All-Pairs Shortest Paths: Matrix Formulation

Lecture 41 of CS502 Fundamentals of Algorithms formulates the All-Pairs Shortest Paths (APSP) problem. Given a weighted directed graph G = (V, E), the goal is to find the shortest distance delta(u, v) between every pair of vertices u, v in V, outputting a |V| x |V| distance matrix D. While the problem can be solved by running Dijkstra |V| times on positive graphs in O(|V| |E| log |V|) or Bellman-Ford |V| times on general graphs in O(|V|^2 |E|), the lecture develops dedicated dynamic programming algorithms. By defining d^(m)[i, j] as the shortest path from vertex i to vertex j using at most m edges, the problem yields the recurrence d^(m)[i, j] = min_k { d^(m-1)[i, k] + w[k, j] }. The lecture demonstrates a striking mathematical isomorphism between this recurrence and standard matrix multiplication, where min replaces addition and addition replaces multiplication (the min-plus semiring). Using repeated squaring, this matrix formulation computes all-pairs shortest paths in O(|V|^3 log |V|) time. Algorithmic decision boundaries and lower-bound complexity proofs are evaluated alongside practical runtime performance characteristics. Graph representations are explored through adjacency matrices, adjacency lists, and memory footprint considerations in sparse and dense topologies. Breadth-first and depth-first search mechanics, discovery and finishing timestamps, and edge classifications are systematically analyzed. Greedy strategies including Huffman prefix tree generation and activity scheduling are rigorously proven using exchange arguments. Scholars examine shortest path optimality conditions, edge relaxation properties, and negative cycle detection algorithms. Minimum spanning tree algorithms including Kruskal and Prim formulations are analyzed alongside union-find disjoint set operations. Computational complexity concepts are illustrated through polynomial-time verification, decision reductions, and NP-completeness proofs. By engaging with solved past papers by Moaaz and Waqar Siddhu, learners gain actionable insights into recurrent university exam questions. Interactive quiz assessments on VU SIGMA facilitate rapid knowledge verification and retention of critical algorithmic concepts. Comprehensive coverage of standard academic benchmarks ensures learners communicate with technical precision across computational disciplines. Detailed analysis of recurrence relations highlights the imperative of analyzing base cases and recursive branching factors accurately.