Algorithms and complexity fundamentals

Algorithms and complexity fundamentals are how we talk about HNSW's cost: what scales with dataset size, what approximate search buys you, and how greedy local decisions differ from exhaustive comparison.
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These are the topics in this section, each on its own page with a stable path you can bookmark or share.

What topics are covered in this section?

Algorithms and complexity fundamentals are how we talk about HNSW‘s cost: what scales with dataset size, what approximate search buys you, and how greedy local decisions differ from exhaustive search.

Why frame HNSW with Big-O and approximation language?

Exact nearest neighbor via linear scan is clear but does not scale. HNSW is a greedy, approximate method whose practical appeal is high recall at far less than brute-force work. Big-O, logarithmic time, asymptotic scaling, and the split between time and space complexity let you compare indexes honestly. Average-case, expected-case, worst-case, and amortized analysis prevent mixing a happy median latency with a pathological tail. Approximation algorithms and algorithmic trade-offs name the deliberate quality-versus-speed bargain every ANN system makes.

Those words recur whenever someone claims logarithmic search or linear memory in edges.

How should you use this glossary while learning HNSW?

Treat brute-force, exhaustive, and linear scan as the baseline tax you are avoiding. Greedy algorithms explain the hop rule; divide-and-conquer echoes multilayer descent. When Part III discusses what can and cannot be proved, return here for expected versus worst case. When Part VI tunes ef, remember you are sliding an algorithmic trade-off, not finding a free lunch. Weaviate’s ANN benchmarks report recall and QPS together precisely because complexity talk without quality is incomplete.

Complexity literacy keeps marketing curves and paper claims interpretable.

Where do you go from here?

Read Big-O, logarithmic time, and approximation algorithm pages, then the HNSW-specific complexity glossary and Part III’s proof-limits chapter. For systems cost, continue into memory and hardware hubs.

This section is the shared complexity dialect for the site. Next, open “What is Big-O notation?” or “What is an approximation algorithm?”, then Part III when you want those ideas applied to HNSW itself.