Testing Expansion in BoundedDegree Graphs Artur Czumaj Shay
Testing Expansion in Bounded-Degree Graphs Artur Czumaj Shay Gershtein Christian Sohler
Model •
Expanders • Sparse • High connectivity • Every subset of vertices has a large boundary • 3 related notions of expansion: § Vertex § Edge § Spectral • Many uses in complexity theory, network design, error correcting codes etc. • known d-regular constructions
Vertex Expansion •
Strong Expansion •
Strong Expansion •
This Talk •
Motivation • Analyses in property testing use expansion properties • Testing problems where input does not contain expanders
Lower Bound •
Main Difficulty • Little is known about bad expanders • Even less is known about graphs far from bad expanders
Main Observation •
Random Walks •
Markov chains •
Mixing Time •
Conductance •
Theorem – expanders mix in log time •
Idea for a Test •
Main Result •
Expansion Tester •
Adjusting Parameters •
Query Complexity •
Analysis •
Analysis •
Accepting Expanders •
Accepting Expanders •
Accepting Expanders •
Rejections - Lemma •
Small Cut - Lemma •
Corollary •
Bad mixing Time • We show that a small ratio cut implies slow convergence to the uniform distribution
Random walk on bad subset •
Random walk on bad subset •
Random walk on bad subset •
Putting everything together •
Putting everything together •
Putting everything together •
- Slides: 38
