893268. Minimum Weight Cycle
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
dpandprevarrays with infinite values. - For each vertex
i, calculate the minimum weight of the path fromito itself, which is simply the weight of the edge incident oni. - For each pair of vertices
(i, j), updatedp[i][j]as the minimum weight of the path fromitoj. Ifdp[i][j]is less than the current minimum weight, update it and setprev[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
prevarray 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
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;
}
}