Given a directed and strongly connected graph with non-negative edge weights. We define the mean weight of a cycle as the summation of all the edge weights of the cycle divided by the no. of edges. Our task is to find the minimum mean weight among all the directed cycles of the graph.
Input : Below Graph
Output : 1.66667
Method to find the smallest mean weight value cycle efficiently
Step 1: Choose first vertex as source. Step 2: Compute the shortest path to all other vertices on a path consisting of k edges 0 <= k <= V where V is number of vertices. This is a simple dp problem which can be computed by the recursive solution dp[k][v] = min(dp[k][v], dp[k-1][u] + weight(u,v) where v is the destination and the edge(u,v) should belong to E Step 3: For each vertex calculate max(dp[n][v]-dp[k][v])/(n-k) where 0<=k<=n-1 Step 4: The minimum of the values calculated above is the required answer.
Please refer to the solution of problem 9.2 here for proof that the above steps find minimum average weight.
Here the graph with no cycle will return value as -1.
Introduction to Algorithms Third Edition page 681 by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein
This article is contributed by Ayush Jha. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.
In case you wish to attend live classes with industry experts, please refer Geeks Classes Live