Number of Walks from source to destination

Given a graph and two vertices src and dest, count the total number of paths from src to dest where the length of the path is k (there should be exactly k edges between them). Note that the graph is represented as an adjacency matrix.

For example, consider the following graph:

The number of paths from vertex 0 to vertex 3 with length 2 is 2 ({0->1->3} and {0->2->3}).

We have already discussed a O(V3 K) approach in count all possible walks from a source to a destination with exactly k edges. In this post, a O(V3 Log K) approach is discussed.

Approach: The idea is to compute a resultant matrix where result = (graph)^k. The total number of paths from source to destination of length k will then be simply result[src][dest]. We use this technique to compute the exponentiation of the adjacency matrix of the given graph.

The recursion tree of power function used here for exponent = 7 looks like below:

Below is the implementation of the above approach:





# Python3 code to count the number of paths 
# from the source to the destination of length
# k in the given graph
# Function to multiply two matrices
def multiply(mat1, mat2):
    result = [[0 for col in range(len(mat2[0]))] \
                    for row in range(len(mat1))]
    for row in range(len(mat1)):
        for col in range(len(mat2[0])):
            for k in range(len(mat2)):
                result[row][col] += mat1[row][k] * mat2[k][col]
    return result
# Function to calculate (matrix)^exp 
def power(matrix, exp):
    if exp == 1:
        return matrix
        tempMatrix = power(matrix, exp//2)
        res = multiply(tempMatrix, tempMatrix)
        if exp % 2 == 0:
            return res
            return multiply(matrix, res)
# Function to calculate total number of paths from 
# source to destination where length of path is k
def countWalks(graph, src, dest, k):
    resultantMatrix = power(graph, k)
    return resultantMatrix[src][dest]
# Driver code
graph = [[0, 1, 1, 1], 
         [0, 0, 0, 1], 
         [0, 0, 0, 1], 
         [0, 0, 0, 0]]
src, dest, k = 0, 3, 2
print("Total number of walks:", countWalks(graph, src, dest, k))



Total number of walks: 2

Time Complexity: O(V^3 logk)

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