Given two number N and M. The task is to find the number of shortest paths to reach the cell(i, j) in the grid of size N × M when the moves started from the bottom-left corner
Note: cell(i, j) represents the ith row and jth column in the grid
Input : N = 3, M = 4 Output : 1 3 6 10 1 2 3 4 1 1 1 1 Input : N = 5, M = 2 Output : 1 5 1 4 1 3 1 2 1 1
Approach : An efficient approach is to compute the grid starting from the bottom-left corner.
- The number of shortest paths to reach cell(n, i) is 1, where, 1 < = i < = M
- The number of shortest paths to reach cell(i, 1) is 1, where, 1 < = i < = N
- The number of shortest paths to reach cell(i, j) are the sum the number of shortest paths of cell(i-1, j) and (i, j+1), where, 1 < = j < = M and 1 < = i < = N
Below is the implementation of the above approach :
1 5 1 4 1 3 1 2 1 1
Time complexity: O(N × M)
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