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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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<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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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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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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<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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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1-key-review" class="md-nav__link">
<a href="#1-key-points" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key Review
1. &nbsp; Key Points
</span>
</a>
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<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#1-key-review" class="md-nav__link">
<a href="#1-key-points" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key Review
1. &nbsp; Key Points
</span>
</a>
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<!-- Page content -->
<h1 id="147-summary">14.7 &nbsp; Summary<a class="headerlink" href="#147-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>
<h3 id="1-key-points">1. &nbsp; Key Points<a class="headerlink" href="#1-key-points" title="Permanent link">&para;</a></h3>
<ul>
<li>Dynamic programming decomposes problems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving computational efficiency.</li>
<li>Without considering time constraints, all dynamic programming problems can be solved using backtracking (brute force search), but the recursion tree contains a large number of overlapping subproblems, resulting in extremely low efficiency. By introducing a memo list, we can store the solutions to all computed subproblems, ensuring that overlapping subproblems are only computed once.</li>
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<li>Subproblem decomposition is a general algorithmic approach, with different properties in divide and conquer, dynamic programming, and backtracking.</li>
<li>Dynamic programming problems have three major characteristics: overlapping subproblems, optimal substructure, and no aftereffects.</li>
<li>If the optimal solution to the original problem can be constructed from the optimal solutions to the subproblems, then it has optimal substructure.</li>
<li>No aftereffects means that for a given state, its future development is only related to that state and has nothing to do with all past states. Many combinatorial optimization problems do not have no aftereffects and cannot be quickly solved using dynamic programming.</li>
<li>No aftereffects means that for a given state, its future development is only related to that state and has nothing to do with all past states. Many combinatorial optimization problems do not satisfy this property and cannot be solved efficiently using dynamic programming.</li>
</ul>
<p><strong>Knapsack problem</strong></p>
<ul>
<li>The knapsack problem is one of the most typical dynamic programming problems, with variants such as the 0-1 knapsack, unbounded knapsack, and multiple knapsack.</li>
<li>The state definition for the 0-1 knapsack is the maximum value among the first <span class="arithmatex">\(i\)</span> items in a knapsack of capacity <span class="arithmatex">\(c\)</span>. Based on the two decisions of not putting an item in the knapsack and putting it in, the optimal substructure can be identified and the state transition equation constructed. In space optimization, since each state depends on the state directly above and to the upper-left, the list needs to be traversed in reverse order to avoid overwriting the upper-left state.</li>
<li>The state definition for the 0-1 knapsack is the maximum value achievable using the first <span class="arithmatex">\(i\)</span> items with a knapsack capacity of <span class="arithmatex">\(c\)</span>. Based on the two decisions of not putting an item in the knapsack and putting it in, the optimal substructure can be identified and the state transition equation constructed. In space optimization, since each state depends on the state directly above and to the upper-left, the list needs to be traversed in reverse order to avoid overwriting the upper-left state.</li>
<li>The unbounded knapsack problem has no limit on the selection quantity of each type of item, so the state transition for choosing to put in an item differs from the 0-1 knapsack problem. Since the state depends on the state directly above and directly to the left, space optimization should use forward traversal.</li>
<li>The coin change problem is a variant of the unbounded knapsack problem. It changes from seeking the "maximum" value to seeking the "minimum" number of coins, so <span class="arithmatex">\(\max()\)</span> in the state transition equation should be changed to <span class="arithmatex">\(\min()\)</span>. It changes from seeking "not exceeding" the knapsack capacity to seeking "exactly" making up the target amount, so <span class="arithmatex">\(amt + 1\)</span> is used to represent the invalid solution of "unable to make up the target amount".</li>
<li>Coin change problem II changes from seeking the "minimum number of coins" to seeking the "number of coin combinations", so the state transition equation correspondingly changes from <span class="arithmatex">\(\min()\)</span> to a summation operator.</li>