BELLMAN FORD

Bellman-Ford's single source shortest path algorithm.
The algorithm calculates the shortest paths in a bottom-up manner. It first calculates the 
shortest distances which have at most one edge in the path. Then, it calculates the shortest 
paths with at-most 2 edges, and so on. After the i-th iteration of the outer loop, the 
shortest paths with at most i edges are calculated. There can be maximum |V| – 1 edges in any 
simple path, that is why the outer loop runs |v| – 1 times. The idea is, assuming that there 
is no negative weight cycle if we have calculated shortest paths with at most i edges, then 
an iteration over all edges guarantees to give the shortest path with at-most (i+1) edges.

Author: Shivam20233

Graph_Algorithms/BELLMAN FORD

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+// Bellman-Ford's single source shortest path algorithm.
+// The algorithm calculates the shortest paths in a bottom-up manner. It first calculates the 
+// shortest distances which have at most one edge in the path. Then, it calculates the shortest 
+// paths with at-most 2 edges, and so on. After the i-th iteration of the outer loop, the 
+// shortest paths with at most i edges are calculated. There can be maximum |V| – 1 edges in any 
+// simple path, that is why the outer loop runs |v| – 1 times. The idea is, assuming that there 
+// is no negative weight cycle if we have calculated shortest paths with at most i edges, then 
+// an iteration over all edges guarantees to give the shortest path with at-most (i+1) edges.
+
+#include <bits/stdc++.h>
+using namespace std;
+
+// a structure to represent a weighted edge in graph
+struct Edge {
+	int src, dest, weight;
+};
+
+// a structure to represent a connected, directed and
+// weighted graph
+struct Graph {
+	// V-> Number of vertices, E-> Number of edges
+	int V, E;
+
+	// graph is represented as an array of edges.
+	struct Edge* edge;
+};
+
+// Creates a graph with V vertices and E edges
+struct Graph* createGraph(int V, int E)
+{
+	struct Graph* graph = new Graph;
+	graph->V = V;
+	graph->E = E;
+	graph->edge = new Edge[E];
+	return graph;
+}
+
+// A utility function used to print the solution
+void printArr(int dist[], int n)
+{
+	printf("Vertex Distance from Source\n");
+	for (int i = 0; i < n; ++i)
+		printf("%d \t\t %d\n", i, dist[i]);
+}
+
+// The main function that finds shortest distances from src
+// to all other vertices using Bellman-Ford algorithm. The
+// function also detects negative weight cycle
+void BellmanFord(struct Graph* graph, int src)
+{
+	int V = graph->V;
+	int E = graph->E;
+	int dist[V];
+
+	// Step 1: Initialize distances from src to all other
+	// vertices as INFINITE
+	for (int i = 0; i < V; i++)
+		dist[i] = INT_MAX;
+	dist[src] = 0;
+
+	// Step 2: Relax all edges |V| - 1 times. A simple
+	// shortest path from src to any other vertex can have
+	// at-most |V| - 1 edges
+	for (int i = 1; i <= V - 1; i++) {
+		for (int j = 0; j < E; j++) {
+			int u = graph->edge[j].src;
+			int v = graph->edge[j].dest;
+			int weight = graph->edge[j].weight;
+			if (dist[u] != INT_MAX
+				&& dist[u] + weight < dist[v])
+				dist[v] = dist[u] + weight;
+		}
+	}
+
+	// Step 3: check for negative-weight cycles. The above
+	// step guarantees shortest distances if graph doesn't
+	// contain negative weight cycle. If we get a shorter
+	// path, then there is a cycle.
+	for (int i = 0; i < E; i++) {
+		int u = graph->edge[i].src;
+		int v = graph->edge[i].dest;
+		int weight = graph->edge[i].weight;
+		if (dist[u] != INT_MAX
+			&& dist[u] + weight < dist[v]) {
+			printf("Graph contains negative weight cycle");
+			return; // If negative cycle is detected, simply
+					// return
+		}
+	}
+
+	printArr(dist, V);
+
+	return;
+}
+
+// Driver's code
+int main()
+{
+	/* Let us create the graph given in above example */
+	int V = 5; // Number of vertices in graph
+	int E = 8; // Number of edges in graph
+	struct Graph* graph = createGraph(V, E);
+
+	// add edge 0-1 (or A-B in above figure)
+	graph->edge[0].src = 0;
+	graph->edge[0].dest = 1;
+	graph->edge[0].weight = -1;
+
+	// add edge 0-2 (or A-C in above figure)
+	graph->edge[1].src = 0;
+	graph->edge[1].dest = 2;
+	graph->edge[1].weight = 4;
+
+	// add edge 1-2 (or B-C in above figure)
+	graph->edge[2].src = 1;
+	graph->edge[2].dest = 2;
+	graph->edge[2].weight = 3;
+
+	// add edge 1-3 (or B-D in above figure)
+	graph->edge[3].src = 1;
+	graph->edge[3].dest = 3;
+	graph->edge[3].weight = 2;
+
+	// add edge 1-4 (or B-E in above figure)
+	graph->edge[4].src = 1;
+	graph->edge[4].dest = 4;
+	graph->edge[4].weight = 2;
+
+	// add edge 3-2 (or D-C in above figure)
+	graph->edge[5].src = 3;
+	graph->edge[5].dest = 2;
+	graph->edge[5].weight = 5;
+
+	// add edge 3-1 (or D-B in above figure)
+	graph->edge[6].src = 3;
+	graph->edge[6].dest = 1;
+	graph->edge[6].weight = 1;
+
+	// add edge 4-3 (or E-D in above figure)
+	graph->edge[7].src = 4;
+	graph->edge[7].dest = 3;
+	graph->edge[7].weight = -3;
+	
+	// Function call
+	BellmanFord(graph, 0);
+
+	return 0;
+}
+
+Time Complexity:  O(V * E).
+Auxiliary Space: O(E).