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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. Given points in a metric space

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lower bounds for nns and metric expansion

Lower Bounds for NNS and Metric Expansion

RinaPanigrahy

KunalTalwar

UdiWieder

Microsoft Research SVC

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AA

nearest neighbor search
Nearest Neighbor Search

Given points in a metric space

Preprocess into a small data structure

Given a query point

Quickly retrieve the closest to

Many Applications

decision version given search radius r
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
Cell Probe Model

Preprocess into data structure with

  • words
  • bits per word

Query algorithm gets charged t

if it probes words of

  • All computation is free

Study tradeoff between and

In this talk

w

m

lower bounds from expansion
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
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 neighbor1
Graphical Nearest Neighbor
  • Assume uniform degree
  • Use a random data set
  • Assume W.h.p the n balls are disjoint.
deterministic bound
Deterministic Bound
  • sdddddddddddddddlklkj
proof idea when t 1 shattering
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
Proof Idea when t=1
  • determines which cell in is read
  • Select a fraction of cells such
  • it is likely that cantains a quarter of the data set points
  • So, and
generalizing for larger t
Generalizing for larger t
  • Select a fraction of each table such
  • Continue as before
    • Non adaptive algorithms
  • Adaptive alg. depend upon content of selected cells
    • Subexp. number of algs
    • Union bound
randomized bounds
Randomized Bounds
  • So far we assumed the algorithm is correct on
    • What if only of are good query point?

Need to relax the definition of vertex expansion

randomized bounds1
Randomized Bounds
  • Robust Expansion
  • N(A) captures all edges from A
  • Expansion =|N(A)|/|A|
  • Capture only ¾ of the edges from A

A

N(A)

robust exapnsion
Robust Exapnsion
  • Small set vertex expansion:
  • In other words:We can cover all the edges incident on with a set of size
  • We can cover of the edges incident on with a set of size
    • Robust expansion is at least the edge expansion
bound for randomized data structure
Bound for Randomized Data Structure
  • Theorem: if is weakly Independent, then a randomized data structure that answers GNS queries with space and queries must satisfy and
proof idea when t 1 shattering1
Proof Idea when t=1 Shattering
  • Most of a random ball is shattered into many parts: about фr
  • фr replication in space
generalizing for larger t1
Generalizing for larger t
  • Sample 1/фr1/t fraction from each table.
  • A random ball, good part survives in all tables.
  • Union bound for adaptive is trickier.
applications
Applications
  • We know how to calculate robust expansion of graphs derived from:
    • when (known)
    • when (new)
    • when (natural input dist.)
  • Don’t know the robust expansion of:
    • when
general upper bound
General Upper Bound
  • Say is a Cayley Graph
  • Take
  • Take with r.e.
  • Use random translations of to define the access function
  • For rand. input success prob. is constant
conclusions and open problems
Conclusions and Open Problems

Unified approach to NNS cell probe lower bounds

    • often characterized by expansion
    • Average case with natural distributions
  • Higher lower bounds?
    • Improve dependency on (very hard)
    • Dynamic NNS, tight bound for special cases shown in the paper
slide26

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randomized bounds2
Randomized Bounds
  • So far we assumed the algorithm is correct on
    • What if only of are good query point?

Need to relax the definition of vertex expansion and independence

is weakly independent if for random it holds that

proof idea
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
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