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.
1class Graph:
2 def __init__(self, n, edges):
3 self.n = n
4 self.dist = [[float('inf')] * n for _ in range(n)]
5 for i in range(n):
6 self.dist[i][i] = 0
7 for u, v, cost in edges:
8 self.dist[u][v] = cost
9 self.floyd_warshall()
10
11 def floyd_warshall(self):
12 for k in range(self.n):
13 for i in range(self.n):
14 for j in range(self.n):
15 if self.dist[i][k] != float('inf') and self.dist[k][j] != float('inf'):
16 self.dist[i][j] = min(self.dist[i][j], self.dist[i][k] + self.dist[k][j])
17
18 def addEdge(self, edge):
19 u, v, cost = edge
20 self.dist[u][v] = cost
21 self.floyd_warshall()
22
23 def shortestPath(self, node1, node2):
24 return -1 if self.dist[node1][node2] == float('inf') else self.dist[node1][node2]
25
26# Demonstration
27if __name__ == "__main__":
28 edges = [[0, 2, 5], [0, 1, 2], [1, 2, 1], [3, 0, 3]]
29 graph = Graph(4, edges)
30 print(graph.shortestPath(3, 2)) # Output: 6
31 print(graph.shortestPath(0, 3)) # Output: -1
32 graph.addEdge([1, 3, 4])
33 print(graph.shortestPath(0, 3)) # Output: 6
34Python computes all pairs’ shortest paths upfront, storing these for efficient query handling, factoring additional edges via necessary resultant matrix updates.