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krahets
2026-03-29 05:07:03 +08:00
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en/chapter_dynamic_programming/dp_solution_pipeline/index.html
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@@ -35,7 +35,7 @@
<link rel="icon" href="../../assets/images/favicon.png">
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.1">
<meta name="generator" content="zensical-0.0.29">
@@ -43,10 +43,15 @@
<link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/palette.ab4e12ef.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/main.6f2fbb28.min.css">
<link rel="stylesheet" href="../../assets/stylesheets/modern/palette.dfe2e883.min.css">
@@ -60,29 +65,20 @@
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Lato:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
<style>:root{--md-text-font:"Lato";--md-code-font:"JetBrains Mono"}</style>
<link rel="stylesheet" href="../../stylesheets/extra.css">
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,t)=>(e<<5)-e+t.charCodeAt(0)),0),__md_get=(e,t=localStorage,a=__md_scope)=>JSON.parse(t.getItem(("__palette"===e?"/":a.pathname)+"."+e)),__md_set=(e,t,a=localStorage,_=__md_scope)=>{try{a.setItem(("__palette"===e?"/":_.pathname)+"."+e,JSON.stringify(t))}catch(e){}},document.documentElement.setAttribute("data-platform",navigator.platform)</script>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
.gslide-desc { color: #666; user-select: text; }
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.gscrollbar-fixer { padding-right: 15px; }
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body[data-md-color-scheme="slate"] .gslide-title { color: var(--md-default-fg-color);}
body[data-md-color-scheme="slate"] .gslide-desc { color: var(--md-default-fg-color);}</style> <script src="../../assets/javascripts/glightbox.min.js"></script></head>
</head>
@@ -97,7 +93,7 @@
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
<label class="md-overlay" for="__drawer"></label>
<label class="md-overlay" for="__drawer" aria-label="Navigation"></label>
<div data-md-component="skip">
@@ -139,12 +135,12 @@
<nav class="md-header__inner md-grid" aria-label="Header">
<a href="../.." title="Hello Algo" class="md-header__button md-logo" aria-label="Hello Algo" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello Algo">
</a>
<label class="md-header__button md-icon" for="__drawer">
<label class="md-header__button md-icon" for="__drawer" aria-label="Navigation">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-menu" viewBox="0 0 24 24"><path d="M4 5h16M4 12h16M4 19h16"/></svg>
</label>
<div class="md-header__title" data-md-component="header-title">
<div class="md-header__ellipsis">
@@ -156,7 +152,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
14.3 Dynamic Programming Problem-Solving Approach
14.3 &nbsp; Dynamic Programming Problem-Solving Approach
</span>
</div>
@@ -169,7 +165,7 @@
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<input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal" aria-label="Dark mode" type="radio" name="__palette" id="__palette_0">
<label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -179,7 +175,7 @@
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<input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal" aria-label="Light mode" type="radio" name="__palette" id="__palette_1">
<label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
@@ -197,7 +193,7 @@
<div class="md-select">
<button class="md-header__button md-icon" aria-label="Select language">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12.87 15.07-2.54-2.51.03-.03A17.5 17.5 0 0 0 14.07 6H17V4h-7V2H8v2H1v2h11.17C11.5 7.92 10.44 9.75 9 11.35 8.07 10.32 7.3 9.19 6.69 8h-2c.73 1.63 1.73 3.17 2.98 4.56l-5.09 5.02L4 19l5-5 3.11 3.11zM18.5 10h-2L12 22h2l1.12-3h4.75L21 22h2zm-2.62 7 1.62-4.33L19.12 17z"/></svg>
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</button>
<div class="md-select__inner">
<ul class="md-select__list">
@@ -240,64 +236,30 @@
<label class="md-header__button md-icon" for="__search">
<label class="md-header__button md-icon" for="__search" aria-label="Search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
<svg xmlns="http://www.w3.org/2000/svg" fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" class="lucide lucide-search" viewBox="0 0 24 24"><path d="m21 21-4.34-4.34"/><circle cx="11" cy="11" r="8"/></svg>
</label>
<div class="md-search" data-md-component="search" role="dialog">
<label class="md-search__overlay" for="__search"></label>
<div class="md-search__inner" role="search">
<form class="md-search__form" name="search">
<input type="text" class="md-search__input" name="query" aria-label="Search" placeholder="Search" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
<label class="md-search__icon md-icon" for="__search">
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
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</label>
<nav class="md-search__options" aria-label="Search">
<a href="javascript:void(0)" class="md-search__icon md-icon" title="Share" aria-label="Share" data-clipboard data-clipboard-text="" data-md-component="search-share" tabindex="-1">
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</a>
<button type="reset" class="md-search__icon md-icon" title="Clear" aria-label="Clear" tabindex="-1">
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<div class="md-search__output">
<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
<div class="md-search-result" data-md-component="search-result">
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Initializing search
</div>
<ol class="md-search-result__list" role="presentation"></ol>
</div>
</div>
</div>
</div>
<div class="md-search" data-md-component="search" role="dialog" aria-label="Search">
<button type="button" class="md-search__button">
Search
</button>
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<div class="md-header__source">
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<a href="https://github.com/krahets/hello-algo" title="Go to repository" class="md-source" data-md-component="source">
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</div>
<div class="md-source__repository">
krahets/hello-algo
</div>
</a>
</div>
</div>
</nav>
</header>
@@ -325,7 +287,7 @@
<label class="md-nav__title" for="__drawer">
<a href="../.." title="Hello Algo" class="md-nav__button md-logo" aria-label="Hello Algo" data-md-component="logo">
<img src="../../assets/images/logo.svg" alt="logo">
<img src="../../assets/images/logo.svg" alt="Hello Algo">
</a>
Hello Algo
@@ -335,7 +297,7 @@
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<p>Given an <span class="arithmatex">\(n \times m\)</span> two-dimensional grid <code>grid</code>, where each cell in the grid contains a non-negative integer representing the cost of that cell. A robot starts from the top-left cell and can only move down or right at each step until reaching the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.</p>
</div>
<p>Figure 14-10 shows an example where the minimum path sum for the given grid is <span class="arithmatex">\(13\)</span>.</p>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Minimum path sum example data" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_example.png" /></a></p>
<p><img alt="Minimum path sum example data" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_example.png" /></p>
<p align="center"> Figure 14-10 &nbsp; Minimum path sum example data </p>
<p><strong>Step 1: Think about the decisions in each round, define the state, and thus obtain the <span class="arithmatex">\(dp\)</span> table</strong></p>
<p>The decision in each round of this problem is to move one step down or right from the current cell. Let the row and column indices of the current cell be <span class="arithmatex">\([i, j]\)</span>. After moving down or right, the indices become <span class="arithmatex">\([i+1, j]\)</span> or <span class="arithmatex">\([i, j+1]\)</span>. Therefore, the state should include two variables, the row index and column index, denoted as <span class="arithmatex">\([i, j]\)</span>.</p>
<p>State <span class="arithmatex">\([i, j]\)</span> corresponds to the subproblem: <strong>the minimum path sum from the starting point <span class="arithmatex">\([0, 0]\)</span> to <span class="arithmatex">\([i, j]\)</span></strong>, denoted as <span class="arithmatex">\(dp[i, j]\)</span>.</p>
<p>From this, we obtain the two-dimensional <span class="arithmatex">\(dp\)</span> matrix shown in Figure 14-11, whose size is the same as the input grid <span class="arithmatex">\(grid\)</span>.</p>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="State definition and dp table" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png" /></a></p>
<p><img alt="State definition and dp table" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png" /></p>
<p align="center"> Figure 14-11 &nbsp; State definition and dp table </p>
<div class="admonition note">
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<div class="arithmatex">\[
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
\]</div>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_solution_state_transition.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Optimal substructure and state transition equation" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_state_transition.png" /></a></p>
<p><img alt="Optimal substructure and state transition equation" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_state_transition.png" /></p>
<p align="center"> Figure 14-12 &nbsp; Optimal substructure and state transition equation </p>
<div class="admonition note">
@@ -4551,7 +4515,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
<p><strong>Step 3: Determine boundary conditions and state transition order</strong></p>
<p>In this problem, states in the first row can only come from the state to their left, and states in the first column can only come from the state above them. Therefore, the first row <span class="arithmatex">\(i = 0\)</span> and first column <span class="arithmatex">\(j = 0\)</span> are boundary conditions.</p>
<p>As shown in Figure 14-13, since each cell is transferred from the cell to its left and the cell above it, we use loops to traverse the matrix, with the outer loop traversing rows and the inner loop traversing columns.</p>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Boundary conditions and state transition order" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png" /></a></p>
<p><img alt="Boundary conditions and state transition order" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png" /></p>
<p align="center"> Figure 14-13 &nbsp; Boundary conditions and state transition order </p>
<div class="admonition note">
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<p>Figure 14-14 shows the recursion tree rooted at <span class="arithmatex">\(dp[2, 1]\)</span>, which includes some overlapping subproblems whose number will increase sharply as the size of grid <code>grid</code> grows.</p>
<p>Essentially, the reason for overlapping subproblems is: <strong>there are multiple paths from the top-left corner to reach a certain cell</strong>.</p>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_dfs.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Brute force search recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs.png" /></a></p>
<p><img alt="Brute force search recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs.png" /></p>
<p align="center"> Figure 14-14 &nbsp; Brute force search recursion tree </p>
<p>Each state has two choices, down and right, so the total number of steps from the top-left corner to the bottom-right corner is <span class="arithmatex">\(m + n - 2\)</span>, giving a worst-case time complexity of <span class="arithmatex">\(O(2^{m + n})\)</span>, where <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span> are the number of rows and columns of the grid, respectively. Note that this calculation does not account for situations near the grid boundaries, where only one choice remains when reaching the grid boundary, so the actual number of paths will be somewhat less.</p>
@@ -5145,7 +5109,7 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
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<p>As shown in Figure 14-15, after introducing memoization, all subproblem solutions only need to be computed once, so the time complexity depends on the total number of states, which is the grid size <span class="arithmatex">\(O(nm)\)</span>.</p>
<p><a class="glightbox" href="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Memoization recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" /></a></p>
<p><img alt="Memoization recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" /></p>
<p align="center"> Figure 14-15 &nbsp; Memoization recursion tree </p>
<h3 id="3-method-3-dynamic-programming">3. &nbsp; Method 3: Dynamic Programming<a class="headerlink" href="#3-method-3-dynamic-programming" title="Permanent link">&para;</a></h3>
@@ -5493,40 +5457,40 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
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