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.
1class Graph {
2 constructor(n, edges) {
3 this.n = n;
4 this.adj = Array.from({ length: n }, () => []);
5 for (const [u, v, cost] of edges) {
6 this.adj[u].push([v, cost]);
7 }
8 }
9
10 addEdge(edge) {
11 const [u, v, cost] = edge;
12 this.adj[u].push([v, cost]);
13 }
14
15 shortestPath(node1, node2) {
16 const dist = new Array(this.n).fill(Infinity);
17 dist[node1] = 0;
18 const pq = new PriorityQueue((a, b) => a[0] < b[0]);
19 pq.enqueue([0, node1]);
20
21 while (!pq.isEmpty()) {
22 const [curDist, u] = pq.dequeue();
23 if (u === node2) return curDist;
24 if (curDist > dist[u]) continue;
25
26 for (const [v, cost] of this.adj[u]) {
27 if (dist[u] + cost < dist[v]) {
28 dist[v] = dist[u] + cost;
29 pq.enqueue([dist[v], v]);
30 }
31 }
32 }
33 return -1;
34 }
35}
36
37class PriorityQueue {
38 constructor(comparator) {
39 this.data = [];
40 this.comparator = comparator;
41 }
42
43 enqueue(value) {
44 this.data.push(value);
45 this.data.sort(this.comparator);
46 }
47
48 dequeue() {
49 return this.data.shift();
50 }
51
52 isEmpty() {
53 return this.data.length === 0;
54 }
55}
56
57// Demonstration
58const edges = [[0, 2, 5], [0, 1, 2], [1, 2, 1], [3, 0, 3]];
59const graph = new Graph(4, edges);
60console.log(graph.shortestPath(3, 2)); // Output: 6
61console.log(graph.shortestPath(0, 3)); // Output: -1
62graph.addEdge([1, 3, 4]);
63console.log(graph.shortestPath(0, 3)); // Output: 6
64JavaScript utilizes a simple priority queue implementation to keep track of and process nodes by minimized distances while computing shortest paths.
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.
1import java.util.*;
2
3class 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[] row : dist)
11 Arrays.fill(row, Integer.MAX_VALUE);
12 for (int i = 0; i < n; i++)
13 dist[i][i] = 0;
14 for (int[] edge : edges) {
15 dist[edge[0]][edge[1]] = edge[2];
16 }
17 floydWarshall();
18 }
19
20 public void floydWarshall() {
21 for (int k = 0; k < n; k++) {
22 for (int i = 0; i < n; i++) {
23 for (int j = 0; j < n; j++) {
24 if (dist[i][k] != Integer.MAX_VALUE && dist[k][j] != Integer.MAX_VALUE)
25 dist[i][j] = Math.min(dist[i][j], dist[i][k] + dist[k][j]);
26 }
27 }
28 }
29 }
30
31 public void addEdge(int[] edge) {
32 dist[edge[0]][edge[1]] = edge[2];
33 floydWarshall();
34 }
35
36 public int shortestPath(int node1, int node2) {
37 return dist[node1][node2] == Integer.MAX_VALUE ? -1 : dist[node1][node2];
38 }
39
40 public static void main(String[] args) {
41 int[][] edges = {{0, 2, 5}, {0, 1, 2}, {1, 2, 1}, {3, 0, 3}};
42 Graph graph = new Graph(4, edges);
43 System.out.println(graph.shortestPath(3, 2)); // Output: 6
44 System.out.println(graph.shortestPath(0, 3)); // Output: -1
45 graph.addEdge(new int[]{1, 3, 4});
46 System.out.println(graph.shortestPath(0, 3)); // Output: 6
47 }
48}Java implements Floyd-Warshall to precompute path efficiencies across the graph's entirety, recalculating these values upon any edge addition to account for potential novel shortest routes.