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Unique Paths LC #62 Medium 2D DP
Problem

A robot on an m × n grid starts at top-left and moves to bottom-right. It can only move right or down. Return the number of unique paths.

Example 1
Input: m = 3, n = 7
Output: 28
Example 2
Input: m = 3, n = 3
Output: 6
Constraints: 1 ≤ m, n ≤ 100
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Approach
2D DP — dp[r][c] = paths to reach (r,c)
Recurrence: dp[r][c] = top + left. Base: dp[0][0]=1. Robot moves right or down only.
Grid Fill
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Variables
cell (r,c)
—
top
—
left
—
paths
—
Step Logic
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Algorithm
1
dp[r][c] = paths to reach (r,c). Only right or down.
2
dp[r][c] = top + left (paths from above + from left)
3
Base: dp[0][0]=1. First row & col = 1.
4
Answer = dp[m-1][n-1]
Time
O(m×n)
Space
O(n)
Why This Works

To reach (r,c) you must come from (r-1,c) or (r,c-1). So paths[r][c] = paths from above + paths from left. First row and column are 1 (one straight path). Fill top-to-bottom, left-to-right so dependencies are ready.