Single-Source Shortest Paths: Dijkstra Algorithm

Lecture 39 of CS502 Fundamentals of Algorithms presents Dijkstra's algorithm, the classic greedy solution for the single-source shortest path problem in graphs with non-negative edge weights (w(u, v) >= 0 for all edges). Dijkstra maintains a set S of vertices whose final shortest-path weights delta(s, u) have already been determined. The algorithm repeatedly selects vertex u in V - S with minimum distance estimate d[u], inserts u into S, and relaxes all outgoing edges (u, v) leaving u. The lecture delivers a formal proof of correctness by induction: because all edge weights are non-negative, any path through unvisited vertices must have weight greater than or equal to d[u], guaranteeing that d[u] = delta(s, u) at the moment u is extracted. However, Dijkstra fails when negative edge weights are present because relaxation of subsequent negative edges could invalidate already finalized vertices in S. Using a min-priority heap, Dijkstra executes in O((|V| + |E|) log |V|) time, serving as the backbone of modern routing protocols like OSPF. Mastering these foundational algorithmic paradigms empowers software engineers to design highly scalable, computationally efficient software applications. Analytical exercises and algorithmic traces help students prepare thoroughly for Virtual University midterm and final term examinations. The curriculum systematically aligns theoretical algorithmic principles with practical implementation paradigms in modern computing systems. Reviewing the accompanying lecture handouts and complexity tables reinforces understanding of asymptotic growth rates and algorithmic bottlenecks. Consistent evaluation of state space trees, dynamic programming matrices, and graph adjacency structures ensures robust preparation for academic assessments. Students investigate the trade-offs between divide-and-conquer strategies, greedy choice properties, and dynamic programming formulations. The discussion highlights how mathematical induction and structural invariants provide formal verification of algorithmic termination and correctness. Key concepts such as optimal substructure, overlapping subproblems, and memoization tables are thoroughly demystified. The lecture details classical algorithmic design techniques demonstrated across standard benchmark problems in computer science. Structured recurrence solving methodologies are demonstrated using the substitution method, recursion tree analysis, and the Master theorem.