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@@ -20,7 +20,7 @@ We make a slight modification to the stair climbing problem to make it more suit
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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 $cost$, where $cost[i]$ represents the cost you need to pay at the $i$-th step, $cost[0]$ is the ground (starting point). What is the minimum cost required to reach the top?
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As shown in the Figure 14-6 , if the costs of the 1st, 2nd, and 3rd steps are $1$, $10$, and $1$ respectively, then the minimum cost to climb to the 3rd step from the ground is $2$.
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As shown in Figure 14-6, if the costs of the 1st, 2nd, and 3rd steps are $1$, $10$, and $1$ respectively, then the minimum cost to climb to the 3rd step from the ground is $2$.
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{ class="animation-figure" }
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@@ -334,7 +334,7 @@ According to the state transition equation, and the initial states $dp[1] = cost
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<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>
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<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 ></a></div>
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The Figure 14-7 shows the dynamic programming process for the above code.
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Figure 14-7 shows the dynamic programming process for the above code.
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{ class="animation-figure" }
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@@ -610,7 +610,7 @@ However, if we add a constraint to the stair climbing problem, the situation cha
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Given a staircase with $n$ steps, you can go up 1 or 2 steps each time, **but you cannot jump 1 step twice in a row**. How many ways are there to climb to the top?
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As shown in the Figure 14-8 , there are only 2 feasible options for climbing to the 3rd step, among which the option of jumping 1 step three times in a row does not meet the constraint condition and is therefore discarded.
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As shown in Figure 14-8, there are only 2 feasible options for climbing to the 3rd step, among which the option of jumping 1 step three times in a row does not meet the constraint condition and is therefore discarded.
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{ class="animation-figure" }
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@@ -625,7 +625,7 @@ For this, we need to expand the state definition: **State $[i, j]$ represents be
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- When the last round was a jump of 1 step, the round before last could only choose to jump 2 steps, that is, $dp[i, 1]$ can only be transferred from $dp[i-1, 2]$.
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- 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, $dp[i, 2]$ can be transferred from $dp[i-2, 1]$ or $dp[i-2, 2]$.
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As shown in the Figure 14-9 , $dp[i, j]$ represents the number of solutions for state $[i, j]$. At this point, the state transition equation is:
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As shown in Figure 14-9, $dp[i, j]$ represents the number of solutions for state $[i, j]$. At this point, the state transition equation is:
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$$
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\begin{cases}
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@@ -39,7 +39,7 @@ To illustrate the problem-solving steps more vividly, we use a classic problem,
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Given an $n \times m$ two-dimensional grid `grid`, each cell in the grid contains a non-negative integer representing the cost of that cell. The robot starts from the top-left cell and can only move down or right at each step until it reaches the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.
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The following figure shows an example, where the given grid's minimum path sum is $13$.
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Figure 14-10 shows an example, where the given grid's minimum path sum is $13$.
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{ class="animation-figure" }
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@@ -51,7 +51,7 @@ Each round of decisions in this problem is to move one step down or right from t
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The state $[i, j]$ corresponds to the subproblem: the minimum path sum from the starting point $[0, 0]$ to $[i, j]$, denoted as $dp[i, j]$.
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Thus, we obtain the two-dimensional $dp$ matrix shown below, whose size is the same as the input grid $grid$.
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Thus, we obtain the two-dimensional $dp$ matrix shown in Figure 14-11, whose size is the same as the input grid $grid$.
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{ class="animation-figure" }
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@@ -67,7 +67,7 @@ Thus, we obtain the two-dimensional $dp$ matrix shown below, whose size is the s
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For the state $[i, j]$, it can only be derived from the cell above $[i-1, j]$ or the cell to the left $[i, j-1]$. Therefore, the optimal substructure is: the minimum path sum to reach $[i, j]$ is determined by the smaller of the minimum path sums of $[i, j-1]$ and $[i-1, j]$.
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Based on the above analysis, the state transition equation shown in the following figure can be derived:
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Based on the above analysis, the state transition equation shown in Figure 14-12 can be derived:
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$$
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dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
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@@ -87,7 +87,7 @@ $$
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In this problem, the states in the first row can only come from the states to their left, and the states in the first column can only come from the states above them, so the first row $i = 0$ and the first column $j = 0$ are the boundary conditions.
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As shown in the Figure 14-13 , since each cell is derived from the cell to its left and the cell above it, we use loops to traverse the matrix, the outer loop iterating over the rows and the inner loop iterating over the columns.
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As shown in Figure 14-13, since each cell is derived from the cell to its left and the cell above it, we use loops to traverse the matrix, the outer loop iterating over the rows and the inner loop iterating over the columns.
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{ class="animation-figure" }
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@@ -398,7 +398,7 @@ Implementation code as follows:
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<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1,%20j%29%20%E5%92%8C%20%28i,%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1,%20j%29%20%E5%92%8C%20%28i,%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
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The following figure shows the recursive tree rooted at $dp[2, 1]$, which includes some overlapping subproblems, the number of which increases sharply as the size of the grid `grid` increases.
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Figure 14-14 shows the recursive tree rooted at $dp[2, 1]$, which includes some overlapping subproblems, the number of which increases sharply as the size of the grid `grid` increases.
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Essentially, the reason for overlapping subproblems is: **there are multiple paths to reach a certain cell from the top-left corner**.
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@@ -771,7 +771,7 @@ We introduce a memo list `mem` of the same size as the grid `grid`, used to reco
|
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<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20%E5%B7%A6%E8%BE%B9%E5%92%8C%E4%B8%8A%E8%BE%B9%E5%8D%95%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&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=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20%E5%B7%A6%E8%BE%B9%E5%92%8C%E4%B8%8A%E8%BE%B9%E5%8D%95%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
As shown in the Figure 14-15 , after introducing memoization, all subproblem solutions only need to be calculated once, so the time complexity depends on the total number of states, i.e., the grid size $O(nm)$.
|
||||
As shown in Figure 14-15, after introducing memoization, all subproblem solutions only need to be calculated once, so the time complexity depends on the total number of states, i.e., the grid size $O(nm)$.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -1157,7 +1157,7 @@ Implement the dynamic programming solution iteratively, code as shown below:
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&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=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%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%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
The following figures show the state transition process of the minimum path sum, traversing the entire grid, **thus the time complexity is $O(nm)$**.
|
||||
Figure 14-16 show the state transition process of the minimum path sum, traversing the entire grid, **thus the time complexity is $O(nm)$**.
|
||||
|
||||
The array `dp` is of size $n \times m$, **therefore the space complexity is $O(nm)$**.
|
||||
|
||||
|
||||
@@ -12,7 +12,7 @@ Edit distance, also known as Levenshtein distance, refers to the minimum number
|
||||
|
||||
You can perform three types of edits on a string: insert a character, delete a character, or replace a character with any other character.
|
||||
|
||||
As shown in the Figure 14-27 , transforming `kitten` into `sitting` requires 3 edits, including 2 replacements and 1 insertion; transforming `hello` into `algo` requires 3 steps, including 2 replacements and 1 deletion.
|
||||
As shown in Figure 14-27, transforming `kitten` into `sitting` requires 3 edits, including 2 replacements and 1 insertion; transforming `hello` into `algo` requires 3 steps, including 2 replacements and 1 deletion.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -20,7 +20,7 @@ As shown in the Figure 14-27 , transforming `kitten` into `sitting` requires 3 e
|
||||
|
||||
**The edit distance problem can naturally be explained with a decision tree model**. Strings correspond to tree nodes, and a round of decision (an edit operation) corresponds to an edge of the tree.
|
||||
|
||||
As shown in the Figure 14-28 , with unrestricted operations, each node can derive many edges, each corresponding to one operation, meaning there are many possible paths to transform `hello` into `algo`.
|
||||
As shown in Figure 14-28, with unrestricted operations, each node can derive many edges, each corresponding to one operation, meaning there are many possible paths to transform `hello` into `algo`.
|
||||
|
||||
From the perspective of the decision tree, the goal of this problem is to find the shortest path between the node `hello` and the node `algo`.
|
||||
|
||||
@@ -47,7 +47,7 @@ From this, we obtain a two-dimensional $dp$ table of size $(i+1) \times (j+1)$.
|
||||
|
||||
**Step two: Identify the optimal substructure and then derive the state transition equation**
|
||||
|
||||
Consider the subproblem $dp[i, j]$, whose corresponding tail characters of the two strings are $s[i-1]$ and $t[j-1]$, which can be divided into three scenarios as shown below.
|
||||
Consider the subproblem $dp[i, j]$, whose corresponding tail characters of the two strings are $s[i-1]$ and $t[j-1]$, which can be divided into three scenarios as shown in Figure 14-29.
|
||||
|
||||
1. Add $t[j-1]$ after $s[i-1]$, then the remaining subproblem is $dp[i, j-1]$.
|
||||
2. Delete $s[i-1]$, then the remaining subproblem is $dp[i-1, j]$.
|
||||
@@ -493,7 +493,7 @@ Observing the state transition equation, solving $dp[i, j]$ depends on the solut
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BC%96%E8%BE%91%E8%B7%9D%E7%A6%BB%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E4%B8%A4%E5%AD%97%E7%AC%A6%E7%9B%B8%E7%AD%89%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%B7%B3%E8%BF%87%E6%AD%A4%E4%B8%A4%E5%AD%97%E7%AC%A6%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%3D%20%E6%8F%92%E5%85%A5%E3%80%81%E5%88%A0%E9%99%A4%E3%80%81%E6%9B%BF%E6%8D%A2%E8%BF%99%E4%B8%89%E7%A7%8D%E6%93%8D%E4%BD%9C%E7%9A%84%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%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%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22%E5%B0%86%20%7Bs%7D%20%E6%9B%B4%E6%94%B9%E4%B8%BA%20%7Bt%7D%20%E6%9C%80%E5%B0%91%E9%9C%80%E8%A6%81%E7%BC%96%E8%BE%91%20%7Bres%7D%20%E6%AD%A5%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&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%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%BC%96%E8%BE%91%E8%B7%9D%E7%A6%BB%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E4%B8%A4%E5%AD%97%E7%AC%A6%E7%9B%B8%E7%AD%89%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%B7%B3%E8%BF%87%E6%AD%A4%E4%B8%A4%E5%AD%97%E7%AC%A6%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%3D%20%E6%8F%92%E5%85%A5%E3%80%81%E5%88%A0%E9%99%A4%E3%80%81%E6%9B%BF%E6%8D%A2%E8%BF%99%E4%B8%89%E7%A7%8D%E6%93%8D%E4%BD%9C%E7%9A%84%E6%9C%80%E5%B0%91%E7%BC%96%E8%BE%91%E6%AD%A5%E6%95%B0%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%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%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22%E5%B0%86%20%7Bs%7D%20%E6%9B%B4%E6%94%B9%E4%B8%BA%20%7Bt%7D%20%E6%9C%80%E5%B0%91%E9%9C%80%E8%A6%81%E7%BC%96%E8%BE%91%20%7Bres%7D%20%E6%AD%A5%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
As shown below, the process of state transition in the edit distance problem is very similar to that in the knapsack problem, which can be seen as filling a two-dimensional grid.
|
||||
As shown in Figure 14-30, the process of state transition in the edit distance problem is very similar to that in the knapsack problem, which can be seen as filling a two-dimensional grid.
|
||||
|
||||
=== "<1>"
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -12,7 +12,7 @@ In this section, we start with a classic problem, first presenting its brute for
|
||||
|
||||
Given a staircase with $n$ steps, where you can climb $1$ or $2$ steps at a time, how many different ways are there to reach the top?
|
||||
|
||||
As shown in the Figure 14-1 , there are $3$ ways to reach the top of a $3$-step staircase.
|
||||
As shown in Figure 14-1, there are $3$ ways to reach the top of a $3$-step staircase.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -456,7 +456,7 @@ $$
|
||||
dp[i] = dp[i-1] + dp[i-2]
|
||||
$$
|
||||
|
||||
This means that in the stair climbing problem, there is a recursive relationship between the subproblems, **the solution to the original problem can be constructed from the solutions to the subproblems**. The following image shows this recursive relationship.
|
||||
This means that in the stair climbing problem, there is a recursive relationship between the subproblems, **the solution to the original problem can be constructed from the solutions to the subproblems**. Figure 14-2 shows this recursive relationship.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -724,13 +724,13 @@ Observe the following code, which, like standard backtracking code, belongs to d
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E5%B7%B2%E7%9F%A5%20dp%5B1%5D%20%E5%92%8C%20dp%5B2%5D%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%29%20%2B%20dfs%28i%20-%202%29%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20return%20dfs%28n%29%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&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%20dfs%28i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E5%B7%B2%E7%9F%A5%20dp%5B1%5D%20%E5%92%8C%20dp%5B2%5D%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%29%20%2B%20dfs%28i%20-%202%29%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20return%20dfs%28n%29%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
The following image shows the recursive tree formed by brute force search. For the problem $dp[n]$, the depth of its recursive tree is $n$, with a time complexity of $O(2^n)$. Exponential order represents explosive growth, and entering a long wait if a relatively large $n$ is input.
|
||||
Figure 14-3 shows the recursive tree formed by brute force search. For the problem $dp[n]$, the depth of its recursive tree is $n$, with a time complexity of $O(2^n)$. Exponential order represents explosive growth, and entering a long wait if a relatively large $n$ is input.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
<p align="center"> Figure 14-3 Recursive tree for climbing stairs </p>
|
||||
|
||||
Observing the above image, **the exponential time complexity is caused by 'overlapping subproblems'**. For example, $dp[9]$ is decomposed into $dp[8]$ and $dp[7]$, $dp[8]$ into $dp[7]$ and $dp[6]$, both containing the subproblem $dp[7]$.
|
||||
Observing Figure 14-3, **the exponential time complexity is caused by 'overlapping subproblems'**. For example, $dp[9]$ is decomposed into $dp[8]$ and $dp[7]$, $dp[8]$ into $dp[7]$ and $dp[6]$, both containing the subproblem $dp[7]$.
|
||||
|
||||
Thus, subproblems include even smaller overlapping subproblems, endlessly. A vast majority of computational resources are wasted on these overlapping subproblems.
|
||||
|
||||
@@ -1103,7 +1103,7 @@ The code is as follows:
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int,%20mem%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E5%B7%B2%E7%9F%A5%20dp%5B1%5D%20%E5%92%8C%20dp%5B2%5D%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20%E8%8B%A5%E5%AD%98%E5%9C%A8%E8%AE%B0%E5%BD%95%20dp%5Bi%5D%20%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20mem%5Bi%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201,%20mem%29%20%2B%20dfs%28i%20-%202,%20mem%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%20dp%5Bi%5D%0A%20%20%20%20mem%5Bi%5D%20%3D%20count%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs_mem%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20mem%5Bi%5D%20%E8%AE%B0%E5%BD%95%E7%88%AC%E5%88%B0%E7%AC%AC%20i%20%E9%98%B6%E7%9A%84%E6%96%B9%E6%A1%88%E6%80%BB%E6%95%B0%EF%BC%8C-1%20%E4%BB%A3%E8%A1%A8%E6%97%A0%E8%AE%B0%E5%BD%95%0A%20%20%20%20mem%20%3D%20%5B-1%5D%20*%20%28n%20%2B%201%29%0A%20%20%20%20return%20dfs%28n,%20mem%29%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs_mem%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&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%20dfs%28i%3A%20int,%20mem%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E5%B7%B2%E7%9F%A5%20dp%5B1%5D%20%E5%92%8C%20dp%5B2%5D%20%EF%BC%8C%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20%E8%8B%A5%E5%AD%98%E5%9C%A8%E8%AE%B0%E5%BD%95%20dp%5Bi%5D%20%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%E4%B9%8B%0A%20%20%20%20if%20mem%5Bi%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201,%20mem%29%20%2B%20dfs%28i%20-%202,%20mem%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%20dp%5Bi%5D%0A%20%20%20%20mem%5Bi%5D%20%3D%20count%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs_mem%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20mem%5Bi%5D%20%E8%AE%B0%E5%BD%95%E7%88%AC%E5%88%B0%E7%AC%AC%20i%20%E9%98%B6%E7%9A%84%E6%96%B9%E6%A1%88%E6%80%BB%E6%95%B0%EF%BC%8C-1%20%E4%BB%A3%E8%A1%A8%E6%97%A0%E8%AE%B0%E5%BD%95%0A%20%20%20%20mem%20%3D%20%5B-1%5D%20*%20%28n%20%2B%201%29%0A%20%20%20%20return%20dfs%28n,%20mem%29%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs_mem%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
Observe the following image, **after memoization, all overlapping subproblems need to be calculated only once, optimizing the time complexity to $O(n)$**, which is a significant leap.
|
||||
Observe Figure 14-4, **after memoization, all overlapping subproblems need to be calculated only once, optimizing the time complexity to $O(n)$**, which is a significant leap.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -1389,7 +1389,7 @@ Since dynamic programming does not include a backtracking process, it only requi
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%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%20n%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%201,%202%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%20dp%5Bi%20-%201%5D%20%2B%20dp%5Bi%20-%202%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%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%20climbing_stairs_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E7%88%AC%E6%A5%BC%E6%A2%AF%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%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%20n%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%201,%202%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%20dp%5Bi%20-%201%5D%20%2B%20dp%5Bi%20-%202%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%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp%28n%29%0A%20%20%20%20print%28f%22%E7%88%AC%20%7Bn%7D%20%E9%98%B6%E6%A5%BC%E6%A2%AF%E5%85%B1%E6%9C%89%20%7Bres%7D%20%E7%A7%8D%E6%96%B9%E6%A1%88%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 ></a></div>
|
||||
|
||||
The image below simulates the execution process of the above code.
|
||||
Figure 14-5 simulates the execution process of the above code.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
|
||||
@@ -12,7 +12,7 @@ In this section, we will first solve the most common 0-1 knapsack problem.
|
||||
|
||||
Given $n$ items, the weight of the $i$-th item is $wgt[i-1]$ and its value is $val[i-1]$, and a knapsack with a capacity of $cap$. Each item can be chosen only once. What is the maximum value of items that can be placed in the knapsack under the capacity limit?
|
||||
|
||||
Observe the following figure, since the item number $i$ starts counting from 1, and the array index starts from 0, thus the weight of item $i$ corresponds to $wgt[i-1]$ and the value corresponds to $val[i-1]$.
|
||||
Observe Figure 14-17, since the item number $i$ starts counting from 1, and the array index starts from 0, thus the weight of item $i$ corresponds to $wgt[i-1]$ and the value corresponds to $val[i-1]$.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -353,7 +353,7 @@ The search code includes the following elements.
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20i%3A%20int,%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%80%89%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E6%97%A0%E5%89%A9%E4%BD%99%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%E4%BB%B7%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E5%8F%AA%E8%83%BD%E9%80%89%E6%8B%A9%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E4%BB%B7%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E4%B8%AD%E4%BB%B7%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E4%B8%AA%0A%20%20%20%20return%20max%28no,%20yes%29%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%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt,%20val,%20n,%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E4%BB%B7%E5%80%BC%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&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%20knapsack_dfs%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20i%3A%20int,%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%80%89%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E6%97%A0%E5%89%A9%E4%BD%99%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%E4%BB%B7%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E5%8F%AA%E8%83%BD%E9%80%89%E6%8B%A9%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E4%BB%B7%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E4%B8%AD%E4%BB%B7%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E4%B8%AA%0A%20%20%20%20return%20max%28no,%20yes%29%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%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt,%20val,%20n,%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E4%BB%B7%E5%80%BC%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
As shown in the Figure 14-18 , since each item generates two search branches of not selecting and selecting, the time complexity is $O(2^n)$.
|
||||
As shown in Figure 14-18, since each item generates two search branches of not selecting and selecting, the time complexity is $O(2^n)$.
|
||||
|
||||
Observing the recursive tree, it is easy to see that there are overlapping sub-problems, such as $dp[1, 10]$, etc. When there are many items and the knapsack capacity is large, especially when there are many items of the same weight, the number of overlapping sub-problems will increase significantly.
|
||||
|
||||
@@ -734,7 +734,7 @@ After introducing memoization, **the time complexity depends on the number of su
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%80%89%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E6%97%A0%E5%89%A9%E4%BD%99%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%E4%BB%B7%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E5%8F%AA%E8%83%BD%E9%80%89%E6%8B%A9%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E4%BB%B7%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E4%B8%AD%E4%BB%B7%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E4%B8%AA%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no,%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%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%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20n,%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E4%BB%B7%E5%80%BC%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&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%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20%E8%83%8C%E5%8C%85%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E9%80%89%E5%AE%8C%E6%89%80%E6%9C%89%E7%89%A9%E5%93%81%E6%88%96%E8%83%8C%E5%8C%85%E6%97%A0%E5%89%A9%E4%BD%99%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%E4%BB%B7%E5%80%BC%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%EF%BC%8C%E5%88%99%E5%8F%AA%E8%83%BD%E9%80%89%E6%8B%A9%E4%B8%8D%E6%94%BE%E5%85%A5%E8%83%8C%E5%8C%85%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%B8%8D%E6%94%BE%E5%85%A5%E5%92%8C%E6%94%BE%E5%85%A5%E7%89%A9%E5%93%81%20i%20%E7%9A%84%E6%9C%80%E5%A4%A7%E4%BB%B7%E5%80%BC%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E4%B8%AD%E4%BB%B7%E5%80%BC%E6%9B%B4%E5%A4%A7%E7%9A%84%E9%82%A3%E4%B8%80%E4%B8%AA%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no,%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%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%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20n,%20cap%29%0A%20%20%20%20print%28f%22%E4%B8%8D%E8%B6%85%E8%BF%87%E8%83%8C%E5%8C%85%E5%AE%B9%E9%87%8F%E7%9A%84%E6%9C%80%E5%A4%A7%E7%89%A9%E5%93%81%E4%BB%B7%E5%80%BC%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
The following figure shows the search branches that are pruned in memoized search.
|
||||
Figure 14-19 shows the search branches that are pruned in memoized search.
|
||||
|
||||
{ class="animation-figure" }
|
||||
|
||||
@@ -742,7 +742,7 @@ The following figure shows the search branches that are pruned in memoized searc
|
||||
|
||||
### 3. Method three: Dynamic programming
|
||||
|
||||
Dynamic programming essentially involves filling the $dp$ table during the state transition, the code is shown below:
|
||||
Dynamic programming essentially involves filling the $dp$ table during the state transition, the code is shown in Figure 14-20:
|
||||
|
||||
=== "Python"
|
||||
|
||||
|
||||
@@ -386,7 +386,7 @@ Comparing the code for the two problems, the state transition changes from $i-1$
|
||||
|
||||
Since the current state comes from the state to the left and above, **the space-optimized solution should perform a forward traversal for each row in the $dp$ table**.
|
||||
|
||||
This traversal order is the opposite of that for the 0-1 knapsack. Please refer to the following figures to understand the difference.
|
||||
This traversal order is the opposite of that for the 0-1 knapsack. Please refer to Figure 14-23 to understand the difference.
|
||||
|
||||
=== "<1>"
|
||||
{ class="animation-figure" }
|
||||
@@ -1200,7 +1200,7 @@ For this reason, we use the number $amt + 1$ to represent an invalid solution, b
|
||||
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%9B%B6%E9%92%B1%E5%85%91%E6%8D%A2%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%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Ba%5D%20%3D%20MAX%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E7%9B%AE%E6%A0%87%E9%87%91%E9%A2%9D%EF%BC%8C%E5%88%99%E4%B8%8D%E9%80%89%E7%A1%AC%E5%B8%81%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%B8%8D%E9%80%89%E5%92%8C%E9%80%89%E7%A1%AC%E5%B8%81%20i%20%E8%BF%99%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E7%9A%84%E8%BE%83%E5%B0%8F%E5%80%BC%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20min%28dp%5Bi%20-%201%5D%5Ba%5D,%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%20if%20dp%5Bn%5D%5Bamt%5D%20!%3D%20MAX%20else%20-1%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%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20coin_change_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22%E5%87%91%E5%88%B0%E7%9B%AE%E6%A0%87%E9%87%91%E9%A2%9D%E6%89%80%E9%9C%80%E7%9A%84%E6%9C%80%E5%B0%91%E7%A1%AC%E5%B8%81%E6%95%B0%E9%87%8F%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&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%20coin_change_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E9%9B%B6%E9%92%B1%E5%85%91%E6%8D%A2%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%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%E9%A6%96%E5%88%97%0A%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Ba%5D%20%3D%20MAX%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8B%A5%E8%B6%85%E8%BF%87%E7%9B%AE%E6%A0%87%E9%87%91%E9%A2%9D%EF%BC%8C%E5%88%99%E4%B8%8D%E9%80%89%E7%A1%AC%E5%B8%81%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E4%B8%8D%E9%80%89%E5%92%8C%E9%80%89%E7%A1%AC%E5%B8%81%20i%20%E8%BF%99%E4%B8%A4%E7%A7%8D%E6%96%B9%E6%A1%88%E7%9A%84%E8%BE%83%E5%B0%8F%E5%80%BC%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20min%28dp%5Bi%20-%201%5D%5Ba%5D,%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%20if%20dp%5Bn%5D%5Bamt%5D%20!%3D%20MAX%20else%20-1%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%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20coin_change_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22%E5%87%91%E5%88%B0%E7%9B%AE%E6%A0%87%E9%87%91%E9%A2%9D%E6%89%80%E9%9C%80%E7%9A%84%E6%9C%80%E5%B0%91%E7%A1%AC%E5%B8%81%E6%95%B0%E9%87%8F%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
|
||||
|
||||
The following images show the dynamic programming process for the coin change problem, which is very similar to the unbounded knapsack problem.
|
||||
Figure 14-25 show the dynamic programming process for the coin change problem, which is very similar to the unbounded knapsack problem.
|
||||
|
||||
=== "<1>"
|
||||
{ class="animation-figure" }
|
||||
|
||||
Reference in New Issue
Block a user