NxN网格中的最小求和下降路径

给定一个大小为NxN的整数的正方形数组A。任务是找到通过A的下降路径的最小和。
下降路径将从第一行中的任何元素开始，到最后一行结束。它从下一行中选择一个元素。下一行的选择必须在与上一行的列最多相差一列的列中。

例子：

Input: N = 2
mat[2][2] = 
{{5, 10},
{25, 15}}
Output: 20
Selected elements are 5, 15.

Input: N = 3
mat[3][3] =
{{1, 2, 3},
{ 4, 5, 6},
{ 7, 8, 9}}
Output: 12
Selected elements are 1, 4, 7.

方法：此问题具有最佳子结构，这意味着子问题的解决方案可用于解决此问题的更大实例。这使得动态编程应运而生。

令dp [R] [C]是从第一行的[R，C]开始到A的最底行的下降路径的最小总权重。

然后， $dp[R][C] = A[R][C] + min(dp[R+1, C-1], dp[R+1, C], dp[R+1, C+1])$ ，答案是第一行的最小值i：e $\underset{C}{min}\; dp(0, C)$ 。

我们将创建一个辅助数组dp来缓存中间值dp [R] [C] 。但是，我们将使用A来缓存这些值。我们的目标是将A的值转换为dp的值。

我们从第二行开始处理每一行。我们设置$A[R][C] = min(A[R+1, C-1], A[R+1, C], A[R+1, C+1])$ ，优雅地处理边界条件。

Explanation of above Approach:
Let’s look at the recursion a little more to get a handle on why it works. For an array like A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], imagine you are at (1, 0) (A[1][0] = 4). You can either go to (2, 0) and get a weight of 7, or (2, 1) and get a weight of 8. Since 7 is lower, we say that the minimum total weight at (1, 0) is dp(1, 0) = 5 + 7 (7 for the original A[R][C].)

After visiting (1, 0), (1, 1), and (1, 2), A [which is storing the values of our dp], looks like [[1, 2, 3], [11, 12, 14], [7, 8, 9]]. We do this procedure again by visiting (0, 0), (0, 1), (0, 2).
We get $A=\begin{bmatrix} 12 & 13 & 15\\ 11 & 12& 14\\ 7 & 8 & 9 \end{bmatrix}$, and the final answer is min(A[0][C]) = 12 for all C in range 0 to n.

下面是上述方法的实现。

C++

// C++ Program to minimum required sum
#include 
using namespace std;
  
const int n = 3;
  
// Function to return minimum path falling sum
int minFallingPathSum(int (&A)[n][n])
{
  
    // R = Row and C = Column
    // We begin from second last row and keep
    // adding maximum sum.
    for (int R = n - 2; R >= 0; --R) {
        for (int C = 0; C < n; ++C) {
  
            // best = min(A[R+1][C-1], A[R+1][C], A[R+1][C+1])
            int best = A[R + 1][C];
            if (C > 0)
                best = min(best, A[R + 1][C - 1]);
            if (C + 1 < n)
                best = min(best, A[R + 1][C + 1]);
            A[R][C] = A[R][C] + best;
        }
    }
  
    int ans = INT_MAX;
    for (int i = 0; i < n; ++i)
        ans = min(ans, A[0][i]);
    return ans;
}
  
// Driver program
int main()
{
  
    int A[n][n] = { { 1, 2, 3 },
                    { 4, 5, 6 },
                    { 7, 8, 9 } };
  
    // function to print required answer
    cout << minFallingPathSum(A);
  
    return 0;
}

Java

// Java Program to minimum required sum
  
import java.io.*;
  
class GFG {
static int n = 3;
  
// Function to return minimum path falling sum
static int minFallingPathSum(int A[][])
{
  
    // R = Row and C = Column
    // We begin from second last row and keep
    // adding maximum sum.
    for (int R = n - 2; R >= 0; --R) {
        for (int C = 0; C < n; ++C) {
  
            // best = min(A[R+1][C-1], A[R+1][C], A[R+1][C+1])
            int best = A[R + 1][C];
            if (C > 0)
                best = Math.min(best, A[R + 1][C - 1]);
            if (C + 1 < n)
                best = Math.min(best, A[R + 1][C + 1]);
            A[R][C] = A[R][C] + best;
        }
    }
  
    int ans = Integer.MAX_VALUE;
    for (int i = 0; i < n; ++i)
        ans = Math.min(ans, A[0][i]);
    return ans;
}
  
// Driver program
public static void main (String[] args) {
            int A[][] = { { 1, 2, 3 },
                    { 4, 5, 6 },
                    { 7, 8, 9 } };
  
    // function to print required answer
    System.out.println( minFallingPathSum(A));
    }
}
// This code is contributed by inder_verma..

Python 3

# Python3 Program to minimum 
# required sum 
import sys
  
n = 3
  
# Function to return minimum 
# path falling sum 
def minFallingPathSum(A) :
  
    # R = Row and C = Column 
    # We begin from second last row and keep 
    # adding maximum sum. 
    for R in range(n - 2, -1, -1) :
        for C in range(n) :
  
            # best = min(A[R+1][C-1], A[R+1][C],
            # A[R+1][C+1]) 
            best = A[R + 1][C]
            if C > 0 :
                best = min(best, A[R + 1][C - 1])
            if C + 1 < n :
                best = min(best, A[R + 1][C + 1])
  
            A[R][C] = A[R][C] + best
  
    ans = sys.maxsize
  
    for i in range(n) :
        ans = min(ans, A[0][i])
          
    return ans
              
  
  
# Driver code
if __name__ == "__main__" :
  
    A = [ [ 1, 2, 3],
        [ 4, 5, 6],
        [ 7, 8, 9] ]
  
    # function to print required answer 
    print(minFallingPathSum(A))
  
# This code is contributed by 
# ANKITRAI1

C#

// C# Program to minimum required sum
  
using System;
  
class GFG {
static int n = 3;
  
// Function to return minimum path falling sum
static int minFallingPathSum(int[,] A)
{
  
    // R = Row and C = Column
    // We begin from second last row and keep
    // adding maximum sum.
    for (int R = n - 2; R >= 0; --R) {
        for (int C = 0; C < n; ++C) {
  
            // best = min(A[R+1,C-1], A[R+1,C], A[R+1,C+1])
            int best = A[R + 1,C];
            if (C > 0)
                best = Math.Min(best, A[R + 1,C - 1]);
            if (C + 1 < n)
                best = Math.Min(best, A[R + 1,C + 1]);
            A[R,C] = A[R,C] + best;
        }
    }
  
    int ans = int.MaxValue;
    for (int i = 0; i < n; ++i)
        ans = Math.Min(ans, A[0,i]);
    return ans;
}
  
// Driver program
public static void Main () {
            int[,] A = { { 1, 2, 3 },
                    { 4, 5, 6 },
                    { 7, 8, 9 } };
  
    // function to print required answer
    Console.WriteLine( minFallingPathSum(A));
    }
}
// This code is contributed by Subhadeep..

输出：

12

时间复杂度： O(N ² )