algorithm/hello-algo
1
0
Fork 0
mirror of https://github.com/krahets/hello-algo.git synced 2025-07-25 03:08:54 +08:00
Code Issues Packages Projects Releases Wiki Activity

deploy

Browse Source
This commit is contained in:
krahets
2023-09-24 16:54:24 +08:00
parent ddab282846
commit 762980261a
9 changed files with 466 additions and 415 deletions
Download Patch File Download Diff File Expand all files Collapse all files

6
chapter_dynamic_programming/dp_problem_features/index.html
View File

@ -3757,10 +3757,10 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
<p>在该问题中，如果上一轮是跳 <span class="arithmatex">\(1\)</span> 阶上来的，那么下一轮就必须跳 <span class="arithmatex">\(2\)</span> 阶。这意味着，<strong>下一步选择不能由当前状态（当前楼梯阶数）独立决定，还和前一个状态（上轮楼梯阶数）有关</strong>。</p>
<p>不难发现，此问题已不满足无后效性，状态转移方程 <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> 也失效了，因为 <span class="arithmatex">\(dp[i-1]\)</span> 代表本轮跳 <span class="arithmatex">\(1\)</span> 阶，但其中包含了许多“上一轮跳 <span class="arithmatex">\(1\)</span> 阶上来的”方案，而为了满足约束，我们就不能将 <span class="arithmatex">\(dp[i-1]\)</span> 直接计入 <span class="arithmatex">\(dp[i]\)</span> 中。</p>
<p>为此，我们需要扩展状态定义：<strong>状态 <span class="arithmatex">\([i, j]\)</span> 表示处在第 <span class="arithmatex">\(i\)</span> 阶、并且上一轮跳了 <span class="arithmatex">\(j\)</span> 阶</strong>，其中 <span class="arithmatex">\(j \in \{1, 2\}\)</span> 。此状态定义有效地区分了上一轮跳了 <span class="arithmatex">\(1\)</span> 阶还是 <span class="arithmatex">\(2\)</span> 阶，我们可以据此来决定下一步该怎么跳。</p>
<p>为此，我们需要扩展状态定义：<strong>状态 <span class="arithmatex">\([i, j]\)</span> 表示处在第 <span class="arithmatex">\(i\)</span> 阶、并且上一轮跳了 <span class="arithmatex">\(j\)</span> 阶</strong>，其中 <span class="arithmatex">\(j \in \{1, 2\}\)</span> 。此状态定义有效地区分了上一轮跳了 <span class="arithmatex">\(1\)</span> 阶还是 <span class="arithmatex">\(2\)</span> 阶，我们可以据此来判断当前状态是从何而来的。</p>
<ul>
<li>当 <span class="arithmatex">\(j\)</span> 等于 <span class="arithmatex">\(1\)</span> ，即上一轮跳了 <span class="arithmatex">\(1\)</span> 阶时，这一轮只能选择跳 <span class="arithmatex">\(2\)</span> 阶。</li>
<li>当 <span class="arithmatex">\(j\)</span> 等于 <span class="arithmatex">\(2\)</span> ，即上一轮跳了 <span class="arithmatex">\(2\)</span> 阶时，这一轮可选择跳 <span class="arithmatex">\(1\)</span> 阶或跳 <span class="arithmatex">\(2\)</span> 阶。</li>
<li>当上一轮跳了 <span class="arithmatex">\(1\)</span> 阶时，上上一轮只能选择跳 <span class="arithmatex">\(2\)</span> 阶，即 <span class="arithmatex">\(dp[i, 1]\)</span> 只能从 <span class="arithmatex">\(dp[i-1, 2]\)</span> 转移过来。</li>
<li>当上一轮跳了 <span class="arithmatex">\(2\)</span> 阶时，上上一轮可选择跳 <span class="arithmatex">\(1\)</span> 阶或跳 <span class="arithmatex">\(2\)</span> 阶，即 <span class="arithmatex">\(dp[i, 2]\)</span> 可以从 <span class="arithmatex">\(dp[i-2, 1]\)</span> 或 <span class="arithmatex">\(dp[i-2, 2]\)</span> 转移过来。</li>
</ul>
<p>如图 14-9 所示，在该定义下，<span class="arithmatex">\(dp[i, j]\)</span> 表示状态 <span class="arithmatex">\([i, j]\)</span> 对应的方案数。此时状态转移方程为：</p>
<div class="arithmatex">\[