Dijkstra's Algorithm

Idea Like You Are 10

Dijkstra helps find the shortest path when roads have distances.

It keeps choosing the not-yet-finished node with the smallest known distance.

Important Rule

Dijkstra works correctly when edge weights are non-negative.

Main Idea

1. Start with source distance 0
2. Other distances are infinity
3. Repeatedly pick the closest unfinished node
4. Try to improve distances of its neighbors

This improving step is called relaxation.

Algorithm

dijkstra(graph, source):
    for each vertex:
        dist[vertex] = infinity
    dist[source] = 0

    priority queue pq
    insert (0, source)

    while pq is not empty:
        (d, u) = extract minimum

        for each edge (u, v, w):
            if dist[u] + w < dist[v]:
                dist[v] = dist[u] + w
                insert (dist[v], v) into pq

Time Complexity

With adjacency list + heap:

O((V + E) log V)

Why?

• heap operations cost about log V
• vertices and edge relaxations cause repeated heap work

With a simple array implementation:

O(V^2)

Space Complexity

O(V + E)

Applications

• GPS navigation
• network routing
• game pathfinding