What is Dijkstra's algorithm and in what cases is it used?

Dijkstra's algorithm is a graph algorithm that finds the shortest paths from a single source vertex to all other vertices in a weighted directed or undirected graph with non-negative edge weights.

Main steps of the algorithm:

1. Initialization: Set the distance to the starting vertex to 0, and to all others to infinity. Create a set of unvisited vertices.
2. Vertex selection: Choose the unvisited vertex with the smallest current distance.
3. Distance update: For each neighbor of the selected vertex, if the path through the current vertex is shorter than the current distance to that neighbor, update the distance.
4. Mark as visited: Move the selected vertex from the unvisited set to the visited set.
5. Repeat: Repeat steps 2-4 until all vertices are visited or the set of unvisited vertices is empty.

Used in the following cases:

• Finding the shortest path: In navigation systems (GPS), network protocols (e.g., OSPF) for determining the optimal route.
• Network routing: To determine the most efficient data transmission paths in computer networks.
• Production planning: To optimize the sequence of operations with minimal time or resource costs.
• Image processing: In some algorithms for finding optimal trajectories or segmentation.
• Games: To find the shortest path for characters or objects across the game field.

Pseudo-code example:

// d - array of distances initialized to infinity, except d[start_node] = 0
// visited - set of visited vertices
// graph - graph represented as an adjacency list with edge weights

function Dijkstra(graph, start_node):
    d[start_node] = 0
    priority_queue = add start_node with priority 0

    while priority_queue is not empty:
        current_node = extract_min(priority_queue)

        if current_node is in visited:
            continue

        add current_node to visited

        for each neighbor of current_node:
            if d[current_node] + weight(current_node, neighbor) < d[neighbor]:
                d[neighbor] = d[current_node] + weight(current_node, neighbor)
                add neighbor to priority_queue with priority d[neighbor]

    return d