Computational Models and 2D Maxima Problem

Lecture 2 of CS502 Fundamentals of Algorithms formalizes computational models and introduces the classic two-dimensional maxima problem. To analyze algorithm performance objectively without hardware-dependent variations, computer scientists rely on idealized models of computation. The primary model utilized throughout this course is the standard Random Access Machine (RAM), which assumes a uniprocessor architecture where instructions are executed sequentially. Under the RAM abstraction, elementary operations including arithmetic additions, variable assignments, logical tests, and pointer dereferences execute in constant unit time O(1), and memory access latency is entirely uniform. Building upon this theoretical model, the lecture formulates the 2D maxima problem. Given a collection of n points situated in a two-dimensional Cartesian plane, point p is said to be dominated by another point q if both the x-coordinate and y-coordinate of q are strictly greater than or equal to those of p. A point is defined as maximal if it is undominated by any other point in the set. The 2D maxima problem requires identifying all maximal points within an arbitrary input configuration, modeling practical real-world trade-offs such as multi-objective decision making. 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. 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.