120. Triangle

Problem:

Given a triangle, find the minimum path sum from top to bottom. Each step you may move to adjacent numbers on the row below.

For example, given the following triangle

[
     [2],
    [3,4],
   [6,5,7],
  [4,1,8,3]
]

The minimum path sum from top to bottom is 11 (i.e., 2 + 3 + 5 + 1 = 11).

Note:

Bonus point if you are able to do this using only O(n) extra space, where n is the total number of rows in the triangle.

Solution:

Define f(i, j) to be the minimum path sum from triangle[0][0] to triangle[i][j].

f(i, 0) = f(i-1, j) + triangle[i][0]
f(i, j) = min( f(i-1, j-1), f(i-1, j) ) + triangle[i][j], 0 < j < i
f(i, i) = f(i-1, i-1) + triangle[i][i], i > 0

Dynamic array can be used.

/**
 * @param {number[][]} triangle
 * @return {number}
 */
let minimumTotal = function (triangle) {
  if (triangle.length <= 0) {
    return 0;
  }

  const dp = [triangle[0][0]];
  for (let i = 1; i < triangle.length; i++) {
    dp[i] = dp[i - 1] + triangle[i][i];
    for (let j = i - 1; j >= 1; j--) {
      dp[j] = Math.min(dp[j], dp[j - 1]) + triangle[i][j];
    }
    dp[0] += triangle[i][0];
  }
  return Math.min(...dp);
};

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