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@@ -35,7 +35,7 @@
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.1">
<meta name="generator" content="zensical-0.0.29">
@@ -43,10 +43,15 @@
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
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@@ -60,29 +65,20 @@
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@@ -97,7 +93,7 @@
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<div data-md-component="skip">
@@ -139,12 +135,12 @@
<nav class="md-header__inner md-grid" aria-label="Header">
<a href="../.." title="Hello Algo" class="md-header__button md-logo" aria-label="Hello Algo" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
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@@ -156,7 +152,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
15.1 Greedy Algorithm
15.1 &nbsp; Greedy Algorithm
</span>
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@@ -169,7 +165,7 @@
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<button class="md-header__button md-icon" aria-label="Select language">
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@@ -240,64 +236,30 @@
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@@ -325,7 +287,7 @@
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@@ -335,7 +297,7 @@
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<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>, with each type of coin available for repeated selection, what is the minimum number of coins needed to make up the target amount? If it is impossible to make up the target amount, return <span class="arithmatex">\(-1\)</span>.</p>
</div>
<p>The greedy strategy adopted for this problem is shown in Figure 15-1. Given a target amount, <strong>we greedily select the coin that is not greater than and closest to it</strong>, and continuously repeat this step until the target amount is reached.</p>
<p><a class="glightbox" href="../greedy_algorithm.assets/coin_change_greedy_strategy.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Greedy strategy for coin change" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_strategy.png" /></a></p>
<p><img alt="Greedy strategy for coin change" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_strategy.png" /></p>
<p align="center"> Figure 15-1 &nbsp; Greedy strategy for coin change </p>
<p>The implementation code is as follows:</p>
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<li><strong>Negative example <span class="arithmatex">\(coins = [1, 20, 50]\)</span></strong>: Suppose <span class="arithmatex">\(amt = 60\)</span>, the greedy algorithm can only find the combination <span class="arithmatex">\(50 + 1 \times 10\)</span>, totaling <span class="arithmatex">\(11\)</span> coins, but dynamic programming can find the optimal solution <span class="arithmatex">\(20 + 20 + 20\)</span>, requiring only <span class="arithmatex">\(3\)</span> coins.</li>
<li><strong>Negative example <span class="arithmatex">\(coins = [1, 49, 50]\)</span></strong>: Suppose <span class="arithmatex">\(amt = 98\)</span>, the greedy algorithm can only find the combination <span class="arithmatex">\(50 + 1 \times 48\)</span>, totaling <span class="arithmatex">\(49\)</span> coins, but dynamic programming can find the optimal solution <span class="arithmatex">\(49 + 49\)</span>, requiring only <span class="arithmatex">\(2\)</span> coins.</li>
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<p><a class="glightbox" href="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Examples where greedy algorithms cannot find the optimal solution" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" /></a></p>
<p><img alt="Examples where greedy algorithms cannot find the optimal solution" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" /></p>
<p align="center"> Figure 15-2 &nbsp; Examples where greedy algorithms cannot find the optimal solution </p>
<p>In other words, for the coin change problem, greedy algorithms cannot guarantee finding the global optimal solution, and may even find very poor solutions. It is better suited for solving with dynamic programming.</p>
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@@ -5020,7 +4944,7 @@ aria-label="Footer"
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