Given a directed graph, which may contain cycles, where every edge has weight, the task is to find the minimum cost of any simple path from a given source vertex ‘s’ to a given destination vertex ‘t’. Simple Path is the path from one vertex to another such that no vertex is visited more than once. If there is no simple path possible then return INF(infinite).
The graph is given as adjacency matrix representation where value of graph[i][j] indicates the weight of an edge from vertex i to vertex j and a value INF(infinite) indicates no edge from i to j.
Input : V = 5, E = 6 s = 0, t = 2 graph = 0 1 2 3 4 0 INF -1 INF 1 INF 1 INF INF -2 INF INF 2 -3 INF INF INF INF 3 INF INF -1 INF INF 4 INF INF INF 2 INF Output : -3 Explanation : The minimum cost simple path between 0 and 2 is given by: 0 -----> 1 ------> 2 whose cost is (-1) + (-2) = (-3). Input : V = 5, E = 6 s = 0, t = 4 graph = 0 1 2 3 4 0 INF -7 INF -2 INF 1 INF INF -11 INF INF 2 INF INF INF INF INF 3 INF INF INF 3 -4 4 INF INF INF INF INF Output : -6 Explanation : The minimum cost simple path between 0 and 2 is given by: 0 -----> 3 ------> 4 whose cost is (-2) + (-4) = (-6).
The main idea to solve the above problem is to traverse through all simple paths from s to t using a modified version of Depth First Search and find the minimum cost path amongst them. One important observation about DFS is that it traverses one path at a time, hence we can traverse separate paths independently using DFS by marking the nodes as unvisited before leaving them.
A simple solution is to start from s, go to all adjacent vertices, and follow recursion for further adjacent vertices until we reach the destination. This algorithm will work even when negative weight cycles or self edges are present in the graph.
Below is the implementation of the above-mentioned approach:
- Minimum Cost Path in a directed graph via given set of intermediate nodes
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- Convert the undirected graph into directed graph such that there is no path of length greater than 1
- Shortest Path in a weighted Graph where weight of an edge is 1 or 2
- Find if there is a path between two vertices in a directed graph | Set 2
- Longest Path in a Directed Acyclic Graph | Set 2
- Shortest Path in Directed Acyclic Graph
- Longest Path in a Directed Acyclic Graph
- Find if there is a path between two vertices in a directed graph
- Shortest path in a directed graph by Dijkstra’s algorithm
- Longest path in a directed Acyclic graph | Dynamic Programming
- Maximum weighted edge in path between two nodes in an N-ary tree using binary lifting
- Maximum cost path in an Undirected Graph such that no edge is visited twice in a row
- Minimum edges required to make a Directed Graph Strongly Connected
- Minimum Cost Graph
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