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14.3 動態規劃解題思路

上兩節介紹了動態規劃問題的主要特徵，接下來我們一起探究兩個更加實用的問題。

如何判斷一個問題是不是動態規劃問題？
求解動態規劃問題該從何處入手，完整步驟是什麼？

14.3.1 問題判斷

總的來說，如果一個問題包含重疊子問題、最優子結構，並滿足無後效性，那麼它通常適合用動態規劃求解。然而，我們很難從問題描述中直接提取出這些特性。因此我們通常會放寬條件，先觀察問題是否適合使用回溯（窮舉）解決。

適合用回溯解決的問題通常滿足“決策樹模型”，這種問題可以使用樹形結構來描述，其中每一個節點代表一個決策，每一條路徑代表一個決策序列。

換句話說，如果問題包含明確的決策概念，並且解是透過一系列決策產生的，那麼它就滿足決策樹模型，通常可以使用回溯來解決。

在此基礎上，動態規劃問題還有一些判斷的“加分項”。

問題包含最大（小）或最多（少）等最最佳化描述。
問題的狀態能夠使用一個串列、多維矩陣或樹來表示，並且一個狀態與其周圍的狀態存在遞推關係。

相應地，也存在一些“減分項”。

問題的目標是找出所有可能的解決方案，而不是找出最優解。
問題描述中有明顯的排列組合的特徵，需要返回具體的多個方案。

如果一個問題滿足決策樹模型，並具有較為明顯的“加分項”，我們就可以假設它是一個動態規劃問題，並在求解過程中驗證它。

14.3.2 問題求解步驟

動態規劃的解題流程會因問題的性質和難度而有所不同，但通常遵循以下步驟：描述決策，定義狀態，建立 dp 表，推導狀態轉移方程，確定邊界條件等。

為了更形象地展示解題步驟，我們使用一個經典問題“最小路徑和”來舉例。

!!! question

給定一個 $n \times m$ 的二維網格 `grid` ，網格中的每個單元格包含一個非負整數，表示該單元格的代價。機器人以左上角單元格為起始點，每次只能向下或者向右移動一步，直至到達右下角單元格。請返回從左上角到右下角的最小路徑和。

圖 14-10 展示了一個例子，給定網格的最小路徑和為 13 。

{ class="animation-figure" }

圖 14-10 最小路徑和示例資料

第一步：思考每輪的決策，定義狀態，從而得到 dp 表

本題的每一輪的決策就是從當前格子向下或向右走一步。設當前格子的行列索引為 [i, j] ，則向下或向右走一步後，索引變為 [i+1, j] 或 [i, j+1] 。因此，狀態應包含行索引和列索引兩個變數，記為 [i, j] 。

狀態 [i, j] 對應的子問題為：從起始點 [0, 0] 走到 [i, j] 的最小路徑和，解記為 dp[i, j] 。

至此，我們就得到了圖 14-11 所示的二維 dp 矩陣，其尺寸與輸入網格 grid 相同。

{ class="animation-figure" }

圖 14-11 狀態定義與 dp 表

!!! note

動態規劃和回溯過程可以描述為一個決策序列，而狀態由所有決策變數構成。它應當包含描述解題進度的所有變數，其包含了足夠的資訊，能夠用來推導出下一個狀態。

每個狀態都對應一個子問題，我們會定義一個 $dp$ 表來儲存所有子問題的解，狀態的每個獨立變數都是 $dp$ 表的一個維度。從本質上看，$dp$ 表是狀態和子問題的解之間的對映。

第二步：找出最優子結構，進而推導出狀態轉移方程

對於狀態 [i, j] ，它只能從上邊格子 [i-1, j] 和左邊格子 [i, j-1] 轉移而來。因此最優子結構為：到達 [i, j] 的最小路徑和由 [i, j-1] 的最小路徑和與 [i-1, j] 的最小路徑和中較小的那一個決定。

根據以上分析，可推出圖 14-12 所示的狀態轉移方程：

$$ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]

{ class="animation-figure" }

圖 14-12 最優子結構與狀態轉移方程

!!! note

根據定義好的 $dp$ 表，思考原問題和子問題的關係，找出透過子問題的最優解來構造原問題的最優解的方法，即最優子結構。

一旦我們找到了最優子結構，就可以使用它來構建出狀態轉移方程。

第三步：確定邊界條件和狀態轉移順序

在本題中，處在首行的狀態只能從其左邊的狀態得來，處在首列的狀態只能從其上邊的狀態得來，因此首行 i = 0 和首列 j = 0 是邊界條件。

如圖 14-13 所示，由於每個格子是由其左方格子和上方格子轉移而來，因此我們使用迴圈來走訪矩陣，外迴圈走訪各行，內迴圈走訪各列。

{ class="animation-figure" }

圖 14-13 邊界條件與狀態轉移順序

!!! note

邊界條件在動態規劃中用於初始化 $dp$ 表，在搜尋中用於剪枝。

狀態轉移順序的核心是要保證在計算當前問題的解時，所有它依賴的更小子問題的解都已經被正確地計算出來。

根據以上分析，我們已經可以直接寫出動態規劃程式碼。然而子問題分解是一種從頂至底的思想，因此按照“暴力搜尋 \rightarrow 記憶化搜尋 \rightarrow 動態規劃”的順序實現更加符合思維習慣。

1. 方法一：暴力搜尋

從狀態 [i, j] 開始搜尋，不斷分解為更小的狀態 [i-1, j] 和 [i, j-1] ，遞迴函式包括以下要素。

遞迴參數：狀態 [i, j] 。
返回值：從 [0, 0] 到 [i, j] 的最小路徑和 dp[i, j] 。
終止條件：當 i = 0 且 j = 0 時，返回代價 grid[0, 0] 。
剪枝：當 i < 0 時或 j < 0 時索引越界，此時返回代價 +\infty ，代表不可行。

實現程式碼如下：

=== "Python"

```python title="min_path_sum.py"
def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -> int:
    """最小路徑和：暴力搜尋"""
    # 若為左上角單元格，則終止搜尋
    if i == 0 and j == 0:
        return grid[0][0]
    # 若行列索引越界，則返回 +∞ 代價
    if i < 0 or j < 0:
        return inf
    # 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    up = min_path_sum_dfs(grid, i - 1, j)
    left = min_path_sum_dfs(grid, i, j - 1)
    # 返回從左上角到 (i, j) 的最小路徑代價
    return min(left, up) + grid[i][j]
```

=== "C++"

```cpp title="min_path_sum.cpp"
/* 最小路徑和：暴力搜尋 */
int minPathSumDFS(vector<vector<int>> &grid, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return INT_MAX;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    int up = minPathSumDFS(grid, i - 1, j);
    int left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
}
```

=== "Java"

```java title="min_path_sum.java"
/* 最小路徑和：暴力搜尋 */
int minPathSumDFS(int[][] grid, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Integer.MAX_VALUE;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    int up = minPathSumDFS(grid, i - 1, j);
    int left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return Math.min(left, up) + grid[i][j];
}
```

=== "C#"

```csharp title="min_path_sum.cs"
/* 最小路徑和：暴力搜尋 */
int MinPathSumDFS(int[][] grid, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return int.MaxValue;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    int up = MinPathSumDFS(grid, i - 1, j);
    int left = MinPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return Math.Min(left, up) + grid[i][j];
}
```

=== "Go"

```go title="min_path_sum.go"
/* 最小路徑和：暴力搜尋 */
func minPathSumDFS(grid [][]int, i, j int) int {
    // 若為左上角單元格，則終止搜尋
    if i == 0 && j == 0 {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return math.MaxInt
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    up := minPathSumDFS(grid, i-1, j)
    left := minPathSumDFS(grid, i, j-1)
    // 返回從左上角到 (i, j) 的最小路徑代價
    return int(math.Min(float64(left), float64(up))) + grid[i][j]
}
```

=== "Swift"

```swift title="min_path_sum.swift"
/* 最小路徑和：暴力搜尋 */
func minPathSumDFS(grid: [[Int]], i: Int, j: Int) -> Int {
    // 若為左上角單元格，則終止搜尋
    if i == 0, j == 0 {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return .max
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)
    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)
    // 返回從左上角到 (i, j) 的最小路徑代價
    return min(left, up) + grid[i][j]
}
```

=== "JS"

```javascript title="min_path_sum.js"
/* 最小路徑和：暴力搜尋 */
function minPathSumDFS(grid, i, j) {
    // 若為左上角單元格，則終止搜尋
    if (i === 0 && j === 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Infinity;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    const up = minPathSumDFS(grid, i - 1, j);
    const left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return Math.min(left, up) + grid[i][j];
}
```

=== "TS"

```typescript title="min_path_sum.ts"
/* 最小路徑和：暴力搜尋 */
function minPathSumDFS(
    grid: Array<Array<number>>,
    i: number,
    j: number
): number {
    // 若為左上角單元格，則終止搜尋
    if (i === 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Infinity;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    const up = minPathSumDFS(grid, i - 1, j);
    const left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return Math.min(left, up) + grid[i][j];
}
```

=== "Dart"

```dart title="min_path_sum.dart"
/* 最小路徑和：暴力搜尋 */
int minPathSumDFS(List<List<int>> grid, int i, int j) {
  // 若為左上角單元格，則終止搜尋
  if (i == 0 && j == 0) {
    return grid[0][0];
  }
  // 若行列索引越界，則返回 +∞ 代價
  if (i < 0 || j < 0) {
    // 在 Dart 中，int 型別是固定範圍的整數，不存在表示“無窮大”的值
    return BigInt.from(2).pow(31).toInt();
  }
  // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
  int up = minPathSumDFS(grid, i - 1, j);
  int left = minPathSumDFS(grid, i, j - 1);
  // 返回從左上角到 (i, j) 的最小路徑代價
  return min(left, up) + grid[i][j];
}
```

=== "Rust"

```rust title="min_path_sum.rs"
/* 最小路徑和：暴力搜尋 */
fn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
    // 若為左上角單元格，則終止搜尋
    if i == 0 && j == 0 {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return i32::MAX;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    let up = min_path_sum_dfs(grid, i - 1, j);
    let left = min_path_sum_dfs(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    std::cmp::min(left, up) + grid[i as usize][j as usize]
}
```

=== "C"

```c title="min_path_sum.c"
/* 最小路徑和：暴力搜尋 */
int minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return INT_MAX;
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    int up = minPathSumDFS(grid, i - 1, j);
    int left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;
}
```

=== "Kotlin"

```kotlin title="min_path_sum.kt"
/* 最小路徑和：暴力搜尋 */
fun minPathSumDFS(grid: Array<IntArray>, i: Int, j: Int): Int {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Int.MAX_VALUE
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    val up = minPathSumDFS(grid, i - 1, j)
    val left = minPathSumDFS(grid, i, j - 1)
    // 返回從左上角到 (i, j) 的最小路徑代價
    return min(left, up) + grid[i][j]
}
```

=== "Ruby"

```ruby title="min_path_sum.rb"
[class]{}-[func]{min_path_sum_dfs}
```

=== "Zig"

```zig title="min_path_sum.zig"
// 最小路徑和：暴力搜尋
fn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 and j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 or j < 0) {
        return std.math.maxInt(i32);
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    var up = minPathSumDFS(grid, i - 1, j);
    var left = minPathSumDFS(grid, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
}
```

??? pythontutor "視覺化執行"

<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%2C%20i%3A%20int%2C%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%91%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E7%82%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%96%AE%E5%85%83%E6%A0%BC%EF%BC%8C%E5%89%87%E7%B5%82%E6%AD%A2%E6%90%9C%E5%B0%8B%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%89%87%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E5%83%B9%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%A8%88%E7%AE%97%E5%BE%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1%2C%20j%29%20%E5%92%8C%20%28i%2C%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid%2C%20i%20-%201%2C%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid%2C%20i%2C%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E5%BE%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i%2C%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20return%20min%28left%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid%2C%20n%20-%201%2C%20m%20-%201%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%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%91%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E7%82%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%96%AE%E5%85%83%E6%A0%BC%EF%BC%8C%E5%89%87%E7%B5%82%E6%AD%A2%E6%90%9C%E5%B0%8B%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%89%87%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E5%83%B9%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%A8%88%E7%AE%97%E5%BE%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1%2C%20j%29%20%E5%92%8C%20%28i%2C%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid%2C%20i%20-%201%2C%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid%2C%20i%2C%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E5%BE%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i%2C%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20return%20min%28left%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E5%B0%8B%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid%2C%20n%20-%201%2C%20m%20-%201%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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">全螢幕觀看 ></a></div>

圖 14-14 給出了以 dp[2, 1] 為根節點的遞迴樹，其中包含一些重疊子問題，其數量會隨著網格 grid 的尺寸變大而急劇增多。

從本質上看，造成重疊子問題的原因為：存在多條路徑可以從左上角到達某一單元格。

{ class="animation-figure" }

圖 14-14 暴力搜尋遞迴樹

每個狀態都有向下和向右兩種選擇，從左上角走到右下角總共需要 m + n - 2 步，所以最差時間複雜度為 O(2^{m + n}) 。請注意，這種計算方式未考慮臨近網格邊界的情況，當到達網路邊界時只剩下一種選擇，因此實際的路徑數量會少一些。

2. 方法二：記憶化搜尋

我們引入一個和網格 grid 相同尺寸的記憶串列 mem ，用於記錄各個子問題的解，並將重疊子問題進行剪枝：

=== "Python"

```python title="min_path_sum.py"
def min_path_sum_dfs_mem(
    grid: list[list[int]], mem: list[list[int]], i: int, j: int
) -> int:
    """最小路徑和：記憶化搜尋"""
    # 若為左上角單元格，則終止搜尋
    if i == 0 and j == 0:
        return grid[0][0]
    # 若行列索引越界，則返回 +∞ 代價
    if i < 0 or j < 0:
        return inf
    # 若已有記錄，則直接返回
    if mem[i][j] != -1:
        return mem[i][j]
    # 左邊和上邊單元格的最小路徑代價
    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)
    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)
    # 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = min(left, up) + grid[i][j]
    return mem[i][j]
```

=== "C++"

```cpp title="min_path_sum.cpp"
/* 最小路徑和：記憶化搜尋 */
int minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return INT_MAX;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    int up = minPathSumDFSMem(grid, mem, i - 1, j);
    int left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
    return mem[i][j];
}
```

=== "Java"

```java title="min_path_sum.java"
/* 最小路徑和：記憶化搜尋 */
int minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Integer.MAX_VALUE;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    int up = minPathSumDFSMem(grid, mem, i - 1, j);
    int left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = Math.min(left, up) + grid[i][j];
    return mem[i][j];
}
```

=== "C#"

```csharp title="min_path_sum.cs"
/* 最小路徑和：記憶化搜尋 */
int MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return int.MaxValue;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    int up = MinPathSumDFSMem(grid, mem, i - 1, j);
    int left = MinPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = Math.Min(left, up) + grid[i][j];
    return mem[i][j];
}
```

=== "Go"

```go title="min_path_sum.go"
/* 最小路徑和：記憶化搜尋 */
func minPathSumDFSMem(grid, mem [][]int, i, j int) int {
    // 若為左上角單元格，則終止搜尋
    if i == 0 && j == 0 {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return math.MaxInt
    }
    // 若已有記錄，則直接返回
    if mem[i][j] != -1 {
        return mem[i][j]
    }
    // 左邊和上邊單元格的最小路徑代價
    up := minPathSumDFSMem(grid, mem, i-1, j)
    left := minPathSumDFSMem(grid, mem, i, j-1)
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]
    return mem[i][j]
}
```

=== "Swift"

```swift title="min_path_sum.swift"
/* 最小路徑和：記憶化搜尋 */
func minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -> Int {
    // 若為左上角單元格，則終止搜尋
    if i == 0, j == 0 {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return .max
    }
    // 若已有記錄，則直接返回
    if mem[i][j] != -1 {
        return mem[i][j]
    }
    // 左邊和上邊單元格的最小路徑代價
    let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)
    let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: j - 1)
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = min(left, up) + grid[i][j]
    return mem[i][j]
}
```

=== "JS"

```javascript title="min_path_sum.js"
/* 最小路徑和：記憶化搜尋 */
function minPathSumDFSMem(grid, mem, i, j) {
    // 若為左上角單元格，則終止搜尋
    if (i === 0 && j === 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Infinity;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] !== -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    const up = minPathSumDFSMem(grid, mem, i - 1, j);
    const left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = Math.min(left, up) + grid[i][j];
    return mem[i][j];
}
```

=== "TS"

```typescript title="min_path_sum.ts"
/* 最小路徑和：記憶化搜尋 */
function minPathSumDFSMem(
    grid: Array<Array<number>>,
    mem: Array<Array<number>>,
    i: number,
    j: number
): number {
    // 若為左上角單元格，則終止搜尋
    if (i === 0 && j === 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Infinity;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    const up = minPathSumDFSMem(grid, mem, i - 1, j);
    const left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = Math.min(left, up) + grid[i][j];
    return mem[i][j];
}
```

=== "Dart"

```dart title="min_path_sum.dart"
/* 最小路徑和：記憶化搜尋 */
int minPathSumDFSMem(List<List<int>> grid, List<List<int>> mem, int i, int j) {
  // 若為左上角單元格，則終止搜尋
  if (i == 0 && j == 0) {
    return grid[0][0];
  }
  // 若行列索引越界，則返回 +∞ 代價
  if (i < 0 || j < 0) {
    // 在 Dart 中，int 型別是固定範圍的整數，不存在表示“無窮大”的值
    return BigInt.from(2).pow(31).toInt();
  }
  // 若已有記錄，則直接返回
  if (mem[i][j] != -1) {
    return mem[i][j];
  }
  // 左邊和上邊單元格的最小路徑代價
  int up = minPathSumDFSMem(grid, mem, i - 1, j);
  int left = minPathSumDFSMem(grid, mem, i, j - 1);
  // 記錄並返回左上角到 (i, j) 的最小路徑代價
  mem[i][j] = min(left, up) + grid[i][j];
  return mem[i][j];
}
```

=== "Rust"

```rust title="min_path_sum.rs"
/* 最小路徑和：記憶化搜尋 */
fn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
    // 若為左上角單元格，則終止搜尋
    if i == 0 && j == 0 {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if i < 0 || j < 0 {
        return i32::MAX;
    }
    // 若已有記錄，則直接返回
    if mem[i as usize][j as usize] != -1 {
        return mem[i as usize][j as usize];
    }
    // 左邊和上邊單元格的最小路徑代價
    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);
    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];
    mem[i as usize][j as usize]
}
```

=== "C"

```c title="min_path_sum.c"
/* 最小路徑和：記憶化搜尋 */
int minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return INT_MAX;
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j];
    }
    // 左邊和上邊單元格的最小路徑代價
    int up = minPathSumDFSMem(grid, mem, i - 1, j);
    int left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;
    return mem[i][j];
}
```

=== "Kotlin"

```kotlin title="min_path_sum.kt"
/* 最小路徑和：記憶化搜尋 */
fun minPathSumDFSMem(
    grid: Array<IntArray>,
    mem: Array<IntArray>,
    i: Int,
    j: Int
): Int {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 && j == 0) {
        return grid[0][0]
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 || j < 0) {
        return Int.MAX_VALUE
    }
    // 若已有記錄，則直接返回
    if (mem[i][j] != -1) {
        return mem[i][j]
    }
    // 左邊和上邊單元格的最小路徑代價
    val up = minPathSumDFSMem(grid, mem, i - 1, j)
    val left = minPathSumDFSMem(grid, mem, i, j - 1)
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[i][j] = min(left, up) + grid[i][j]
    return mem[i][j]
}
```

=== "Ruby"

```ruby title="min_path_sum.rb"
[class]{}-[func]{min_path_sum_dfs_mem}
```

=== "Zig"

```zig title="min_path_sum.zig"
// 最小路徑和：記憶化搜尋
fn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {
    // 若為左上角單元格，則終止搜尋
    if (i == 0 and j == 0) {
        return grid[0][0];
    }
    // 若行列索引越界，則返回 +∞ 代價
    if (i < 0 or j < 0) {
        return std.math.maxInt(i32);
    }
    // 若已有記錄，則直接返回
    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {
        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
    }
    // 計算從左上角到 (i-1, j) 和 (i, j-1) 的最小路徑代價
    var up = minPathSumDFSMem(grid, mem, i - 1, j);
    var left = minPathSumDFSMem(grid, mem, i, j - 1);
    // 返回從左上角到 (i, j) 的最小路徑代價
    // 記錄並返回左上角到 (i, j) 的最小路徑代價
    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
}
```

??? pythontutor "視覺化執行"

<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%2C%20mem%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%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%91%E5%92%8C%EF%BC%9A%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E7%82%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%96%AE%E5%85%83%E6%A0%BC%EF%BC%8C%E5%89%87%E7%B5%82%E6%AD%A2%E6%90%9C%E5%B0%8B%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%89%87%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E5%83%B9%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%A8%98%E9%8C%84%EF%BC%8C%E5%89%87%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20%21%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%E9%82%8A%E5%92%8C%E4%B8%8A%E9%82%8A%E5%96%AE%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20i%20-%201%2C%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20i%2C%20j%20-%201%29%0A%20%20%20%20%23%20%E8%A8%98%E9%8C%84%E4%B8%A6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i%2C%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20n%20-%201%2C%20m%20-%201%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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%2C%20mem%3A%20list%5Blist%5Bint%5D%5D%2C%20i%3A%20int%2C%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%91%E5%92%8C%EF%BC%9A%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E7%82%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%96%AE%E5%85%83%E6%A0%BC%EF%BC%8C%E5%89%87%E7%B5%82%E6%AD%A2%E6%90%9C%E5%B0%8B%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%89%87%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E5%83%B9%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%A8%98%E9%8C%84%EF%BC%8C%E5%89%87%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20%21%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%E9%82%8A%E5%92%8C%E4%B8%8A%E9%82%8A%E5%96%AE%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20i%20-%201%2C%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20i%2C%20j%20-%201%29%0A%20%20%20%20%23%20%E8%A8%98%E9%8C%84%E4%B8%A6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i%2C%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%91%E4%BB%A3%E5%83%B9%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%A8%98%E6%86%B6%E5%8C%96%E6%90%9C%E5%B0%8B%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid%2C%20mem%2C%20n%20-%201%2C%20m%20-%201%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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">全螢幕觀看 ></a></div>

如圖 14-15 所示，在引入記憶化後，所有子問題的解只需計算一次，因此時間複雜度取決於狀態總數，即網格尺寸 O(nm) 。

{ class="animation-figure" }

圖 14-15 記憶化搜尋遞迴樹

3. 方法三：動態規劃

基於迭代實現動態規劃解法，程式碼如下所示：

=== "Python"

```python title="min_path_sum.py"
def min_path_sum_dp(grid: list[list[int]]) -> int:
    """最小路徑和：動態規劃"""
    n, m = len(grid), len(grid[0])
    # 初始化 dp 表
    dp = [[0] * m for _ in range(n)]
    dp[0][0] = grid[0][0]
    # 狀態轉移：首行
    for j in range(1, m):
        dp[0][j] = dp[0][j - 1] + grid[0][j]
    # 狀態轉移：首列
    for i in range(1, n):
        dp[i][0] = dp[i - 1][0] + grid[i][0]
    # 狀態轉移：其餘行和列
    for i in range(1, n):
        for j in range(1, m):
            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
    return dp[n - 1][m - 1]
```

=== "C++"

```cpp title="min_path_sum.cpp"
/* 最小路徑和：動態規劃 */
int minPathSumDP(vector<vector<int>> &grid) {
    int n = grid.size(), m = grid[0].size();
    // 初始化 dp 表
    vector<vector<int>> dp(n, vector<int>(m));
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (int j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (int i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < m; j++) {
            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    return dp[n - 1][m - 1];
}
```

=== "Java"

```java title="min_path_sum.java"
/* 最小路徑和：動態規劃 */
int minPathSumDP(int[][] grid) {
    int n = grid.length, m = grid[0].length;
    // 初始化 dp 表
    int[][] dp = new int[n][m];
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (int j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (int i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < m; j++) {
            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    return dp[n - 1][m - 1];
}
```

=== "C#"

```csharp title="min_path_sum.cs"
/* 最小路徑和：動態規劃 */
int MinPathSumDP(int[][] grid) {
    int n = grid.Length, m = grid[0].Length;
    // 初始化 dp 表
    int[,] dp = new int[n, m];
    dp[0, 0] = grid[0][0];
    // 狀態轉移：首行
    for (int j = 1; j < m; j++) {
        dp[0, j] = dp[0, j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (int i = 1; i < n; i++) {
        dp[i, 0] = dp[i - 1, 0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < m; j++) {
            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];
        }
    }
    return dp[n - 1, m - 1];
}
```

=== "Go"

```go title="min_path_sum.go"
/* 最小路徑和：動態規劃 */
func minPathSumDP(grid [][]int) int {
    n, m := len(grid), len(grid[0])
    // 初始化 dp 表
    dp := make([][]int, n)
    for i := 0; i < n; i++ {
        dp[i] = make([]int, m)
    }
    dp[0][0] = grid[0][0]
    // 狀態轉移：首行
    for j := 1; j < m; j++ {
        dp[0][j] = dp[0][j-1] + grid[0][j]
    }
    // 狀態轉移：首列
    for i := 1; i < n; i++ {
        dp[i][0] = dp[i-1][0] + grid[i][0]
    }
    // 狀態轉移：其餘行和列
    for i := 1; i < n; i++ {
        for j := 1; j < m; j++ {
            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]
        }
    }
    return dp[n-1][m-1]
}
```

=== "Swift"

```swift title="min_path_sum.swift"
/* 最小路徑和：動態規劃 */
func minPathSumDP(grid: [[Int]]) -> Int {
    let n = grid.count
    let m = grid[0].count
    // 初始化 dp 表
    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)
    dp[0][0] = grid[0][0]
    // 狀態轉移：首行
    for j in 1 ..< m {
        dp[0][j] = dp[0][j - 1] + grid[0][j]
    }
    // 狀態轉移：首列
    for i in 1 ..< n {
        dp[i][0] = dp[i - 1][0] + grid[i][0]
    }
    // 狀態轉移：其餘行和列
    for i in 1 ..< n {
        for j in 1 ..< m {
            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
        }
    }
    return dp[n - 1][m - 1]
}
```

=== "JS"

```javascript title="min_path_sum.js"
/* 最小路徑和：動態規劃 */
function minPathSumDP(grid) {
    const n = grid.length,
        m = grid[0].length;
    // 初始化 dp 表
    const dp = Array.from({ length: n }, () =>
        Array.from({ length: m }, () => 0)
    );
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (let j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (let i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (let i = 1; i < n; i++) {
        for (let j = 1; j < m; j++) {
            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    return dp[n - 1][m - 1];
}
```

=== "TS"

```typescript title="min_path_sum.ts"
/* 最小路徑和：動態規劃 */
function minPathSumDP(grid: Array<Array<number>>): number {
    const n = grid.length,
        m = grid[0].length;
    // 初始化 dp 表
    const dp = Array.from({ length: n }, () =>
        Array.from({ length: m }, () => 0)
    );
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (let j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (let i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (let i = 1; i < n; i++) {
        for (let j: number = 1; j < m; j++) {
            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    return dp[n - 1][m - 1];
}
```

=== "Dart"

```dart title="min_path_sum.dart"
/* 最小路徑和：動態規劃 */
int minPathSumDP(List<List<int>> grid) {
  int n = grid.length, m = grid[0].length;
  // 初始化 dp 表
  List<List<int>> dp = List.generate(n, (i) => List.filled(m, 0));
  dp[0][0] = grid[0][0];
  // 狀態轉移：首行
  for (int j = 1; j < m; j++) {
    dp[0][j] = dp[0][j - 1] + grid[0][j];
  }
  // 狀態轉移：首列
  for (int i = 1; i < n; i++) {
    dp[i][0] = dp[i - 1][0] + grid[i][0];
  }
  // 狀態轉移：其餘行和列
  for (int i = 1; i < n; i++) {
    for (int j = 1; j < m; j++) {
      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
    }
  }
  return dp[n - 1][m - 1];
}
```

=== "Rust"

```rust title="min_path_sum.rs"
/* 最小路徑和：動態規劃 */
fn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {
    let (n, m) = (grid.len(), grid[0].len());
    // 初始化 dp 表
    let mut dp = vec![vec![0; m]; n];
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for j in 1..m {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for i in 1..n {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for i in 1..n {
        for j in 1..m {
            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    dp[n - 1][m - 1]
}
```

=== "C"

```c title="min_path_sum.c"
/* 最小路徑和：動態規劃 */
int minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {
    // 初始化 dp 表
    int **dp = malloc(n * sizeof(int *));
    for (int i = 0; i < n; i++) {
        dp[i] = calloc(m, sizeof(int));
    }
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (int j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (int i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (int i = 1; i < n; i++) {
        for (int j = 1; j < m; j++) {
            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    int res = dp[n - 1][m - 1];
    // 釋放記憶體
    for (int i = 0; i < n; i++) {
        free(dp[i]);
    }
    return res;
}
```

=== "Kotlin"

```kotlin title="min_path_sum.kt"
/* 最小路徑和：動態規劃 */
fun minPathSumDP(grid: Array<IntArray>): Int {
    val n = grid.size
    val m = grid[0].size
    // 初始化 dp 表
    val dp = Array(n) { IntArray(m) }
    dp[0][0] = grid[0][0]
    // 狀態轉移：首行
    for (j in 1..<m) {
        dp[0][j] = dp[0][j - 1] + grid[0][j]
    }
    // 狀態轉移：首列
    for (i in 1..<n) {
        dp[i][0] = dp[i - 1][0] + grid[i][0]
    }
    // 狀態轉移：其餘行和列
    for (i in 1..<n) {
        for (j in 1..<m) {
            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
        }
    }
    return dp[n - 1][m - 1]
}
```

=== "Ruby"

```ruby title="min_path_sum.rb"
[class]{}-[func]{min_path_sum_dp}
```

=== "Zig"

```zig title="min_path_sum.zig"
// 最小路徑和：動態規劃
fn minPathSumDP(comptime grid: anytype) i32 {
    comptime var n = grid.len;
    comptime var m = grid[0].len;
    // 初始化 dp 表
    var dp = [_][m]i32{[_]i32{0} ** m} ** n;
    dp[0][0] = grid[0][0];
    // 狀態轉移：首行
    for (1..m) |j| {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
    }
    // 狀態轉移：首列
    for (1..n) |i| {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
    }
    // 狀態轉移：其餘行和列
    for (1..n) |i| {
        for (1..m) |j| {
            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
    }
    return dp[n - 1][m - 1];
}
```

??? pythontutor "視覺化執行"

<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%91%E5%92%8C%EF%BC%9A%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%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%2A%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281%2C%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281%2C%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281%2C%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%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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%91%E5%92%8C%EF%BC%9A%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%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%2A%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281%2C%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281%2C%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%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281%2C%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%2C%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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">全螢幕觀看 ></a></div>

圖 14-16 展示了最小路徑和的狀態轉移過程，其走訪了整個網格，因此時間複雜度為 $O(nm)$ 。

陣列 dp 大小為 n \times m ，因此空間複雜度為 $O(nm)$ 。

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=== "<11>" { class="animation-figure" }

=== "<12>" { class="animation-figure" }

圖 14-16 最小路徑和的動態規劃過程

4. 空間最佳化

由於每個格子只與其左邊和上邊的格子有關，因此我們可以只用一個單行陣列來實現 dp 表。

請注意，因為陣列 dp 只能表示一行的狀態，所以我們無法提前初始化首列狀態，而是在走訪每行時更新它：

=== "Python"

```python title="min_path_sum.py"
def min_path_sum_dp_comp(grid: list[list[int]]) -> int:
    """最小路徑和：空間最佳化後的動態規劃"""
    n, m = len(grid), len(grid[0])
    # 初始化 dp 表
    dp = [0] * m
    # 狀態轉移：首行
    dp[0] = grid[0][0]
    for j in range(1, m):
        dp[j] = dp[j - 1] + grid[0][j]
    # 狀態轉移：其餘行
    for i in range(1, n):
        # 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0]
        # 狀態轉移：其餘列
        for j in range(1, m):
            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
    return dp[m - 1]
```

=== "C++"

```cpp title="min_path_sum.cpp"
/* 最小路徑和：空間最佳化後的動態規劃 */
int minPathSumDPComp(vector<vector<int>> &grid) {
    int n = grid.size(), m = grid[0].size();
    // 初始化 dp 表
    vector<int> dp(m);
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (int j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (int i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (int j = 1; j < m; j++) {
            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

=== "Java"

```java title="min_path_sum.java"
/* 最小路徑和：空間最佳化後的動態規劃 */
int minPathSumDPComp(int[][] grid) {
    int n = grid.length, m = grid[0].length;
    // 初始化 dp 表
    int[] dp = new int[m];
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (int j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (int i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (int j = 1; j < m; j++) {
            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

=== "C#"

```csharp title="min_path_sum.cs"
/* 最小路徑和：空間最佳化後的動態規劃 */
int MinPathSumDPComp(int[][] grid) {
    int n = grid.Length, m = grid[0].Length;
    // 初始化 dp 表
    int[] dp = new int[m];
    dp[0] = grid[0][0];
    // 狀態轉移：首行
    for (int j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (int i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (int j = 1; j < m; j++) {
            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

=== "Go"

```go title="min_path_sum.go"
/* 最小路徑和：空間最佳化後的動態規劃 */
func minPathSumDPComp(grid [][]int) int {
    n, m := len(grid), len(grid[0])
    // 初始化 dp 表
    dp := make([]int, m)
    // 狀態轉移：首行
    dp[0] = grid[0][0]
    for j := 1; j < m; j++ {
        dp[j] = dp[j-1] + grid[0][j]
    }
    // 狀態轉移：其餘行和列
    for i := 1; i < n; i++ {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0]
        // 狀態轉移：其餘列
        for j := 1; j < m; j++ {
            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]
        }
    }
    return dp[m-1]
}
```

=== "Swift"

```swift title="min_path_sum.swift"
/* 最小路徑和：空間最佳化後的動態規劃 */
func minPathSumDPComp(grid: [[Int]]) -> Int {
    let n = grid.count
    let m = grid[0].count
    // 初始化 dp 表
    var dp = Array(repeating: 0, count: m)
    // 狀態轉移：首行
    dp[0] = grid[0][0]
    for j in 1 ..< m {
        dp[j] = dp[j - 1] + grid[0][j]
    }
    // 狀態轉移：其餘行
    for i in 1 ..< n {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0]
        // 狀態轉移：其餘列
        for j in 1 ..< m {
            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
        }
    }
    return dp[m - 1]
}
```

=== "JS"

```javascript title="min_path_sum.js"
/* 最小路徑和：狀態壓縮後的動態規劃 */
function minPathSumDPComp(grid) {
    const n = grid.length,
        m = grid[0].length;
    // 初始化 dp 表
    const dp = new Array(m);
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (let j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (let i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (let j = 1; j < m; j++) {
            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

=== "TS"

```typescript title="min_path_sum.ts"
/* 最小路徑和：狀態壓縮後的動態規劃 */
function minPathSumDPComp(grid: Array<Array<number>>): number {
    const n = grid.length,
        m = grid[0].length;
    // 初始化 dp 表
    const dp = new Array(m);
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (let j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (let i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (let j = 1; j < m; j++) {
            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

=== "Dart"

```dart title="min_path_sum.dart"
/* 最小路徑和：空間最佳化後的動態規劃 */
int minPathSumDPComp(List<List<int>> grid) {
  int n = grid.length, m = grid[0].length;
  // 初始化 dp 表
  List<int> dp = List.filled(m, 0);
  dp[0] = grid[0][0];
  for (int j = 1; j < m; j++) {
    dp[j] = dp[j - 1] + grid[0][j];
  }
  // 狀態轉移：其餘行
  for (int i = 1; i < n; i++) {
    // 狀態轉移：首列
    dp[0] = dp[0] + grid[i][0];
    // 狀態轉移：其餘列
    for (int j = 1; j < m; j++) {
      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
    }
  }
  return dp[m - 1];
}
```

=== "Rust"

```rust title="min_path_sum.rs"
/* 最小路徑和：空間最佳化後的動態規劃 */
fn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {
    let (n, m) = (grid.len(), grid[0].len());
    // 初始化 dp 表
    let mut dp = vec![0; m];
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for j in 1..m {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for i in 1..n {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for j in 1..m {
            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    dp[m - 1]
}
```

=== "C"

```c title="min_path_sum.c"
/* 最小路徑和：空間最佳化後的動態規劃 */
int minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {
    // 初始化 dp 表
    int *dp = calloc(m, sizeof(int));
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (int j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (int i = 1; i < n; i++) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        // 狀態轉移：其餘列
        for (int j = 1; j < m; j++) {
            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    int res = dp[m - 1];
    // 釋放記憶體
    free(dp);
    return res;
}
```

=== "Kotlin"

```kotlin title="min_path_sum.kt"
/* 最小路徑和：空間最佳化後的動態規劃 */
fun minPathSumDPComp(grid: Array<IntArray>): Int {
    val n = grid.size
    val m = grid[0].size
    // 初始化 dp 表
    val dp = IntArray(m)
    // 狀態轉移：首行
    dp[0] = grid[0][0]
    for (j in 1..<m) {
        dp[j] = dp[j - 1] + grid[0][j]
    }
    // 狀態轉移：其餘行
    for (i in 1..<n) {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0]
        // 狀態轉移：其餘列
        for (j in 1..<m) {
            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
        }
    }
    return dp[m - 1]
}
```

=== "Ruby"

```ruby title="min_path_sum.rb"
[class]{}-[func]{min_path_sum_dp_comp}
```

=== "Zig"

```zig title="min_path_sum.zig"
// 最小路徑和：空間最佳化後的動態規劃
fn minPathSumDPComp(comptime grid: anytype) i32 {
    comptime var n = grid.len;
    comptime var m = grid[0].len;
    // 初始化 dp 表
    var dp = [_]i32{0} ** m;
    // 狀態轉移：首行
    dp[0] = grid[0][0];
    for (1..m) |j| {
        dp[j] = dp[j - 1] + grid[0][j];
    }
    // 狀態轉移：其餘行
    for (1..n) |i| {
        // 狀態轉移：首列
        dp[0] = dp[0] + grid[i][0];
        for (1..m) |j| {
            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];
        }
    }
    return dp[m - 1];
}
```

??? pythontutor "視覺化執行"

<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_comp%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%91%E5%92%8C%EF%BC%9A%E7%A9%BA%E9%96%93%E6%9C%80%E4%BD%B3%E5%8C%96%E5%BE%8C%E7%9A%84%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%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%5B0%5D%20%2A%20m%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281%2C%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E8%A1%8C%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E5%88%97%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281%2C%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D%2C%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E7%A9%BA%E9%96%93%E6%9C%80%E4%BD%B3%E5%8C%96%E5%BE%8C%E7%9A%84%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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_comp%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%91%E5%92%8C%EF%BC%9A%E7%A9%BA%E9%96%93%E6%9C%80%E4%BD%B3%E5%8C%96%E5%BE%8C%E7%9A%84%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%22%22%22%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%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%5B0%5D%20%2A%20m%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281%2C%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E8%A1%8C%0A%20%20%20%20for%20i%20in%20range%281%2C%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20%E7%8B%80%E6%85%8B%E8%BD%89%E7%A7%BB%EF%BC%9A%E5%85%B6%E9%A4%98%E5%88%97%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281%2C%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D%2C%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%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%2C%203%2C%201%2C%205%5D%2C%20%5B2%2C%202%2C%204%2C%202%5D%2C%20%5B5%2C%203%2C%202%2C%201%5D%2C%20%5B4%2C%203%2C%205%2C%202%5D%5D%0A%20%20%20%20n%2C%20m%20%3D%20len%28grid%29%2C%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E7%A9%BA%E9%96%93%E6%9C%80%E4%BD%B3%E5%8C%96%E5%BE%8C%E7%9A%84%E5%8B%95%E6%85%8B%E8%A6%8F%E5%8A%83%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22%E5%BE%9E%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%91%E5%92%8C%E7%82%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">全螢幕觀看 ></a></div>

71 KiB Raw Blame History Unescape Escape

14.3 動態規劃解題思路

14.3.1 問題判斷

14.3.2 問題求解步驟

1. 方法一：暴力搜尋

2. 方法二：記憶化搜尋

3. 方法三：動態規劃

4. 空間最佳化

71 KiB

Raw Blame History