62. Unique Paths

Problem:

A robot is located at the top-left corner of a m x n grid (marked ‘Start’ in the diagram below).

The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked ‘Finish’ in the diagram below).

How many possible unique paths are there?

Above is a 7 x 3 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

Example 1:

Input: m = 3, n = 2
Output: 3
Explanation:
From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
1. Right -> Right -> Down
2. Right -> Down -> Right
3. Down -> Right -> Right

Example 2:

Input: m = 7, n = 3
Output: 28

Solution:

DP.

Define f(i, j) to be the number of total unique paths from (0, 0) to (i, j).

f(i, 0) = 1
f(0, j) = 1
f(i, j) = f(i-1, j) + f(i, j-1)

Only two previous states are dependant. Use dynamic array to reduce memory allocation.

/**
 * @param {number} m
 * @param {number} n
 * @return {number}
 */
let uniquePaths = function (m, n) {
  const dp = new Array(m).fill(1);
  while (--n > 0) {
    for (let i = 1; i < m; i++) {
      dp[i] += dp[i - 1];
    }
  }
  return dp[m - 1] || 1;
};

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