Computing Shortest Paths Using Dijkstra's Algorithm

In computational applications, Dijkstra's algorithm is widely used for finding shortest paths in graphs. The core concept involves labeling each node with a pair of values (dj, pj). Here, dj represents the shortest distance from source node s to node j—note that the path from a vertex to itself has zero length with no connecting edges. Meanwhile, pj indicates the immediate predecessor node of j in the shortest path from s to j. The algorithm implementation typically follows these computational steps:

1. Initialize the known shortest-path set T with only the source node s. In code, this involves creating a visited nodes set and initializing it with the starting vertex.

2. Set initial distances for all remaining nodes: assign infinity to all dj values and null to all pj values. Programmatically, this is achieved through distance array initialization using a large numeric value (e.g., INT_MAX in C++ or float('inf') in Python) for unreachable nodes.

3. Iteratively select the unvisited node k with the minimum dj value using priority queue optimization, add it to set T, and update distance labels for all adjacent nodes. The key operation involves checking if dj[k] + edge_weight(k,j) < dj[j] for each neighbor j, then updating dj[j] and setting pj[j] = k.

4. Repeat step 3 until all nodes are incorporated into set T, ensuring optimal path calculation through systematic neighbor relaxation.

This algorithmic flow efficiently computes shortest paths from source s to all reachable nodes. The implementation complexity is O((V+E) log V) when using min-heap data structures, making it suitable for network routing and graph analysis applications.