Here is the detailed Markdown blog post for the Minimum Weight Cycle problem:

Minimum Weight Cycle

Summary

The Minimum Weight Cycle problem involves finding the minimum weight cycle in a graph. A cycle is a path that starts and ends at the same vertex, while the weight of a cycle is the sum of its edge weights. The goal is to find the cycle with the minimum total weight.

Detailed Explanation

To solve this problem, we'll use a dynamic programming approach. We'll create a 2D array dp where dp[i][j] represents the minimum weight of the path from vertex i to vertex j. We'll also maintain an array prev to keep track of the previous vertex in the optimal cycle.

Here's the step-by-step breakdown:

Initialize dp and prev arrays with infinite values.
For each vertex i, calculate the minimum weight of the path from i to itself, which is simply the weight of the edge incident on i.
For each pair of vertices (i, j), update dp[i][j] as the minimum weight of the path from i to j. If dp[i][j] is less than the current minimum weight, update it and set prev[j] = i.
To find the minimum weight cycle, iterate through the vertices and keep track of the vertex with the minimum weight in the cycle.
Use prev array to construct the minimum weight cycle.

Here's an ASCII art diagram illustrating the dynamic programming approach:

  +--------+
  |   i    |
  +--------+
       |
       v
  +--------+
  |  dp[i][j] |
  |  (min weight)|
  +--------+
       ^
       |
  +--------+
  |  prev[j] = i |
  +--------+

Time complexity: O(|E| * |V|), where |E| is the number of edges and |V| is the number of vertices. Space complexity: O(|V|^2).

Optimized Solutions

Java

View GeeksforGeeks Problem

public class MinimumWeightCycle {
    public static int minWeightCycle(int[][] graph) {
        int n = graph.length;
        int[] dp = new int[n];
        int[] prev = new int[n];

        for (int i = 0; i < n; i++) {
            dp[i] = Integer.MAX_VALUE;
            prev[i] = -1;
        }

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (graph[i][j] != 0) {
                    if (dp[j] > dp[i] + graph[i][j]) {
                        dp[j] = dp[i] + graph[i][j];
                        prev[j] = i;
                    }
                }
            }
        }

        int minWeight = Integer.MAX_VALUE;
        int startVertex = -1;

        for (int i = 0; i < n; i++) {
            if (dp[i] < minWeight) {
                minWeight = dp[i];
                startVertex = i;
            }
        }

        return minWeight;
    }
}

Code Editor (Testing phase)

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893268. Minimum Weight Cycle