Is Unique Paths easy or hard?

Unique Paths is generally considered a medium-level problem. The DP solution is straightforward once the recurrence relation is identified, but recognizing the combinatorial shortcut requires stronger mathematical insight.

What is the best approach for Unique Paths?

Dynamic Programming is the most common approach because it models the grid directly with the transition dp[i][j] = dp[i-1][j] + dp[i][j-1]. It runs in O(m*n) time and clearly shows how each path builds from previous cells. A combinatorial solution using binomial coefficients is more optimal in space and runs in O(min(m,n)) time.

What data structure is used in Unique Paths?

The dynamic programming solution typically uses a 2D array (or a 1D optimized array) to store the number of paths to each grid cell. The combinatorial solution instead relies on mathematical computation without additional data structures.

What is the time complexity of Unique Paths?

The standard dynamic programming solution runs in O(m*n) time because every cell in the grid is computed once. The combinatorial math approach improves this to O(min(m,n)) by calculating the binomial coefficient C(m+n-2, m-1) iteratively.

Unique Paths Python or Java solution approach?

Both Python and Java implementations usually follow the DP grid approach where a matrix is filled using dp[i][j] = dp[i-1][j] + dp[i][j-1]. Many optimized solutions use a single 1D array to reduce space from O(m*n) to O(n).

How to solve Unique Paths in O(n) or better time?

Recognize that the robot must take exactly m-1 down moves and n-1 right moves. The number of valid sequences equals C(m+n-2, m-1). Computing this binomial coefficient iteratively takes O(min(m,n)) time and O(1) space without building the grid.

Is Unique Paths asked at Google, Amazon, or Meta?

Unique Paths is a common grid dynamic programming question asked across major tech companies including Amazon, Google, and Meta. It tests understanding of DP state transitions and recognizing combinatorial patterns.

Description Solution Video 10:48

62. Unique Paths

Medium67.3% acceptance21 companies

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 * 10⁹.

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: