Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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Comparison sorts: quicksort O(n log n) avg, mergesort O(n log n) worst, heapsort O(n log n). Non-comparison: counting sort O(n+k), radix sort O(nk), bucket sort O(n) avg. Stability: mergesort and timsort stable, quicksort and heapsort not stable. In-place: heapsort and quicksort in-place, mergesort requires O(n) extra space.
BFS: shortest path in unweighted graphs, O(V+E), uses queue. DFS: topological sort, cycle detection, SCC, O(V+E), uses stack or recursion. Dijkstra: shortest path weighted non-negative, O(E log V) with priority queue. Bellman-Ford: handles negative weights, detects negative cycles, O(VE). Floyd-Warshall: all-pairs shortest path, O(V3), dynamic programming. Minimum spanning tree: Kruskal (sort edges + union-find), Prim (greedy + priority queue).
Optimal substructure: optimal solution contains optimal solutions to subproblems. Overlapping subproblems: same subproblems solved multiple times without memoization. Top-down memoization: recursive with cache. Bottom-up tabulation: iterative filling of DP table. Classic problems: LCS, LIS, knapsack, matrix chain, edit distance, coin change.
Big-O classes: O(1) constant, O(log n) logarithmic, O(n) linear, O(n log n), O(n2), O(2n). P vs NP: P decidable in polynomial time, NP verifiable in polynomial time. NP-complete: NP-hard and in NP. First: SAT. Others via reduction. Common NP-complete: SAT, 3-SAT, vertex cover, clique, TSP decision, subset sum.
| Pitfall | Fix |
|---|---|
| Off-by-one in DP table | Carefully define base cases and indices |
| Dijkstra on negative edges | Use Bellman-Ford instead |
| Ignoring integer overflow | Use long or BigInteger for large inputs |
| Wrong complexity analysis | Account for all loops and recursive calls |
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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level analytical chemistry knowledge. Use when working with chromatography, spectroscopy, electroanalytical methods, sample preparation, quantitative analysis, calibration, statistics, or instrumental analysis. Also use when the user mentions 'HPLC', 'GC', 'mass spectrometry', 'atomic absorption', 'ICP', 'titration', 'calibration curve', 'detection limit', 'NMR', 'chromatography', 'electrochemistry', 'sample preparation', or 'method validation'.
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