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 heapq
2
3class Graph:
4 def __init__(self, n, edges):
5 self.n = n
6 self.adj = [[] for _ in range(n)]
7 for u, v, cost in edges:
8 self.adj[u].append((v, cost))
9
10 def addEdge(self, edge):
11 u, v, cost = edge
12 self.adj[u].append((v, cost))
13
14 def shortestPath(self, node1, node2):
15 dist = [float('inf')] * self.n
16 dist[node1] = 0
17 pq = [(0, node1)]
18
19 while pq:
20 curDist, u = heapq.heappop(pq)
21 if u == node2:
22 return curDist
23 if curDist > dist[u]:
24 continue
25 for v, cost in self.adj[u]:
26 if dist[u] + cost < dist[v]:
27 dist[v] = dist[u] + cost
28 heapq.heappush(pq, (dist[v], v))
29 return -1
30
31# Demonstration
32if __name__ == "__main__":
33 edges = [[0, 2, 5], [0, 1, 2], [1, 2, 1], [3, 0, 3]]
34 graph = Graph(4, edges)
35 print(graph.shortestPath(3, 2)) # Output: 6
36 print(graph.shortestPath(0, 3)) # Output: -1
37 graph.addEdge([1, 3, 4])
38 print(graph.shortestPath(0, 3)) # Output: 6
39Python implementation utilizes `heapq` for the priority queue, allowing Dijkstra's approach to effectively evaluate the shortest path by updating minimum costs as graph exploration progresses.
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.