Number of paths from source to destination in a directed acyclic graph
Given a Directed Acyclic Graph with n vertices and m edges. The task is to find the number of different paths that exist from a source vertex to destination vertex.
Input: source = 0, destination = 4
0 -> 2 -> 3 -> 4
0 -> 3 -> 4
0 -> 4
Input: source = 0, destination = 1
Explanation: There exists only one path 0->1
Approach: Let f(u) be the number of ways one can travel from node u to destination vertex. Hence, f(source) is required answer. As f(destination) = 1 here so there is just one path from destination to itself. One can observe, f(u) depends on nothing other than the f values of all the nodes which are possible to travel from u. It makes sense because the number of different paths from u to the destination is the sum of all different paths from v1, v2, v3… v-n to destination vertex where v1 to v-n are all the vertices that have a direct path from vertex u. This approach, however, is too slow to be useful. Each function call branches out into further calls, and that branches into further calls, until each and every path is explored once.
The problem with this approach is the calculation of f(u) again and again each time the function is called with argument u. Since this problem exhibits both overlapping subproblems and optimal substructure, dynamic programming is applicable here. In order to evaluate f(u) for each u just once, evaluate f(v) for all v that can be visited from u before evaluating f(u). This condition is satisfied by reverse topological sorted order of the nodes of the graph.
Below is the implementation of the above approach:
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