Dijkstra's Algorithm is a greedy algorithm used to find the shortest path from a single source node to all other nodes in a weighted graph with non-negative weights. We can adapt it to efficiently find the shortest path for this problem.
Time Complexity: O(V^2), where V is the number of vertices since we're using an adjacency matrix and no priority queue to obtain the minimum distance vertex.
Space Complexity: O(V^2) due to the adjacency matrix storage requirements.
1import java.util.*;
2
3class Graph {
4 private int n;
5 private List<List<int[]>> adj;
6
7 public Graph(int n, int[][] edges) {
8 this.n = n;
9 this.adj = new ArrayList<>();
10 for (int i = 0; i < n; i++) {
11 adj.add(new ArrayList<>());
12 }
13 for (int[] edge : edges) {
14 adj.get(edge[0]).add(new int[]{edge[1], edge[2]});
15 }
16 }
17
18 public void addEdge(int[] edge) {
19 adj.get(edge[0]).add(new int[]{edge[1], edge[2]});
20 }
21
22 public int shortestPath(int node1, int node2) {
23 int[] dist = new int[n];
24 Arrays.fill(dist, Integer.MAX_VALUE);
25 dist[node1] = 0;
26 PriorityQueue<int[]> pq = new PriorityQueue<>(Comparator.comparingInt(a -> a[1]));
27 pq.offer(new int[]{node1, 0});
28
29 while (!pq.isEmpty()) {
30 int[] current = pq.poll();
31 int u = current[0];
32 int curDist = current[1];
33
34 if (u == node2) return curDist;
35 if (curDist > dist[u]) continue;
36
37 for (int[] neighbor : adj.get(u)) {
38 int v = neighbor[0];
39 int weight = neighbor[1];
40 if (dist[u] + weight < dist[v]) {
41 dist[v] = dist[u] + weight;
42 pq.offer(new int[]{v, dist[v]});
43 }
44 }
45 }
46 return -1;
47 }
48
49 public static void main(String[] args) {
50 int[][] edges = {{0, 2, 5}, {0, 1, 2}, {1, 2, 1}, {3, 0, 3}};
51 Graph graph = new Graph(4, edges);
52 System.out.println(graph.shortestPath(3, 2)); // Output: 6
53 System.out.println(graph.shortestPath(0, 3)); // Output: -1
54 graph.addEdge(new int[]{1, 3, 4});
55 System.out.println(graph.shortestPath(0, 3)); // Output: 6
56 }
57}Java solution uses a priority queue implemented by `PriorityQueue` class for Dijkstra's algorithm to ensure O(log V) minimum distance updates.
The Floyd-Warshall Algorithm is a dynamic programming algorithm used to find shortest paths between all pairs of vertices in a weighted graph. This approach is more suited when there are frequent shortest path queries between multiple different node pairs.
Time Complexity: O(n^3), where n is the number of vertices, arising from the three nested loops.
Space Complexity: O(n^2) due to the distance matrix storage covering all node pairs.
1using System;
2
3public class Graph {
4 private int n;
5 private int[,] dist;
6
7 public Graph(int n, int[][] edges) {
8 this.n = n;
9 dist = new int[n, n];
10 for (int i = 0; i < n; i++)
11 for (int j = 0; j < n; j++)
12 dist[i, j] = (i == j) ? 0 : int.MaxValue;
13 foreach (var edge in edges)
14 dist[edge[0], edge[1]] = edge[2];
15 FloydWarshall();
16 }
17
18 private void FloydWarshall() {
19 for (int k = 0; k < n; k++) {
20 for (int i = 0; i < n; i++) {
21 for (int j = 0; j < n; j++) {
22 if (dist[i, k] != int.MaxValue && dist[k, j] != int.MaxValue)
23 dist[i, j] = Math.Min(dist[i, j], dist[i, k] + dist[k, j]);
24 }
25 }
26 }
27 }
28
29 public void AddEdge(int[] edge) {
30 dist[edge[0], edge[1]] = edge[2];
31 FloydWarshall();
32 }
33
34 public int ShortestPath(int node1, int node2) {
35 return dist[node1, node2] == int.MaxValue ? -1 : dist[node1, node2];
36 }
37
38 public static void Main(String[] args) {
39 int[][] edges = new int[][] { new int[] { 0, 2, 5 }, new int[] { 0, 1, 2 }, new int[] { 1, 2, 1 }, new int[] { 3, 0, 3 } };
40 Graph graph = new Graph(4, edges);
41 Console.WriteLine(graph.ShortestPath(3, 2)); // Output: 6
42 Console.WriteLine(graph.ShortestPath(0, 3)); // Output: -1
43 graph.AddEdge(new int[] { 1, 3, 4 });
44 Console.WriteLine(graph.ShortestPath(0, 3)); // Output: 6
45 }
46}Floyd-Warshall precomputes full path pair information upfront for C#, with each edge insertion prompting algorithm re-execution, ensuring all shortest distances remain current.