Greedy Algorithms Paradigm and Coin Change

Lecture 23 of CS502 Fundamentals of Algorithms introduces the Greedy algorithmic paradigm, an intuitive and computationally efficient approach for solving optimization problems. Unlike dynamic programming which evaluates all possible subproblem choices and solves them from the bottom up, a greedy algorithm makes whatever choice seems best at the immediate moment—a locally optimal choice—in the hope that this choice leads to a globally optimal solution. The lecture formalizes the two essential properties required for a greedy algorithm to yield guaranteed optimal solutions: optimal substructure, and the greedy choice property, which proves that a globally optimal solution can be assembled by iteratively making locally optimal choices without reconsidering previous decisions. The paradigm is demonstrated using the Coin Change problem: making change for an amount n using the fewest coins possible. While standard currency systems (like US coins: 25, 10, 5, 1) are canonical and guarantee greedy optimality, the lecture presents non-canonical denominations (such as coins 4, 3, 1 for change of 6) where greedy yields suboptimal solutions (4+1+1 = 3 coins vs 3+3 = 2 coins). 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. 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.