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<h1 id="152">15.2 &nbsp; 分数背包问题<a class="headerlink" href="#152" title="Permanent link">&para;</a></h1>
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<p><img alt="分数背包问题的几何表示" src="../fractional_knapsack_problem.assets/fractional_knapsack_area_chart.png" /></p>
<p align="center"> 图 15-6 &nbsp; 分数背包问题的几何表示 </p>
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<h1 id="151">15.1 &nbsp; 贪心算法<a class="headerlink" href="#151" title="Permanent link">&para;</a></h1>
<p>「贪心算法 greedy algorithm」是一种常见的解决优化问题的算法，其基本思想是在问题的每个决策阶段，都选择当前看起来最优的选择，即贪心地做出局部最优的决策，以期望获得全局最优解。贪心算法简洁且高效，在许多实际问题中都有着广泛的应用。</p>
<p>贪心算法和动态规划都常用于解决优化问题。它们之间存在一些相似之处，比如都依赖最优子结构性质，但工作原理是不同的。</p>
@@ -3795,11 +3827,73 @@
<li><strong>Dijkstra 算法</strong>：它是一种解决给定源顶点到其余各顶点的最短路径问题的贪心算法。</li>
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@@ -3791,11 +3823,73 @@ cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1]
<p>观察发现，<strong>这些被跳过的状态实际上就是将长板 <span class="arithmatex">\(j\)</span> 向内移动的所有状态</strong>。而在第二步中，我们已经证明内移长板一定会导致容量变小。也就是说，被跳过的状态都不可能是最优解，<strong>跳过它们不会导致错过最优解</strong>。</p>
<p>以上的分析说明，<strong>移动短板的操作是“安全”的</strong>，贪心策略是有效的。</p>
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<li><strong>切分方案最多包含两个 <span class="arithmatex">\(2\)</span></strong> ：假设最优切分方案中包含三个 <span class="arithmatex">\(2\)</span> ，那么一定可以替换为两个 <span class="arithmatex">\(3\)</span> ，乘积更大。这与假设矛盾。</li>
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