Unique Paths

Description

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?

 

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

 

Constraints:

• 1 <= m, n <= 100
• It's guaranteed that the answer will be less than or equal to 2 * 10 ^ 9.

Solution(javascript)

// Recursion with memo
const uniquePaths = (m, n) => {
  const memo = {}
  const aux = (rowIndex, columnIndex) => {
    if (memo[rowIndex] !== undefined && memo[rowIndex][columnIndex] !== undefined) {
      return memo[rowIndex][columnIndex]
    }
    if (rowIndex >= m || columnIndex >= n) {
      return 0
    }
    if (rowIndex === m - 1 && columnIndex === n - 1) {
      return 1
    }
    memo[rowIndex] = memo[rowIndex] || {}
    memo[rowIndex][columnIndex] = aux(rowIndex + 1, columnIndex) + aux(rowIndex, columnIndex + 1)
    return memo[rowIndex][columnIndex]
  }
  return aux(0, 0)
}