Minimum Spanning Trees: Prim Algorithm

Lecture 37 of CS502 Fundamentals of Algorithms covers Prim's algorithm for computing a Minimum Spanning Tree. Unlike Kruskal's algorithm which merges disjoint trees across a forest, Prim's algorithm grows a single tree continuously from an arbitrary starting root vertex r. At each step, the set of vertices already in tree A forms a cut (V_A, V - V_A). Prim's algorithm greedily selects a light edge crossing this cut, adding the adjacent vertex into V_A. To identify the light edge efficiently without searching all crossing edges, all vertices outside the tree (V - V_A) reside in a min-priority queue Q. Each vertex v stores an attribute key[v], recording the minimum weight of any edge connecting v to a vertex in the growing tree, and a predecessor pointer pi[v]. In each iteration, Extract-Min retrieves the vertex u with smallest key[u], adds it to the tree, and relaxes the keys of all neighbors incident to u. Implemented using a standard binary heap, Prim runs in O(|E| log |V|) time; with a Fibonacci heap, it achieves O(|E| + |V| log |V|). Students evaluate the trade-offs between comparison-based sorting boundaries and specialized linear-time integer distribution sorting techniques. The lesson explores intractable computational problems by examining formal polynomial reductions to known NP-complete cores. Practical code illustrations demonstrate iterative and recursive implementations, memoized table lookups, and pointer-based data structures. Geometric computational techniques including sweep-line approaches are investigated to demonstrate spatial dimension reduction in algorithms. Scholars analyze matrix multiplication parenthesization and edit distance dynamic tables to internalize two-dimensional state memoization. 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.