role-algorithmsalgorithm-design

name: role-algorithms:algorithm-design description: Designs algorithms with formal analysis — Big-O/Theta/Omega, amortized analysis, recurrence relations (Master theorem), correctness proofs (loop invariants, induction, reduction), and paradigm selection (greedy, divide-and-conquer, dynamic programming, backtracking). Use when analyzing efficiency, proving correctness, comparing approaches, or selecting optimal algorithms for given constraints. allowed-tools: Read, Grep, Glob, Bash

Algorithm Design

When to use

• Analyzing algorithm efficiency and comparing time/space bounds
• Selecting the right paradigm: greedy, divide-and-conquer, DP, or backtracking
• Proving algorithm correctness with loop invariants or induction
• Solving recurrence relations and applying the Master theorem
• Evaluating space-time trade-offs under memory or latency constraints
• Explaining why a greedy choice is (or is not) valid

Core principles

1. Prove before you code — choose a paradigm and establish correctness argument before implementation
2. Tight bounds over loose bounds — Big-Theta tells the truth; Big-O alone can mislead
3. Constants matter at small n — O(n log n) with large constants can lose to O(n²) for n < 10,000
4. Amortized cost is not average cost — a single expensive operation does not break O(1) amortized
5. Greedy requires proof — exchange argument or cut property; intuition is not a proof

Reference Files

• references/asymptotic-analysis.md — notation (O/Ω/Θ/o), practical interpretation, amortized methods (aggregate, accounting, potential), recurrence relations and Master theorem
• references/design-paradigms.md — greedy (exchange argument, classic examples), divide-and-conquer (split/merge decisions), dynamic programming (when DP beats greedy), backtracking (pruning strategies)
• references/correctness-and-tradeoffs.md — loop invariants, structural induction, reduction proofs, space-time trade-off decision matrix