Graph Representations: Matrices and Lists

Lecture 28 of CS502 Fundamentals of Algorithms examines the two standard data structures used to represent graphs in computer memory: Adjacency Matrices and Adjacency Lists. An Adjacency Matrix represents a graph G = (V, E) as a |V| x |V| boolean or weight matrix A, where entry A[u, v] = 1 if an edge (u, v) exists, and 0 otherwise. While an adjacency matrix enables instantaneous O(1) edge existence queries, it requires Theta(|V|^2) memory regardless of the number of edges, proving prohibitively wasteful for sparse graphs where |E| << |V|^2. Conversely, an Adjacency List maintains an array of |V| linked lists, where list Adj[u] contains all vertices adjacent to vertex u. Adjacency lists require compact storage of Theta(|V| + |E|), optimizing memory consumption and enabling rapid iteration over neighbors in O(degree(u)) time. The lecture discusses the practical trade-offs between matrices and lists across dense, sparse, and weighted network applications. 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. 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.