Shortest Paths: Single-Source Overview

Lecture 38 of CS502 Fundamentals of Algorithms introduces the Single-Source Shortest Path (SSSP) problem in weighted directed graphs G = (V, E). Given a source vertex s, the objective is to find a path of minimum total weight delta(s, v) to every reachable vertex v in V. The lecture proves that shortest paths exhibit optimal substructure: any subpath of a shortest path is itself a shortest path between its endpoints. Students investigate the implications of edge weights: while positive weights guarantee well-defined finite shortest paths, negative-weight edges introduce complexities. If a graph contains a negative-weight cycle reachable from source s, the shortest path distance to any vertex reachable from that cycle is -infinity, as traversing the cycle infinitely reduces path weight. The central operational primitive across all shortest path algorithms is Relaxation: for an edge (u, v) with weight w, testing whether d[v] > d[u] + w(u, v) and updating d[v] = d[u] + w(u, v) and pi[v] = u decreases upper bounds toward true distances. Network flow principles and bipartite matching formulations are examined as extensions of core graph optimization methodologies. The relationship between decision problems, certificate verification, and non-deterministic polynomial time is reinforced through formal definitions. In this lecture, students examine formal computational models, asymptotic behavior, and mathematical proofs of algorithmic correctness. Through detailed theoretical breakdowns, the lesson explores running time recurrence equations, recursion trees, and asymptotic bounds. Special emphasis is placed on worst-case execution bounds, operational space requirements, and optimal substructure formulations. 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.