What is Dijkstra's algorithm?

Dijkstra's algorithm is a method for finding the shortest path from a starting vertex to all other vertices in a weighted directed or undirected graph with non-negative edge weights.

Main principles:

• The algorithm operates in a step-by-step manner.
• At each step, the unprocessed vertex with the smallest current distance from the start is selected.
• The distances to adjacent vertices are updated if a shorter path is found through the current vertex.

Algorithm steps:

1. Initialize distances to all vertices as infinity, except for the starting vertex, which is set to 0.
2. Create a set of vertices that have not yet been processed.
3. Repeat while there are unprocessed vertices: a. Select the unprocessed vertex u with the minimum distance. b. Mark u as processed. c. For each adjacent vertex v of u: i. If the distance to v is greater than the distance to u plus the weight of the edge u->v, update the distance to v.

Example implementation (pseudocode):

// Dijkstra's function
def dijkstra(graph, start_node):
    distances = {} // Dictionary to store shortest distances
    unvisited_nodes = set(graph.get_nodes()) // Set of unprocessed vertices

    // Initialize distances
    for node in graph.get_nodes():
        distances[node] = float('inf')
    distances[start_node] = 0

    while unvisited_nodes:
        // Find the unprocessed vertex with the minimum distance
        current_node = get_node_with_min_distance(unvisited_nodes, distances)

        if distances[current_node] == float('inf'):
            break // Disconnected vertices

        unvisited_nodes.remove(current_node)

        // Update distances for adjacent vertices
        for neighbor, weight in graph.get_neighbors(current_node):
            new_distance = distances[current_node] + weight
            if new_distance < distances[neighbor]:
                distances[neighbor] = new_distance

    return distances

// Helper function to find the vertex with the minimum distance
def get_node_with_min_distance(unvisited_nodes, distances):
    min_distance = float('inf')
    min_node = None
    for node in unvisited_nodes:
        if distances[node] < min_distance:
            min_distance = distances[node]
            min_node = node
    return min_node

Algorithm complexity:

• Using lists and linear search for the minimum distance: O(V^2), where V is the number of vertices.
• Using a priority queue (e.g., binary heap): O((V + E) log V), where E is the number of edges.

Applications:

• Finding shortest routes on maps (GPS navigation).
• Routing in computer networks.
• Finding shortest paths in dependency graphs.