Unique Paths

LeetCode 62 | Difficulty: Medium

Medium

Problem Description

There is a robot on an m x n grid. The robot is initially located at the top-left corner (i.e., grid[0][0]). The robot tries to move to the bottom-right corner (i.e., grid[m - 1][n - 1]). The robot can only move either down or right at any point in time.

Given the two integers m and n, return the number of possible unique paths that the robot can take to reach the bottom-right corner.

The test cases are generated so that the answer will be less than or equal to 2 * 10^9.

Example 1:

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

Example 2:

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 -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

Constraints:

- `1 <= m, n <= 100`

Topics: Math, Dynamic Programming, Combinatorics

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

Mathematical

Look for mathematical patterns or formulas. Consider: modular arithmetic, GCD/LCM, prime factorization, combinatorics, or geometric properties.

When to use

Problems with clear mathematical structure, counting, number properties.

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Solutions

Solution 1: C# (Best: 48 ms)

Metric	Value
Runtime	48 ms
Memory	N/A
Date	2018-03-06

Solution

public class Solution {
    public int UniquePaths(int m, int n) {
        
 int[,] dp = new int[m,n];
    for (int i = 0; i < m; i++)
    {
        for (int j = 0; j < n; j++)
        {
            if(i==0|| j==0) dp[i,j] = 1;
            else dp[i,j] = dp[i-1,j]+dp[i,j-1];
        }
    }
    return dp[m-1,n-1];
}
}

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Complexity Analysis

Approach	Time	Space
DP (2D)	$O(n × m)$	$O(n × m)$

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Interview Tips

Key Points

• Discuss the brute force approach first, then optimize. Explain your thought process.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.