This is a Python solution for the following algorithmic problem that can be found on LeetCode website (LeetCode problem 62, Unique paths).
The core ideas behind this solution:
- This can be solved in a few different ways, including backtracking and dynamic programming.
- Our code example is using dynamic programming
- Solutions for up to m==2 , n==2 can be used as a basis
- Every element after 2x2 grid is calculated as follows: if it is start of the row or start of the column, its value is 1. Otherwise, its value is a sum of the element above it and the one left to it.
- Element values are stored in a table (tabulation technique) and the final solution is the table value at coordinates (m-1,n-1)
Video example for 4x4 grid:
''' # Unique paths in a grid A small example of using dynamic programming techniques for finding out the total number of unique paths between two points in a 2D grid. Author: Goran Trlin Find more tutorials and code samples on: https://playandlearntocode.com ''' class Solution: # camelCase is used at the LeetCode website # def uniquePaths(self, m: int, n: int) -> int: def unique_paths(self, m: int, n: int) -> int: ''' Dynamic programming solution On paper, it's just about summation of left and above at each new element An alternative would be backtracking. ''' if m == 0 or n == 0: return 0 if m == 1 and n == 1: return 1 if m == 2 and n == 1: return 1 if m == 1 and n == 2: return 1 if m == 2 and n == 2: return 2 cur_m = 1 cur_n = 1 if cur_m > m - 1: cur_m = m - 1 if cur_n > n - 1: cur_n = n - 1 table = [[0 for x in range(n)] for y in range(m)] # create table done_m = False done_n = False while cur_m < m and cur_n < n: temp_m = 0 temp_n = 0 # go right: if done_m == False: while temp_n <= cur_n: if temp_n == 0: table[cur_m][temp_n] = 1 else: table[cur_m][temp_n] = table[cur_m][temp_n - 1] + table[cur_m - 1][temp_n] temp_n += 1 # go down: if done_n == False: while temp_m <= cur_m: if temp_m == 0: table[temp_m][cur_n] = 1 else: table[temp_m][cur_n] = table[temp_m][cur_n - 1] + table[temp_m - 1][cur_n] temp_m += 1 if cur_m == m - 1 and cur_n == n - 1: return table[cur_m][cur_n] # done cur_m += 1 cur_n += 1 if cur_m > m - 1: done_m = True cur_m = m - 1 if cur_n > n - 1: done_n = True cur_n = n - 1 return 0
The code for this project can be found on GitHub.