Dijkstra's Algorithm Trace

Graph Theory · Axiom Academy

EXAMPLE Dijkstra's Algorithm Trace Step-by-step execution finding shortest paths from vertex A to all other vertices Goal: Find shortest paths from source vertex A to all other vertices Excellent work! You've traced through Dijkstra's algorithm. Here's what we learned: Greedy Selection: Dijkstra's algorithm always extracts the vertex with minimum distance from the priority queue, ensuring optimal substructure. Relaxation Process: For each extracted vertex, we check all neighbors and update their distances if we find a shorter path (dist[u] + weight(u,v) < dist[v]). Priority Queue Efficiency: Using a min-heap priority queue allows efficient extraction of minimum-distance vertices in O(log V) time. Visited Set: Once a vertex is extracted from the priority queue, its shortest distance is finalized and won't change. Time Complexity: With a binary heap, Dijkstra's runs in O((V + E) log V), where V is vertices and E is edges. Limitation: This algorithm only works correctly with non-negative edge weights. For negative weights, use Bellman-Ford instead. Dijkstra's algorithm is fundamental to GPS navigation, network routing protocols, and many optimization problems. Practice tracing it on different graphs to build intuition!

This is the written version of the interactive lesson above. See the full Graph Theory course.