Added Bellman Ford Algorithm

Author: RashmitTopG

Java/algorithms/graph_algorithms/BellmanFord.java

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+/*
+ * Bellman–Ford — Single-Source Shortest Paths (+ Negative-Cycle Check)
+ * ===================================================================
+ * ✍️ Algorithm Description
+ * -----------------------
+ * Bellman–Ford computes shortest path distances from a single source in a
+ * weighted directed graph — even when there are negative edge weights.
+ * It can also detect the presence of any negative-weight cycle reachable
+ * from the source (or anywhere, using a super-source trick).
+ *
+ * Core idea:
+ * 1) Initialize dist[src] = 0, all others = +∞.
+ * 2) Relax all edges V-1 times (where V is #vertices).
+ *    - A "relax" on edge (u -> v, w) does: dist[v] = min(dist[v], dist[u] + w).
+ * 3) Do one more pass over all edges:
+ *    - If any distance can still be improved, a negative-weight cycle exists.
+ *
+ * Use cases:
+ * - Shortest paths with negative edges (no negative cycles reachable from src)
+ * - Financial arbitrage detection (negative cycles)
+ * - As a subroutine in Johnson’s algorithm (reweighting for all-pairs)
+ *
+ * 📊 Complexity Analysis
+ * ----------------------
+ * Let V = number of vertices, E = number of edges.
+ * Time:  O(V * E) — V-1 relaxation rounds plus a final scan.
+ * Space: O(V)     — distance array (+ optional parent array).
+ *
+ * 💡 Notes
+ * - `bellmanFord1` (array distances) is fastest when vertices are 0..V-1.
+ * - `bellmanFordMap` (HashMap distances) is handy if vertex ids are sparse or labeled.
+ * - `hasNegativeCycleAnywhere` adds a super-source → detects ANY neg cycle in the graph.
+ * - Distances use `long` to reduce overflow risk. Use `INF` as a large sentinel.
+ */
+
+ import java.util.*;
+
+ class Solution {
+ 
+     // Use a large INF sentinel to avoid overflow when adding weights.
+     private static final long INF = (long)1e18;
+ 
+     /**
+      * Bellman–Ford with array distances (vertices assumed 0..V-1).
+      *
+      * @param V     number of vertices
+      * @param edges directed weighted edges as [u, v, w]
+      * @param src   source vertex
+      * @return      distances array; if a negative cycle is reachable from src,
+      *              returns null to signal "no well-defined shortest paths"
+      */
+     public static long[] bellmanFord1(int V, int[][] edges, int src) {
+         long[] dist = new long[V];
+         Arrays.fill(dist, INF);
+         dist[src] = 0;
+ 
+         // Relax edges V-1 times
+         for (int i = 1; i <= V - 1; i++) {
+             boolean changed = false;
+             for (int[] e : edges) {
+                 int u = e[0], v = e[1];
+                 long w = e[2];
+                 if (dist[u] != INF && dist[u] + w < dist[v]) {
+                     dist[v] = dist[u] + w;
+                     changed = true;
+                 }
+             }
+             // Early exit if no update in a round
+             if (!changed) break;
+         }
+ 
+         // Check for a negative cycle reachable from src
+         for (int[] e : edges) {
+             int u = e[0], v = e[1];
+             long w = e[2];
+             if (dist[u] != INF && dist[u] + w < dist[v]) {
+                 return null; // Negative cycle detected (reachable from src)
+             }
+         }
+         return dist;
+     }
+ 
+     /**
+      * Bellman–Ford with HashMap distances (good for sparse / labeled vertices).
+      * Vertices are still 0..V-1 here, but you can adapt to arbitrary ids
+      * by mapping them to [0..n-1] and keeping only present keys in the map.
+      *
+      * @param V     number of vertices
+      * @param edges directed weighted edges as [u, v, w]
+      * @param src   source vertex
+      * @return      distance map; null if a negative cycle is reachable from src
+      */
+     public static HashMap<Integer, Long> bellmanFordMap(int V, int[][] edges, int src) {
+         HashMap<Integer, Long> dist = new HashMap<>();
+         for (int i = 0; i < V; i++) dist.put(i, INF);
+         dist.put(src, 0L);
+ 
+         for (int i = 1; i <= V - 1; i++) {
+             boolean changed = false;
+             for (int[] e : edges) {
+                 int u = e[0], v = e[1];
+                 long w = e[2];
+                 long du = dist.get(u);
+                 long dv = dist.get(v);
+                 if (du != INF && du + w < dv) {
+                     dist.put(v, du + w);
+                     changed = true;
+                 }
+             }
+             if (!changed) break;
+         }
+ 
+         for (int[] e : edges) {
+             int u = e[0], v = e[1];
+             long w = e[2];
+             long du = dist.get(u);
+             long dv = dist.get(v);
+             if (du != INF && du + w < dv) {
+                 return null; // Negative cycle detected (reachable from src)
+             }
+         }
+         return dist;
+     }
+ 
+     /**
+      * Detect ANY negative-weight cycle in the graph (not just reachable from a given src).
+      * Technique: add a super-source S connected to all nodes with 0-weight edges, run BF.
+      *
+      * @param V     number of vertices
+      * @param edges directed weighted edges as [u, v, w]
+      * @return      true if any negative-weight cycle exists; false otherwise
+      */
+     public static boolean hasNegativeCycleAnywhere(int V, int[][] edges) {
+         // Build a new edge list with an extra super-source (-1) conceptually.
+         // Implementation trick: initialize all dist[i]=0 and run V rounds.
+         long[] dist = new long[V];
+         Arrays.fill(dist, 0L); // Equivalent to relaxing from a super-source with 0 edges to all
+ 
+         // Relax V times; if we can still relax on the V-th time, a neg cycle exists anywhere.
+         for (int i = 1; i <= V; i++) {
+             boolean changed = false;
+             for (int[] e : edges) {
+                 int u = e[0], v = e[1];
+                 long w = e[2];
+                 if (dist[u] + w < dist[v]) {
+                     dist[v] = dist[u] + w;
+                     changed = true;
+                     if (i == V) return true; // improvement on the V-th round => negative cycle
+                 }
+             }
+             if (!changed) break;
+         }
+         return false;
+     }
+ 
+     // ----------------------------
+     // ✅ Test Cases / Examples
+     // ----------------------------
+     public static void main(String[] args) {
+         // 1) Positive weights only
+         int V1 = 5;
+         int[][] E1 = {
+             {0,1,6}, {0,3,7},
+             {1,2,5}, {1,3,8}, {1,4,-4},
+             {2,1,-2}, {3,2,-3}, {3,4,9},
+             {4,0,2}, {4,2,7}
+         };
+         System.out.println("Case 1: Positive & some negative edges (no neg cycle, src=0)");
+         System.out.println("hasNegativeCycleAnywhere: " + hasNegativeCycleAnywhere(V1, E1)); // false
+         System.out.println("bellmanFord1 distances from 0: " + Arrays.toString(bellmanFord1(V1, E1, 0)));
+ 
+         // 2) Negative edge but no negative cycle
+         int V2 = 4;
+         int[][] E2 = {
+             {0,1,1}, {1,2,-1}, {2,3,-1}
+         };
+         System.out.println("\nCase 2: Negative edges, no cycle (src=0)");
+         System.out.println("hasNegativeCycleAnywhere: " + hasNegativeCycleAnywhere(V2, E2)); // false
+         System.out.println("bellmanFordMap distances from 0: " + bellmanFordMap(V2, E2, 0));
+ 
+         // 3) Graph with a negative cycle reachable from src
+         //    Cycle: 1 -> 2 -> 3 -> 1 with total weight -3
+         int V3 = 4;
+         int[][] E3 = {
+             {0,1,2}, {1,2,-2}, {2,3,-2}, {3,1,1}
+         };
+         System.out.println("\nCase 3: Negative cycle reachable from src=0");
+         System.out.println("hasNegativeCycleAnywhere: " + hasNegativeCycleAnywhere(V3, E3)); // true
+         long[] d3 = bellmanFord1(V3, E3, 0);
+         System.out.println("bellmanFord1 distances from 0: " + (d3 == null ? "NEGATIVE CYCLE" : Arrays.toString(d3)));
+ 
+         // 4) Disconnected graph (unreachable nodes remain INF)
+         int V4 = 6;
+         int[][] E4 = {
+             {0,1,3}, {1,2,4},          // component 1
+             {3,4,1}, {4,5,1}           // component 2 (disconnected from src=0)
+         };
+         System.out.println("\nCase 4: Disconnected graph (src=0)");
+         System.out.println("hasNegativeCycleAnywhere: " + hasNegativeCycleAnywhere(V4, E4)); // false
+         long[] d4 = bellmanFord1(V4, E4, 0);
+         System.out.println("bellmanFord1 distances from 0: " + Arrays.toString(d4));
+         // Expect: [0, 3, 7, INF, INF, INF]
+ 
+         // 5) Single vertex, no edges
+         int V5 = 1;
+         int[][] E5 = {};
+         System.out.println("\nCase 5: Single node (src=0)");
+         System.out.println("hasNegativeCycleAnywhere: " + hasNegativeCycleAnywhere(V5, E5)); // false
+         System.out.println("bellmanFord1 distances from 0: " + Arrays.toString(bellmanFord1(V5, E5, 0)));
+     }
+ }
+