Given a directed and strongly connected graph with non negative edge weighs. We define 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.

Examples:

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 solution of problem 9.2 here for proof that above steps find minimum average weight.

`// C++ program to find minimum average ` `// weight of a cycle in connected and ` `// directed graph. ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `const` `int` `V = 4; ` ` ` `// a struct to represent edges ` `struct` `edge ` `{ ` ` ` `int` `from, weight; ` `}; ` ` ` `// vector to store edges ` `vector <edge> edges[V]; ` ` ` `void` `addedge(` `int` `u,` `int` `v,` `int` `w) ` `{ ` ` ` `edges[v].push_back({u, w}); ` `} ` ` ` `// calculates the shortest path ` `void` `shortestpath(` `int` `dp[][V]) ` `{ ` ` ` `// initializing all distances as -1 ` ` ` `for` `(` `int` `i=0; i<=V; i++) ` ` ` `for` `(` `int` `j=0; j<V; j++) ` ` ` `dp[i][j] = -1; ` ` ` ` ` `// shortest distance from first vertex ` ` ` `// to in tself consisting of 0 edges ` ` ` `dp[0][0] = 0; ` ` ` ` ` `// filling up the dp table ` ` ` `for` `(` `int` `i=1; i<=V; i++) ` ` ` `{ ` ` ` `for` `(` `int` `j=0; j<V; j++) ` ` ` `{ ` ` ` `for` `(` `int` `k=0; k<edges[j].size(); k++) ` ` ` `{ ` ` ` `if` `(dp[i-1][edges[j][k].from] != -1) ` ` ` `{ ` ` ` `int` `curr_wt = dp[i-1][edges[j][k].from] + ` ` ` `edges[j][k].weight; ` ` ` `if` `(dp[i][j] == -1) ` ` ` `dp[i][j] = curr_wt; ` ` ` `else` ` ` `dp[i][j] = min(dp[i][j], curr_wt); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` `} ` `} ` ` ` `// Returns minimum value of average weight of a ` `// cycle in graph. ` `double` `minAvgWeight() ` `{ ` ` ` `int` `dp[V+1][V]; ` ` ` `shortestpath(dp); ` ` ` ` ` `// array to store the avg values ` ` ` `double` `avg[V]; ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `avg[i] = -1; ` ` ` ` ` `// Compute average values for all vertices using ` ` ` `// weights of shortest paths store in dp. ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `{ ` ` ` `if` `(dp[V][i] != -1) ` ` ` `{ ` ` ` `for` `(` `int` `j=0; j<V; j++) ` ` ` `if` `(dp[j][i] != -1) ` ` ` `avg[i] = max(avg[i], ` ` ` `((` `double` `)dp[V][i]-dp[j][i])/(V-j)); ` ` ` `} ` ` ` `} ` ` ` ` ` `// Find minimum value in avg[] ` ` ` `double` `result = avg[0]; ` ` ` `for` `(` `int` `i=0; i<V; i++) ` ` ` `if` `(avg[i] != -1 && avg[i] < result) ` ` ` `result = avg[i]; ` ` ` ` ` `return` `result; ` `} ` ` ` `// Driver function ` `int` `main() ` `{ ` ` ` `addedge(0, 1, 1); ` ` ` `addedge(0, 2, 10); ` ` ` `addedge(1, 2, 3); ` ` ` `addedge(2, 3, 2); ` ` ` `addedge(3, 1, 0); ` ` ` `addedge(3, 0, 8); ` ` ` ` ` `cout << minAvgWeight(); ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

Output:

1.66667

Here the graph with no cycle will return value as -1.

**Reference:**

https://courses.csail.mit.edu/6.046/fall01/handouts/ps9sol.pdf

https://www.hackerearth.com/practice/notes/karp-minimum-mean-weighted-cycle/

Introduction to Algorithms Third Edition page 681 by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein

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