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<p class="admonition-title">Question</p>
<p>给定 <span class="arithmatex">\(n\)</span> 种硬币，第 <span class="arithmatex">\(i\)</span> 个硬币的面值为 <span class="arithmatex">\(coins[i - 1]\)</span> ，目标金额为 <span class="arithmatex">\(amt\)</span> ，每种硬币可以重复选取，问能够凑出目标金额的最少硬币个数。如果无法凑出目标金额则返回 <span class="arithmatex">\(-1\)</span> 。</p>
</div>
<p>贪心算法会迭代地做出一个又一个的贪心选择，每轮都将问题转化成一个规模更小的子问题，直到问题被解决。</p>
<p>这道题的贪心策略在生活中很常见：给定目标金额，<strong>我们贪心地选择不大于且最接近它的硬币</strong>，不断循环该步骤，直至凑出目标金额为止。</p>
<p><img alt="零钱兑换的贪心策略" src="../greedy_algorithm.assets/coin_change_greedy_strategy.png" /></p>
<p align="center"> Fig. 零钱兑换的贪心策略 </p>
@ -3512,7 +3541,7 @@
</ol>
<h2 id="1512">15.1.2. &nbsp; 贪心算法特性<a class="headerlink" href="#1512" title="Permanent link">&para;</a></h2>
<p>那么问题来了，什么样的问题适合用贪心算法求解呢？或者说，贪心算法在什么情况下可以保证找到最优解？</p>
<p>相较于动态规划，贪心算法的使用条件更加苛刻，其主要关心问题的两个性质：</p>
<p>相较于动态规划，贪心算法的使用条件更加苛刻，其主要关注问题的两个性质：</p>
<ul>
<li><strong>贪心选择性质</strong>：只有当局部最优选择始终可以导致全局最优解时，贪心算法才能保证得到最优解。</li>
<li><strong>最优子结构</strong>：原问题的最优解包含子问题的最优解。值得注意的是，一些问题的最优子结构并不明显，但仍然可使用贪心算法解决。</li>

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<h1>Summary</h1>
<h1 id="155">15.5. &nbsp; 小结<a class="headerlink" href="#155" title="Permanent link">&para;</a></h1>
<ul>
<li>贪心算法通常用于解决最优化问题，其原理是在每个决策阶段都做出局部最优的决策，以期望获得全局最优解。</li>
<li>贪心算法会迭代地做出一个又一个的贪心选择，每轮都将问题转化成一个规模更小的子问题，直到问题被解决。</li>
<li>贪心算法不仅实现简单，还具有很高的解题效率。相比于动态规划，贪心算法的时间复杂度通常低一个数量级。</li>
<li>在零钱兑换问题中，对于某些硬币组合，贪心算法可以保证找到最优解；对于另外一些硬币组合则不然，贪心算法可能找到很差的解。</li>
<li>贪心问题具有两大性质：贪心选择性质和最优子结构。贪心选择性质代表贪心策略的有效性。</li>
<li>对于某些复杂问题，贪心选择性质的证明并不简单。相对来说，证伪更加容易，例如零钱兑换问题。</li>
<li>求解贪心问题主要分为三步：问题分析、贪心策略确定、正确性证明。其中，贪心策略确定是核心步骤，而正确性证明是难点。</li>
<li>分数背包问题在 0-1 背包的基础上，允许选择物品的一部分，因此可使用贪心算法求解。可以采用反证法证明贪心策略的正确性。</li>
<li>最大容量问题可使用穷举法求解，时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> 。通过设计贪心策略，每轮向内移动短板，可将时间复杂度优化至 <span class="arithmatex">\(O(n)\)</span> 。</li>
<li>在最大切分乘积问题中，我们先后推理出两个贪心策略：<span class="arithmatex">\(\geq 4\)</span> 的整数都应该继续切分、最优切分因子为 <span class="arithmatex">\(3\)</span> ，从而得到贪心解法。代码中包含幂运算，时间复杂度取决于幂运算实现方法，通常为 <span class="arithmatex">\(O(1)\)</span> 或 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
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