How do you solve Unique Paths on LeetCode?

Unique Paths (LeetCode #62) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(1) space complexity. Key insight: Understanding the combinatorial nature of the problem can lead to a much more efficient solution.

What is the brute force solution for Unique Paths?

The brute force approach for Unique Paths is Brute Force, with O(2^(m+n)) time complexity and O(m+n) space complexity. The brute force approach involves exploring all possible paths the robot can take to reach the bottom-right corner. This means recursively trying every combination of moves down and right until we reach the destination. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Unique Paths?

Unique Paths uses the following concepts: Math, Dynamic Programming, Combinatorics. The recommended patterns to study are: Dynamic Programming, Combinatorics.

What are the interview tips for Unique Paths?

Always start with a brute force solution to understand the problem before optimizing. Be ready to explain your thought process, especially when transitioning from brute force to optimal solutions. Practice calculating combinations and factorials, as they often come up in grid-based problems.

What are common mistakes when solving Unique Paths?

Not considering the base cases in recursive solutions, leading to infinite recursion. Overcomplicating the problem by trying to track every path instead of recognizing the combinatorial nature.

#62

Unique Paths

Medium

MathDynamic ProgrammingCombinatoricsDynamic ProgrammingCombinatorics

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(2^(m+n))	O(n)
Space	O(m+n)	O(1)

💡

Intuition

Time O(n)Space O(1)

The optimal solution leverages combinatorial mathematics. The robot needs to make exactly (m-1) down moves and (n-1) right moves, regardless of the order. The total number of unique paths can be calculated using combinations.

⚙️

Algorithm

3 steps

1Step 1: Calculate the total number of moves required: totalMoves = (m - 1) + (n - 1).
2Step 2: Calculate the number of down moves needed: downMoves = m - 1.
3Step 3: Use the combination formula C(totalMoves, downMoves) = totalMoves! / (downMoves! * (totalMoves - downMoves)!).

solution.py6 lines

1import math
2
3def uniquePaths(m, n):
4    totalMoves = (m - 1) + (n - 1)
5    downMoves = m - 1
6    return math.comb(totalMoves, downMoves)

ℹ

Complexity note: The time complexity is O(n) due to the factorial calculations, but since they can be computed in constant space, the space complexity is O(1).

1Understanding the combinatorial nature of the problem can lead to a much more efficient solution.
2Recursive solutions can be intuitive but often lead to performance issues with larger inputs.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.