From 571d05caa8b851461d490cdfbba9124ee7af598a Mon Sep 17 00:00:00 2001 From: Muhammad Ezzat Date: Mon, 5 May 2025 18:09:28 +0300 Subject: [PATCH] Add Karger's minimum cut algorithm (#6233) --- .../randomized/KargerMinCut.java | 195 ++++++++++++++++++ .../randomized/KargerMinCutTest.java | 114 ++++++++++ 2 files changed, 309 insertions(+) create mode 100644 src/main/java/com/thealgorithms/randomized/KargerMinCut.java create mode 100644 src/test/java/com/thealgorithms/randomized/KargerMinCutTest.java diff --git a/src/main/java/com/thealgorithms/randomized/KargerMinCut.java b/src/main/java/com/thealgorithms/randomized/KargerMinCut.java new file mode 100644 index 000000000..14f1f9745 --- /dev/null +++ b/src/main/java/com/thealgorithms/randomized/KargerMinCut.java @@ -0,0 +1,195 @@ +package com.thealgorithms.randomized; + +import java.util.ArrayList; +import java.util.Collection; +import java.util.HashSet; +import java.util.List; +import java.util.Random; +import java.util.Set; + +/** + * Implementation of Karger's Minimum Cut algorithm. + * + *

Karger's algorithm is a randomized algorithm to compute the minimum cut of a connected graph. + * A minimum cut is the smallest set of edges that, if removed, would split the graph into two + * disconnected components. + * + *

The algorithm works by repeatedly contracting random edges in the graph until only two + * nodes remain. The edges between these two nodes represent a cut. By running the algorithm + * multiple times and keeping track of the smallest cut found, the probability of finding the + * true minimum cut increases. + * + *

Key steps of the algorithm: + *

    + *
  1. Randomly select an edge and contract it, merging the two nodes into one.
    2. + *
  2. Repeat the contraction process until only two nodes remain.
    3. + *
  3. Count the edges between the two remaining nodes to determine the cut size.
    4. + *
  4. Repeat the process multiple times to improve the likelihood of finding the true minimum cut.
    5. + *
+ *

+ * See more: Karger's algorithm + * + * @author MuhammadEzzatHBK + */ +public final class KargerMinCut { + + /** + * Output of the Karger algorithm. + * + * @param first The first set of nodes in the cut. + * @param second The second set of nodes in the cut. + * @param minCut The size of the minimum cut. + */ + public record KargerOutput(Set first, Set second, int minCut) { + } + + private KargerMinCut() { + } + + public static KargerOutput findMinCut(Collection nodeSet, List edges) { + return findMinCut(nodeSet, edges, 100); + } + + /** + * Finds the minimum cut of a graph using Karger's algorithm. + * + * @param nodeSet: Input graph nodes + * @param edges: Input graph edges + * @param iterations: Iterations to run the algorithms for, more iterations = more accuracy + * @return A KargerOutput object containing the two sets of nodes and the size of the minimum cut. + */ + public static KargerOutput findMinCut(Collection nodeSet, List edges, int iterations) { + Graph graph = new Graph(nodeSet, edges); + KargerOutput minCut = new KargerOutput(new HashSet<>(), new HashSet<>(), Integer.MAX_VALUE); + KargerOutput output; + + // Run the algorithm multiple times to increase the probability of finding + for (int i = 0; i < iterations; i++) { + Graph clone = graph.copy(); + output = clone.findMinCut(); + if (output.minCut < minCut.minCut) { + minCut = output; + } + } + return minCut; + } + + private static class DisjointSetUnion { + private final int[] parent; + int setCount; + + DisjointSetUnion(int size) { + parent = new int[size]; + for (int i = 0; i < size; i++) { + parent[i] = i; + } + setCount = size; + } + + int find(int i) { + // If it's not its own parent, then it's not the root of its set + if (parent[i] != i) { + // Recursively find the root of its parent + // and update i's parent to point directly to the root (path compression) + parent[i] = find(parent[i]); + } + + // Return the root (representative) of the set + return parent[i]; + } + + void union(int u, int v) { + // Find the root of each node + int rootU = find(u); + int rootV = find(v); + + // If they belong to different sets, merge them + if (rootU != rootV) { + // Make rootV point to rootU — merge the two sets + parent[rootV] = rootU; + + // Reduce the count of disjoint sets by 1 + setCount--; + } + } + + boolean inSameSet(int u, int v) { + return find(u) == find(v); + } + + /* + This is a verbosity method, it's not a part of the core algorithm, + But it helps us provide more useful output. + */ + Set getAnySet() { + int aRoot = find(0); // Get one of the two roots + + Set set = new HashSet<>(); + for (int i = 0; i < parent.length; i++) { + if (find(i) == aRoot) { + set.add(i); + } + } + + return set; + } + } + + private static class Graph { + private final List nodes; + private final List edges; + + Graph(Collection nodeSet, List edges) { + this.nodes = new ArrayList<>(nodeSet); + this.edges = new ArrayList<>(); + for (int[] e : edges) { + this.edges.add(new int[] {e[0], e[1]}); + } + } + + Graph copy() { + return new Graph(this.nodes, this.edges); + } + + KargerOutput findMinCut() { + DisjointSetUnion dsu = new DisjointSetUnion(nodes.size()); + List workingEdges = new ArrayList<>(edges); + + Random rand = new Random(); + + while (dsu.setCount > 2) { + int[] e = workingEdges.get(rand.nextInt(workingEdges.size())); + if (!dsu.inSameSet(e[0], e[1])) { + dsu.union(e[0], e[1]); + } + } + + int cutEdges = 0; + for (int[] e : edges) { + if (!dsu.inSameSet(e[0], e[1])) { + cutEdges++; + } + } + + return collectResult(dsu, cutEdges); + } + + /* + This is a verbosity method, it's not a part of the core algorithm, + But it helps us provide more useful output. + */ + private KargerOutput collectResult(DisjointSetUnion dsu, int cutEdges) { + Set firstIndices = dsu.getAnySet(); + Set firstSet = new HashSet<>(); + Set secondSet = new HashSet<>(); + for (int i = 0; i < nodes.size(); i++) { + if (firstIndices.contains(i)) { + firstSet.add(nodes.get(i)); + } else { + secondSet.add(nodes.get(i)); + } + } + return new KargerOutput(firstSet, secondSet, cutEdges); + } + } +} diff --git a/src/test/java/com/thealgorithms/randomized/KargerMinCutTest.java b/src/test/java/com/thealgorithms/randomized/KargerMinCutTest.java new file mode 100644 index 000000000..876b6bf45 --- /dev/null +++ b/src/test/java/com/thealgorithms/randomized/KargerMinCutTest.java @@ -0,0 +1,114 @@ +package com.thealgorithms.randomized; + +import static org.junit.jupiter.api.Assertions.assertEquals; +import static org.junit.jupiter.api.Assertions.assertTrue; + +import java.util.Arrays; +import java.util.Collection; +import java.util.List; +import org.junit.jupiter.api.Test; + +public class KargerMinCutTest { + + @Test + public void testSimpleGraph() { + // Graph: 0 -- 1 + Collection nodes = Arrays.asList(0, 1); + List edges = List.of(new int[] {0, 1}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(1, result.minCut()); + assertTrue(result.first().contains(0) || result.first().contains(1)); + assertTrue(result.second().contains(0) || result.second().contains(1)); + } + + @Test + public void testTriangleGraph() { + // Graph: 0 -- 1 -- 2 -- 0 + Collection nodes = Arrays.asList(0, 1, 2); + List edges = List.of(new int[] {0, 1}, new int[] {1, 2}, new int[] {2, 0}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(2, result.minCut()); + } + + @Test + public void testSquareGraph() { + // Graph: 0 -- 1 + // | | + // 3 -- 2 + Collection nodes = Arrays.asList(0, 1, 2, 3); + List edges = List.of(new int[] {0, 1}, new int[] {1, 2}, new int[] {2, 3}, new int[] {3, 0}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(2, result.minCut()); + } + + @Test + public void testDisconnectedGraph() { + // Graph: 0 -- 1 2 -- 3 + Collection nodes = Arrays.asList(0, 1, 2, 3); + List edges = List.of(new int[] {0, 1}, new int[] {2, 3}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(0, result.minCut()); + } + + @Test + public void testCompleteGraph() { + // Complete Graph: 0 -- 1 -- 2 -- 3 (all nodes connected to each other) + Collection nodes = Arrays.asList(0, 1, 2, 3); + List edges = List.of(new int[] {0, 1}, new int[] {0, 2}, new int[] {0, 3}, new int[] {1, 2}, new int[] {1, 3}, new int[] {2, 3}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(3, result.minCut()); + } + + @Test + public void testSingleNodeGraph() { + // Graph: Single node with no edges + Collection nodes = List.of(0); + List edges = List.of(); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(0, result.minCut()); + assertTrue(result.first().contains(0)); + assertTrue(result.second().isEmpty()); + } + + @Test + public void testTwoNodesNoEdge() { + // Graph: 0 1 (no edges) + Collection nodes = Arrays.asList(0, 1); + List edges = List.of(); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + assertEquals(0, result.minCut()); + assertTrue(result.first().contains(0) || result.first().contains(1)); + assertTrue(result.second().contains(0) || result.second().contains(1)); + } + + @Test + public void testComplexGraph() { + // Nodes: 0, 1, 2, 3, 4, 5, 6, 7, 8 + // Edges: Fully connected graph with additional edges for complexity + Collection nodes = Arrays.asList(0, 1, 2, 3, 4, 5, 6, 7, 8); + List edges = List.of(new int[] {0, 1}, new int[] {0, 2}, new int[] {0, 3}, new int[] {0, 4}, new int[] {0, 5}, new int[] {1, 2}, new int[] {1, 3}, new int[] {1, 4}, new int[] {1, 5}, new int[] {1, 6}, new int[] {2, 3}, new int[] {2, 4}, new int[] {2, 5}, new int[] {2, 6}, + new int[] {2, 7}, new int[] {3, 4}, new int[] {3, 5}, new int[] {3, 6}, new int[] {3, 7}, new int[] {3, 8}, new int[] {4, 5}, new int[] {4, 6}, new int[] {4, 7}, new int[] {4, 8}, new int[] {5, 6}, new int[] {5, 7}, new int[] {5, 8}, new int[] {6, 7}, new int[] {6, 8}, new int[] {7, 8}, + new int[] {0, 6}, new int[] {1, 7}, new int[] {2, 8}); + + KargerMinCut.KargerOutput result = KargerMinCut.findMinCut(nodes, edges); + + // The exact minimum cut value depends on the randomization, but it should be consistent + // for this graph structure. For a fully connected graph, the minimum cut is typically + // determined by the smallest number of edges connecting two partitions. + assertTrue(result.minCut() > 0); + } +}