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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
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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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">
10.5 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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<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>
@@ -4356,14 +4356,14 @@
<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>
en/chapter_divide_and_conquer/build_binary_tree_problem/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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Chapter 4. Array and Linked List
Chapter 4. Arrays and Linked Lists
</label>
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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
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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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">
10.5 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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<p align="center"> Figure 12-6 &nbsp; Dividing subtrees in preorder and inorder traversals </p>
<h3 id="3-describing-subtree-intervals-based-on-variables">3. &nbsp; Describing Subtree Intervals Based on Variables<a class="headerlink" href="#3-describing-subtree-intervals-based-on-variables" title="Permanent link">&para;</a></h3>
<p>Based on the above division method, <strong>we have obtained the index intervals of the root node, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code></strong>. To describe these index intervals, we need to use several pointer variables.</p>
<p>Based on the above division method, <strong>we have obtained the index intervals of the root node, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code></strong>. To describe these index intervals, we need to use several index variables.</p>
<ul>
<li>Denote the index of the current tree's root node in <code>preorder</code> as <span class="arithmatex">\(i\)</span>.</li>
<li>Denote the index of the current tree's root node in <code>inorder</code> as <span class="arithmatex">\(m\)</span>.</li>
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<a
href="../hanota_problem/"
class="md-footer__link md-footer__link--next"
aria-label="Next: 12.4 Hanoi Tower Problem"
aria-label="Next: 12.4 Hanota Problem"
rel="next"
>
<div class="md-footer__title">
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</span>
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12.4 Hanoi Tower Problem
12.4 Hanota Problem
</div>
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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">
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<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
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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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">
10.5 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="121-divide-and-conquer-algorithms">12.1 &nbsp; Divide and Conquer Algorithms<a class="headerlink" href="#121-divide-and-conquer-algorithms" title="Permanent link">&para;</a></h1>
<p><u>Divide and conquer</u> is a very important and common algorithm strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: "divide" and "conquer".</p>
<p><u>Divide and conquer</u> is a very important and common algorithmic strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: "divide" and "conquer".</p>
<ol>
<li><strong>Divide (partition phase)</strong>: Recursively divide the original problem into two or more subproblems until the smallest subproblem is reached.</li>
<li><strong>Conquer (merge phase)</strong>: Starting from the smallest subproblems with known solutions, merge the solutions of subproblems from bottom to top to construct the solution to the original problem.</li>
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<li><strong>Solutions of subproblems can be merged</strong>: Two sorted subarrays (solutions of subproblems) can be merged into one sorted array (solution of the original problem).</li>
</ol>
<h2 id="1212-improving-efficiency-through-divide-and-conquer">12.1.2 &nbsp; Improving Efficiency Through Divide and Conquer<a class="headerlink" href="#1212-improving-efficiency-through-divide-and-conquer" title="Permanent link">&para;</a></h2>
<p><strong>Divide and conquer can not only effectively solve algorithmic problems but often also improve algorithm efficiency</strong>. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.</p>
<p><strong>Divide and conquer can not only effectively solve algorithmic problems, but can often also improve algorithmic efficiency</strong>. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.</p>
<p>This raises the question: <strong>Why can divide and conquer improve algorithm efficiency, and what is the underlying logic</strong>? In other words, why is dividing a large problem into multiple subproblems, solving the subproblems, and merging their solutions more efficient than directly solving the original problem? This question can be discussed from two aspects: operation count and parallel computation.</p>
<h3 id="1-operation-count-optimization">1. &nbsp; Operation Count Optimization<a class="headerlink" href="#1-operation-count-optimization" title="Permanent link">&para;</a></h3>
<p>Taking "bubble sort" as an example, processing an array of length <span class="arithmatex">\(n\)</span> requires <span class="arithmatex">\(O(n^2)\)</span> time. Suppose we divide the array into two subarrays from the midpoint as shown in Figure 12-2, the division requires <span class="arithmatex">\(O(n)\)</span> time, sorting each subarray requires <span class="arithmatex">\(O((n / 2)^2)\)</span> time, and merging the two subarrays requires <span class="arithmatex">\(O(n)\)</span> time, resulting in an overall time complexity of:</p>
<p>Taking "bubble sort" as an example, processing an array of length <span class="arithmatex">\(n\)</span> requires <span class="arithmatex">\(O(n^2)\)</span> time. Suppose we divide the array at the midpoint into two subarrays, as shown in Figure 12-2. The division requires <span class="arithmatex">\(O(n)\)</span> time, sorting each subarray requires <span class="arithmatex">\(O((n / 2)^2)\)</span> time, and merging the two subarrays requires <span class="arithmatex">\(O(n)\)</span> time, resulting in an overall time complexity of:</p>
<div class="arithmatex">\[
O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
\]</div>
@@ -4484,14 +4484,14 @@ n(n - 4) &amp; &gt; 0
<p>Going further, <strong>what if we continuously divide the subarrays from their midpoints into two subarrays</strong> until the subarrays have only one element? This approach is actually "merge sort", with a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>.</p>
<p>Thinking further, <strong>what if we set multiple division points</strong> and evenly divide the original array into <span class="arithmatex">\(k\)</span> subarrays? This situation is very similar to "bucket sort", which is well-suited for sorting massive amounts of data, with a theoretical time complexity of <span class="arithmatex">\(O(n + k)\)</span>.</p>
<h3 id="2-parallel-computation-optimization">2. &nbsp; Parallel Computation Optimization<a class="headerlink" href="#2-parallel-computation-optimization" title="Permanent link">&para;</a></h3>
<p>We know that the subproblems generated by divide and conquer are independent of each other, <strong>so they can typically be solved in parallel</strong>. This means divide and conquer can not only reduce the time complexity of algorithms, <strong>but also benefits from parallel optimization by operating systems</strong>.</p>
<p>We know that the subproblems generated by divide and conquer are independent of each other, <strong>so they can typically be solved in parallel</strong>. This means divide and conquer can not only reduce the time complexity of algorithms, <strong>but is also amenable to parallel optimization by the operating system</strong>.</p>
<p>Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can simultaneously handle multiple subproblems, making fuller use of computing resources and significantly reducing overall runtime.</p>
<p>For example, in the "bucket sort" shown in Figure 12-3, we evenly distribute massive data into various buckets, and the sorting tasks for all buckets can be distributed to various computing units. After completion, the results are merged.</p>
<p><img alt="Parallel computation in bucket sort" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></p>
<p align="center"> Figure 12-3 &nbsp; Parallel computation in bucket sort </p>
<h2 id="1213-common-applications-of-divide-and-conquer">12.1.3 &nbsp; Common Applications of Divide and Conquer<a class="headerlink" href="#1213-common-applications-of-divide-and-conquer" title="Permanent link">&para;</a></h2>
<p>On one hand, divide and conquer can be used to solve many classic algorithmic problems.</p>
<p>On the one hand, divide and conquer can be used to solve many classic algorithmic problems.</p>
<ul>
<li><strong>Finding the closest pair of points</strong>: This algorithm first divides the point set into two parts, then finds the closest pair of points in each part separately, and finally finds the closest pair of points that spans both parts.</li>
<li><strong>Large integer multiplication</strong>: For example, the Karatsuba algorithm, which decomposes large integer multiplication into several smaller integer multiplications and additions.</li>
@@ -4501,15 +4501,15 @@ n(n - 4) &amp; &gt; 0
</ul>
<p>On the other hand, divide and conquer is widely applied in the design of algorithms and data structures.</p>
<ul>
<li><strong>Binary search</strong>: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary operation on the remaining interval.</li>
<li><strong>Binary search</strong>: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary-search step on the remaining interval.</li>
<li><strong>Merge sort</strong>: Already introduced at the beginning of this section, no further elaboration needed.</li>
<li><strong>Quick sort</strong>: Quick sort selects a pivot value, then divides the array into two subarrays, one with elements smaller than the pivot and the other with elements larger than the pivot, then performs the same division operation on these two parts until the subarrays have only one element.</li>
<li><strong>Bucket sort</strong>: The basic idea of bucket sort is to scatter data into multiple buckets, then sort the elements within each bucket, and finally extract the elements from each bucket in sequence to obtain a sorted array.</li>
<li><strong>Trees</strong>: For example, binary search trees, AVL trees, red-black trees, B-trees, B+ trees, etc. Their search, insertion, and deletion operations can all be viewed as applications of the divide and conquer strategy.</li>
<li><strong>Heaps</strong>: A heap is a special complete binary tree, and its various operations, such as insertion, deletion, and heapify, actually imply the divide and conquer idea.</li>
<li><strong>Hash tables</strong>: Although hash tables do not directly apply divide and conquer, some hash collision resolution solutions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve query efficiency.</li>
<li><strong>Hash tables</strong>: Although hash tables do not directly apply divide and conquer, some methods for resolving hash collisions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve lookup efficiency.</li>
</ul>
<p>It can be seen that <strong>divide and conquer is a "subtly pervasive" algorithmic idea</strong>, embedded in various algorithms and data structures.</p>
<p>It can be seen that <strong>divide and conquer is a "quietly pervasive" algorithmic idea</strong>, embedded in various algorithms and data structures.</p>
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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">
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<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
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 5. Stack and Queue
Chapter 5. Stacks and Queues
</label>
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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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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hashing
Chapter 6. Hash Table
</label>
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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">
10.5 Search Algorithms Revisited
10.5 Searching Algorithms Revisited
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11.1 Sorting Algorithms
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<div class="admonition quote">
<p class="admonition-title">Quote</p>
<p>The hanota problem originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and <span class="arithmatex">\(64\)</span> golden discs of different sizes. The monks continuously moved the discs, believing that when the last disc was correctly placed, the world would come to an end.</p>
<p>However, even if the monks moved one disc per second, it would take approximately <span class="arithmatex">\(2^{64} \approx 1.84×10^{19}\)</span> seconds, which is about <span class="arithmatex">\(5850\)</span> billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.</p>
<p>However, even if the monks moved one disc per second, it would take approximately <span class="arithmatex">\(2^{64} \approx 1.84×10^{19}\)</span> seconds, which is about <span class="arithmatex">\(585\)</span> billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.</p>
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<meta name="author" content="krahets">
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Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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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 *
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Chapter 5. Stacks and Queues
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Chapter 6. Hash Table
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Chapter 6. Hashing
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<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
<p>Difficult problems are decomposed layer by layer, with each decomposition making them simpler.</p>
<p>Divide and conquer reveals an important truth: start with simplicity, and nothing remains complex.</p>
<p>Divide and conquer reveals an important truth: start with what is simple, and nothing remains complex.</p>
</div>
<h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
<ul>
<li><a href="divide_and_conquer/">12.1 &nbsp; Divide and Conquer Algorithms</a></li>
<li><a href="binary_search_recur/">12.2 &nbsp; Divide and Conquer Search Strategy</a></li>
<li><a href="build_binary_tree_problem/">12.3 &nbsp; Building a Binary Tree Problem</a></li>
<li><a href="hanota_problem/">12.4 &nbsp; Hanoi Tower Problem</a></li>
<li><a href="hanota_problem/">12.4 &nbsp; Hanota Problem</a></li>
<li><a href="summary/">12.5 &nbsp; Summary</a></li>
</ul>
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<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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Chapter 1. Encounter With Algorithms
Chapter 1. Encounter with Algorithms
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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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5.3 Deque
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Chapter 6. Hashing
Chapter 6. Hash Table
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Chapter 6. Hashing
Chapter 6. Hash Table
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8.2 Heap Construction Operation
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8.3 Top-k Problem
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10.2 Binary Search Insertion Point
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11.1 Sorting Algorithms
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16.3 Glossary
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<h1 id="125-summary">12.5 &nbsp; Summary<a class="headerlink" href="#125-summary" title="Permanent link">&para;</a></h1>
<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Divide and conquer is a common algorithm design strategy, consisting of two phases: divide (partition) and conquer (merge), typically implemented based on recursion.</li>
<li>Divide and conquer is a common algorithm design strategy consisting of two phases, divide (partition) and conquer (merge), and is typically implemented recursively.</li>
<li>The criteria for determining whether a problem is a divide and conquer problem include: whether the problem can be decomposed, whether subproblems are independent, and whether subproblems can be merged.</li>
<li>Merge sort is a typical application of the divide and conquer strategy. It recursively divides an array into two equal-length subarrays until only one element remains, then merges them layer by layer to complete the sorting.</li>
<li>Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, the divide and conquer strategy reduces the number of operations; on the other hand, it facilitates parallel optimization of the system after division.</li>
<li>Divide and conquer can both solve many algorithmic problems and is widely applied in data structure and algorithm design, appearing everywhere.</li>
<li>Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, it reduces the number of operations; on the other hand, it makes parallel optimization by the system easier.</li>
<li>Divide and conquer can solve many algorithmic problems and is also widely used in data structures and algorithm design, making it ubiquitous.</li>
<li>Compared to brute-force search, adaptive search is more efficient. Search algorithms with time complexity of <span class="arithmatex">\(O(\log n)\)</span> are typically implemented based on the divide and conquer strategy.</li>
<li>Binary search is another typical application of divide and conquer. It does not include the step of merging solutions of subproblems. We can implement binary search through recursive divide and conquer.</li>
<li>In the problem of building a binary tree, building the tree (original problem) can be divided into building the left subtree and right subtree (subproblems), which can be achieved by dividing the index intervals of the preorder and inorder traversals.</li>
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