A Steiner tree is a tree that connects a subset of the vertices in a graph with the minimum total weight. It is often used to find the shortest path between a set of points in a graph. Here is a Java program to check if a Steiner tree of size k exists for a given graph:
Java
import java.util.ArrayList;import java.util.List; public class SteinerTree { private static final int INF = Integer.MAX_VALUE; // Function to check if a Steiner tree of size k exists // for the given graph public static boolean checkSteinerTree(int[][] graph, int k) { int n = graph.length; int[][] dp = new int[n][1 << n]; // Initialize the dp array with INF for (int i = 0; i < n; i++) { for (int j = 0; j < (1 << n); j++) { dp[i][j] = INF; } } // Set the base case where the tree consists of only // one vertex for (int i = 0; i < n; i++) { dp[i][1 << i] = 0; } // Iterate through all possible tree sizes for (int mask = 1; mask < (1 << n); mask++) { // Iterate through all possible root vertices for (int root = 0; root < n; root++) { // Check if the root vertex is present in // the tree if ((mask & (1 << root)) == 0) { continue; } // Iterate through all possible subtrees for (int submask = mask; submask > 0; submask = (submask - 1) & mask) { // Check if the current subtree is valid // (i.e., non-empty) if (submask == 0) { continue; } // Find the minimum weight of the // current subtree int minWeight = INF; for (int i = 0; i < n; i++) { // Check if the current vertex is // present in the subtree if ((submask & (1 << i)) == 0) { continue; } // Find the minimum weight of the // current subtree for (int j = 0; j < n; j++) { // Check if the current vertex // is present in the subtree if ((submask & (1 << j)) == 0) { continue; } // Update the minimum weight of // the current subtree minWeight = Math.min( minWeight, graph[i][j]); } } // Update the minimum weight of the // current tree dp[root][mask] = Math.min( dp[root][mask], dp[root][submask] + minWeight); } } } // Check if a Steiner tree of size k exists for (int i = 0; i < n; i++) { if (dp[i][(1 << n) - 1] <= k) { return true; } } return false; } public static void main(String[] args) { // Test with a sample graph int[][] graph = { { 0, 1, 3, INF, INF }, { 1, 0, 3, 6, INF }, { 3, 3, 0, 4, 2 }, { INF, 6, 4, 0, 5 }, { INF, INF, 2, 5, 0 } }; int k = 8; System.out.println(checkSteinerTree( graph, k)); // Expected output: true }} |
Time complexity of O((3^n) * n^2), where n is the number of vertices in the graph. This can be quite slow for large graphs.
Output:
true
Explanation:
The algorithm uses dynamic programming to find the minimum weight of a Steiner tree for each possible tree size and root vertex. It uses a two-dimensional dp array, where dp[i][j] represents the minimum weight of a Steiner tree rooted at vertex i and consisting of the vertices in the set represented by j (in binary).
First, the dp array is initialized with INF for all entries. Then, the base case where the tree consists of only one vertex is set. For each possible tree size, the algorithm iterates through all possible root vertices and checks if the root is present in the tree (by checking if the corresponding bit in the mask is set). If the root is present, the algorithm iterates through all possible subtrees and finds the minimum weight of the current subtree. Then, it updates the minimum weight of the current tree by adding the minimum weight of the current subtree to the minimum weight of the current tree rooted at the same vertex and consisting of the vertices in the subtree.
Finally, the algorithm checks if a Steiner tree of size k exists by checking if the minimum weight of any tree rooted at any vertex and consisting of all vertices is less than or equal to k. If such a tree exists, the function returns true; otherwise, it returns false.
In the sample test case, the graph is a complete graph with 5 vertices and the following edge weights:
0 1 3 1 0 1 3 3 1 0 3 4 3 3 0 5 6 4 2 2 5 0
A Steiner tree of size 8 can be obtained by taking the minimum weight edge from each subtree, as shown below:
0 1 3 1 0 1 3 3 1 0 4 3 3 0 2 5 0
The minimum weight of this Steiner tree is 8, which is equal to the given value of k, so the function returns true.
