heap Construction Operation
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<span class="md-ellipsis">
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8.3 Top-K Problem
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8.3 Top-k Problem
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<span class="md-ellipsis">
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10.2 Binary Search Insertion
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10.2 Binary Search Insertion Point
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<span class="md-ellipsis">
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10.3 Binary Search Edge Cases
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10.3 Binary Search Boundaries
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<span class="md-ellipsis">
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10.5 Search Algorithms Revisited
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10.5 Searching Algorithms Revisited
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<span class="md-ellipsis">
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11.1 Sorting Algorithms
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11.1 Sorting Algorithm
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<span class="md-ellipsis">
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12.4 Hanoi Tower Problem
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12.4 Hanota Problem
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<span class="md-ellipsis">
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16.3 Terminology Table
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16.3 Glossary
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<!-- Page content -->
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<h1 id="113-bubble-sort">11.3 Bubble Sort<a class="headerlink" href="#113-bubble-sort" title="Permanent link">¶</a></h1>
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<p><u>Bubble sort (bubble sort)</u> achieves sorting by continuously comparing and swapping adjacent elements. This process is like bubbles rising from the bottom to the top, hence the name bubble sort.</p>
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<p>As shown in Figure 11-4, the bubbling process can be simulated using element swap operations: starting from the leftmost end of the array and traversing to the right, compare the size of adjacent elements, and if "left element > right element", swap them. After completing the traversal, the largest element will be moved to the rightmost end of the array.</p>
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<p><u>Bubble sort</u> sorts an array by continuously comparing and swapping adjacent elements. This process resembles bubbles rising from the bottom to the top, hence the name bubble sort.</p>
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<p>As shown in Figure 11-4, the bubbling process can be simulated using element swaps: starting from the leftmost end of the array and traversing to the right, compare each pair of adjacent elements, and if "left element > right element", swap them. After the traversal is complete, the largest element is moved to the rightmost end of the array.</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="1:7"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1"><1></label><label for="__tabbed_1_2"><2></label><label for="__tabbed_1_3"><3></label><label for="__tabbed_1_4"><4></label><label for="__tabbed_1_5"><5></label><label for="__tabbed_1_6"><6></label><label for="__tabbed_1_7"><7></label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><img alt="Simulating bubble using element swap operation" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step1.png" /></p>
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<p><img alt="Simulating bubble sort using element swaps" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step1.png" /></p>
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</div>
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<div class="tabbed-block">
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<p><img alt="bubble_operation_step2" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step2.png" /></p>
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</div>
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</div>
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</div>
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<p align="center"> Figure 11-4 Simulating bubble using element swap operation </p>
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<p align="center"> Figure 11-4 Simulating bubble sort using element swaps </p>
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<h2 id="1131-algorithm-flow">11.3.1 Algorithm Flow<a class="headerlink" href="#1131-algorithm-flow" title="Permanent link">¶</a></h2>
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<p>Assume the array has length <span class="arithmatex">\(n\)</span>. The steps of bubble sort are shown in Figure 11-5.</p>
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</div>
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</div>
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<h2 id="1132-efficiency-optimization">11.3.2 Efficiency Optimization<a class="headerlink" href="#1132-efficiency-optimization" title="Permanent link">¶</a></h2>
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<p>We notice that if no swap operations are performed during a certain round of "bubbling", it means the array has already completed sorting and can directly return the result. Therefore, we can add a flag <code>flag</code> to monitor this situation and return immediately once it occurs.</p>
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<p>After optimization, the worst-case time complexity and average time complexity of bubble sort remain <span class="arithmatex">\(O(n^2)\)</span>; but when the input array is completely ordered, the best-case time complexity can reach <span class="arithmatex">\(O(n)\)</span>.</p>
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<p>We can observe that if no swaps occur during a round of "bubbling", the array is already sorted and the algorithm can return immediately. Therefore, we can add a flag <code>flag</code> to detect this situation and terminate as soon as it occurs.</p>
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<p>After this optimization, the worst-case and average-case time complexities of bubble sort remain <span class="arithmatex">\(O(n^2)\)</span>; however, when the input array is already sorted, the best-case time complexity becomes <span class="arithmatex">\(O(n)\)</span>.</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
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</div>
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<h2 id="1133-algorithm-characteristics">11.3.3 Algorithm Characteristics<a class="headerlink" href="#1133-algorithm-characteristics" title="Permanent link">¶</a></h2>
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<ul>
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<li><strong>Time complexity of <span class="arithmatex">\(O(n^2)\)</span>, adaptive sorting</strong>: The array lengths traversed in each round of "bubbling" are <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, totaling <span class="arithmatex">\((n - 1) n / 2\)</span>. After introducing the <code>flag</code> optimization, the best-case time complexity can reach <span class="arithmatex">\(O(n)\)</span>.</li>
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<li><strong>Time complexity is <span class="arithmatex">\(O(n^2)\)</span>; adaptive</strong>: In successive rounds of "bubbling", the traversed portion of the array has lengths <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, for a total of <span class="arithmatex">\((n - 1) n / 2\)</span>. After introducing the <code>flag</code> optimization, the best-case time complexity can reach <span class="arithmatex">\(O(n)\)</span>.</li>
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<li><strong>Space complexity of <span class="arithmatex">\(O(1)\)</span>, in-place sorting</strong>: Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use a constant amount of extra space.</li>
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<li><strong>Stable sorting</strong>: Since equal elements are not swapped during "bubbling".</li>
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<li><strong>Stable sorting</strong>: Equal elements are not swapped during "bubbling".</li>
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</ul>
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<!-- Source file information -->
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