Added: DP Algortihms - Bellman-Ford Algorithm

dsa/Algorithms/Dynamic Programming/11-Bellman-Ford-Algorithm.md

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+---
+id: bellman-ford-algorithm
+title: Bellman-Ford Algorithm
+sidebar_label: Bellman-Ford Algorithm
+tags: [python, java, c++, programming, algorithms, dynamic programming, graph, tutorial, in-depth]
+description: In this tutorial, we will learn about the Bellman-Ford Algorithm and its solution using Dynamic Programming in Python, Java, and C++ with detailed explanations and examples.
+---
+
+# Bellman-Ford Algorithm
+
+The Bellman-Ford algorithm is a classic algorithm for finding the shortest paths in a weighted graph with negative weights. It is capable of handling graphs with negative edge weights and can also detect negative weight cycles.
+
+## Problem Statement
+
+Given a graph and a source vertex, find the shortest paths from the source vertex to all other vertices in the graph.
+
+### Intuition
+
+The algorithm iteratively relaxes the edges of the graph. The idea is to improve the estimate of the shortest path step by step. It takes up to `|V| - 1` iterations, where `|V|` is the number of vertices, to ensure that the shortest paths are found. If we perform one more iteration and still find a shorter path, it indicates the presence of a negative weight cycle.
+
+## Dynamic Programming Approach
+
+Using dynamic programming, we maintain an array `dist` where `dist[i]` holds the shortest distance from the source vertex to vertex `i`.
+
+## Pseudocode for Bellman-Ford Algorithm using DP
+
+#### Initialize:
+
+```markdown
+dist[source] = 0
+for i from 1 to |V| - 1:
+    for each edge (u, v) with weight w:
+        if dist[u] + w < dist[v]:
+            dist[v] = dist[u] + w
+
+for each edge (u, v) with weight w:
+    if dist[u] + w < dist[v]:
+        print("Graph contains a negative weight cycle")
+        return
+```
+
+### Example Output:
+
+Given the graph:
+
+- `Vertices: {0, 1, 2, 3}`
+- `Edges: {(0, 1, 1), (1, 2, 3), (2, 3, 2), (3, 1, -6)}`
+
+The set can be partitioned into two subsets with equal sum.
+
+### Output Explanation:
+
+The shortest distances from the source vertex 0 to all other vertices are:
+
+- `0 -> 1: 1`
+- `0 -> 2: 4`
+- `0 -> 3: 6`
+
+By following the Bellman-Ford algorithm, the shortest path distances from the source vertex 0 to vertices 1, 2, and 3 are found to be 1, 4, and 6 respectively.
+
+## Implementing Bellman-Ford Algorithm using DP
+
+### Python Implementation
+
+```python
+class Graph:
+    def __init__(self, vertices):
+        self.V = vertices
+        self.graph = []
+
+    def add_edge(self, u, v, w):
+        self.graph.append([u, v, w])
+
+    def bellman_ford(self, src):
+        dist = [float("Inf")] * self.V
+        dist[src] = 0
+
+        for _ in range(self.V - 1):
+            for u, v, w in self.graph:
+                if dist[u] != float("Inf") and dist[u] + w < dist[v]:
+                    dist[v] = dist[u] + w
+
+        for u, v, w in self.graph:
+            if dist[u] != float("Inf") and dist[u] + w < dist[v]:
+                print("Graph contains a negative weight cycle")
+                return
+
+        print("Vertex Distance from Source")
+        for i in range(self.V):
+            print(f"{i}\t\t{dist[i]}")
+
+g = Graph(4)
+g.add_edge(0, 1, 1)
+g.add_edge(1, 2, 3)
+g.add_edge(2, 3, 2)
+g.add_edge(3, 1, -6)
+
+g.bellman_ford(0)
+```
+
+### Java Implementation
+
+```java
+import java.util.*;
+
+class Graph {
+    class Edge {
+        int src, dest, weight;
+        Edge() {
+            src = dest = weight = 0;
+        }
+    };
+
+    int V, E;
+    Edge edge[];
+
+    Graph(int v, int e) {
+        V = v;
+        E = e;
+        edge = new Edge[e];
+        for (int i = 0; i < e; ++i)
+            edge[i] = new Edge();
+    }
+
+    void BellmanFord(Graph graph, int src) {
+        int V = graph.V, E = graph.E;
+        int dist[] = new int[V];
+
+        Arrays.fill(dist, Integer.MAX_VALUE);
+        dist[src] = 0;
+
+        for (int i = 1; i < V; ++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] != Integer.MAX_VALUE && dist[u] + weight < dist[v])
+                    dist[v] = dist[u] + weight;
+            }
+        }
+
+        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] != Integer.MAX_VALUE && dist[u] + weight < dist[v])
+                System.out.println("Graph contains a negative weight cycle");
+        }
+
+        printArr(dist, V);
+    }
+
+    void printArr(int dist[], int V) {
+        System.out.println("Vertex Distance from Source");
+        for (int i = 0; i < V; ++i)
+            System.out.println(i + "\t\t" + dist[i]);
+    }
+
+    public static void main(String[] args) {
+        Graph graph = new Graph(4, 4);
+
+        graph.edge[0].src = 0;
+        graph.edge[0].dest = 1;
+        graph.edge[0].weight = 1;
+
+        graph.edge[1].src = 1;
+        graph.edge[1].dest = 2;
+        graph.edge[1].weight = 3;
+
+        graph.edge[2].src = 2;
+        graph.edge[2].dest = 3;
+        graph.edge[2].weight = 2;
+
+        graph.edge[3].src = 3;
+        graph.edge[3].dest = 1;
+        graph.edge[3].weight = -6;
+
+        graph.BellmanFord(graph, 0);
+    }
+}
+```
+### C++ Implementation
+
+```cpp
+#include <bits/stdc++.h>
+using namespace std;
+
+struct Edge {
+    int src, dest, weight;
+};
+
+struct Graph {
+    int V, E;
+    struct Edge* edge;
+};
+
+struct Graph* createGraph(int V, int E) {
+    struct Graph* graph = (struct Graph*) malloc(sizeof(struct Graph));
+    graph->V = V;
+    graph->E = E;
+    graph->edge = (struct Edge*) malloc(graph->E * sizeof(struct Edge));
+    return graph;
+}
+
+void printArr(int dist[], int n) {
+    cout << "Vertex Distance from Source" << endl;
+    for (int i = 0; i < n; ++i)
+        cout << i << "\t\t" << dist[i] << endl;
+}
+
+void BellmanFord(struct Graph* graph, int src) {
+    int V = graph->V;
+    int E = graph->E;
+    int dist[V];
+
+    for (int i = 0; i < V; i++)
+        dist[i] = INT_MAX;
+    dist[src] = 0;
+
+    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;
+        }
+    }
+
+    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]) {
+            cout << "Graph contains a negative weight cycle" << endl;
+            return;
+        }
+    }
+
+    printArr(dist, V);
+}
+
+int main() {
+    int V = 4;
+    int E = 4;
+    struct Graph* graph = createGraph(V, E);
+
+    graph->edge[0].src = 0;
+    graph->edge[0].dest = 1;
+    graph->edge[0].weight = 1;
+
+    graph->edge[1].src = 1;
+    graph->edge[1].dest = 2;
+    graph->edge[1].weight = 3;
+
+    graph->edge[2].src = 2;
+    graph->edge[2].dest = 3;
+    graph->edge[2].weight = 2;
+
+    graph->edge[3].src = 3;
+    graph->edge[3].dest = 1;
+    graph->edge[3].weight = -6;
+
+    BellmanFord(graph, 0);
+
+    return 0;
+}
+```
+
+## Complexity Analysis
+
+- Time Complexity: $O(V \times E)$, where V is the number of vertices and E is the number of edges.
+- Space Complexity: $O(V)$, for storing the distance array.
+
+## Conclusion
+
+The Bellman-Ford algorithm provides an efficient solution for finding the shortest paths in a graph with negative weights and can also detect negative weight cycles. This technique is useful in various applications, including network routing and financial modeling.