Shortest Paths: Bellman-Ford Algorithm

Lecture 40 of CS502 Fundamentals of Algorithms covers the Bellman-Ford algorithm, a versatile dynamic programming algorithm that solves the single-source shortest path problem in general graphs containing arbitrary positive and negative edge weights. Unlike Dijkstra which greedily finalizes vertices one by one, Bellman-Ford operates globally: it performs |V| - 1 successive passes over all |E| edges in the graph, relaxing every edge (u, v) during each pass. The mathematical justification relies on the simple path property: any shortest path in a graph with no negative-weight cycles contains at most |V| - 1 edges. Therefore, |V| - 1 passes of relaxation are sufficient to propagate shortest distances from the source to every reachable vertex. After completing |V| - 1 passes, Bellman-Ford conducts a final check: it inspects every edge (u, v); if d[v] > d[u] + w(u, v) can still be relaxed, the algorithm concludes that a negative-weight cycle exists and returns false. Running in O(|V| * |E|) time, Bellman-Ford forms the basis of Distance Vector routing protocols. 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. 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.