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<p class="admonition-title">Minimum cost of climbing stairs</p>
<p>Given a staircase, you can step up 1 or 2 steps at a time, and each step on the staircase has a non-negative integer representing the cost you need to pay at that step. Given a non-negative integer array <span class="arithmatex">\(cost\)</span>, where <span class="arithmatex">\(cost[i]\)</span> represents the cost you need to pay at the <span class="arithmatex">\(i\)</span>-th step, <span class="arithmatex">\(cost[0]\)</span> is the ground (starting point). What is the minimum cost required to reach the top?</p>
</div>
<p>As shown in the Figure 14-6 , if the costs of the 1<sup>st</sup>, 2<sup>nd</sup>, and 3<sup>rd</sup> steps are <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(1\)</span> respectively, then the minimum cost to climb to the 3<sup>rd</sup> step from the ground is <span class="arithmatex">\(2\)</span>.</p>
<p>As shown in Figure 14-6, if the costs of the 1<sup>st</sup>, 2<sup>nd</sup>, and 3<sup>rd</sup> steps are <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(1\)</span> respectively, then the minimum cost to climb to the 3<sup>rd</sup> step from the ground is <span class="arithmatex">\(2\)</span>.</p>
<p><a class="glightbox" href="../dp_problem_features.assets/min_cost_cs_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Minimum cost to climb to the 3rd step" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_example.png" /></a></p>
<p align="center"> Figure 14-6 &nbsp; Minimum cost to climb to the 3rd step </p>
@@ -3886,7 +3886,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%E6%9C%80%E5%B0%8F%E4%BB%A3%E4%BB%B7%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%EF%BC%8C%E7%94%A8%E4%BA%8E%E5%AD%98%E5%82%A8%E5%AD%90%E9%97%AE%E9%A2%98%E7%9A%84%E8%A7%A3%0A%20%20%20%20dp%20%3D%20%5B0%5D%20*%20%28n%20%2B%201%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E7%8A%B6%E6%80%81%EF%BC%9A%E9%A2%84%E8%AE%BE%E6%9C%80%E5%B0%8F%E5%AD%90%E9%97%AE%E9%A2%98%E7%9A%84%E8%A7%A3%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E4%BB%8E%E8%BE%83%E5%B0%8F%E5%AD%90%E9%97%AE%E9%A2%98%E9%80%90%E6%AD%A5%E6%B1%82%E8%A7%A3%E8%BE%83%E5%A4%A7%E5%AD%90%E9%97%AE%E9%A2%98%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20min%28dp%5Bi%20-%201%5D,%20dp%5Bi%20-%202%5D%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22%E8%BE%93%E5%85%A5%E6%A5%BC%E6%A2%AF%E7%9A%84%E4%BB%A3%E4%BB%B7%E5%88%97%E8%A1%A8%E4%B8%BA%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp%28cost%29%0A%20%20%20%20print%28f%22%E7%88%AC%E5%AE%8C%E6%A5%BC%E6%A2%AF%E7%9A%84%E6%9C%80%E4%BD%8E%E4%BB%A3%E4%BB%B7%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%E6%9C%80%E5%B0%8F%E4%BB%A3%E4%BB%B7%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%EF%BC%8C%E7%94%A8%E4%BA%8E%E5%AD%98%E5%82%A8%E5%AD%90%E9%97%AE%E9%A2%98%E7%9A%84%E8%A7%A3%0A%20%20%20%20dp%20%3D%20%5B0%5D%20*%20%28n%20%2B%201%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E7%8A%B6%E6%80%81%EF%BC%9A%E9%A2%84%E8%AE%BE%E6%9C%80%E5%B0%8F%E5%AD%90%E9%97%AE%E9%A2%98%E7%9A%84%E8%A7%A3%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E4%BB%8E%E8%BE%83%E5%B0%8F%E5%AD%90%E9%97%AE%E9%A2%98%E9%80%90%E6%AD%A5%E6%B1%82%E8%A7%A3%E8%BE%83%E5%A4%A7%E5%AD%90%E9%97%AE%E9%A2%98%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20min%28dp%5Bi%20-%201%5D,%20dp%5Bi%20-%202%5D%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22%E8%BE%93%E5%85%A5%E6%A5%BC%E6%A2%AF%E7%9A%84%E4%BB%A3%E4%BB%B7%E5%88%97%E8%A1%A8%E4%B8%BA%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp%28cost%29%0A%20%20%20%20print%28f%22%E7%88%AC%E5%AE%8C%E6%A5%BC%E6%A2%AF%E7%9A%84%E6%9C%80%E4%BD%8E%E4%BB%A3%E4%BB%B7%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
</details>
<p>The Figure 14-7 shows the dynamic programming process for the above code.</p>
<p>Figure 14-7 shows the dynamic programming process for the above code.</p>
<p><a class="glightbox" href="../dp_problem_features.assets/min_cost_cs_dp.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Dynamic programming process for minimum cost of climbing stairs" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_dp.png" /></a></p>
<p align="center"> Figure 14-7 &nbsp; Dynamic programming process for minimum cost of climbing stairs </p>
@@ -4131,7 +4131,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
<p class="admonition-title">Stair climbing with constraints</p>
<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, you can go up 1 or 2 steps each time, <strong>but you cannot jump 1 step twice in a row</strong>. How many ways are there to climb to the top?</p>
</div>
<p>As shown in the Figure 14-8 , there are only 2 feasible options for climbing to the 3<sup>rd</sup> step, among which the option of jumping 1 step three times in a row does not meet the constraint condition and is therefore discarded.</p>
<p>As shown in Figure 14-8, there are only 2 feasible options for climbing to the 3<sup>rd</sup> step, among which the option of jumping 1 step three times in a row does not meet the constraint condition and is therefore discarded.</p>
<p><a class="glightbox" href="../dp_problem_features.assets/climbing_stairs_constraint_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Number of feasible options for climbing to the 3rd step with constraints" class="animation-figure" src="../dp_problem_features.assets/climbing_stairs_constraint_example.png" /></a></p>
<p align="center"> Figure 14-8 &nbsp; Number of feasible options for climbing to the 3rd step with constraints </p>
@@ -4142,7 +4142,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
<li>When the last round was a jump of 1 step, the round before last could only choose to jump 2 steps, that is, <span class="arithmatex">\(dp[i, 1]\)</span> can only be transferred from <span class="arithmatex">\(dp[i-1, 2]\)</span>.</li>
<li>When the last round was a jump of 2 steps, the round before last could choose to jump 1 step or 2 steps, that is, <span class="arithmatex">\(dp[i, 2]\)</span> can be transferred from <span class="arithmatex">\(dp[i-2, 1]\)</span> or <span class="arithmatex">\(dp[i-2, 2]\)</span>.</li>
</ul>
<p>As shown in the Figure 14-9 , <span class="arithmatex">\(dp[i, j]\)</span> represents the number of solutions for state <span class="arithmatex">\([i, j]\)</span>. At this point, the state transition equation is:</p>
<p>As shown in Figure 14-9, <span class="arithmatex">\(dp[i, j]\)</span> represents the number of solutions for state <span class="arithmatex">\([i, j]\)</span>. At this point, the state transition equation is:</p>
<div class="arithmatex">\[
\begin{cases}
dp[i, 1] = dp[i-1, 2] \\