Lower Bounds for NNS and Metric Expansion Rina Panigrahy Kunal Talwar Udi Wieder Microsoft Research SVC TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

Nearest Neighbor Search

Decision Version. Given search radius r Find a point in distance r of query point Relation to Approximate NNS: – If second neighbor is at distance cr – Then this is also a c-approximate NN r cr

Cell Probe Model m w

Many different lower bounds Metric spaceApproximationRandomized?Ref ExactyesPT[06], BR[02] noPT[06], Liu[04] yesAIP[06] yesPTW[08] noACP[08] n.exp(ϵ 3 d)

Lower bounds from Expansion Show a unified approach for proving cell probe lower bounds for near neighbor and other similar problems. Show that all lower bounds stem from the same combinatorial property of the metric space Expansion : |number of points near A|/|A| (show some new lower bounds)

Graphical Nearest Neighbor Convert metric space to Graph Place an edge if nodes are within distance r Return a neighbor of the query. Now r=1

Graphical Nearest Neighbor Assume uniform degree Use a random data set Assume W.h.p the n balls are disjoint.

Deterministic Bounds via Expansion

Deterministic Bound sdddddddddddddddlklkj

Example Application n. exp( ϵ 2 d)

Proof Idea when t=1 Shattering F : V → [m] partitions V into m regions Split large regions A random ball is shattered into many parts: about ф(G) ф(G) replication in space

Proof Idea when t=1

Generalizing for larger t

Randomized Bounds Need to relax the definition of vertex expansion

Randomized Bounds Robust Expansion A N(A) N(A) captures all edges from A Expansion =|N(A)|/|A| Capture only ¾ of the edges from A

Robust Exapnsion

Bound for Randomized Data Structure

Proof Idea when t=1 Shattering Most of a random ball is shattered into many parts: about ф r ф r replication in space

Generalizing for larger t Sample 1/ ф r 1/t fraction from each table. A random ball, good part survives in all tables. Union bound for adaptive is trickier.

Applications

General Upper Bound

Conclusions and Open Problems

Approximate Near Neighbor Search sdfsdfsffjlaskdjffj

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Graphical Nearest Neighbor

Randomized Bounds Need to relax the definition of vertex expansion and independence

Deterministic Bounds via Expansion

Proof Idea Can we plug the new definitions in the old proof? – Conceptually – yes! – Actually….well no Dependencies everywhere – the set of good neighbors of a data point depends upon the rest of the data set Solving this is the technical crux of the paper