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.
1#include <stdio.h>
2#include <stdlib.h>
3#include <limits.h>
4
5#define INF INT_MAX
6
7typedef struct {
8 int n;
9 int** dist;
10} Graph;
11
12Graph* createGraph(int n, int edges[][3], int edgesSize) {
13 Graph* graph = (Graph*)malloc(sizeof(Graph));
14 graph->n = n;
15 graph->dist = (int**)malloc(n * sizeof(int*));
16 for (int i = 0; i < n; i++) {
17 graph->dist[i] = (int*)malloc(n * sizeof(int));
18 for (int j = 0; j < n; j++)
19 graph->dist[i][j] = (i == j) ? 0 : INF;
20 }
21 for (int i = 0; i < edgesSize; i++) {
22 int u = edges[i][0];
23 int v = edges[i][1];
24 graph->dist[u][v] = edges[i][2];
25 }
26 for (int k = 0; k < n; k++) {
27 for (int i = 0; i < n; i++) {
28 for (int j = 0; j < n; j++) {
29 if (graph->dist[i][k] != INF && graph->dist[k][j] != INF)
30 graph->dist[i][j] = (graph->dist[i][j] > graph->dist[i][k] + graph->dist[k][j]) ? graph->dist[i][k] + graph->dist[k][j] : graph->dist[i][j];
31 }
32 }
33 }
34
35 return graph;
36}
37
38void addEdge(Graph* graph, int u, int v, int cost) {
39 graph->dist[u][v] = cost;
40 // Re-run Floyd-Warshall
41 for (int k = 0; k < graph->n; k++) {
42 for (int i = 0; i < graph->n; i++) {
43 for (int j = 0; j < graph->n; j++) {
44 if (graph->dist[i][k] != INF && graph->dist[k][j] != INF)
45 graph->dist[i][j] = (graph->dist[i][j] > graph->dist[i][k] + graph->dist[k][j]) ? graph->dist[i][k] + graph->dist[k][j] : graph->dist[i][j];
46 }
47 }
48 }
49}
50
51int shortestPath(Graph* graph, int node1, int node2) {
52 if(graph->dist[node1][node2] == INF)
53 return -1;
54 return graph->dist[node1][node2];
55}
56
57int main() {
58 int edges[4][3] = {{0, 2, 5}, {0, 1, 2}, {1, 2, 1}, {3, 0, 3}};
59 Graph* graph = createGraph(4, edges, 4);
60 printf("%d\n", shortestPath(graph, 3, 2)); // Output: 6
61 printf("%d\n", shortestPath(graph, 0, 3)); // Output: -1
62 addEdge(graph, 1, 3, 4);
63 printf("%d\n", shortestPath(graph, 0, 3)); // Output: 6
64
65 // Free the memory
66 for(int i = 0; i < graph->n; i++)
67 free(graph->dist[i]);
68 free(graph->dist);
69 free(graph);
70 return 0;
71}
This implementation uses Floyd-Warshall Algorithm, which precomputes shortest paths between all pairs of nodes. Post addEdge operations, the algorithm recomputes paths by potentially updating the distance matrix.