1289. Minimum Falling Path Sum II

Description

Given an n x n integer matrix grid, return the minimum sum of a falling path with non-zero shifts.

A falling path with non-zero shifts is a choice of exactly one element from each row of grid such that no two elements chosen in adjacent rows are in the same column.

 

Example 1:

Input: grid = [[1,2,3],[4,5,6],[7,8,9]]
Output: 13
Explanation: 
The possible falling paths are:
[1,5,9], [1,5,7], [1,6,7], [1,6,8],
[2,4,8], [2,4,9], [2,6,7], [2,6,8],
[3,4,8], [3,4,9], [3,5,7], [3,5,9]
The falling path with the smallest sum is [1,5,7], so the answer is 13.

Example 2:

Input: grid = [[7]]
Output: 7

 

Constraints:

• n == grid.length == grid[i].length
• 1 <= n <= 200
• -99 <= grid[i][j] <= 99

 

Solutions

Solution: Dynamic Programming

• Time complexity: O(n²)
• Space complexity: O(n)

 

JavaScript

/**
 * @param {number[][]} grid
 * @return {number}
 */
const minFallingPathSum = function (grid) {
  const n = grid.length;
  const dp = [...grid[0]];

  for (let row = 1; row < n; row++) {
    let first = (second = Number.MAX_SAFE_INTEGER);
    let index = -1;

    for (let col = 0; col < n; col++) {
      if (dp[col] < first) {
        second = first;
        first = dp[col];
        index = col;
      } else if (dp[col] < second) second = dp[col];
    }
    for (let col = 0; col < n; col++) {
      const value = grid[row][col];

      dp[col] = index === col ? value + second : value + first;
    }
  }
  return Math.min(...dp);
};