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en/chapter_dynamic_programming/unbounded_knapsack_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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<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
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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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<a href="#1453-coin-change-problem-ii" class="md-nav__link">
<span class="md-ellipsis">
14.5.3 &nbsp; Coin Change Problem Ii
14.5.3 &nbsp; Coin Change Problem II
</span>
</a>
<nav class="md-nav" aria-label="14.5.3 Coin Change Problem Ii">
<nav class="md-nav" aria-label="14.5.3 Coin Change Problem II">
<ul class="md-nav__list">
<li class="md-nav__item">
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<span class="md-ellipsis">
16.3 Terminology Table
16.3 Glossary
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<a href="#1453-coin-change-problem-ii" class="md-nav__link">
<span class="md-ellipsis">
14.5.3 &nbsp; Coin Change Problem Ii
14.5.3 &nbsp; Coin Change Problem II
</span>
</a>
<nav class="md-nav" aria-label="14.5.3 Coin Change Problem Ii">
<nav class="md-nav" aria-label="14.5.3 Coin Change Problem II">
<ul class="md-nav__list">
<li class="md-nav__item">
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<p>This problem differs from the unbounded knapsack problem in the following two aspects regarding the state transition equation.</p>
<ul>
<li>This problem seeks the minimum value, so the operator <span class="arithmatex">\(\max()\)</span> needs to be changed to <span class="arithmatex">\(\min()\)</span>.</li>
<li>The optimization target is the number of coins rather than item value, so when a coin is selected, simply execute <span class="arithmatex">\(+1\)</span>.</li>
<li>The optimization target is the number of coins rather than item value, so when a coin is selected, simply add <span class="arithmatex">\(1\)</span>.</li>
</ul>
<div class="arithmatex">\[
dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
@@ -5309,7 +5309,7 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
<p>When the target amount is <span class="arithmatex">\(0\)</span>, the minimum number of coins needed to make it up is <span class="arithmatex">\(0\)</span>, so all <span class="arithmatex">\(dp[i, 0]\)</span> in the first column equal <span class="arithmatex">\(0\)</span>.</p>
<p>When there are no coins, <strong>it is impossible to make up any amount <span class="arithmatex">\(&gt; 0\)</span></strong>, which is an invalid solution. To enable the <span class="arithmatex">\(\min()\)</span> function in the state transition equation to identify and filter out invalid solutions, we consider using <span class="arithmatex">\(+ \infty\)</span> to represent them, i.e., set all <span class="arithmatex">\(dp[0, a]\)</span> in the first row to <span class="arithmatex">\(+ \infty\)</span>.</p>
<h3 id="2-code-implementation_1">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
<p>Most programming languages do not provide a <span class="arithmatex">\(+ \infty\)</span> variable, and can only use the maximum value of integer type <code>int</code> as a substitute. However, this can lead to large number overflow: the <span class="arithmatex">\(+ 1\)</span> operation in the state transition equation may cause overflow.</p>
<p>Most programming languages do not provide a <span class="arithmatex">\(+ \infty\)</span> variable, and can only use the maximum value of integer type <code>int</code> as a substitute. However, this can lead to integer overflow: the <span class="arithmatex">\(+ 1\)</span> operation in the state transition equation may cause overflow.</p>
<p>For this reason, we use the number <span class="arithmatex">\(amt + 1\)</span> to represent invalid solutions, because the maximum number of coins needed to make up <span class="arithmatex">\(amt\)</span> is at most <span class="arithmatex">\(amt\)</span>. Before returning, check whether <span class="arithmatex">\(dp[n, amt]\)</span> equals <span class="arithmatex">\(amt + 1\)</span>; if so, return <span class="arithmatex">\(-1\)</span>, indicating that the target amount cannot be made up. The code is as follows:</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
<div class="tabbed-content">
@@ -6071,7 +6071,7 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
</div>
</div>
</div>
<h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin Change Problem Ii<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
<h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin Change Problem II<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i\)</span>-th type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. Each type of coin can be selected multiple times. <strong>What is the number of coin combinations that can make up the target amount?</strong> An example is shown in Figure 14-26.</p>