Linear-Time Sorting: Counting and Radix Sort

Lecture 16 of CS502 Fundamentals of Algorithms introduces non-comparison sorting algorithms that break the Omega(n log n) lower bound by making specific structural assumptions about input keys. Counting Sort assumes that each of the n input elements is an integer in the range 0 to k. By using array indices to count key frequencies and computing running prefix sums, Counting Sort determines the exact output position of each element in O(n + k) time. Crucially, Counting Sort is stable, meaning elements with equal keys maintain their original relative input order. Radix Sort leverages this stability to sort d-digit numbers by executing d passes of a stable sub-sort from the least significant digit to the most significant digit, achieving an overall running time of O(d * (n + k)). Bucket Sort partitions elements uniformly across a set of bins or buckets, sorts each bucket individually, and concatenates results in linear O(n) average-case time. 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. Dynamic programming state transitions are contrasted with greedy heuristics to evaluate exactness versus computational overhead. 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.