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.
1using System;
2using System.Collections.Generic;
3
4public class Graph {
5 private int n;
6 private List<List<(int, int)>> adj;
7
8 public Graph(int n, int[][] edges) {
9 this.n = n;
10 this.adj = new List<List<(int, int)>>();
11 for (int i = 0; i < n; i++) {
12 adj.Add(new List<(int, int)>());
13 }
14 foreach (var edge in edges) {
15 adj[edge[0]].Add((edge[1], edge[2]));
16 }
17 }
18
19 public void AddEdge(int[] edge) {
20 adj[edge[0]].Add((edge[1], edge[2]));
21 }
22
23 public int ShortestPath(int node1, int node2) {
24 int[] dist = new int[n];
25 Array.Fill(dist, int.MaxValue);
26 dist[node1] = 0;
27
28 var pq = new SortedSet<(int Distance, int Vertex)>(Comparer<(int, int)>.Create((x, y) => {
29 int result = x.Distance.CompareTo(y.Distance);
30 return result == 0 ? x.Vertex.CompareTo(y.Vertex) : result;
31 }));
32 pq.Add((0, node1));
33
34 while (pq.Count > 0) {
35 var current = pq.Min;
36 pq.Remove(current);
37 int u = current.Vertex;
38 int curDist = current.Distance;
39
40 if (u == node2) return curDist;
41 if (curDist > dist[u]) continue;
42
43 foreach (var (v, cost) in adj[u]) {
44 if (dist[u] + cost < dist[v]) {
45 dist[v] = dist[u] + cost;
46 pq.Add((dist[v], v));
47 }
48 }
49 }
50 return -1;
51 }
52
53 public static void Main() {
54 int[][] edges = new int[][] { new int[] { 0, 2, 5 }, new int[] { 0, 1, 2 }, new int[] { 1, 2, 1 }, new int[] { 3, 0, 3 } };
55 Graph g = new Graph(4, edges);
56 Console.WriteLine(g.ShortestPath(3, 2)); // Output: 6
57 Console.WriteLine(g.ShortestPath(0, 3)); // Output: -1
58 g.AddEdge(new int[] { 1, 3, 4 });
59 Console.WriteLine(g.ShortestPath(0, 3)); // Output: 6
60 }
61}The C# implementation leverages a `SortedSet` for prioritizing graph nodes based on their cumulative path distance, thereby optimizing time complexity when executing Dijkstra's shortest path algorithm.
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.