62 Unique Paths

This is a classic dynamic programming problem. We need to count how many ways exist when we start from the top-left cell in the grid and make way till the bottom-right cell.

Let’s think about reduction. Reduction is the process of breaking down a problem into smaller sub-problems. To calculate the number of ways to reach cell (m, n) - is basically the number of ways to reach cell (m-1, n) PLUS the number of ways to reach cell (m, n-1).

Note: the number of ways to go from (0,0) to (m, n) == the number of ways to go from (m, n) to (0, 0).

Code

Python3

  def uniquePaths(self, m: int, n: int) -> int:
 
      # (m-1, n) OR (m, n-1)
 
      @functools.cache
      def compute(i, j):
          if i == 0 or j == 0:
              return 1
 
          return compute(i-1, j) + compute(i, j-1)
 
 
      return compute(m-1, n-1)

Big O Analysis

• Runtime

  The runtime complexity here is $O(N)$ with the caching - without caching it’s $O(2^N)$.

• Memory

  The memory usage is $O(N)$ since the caching mechanism would require some sort of dictionary.

— A

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