What is the optimal time complexity of solving Unique Paths?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Unique Paths?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Unique Paths?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Unique Paths?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Unique Paths

Learn how to solve the 'Unique Paths' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Medium

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.

Write a function uniquePaths(m: int, n: int) -> int.

Constraints

•1 <= m, n <= 100

Examples

Example 1

Input

m = 3, n = 7

Output

Explanation

There are 28 unique paths to go from top-left to bottom-right.

Example 2

Input

m = 3, n = 2

Output

Explanation

From the top-left corner, there are 3 ways: Right -> Down -> Down, Down -> Down -> Right, or Down -> Right -> Down.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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