Approximations and Streaming Algorithms for Geometric Problems

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Approximations and Streaming Algorithms for

Geometric Problems

Piotr IndykMIT

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Computational Model Single* pass over the data: e1, e2,

…,en Bounded storage Fast processing time per element

*For the purpose of this talk

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Streaming Data Types Vector problems:

Stream defines an array of numbers Maintain stats of the array, e.g., median

Metric problems Clustering

Graph problems, Text problems Geometric Problems [this talk]

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Geometric Data Stream Algorithms as Data Structures Data structures that support:

Insert(p) to P Possibly: Delete(p) from P Compute(P)

Use space that is sub-linear in |P|

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Insertions-only

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Clustering in Geometric Spaces Problems:

k-center [Charikar-Chekuri-Feder-Motwani’97] k-median [Guha-Mishra-Motwani-O’Callaghan’00,

Meyerson’01, Charikar-O’Callaghan-Panigrahy’03]

Bounds: poly(k,log n) space O(1)-approximation

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k-median/k-center

• k is given• Goal: choose k medians/centers to minimize:

• k-median: the sum of the distances• k-center: the max distance

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Geometric Space Bounds:

poly(k,log n) space (1+)-approximation

Problems: Diameter, Minimum Enclosing Ball [Agarwal-Har-Peled’01,

Feigenbaum-Kannan-Zhang’02, Cormode-Muthukrishnan’02, Hershberger-Suri’04]

k-center [Agarwal-HarPeled’01, Agarwal-HarPeled-Varadarajan’04] k-median [HarPeled-Mazumdar’04] Range searching via -approximations:

[Suri-Toth-Zhou’04] [Bagchi-Chaudhary-Eppstein-Goodrich’04]

…

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Dominant Approach: Merge and Reduce Main ideas:

Design an (off-line) algorithm that converts the input into a “sketch”:

Small size Sufficient to solve the problem A sketch of sketches is a sketch

Partition the input in a tree-like fashion Simulate tree computation in small space

Technique can traced back to ancient times

i.e., 80’s [Munro-Paterson’80]

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Tree Computation

p1 p2 p3 p4 p5 p7p6 p8 p9 p10 p11 p12 p13 p15p14 p16

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Analysis Space: (sketch size)*log n Time: sketch computation time Question: Where do sketches come

from ?

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Idea I: solution=sketch Consider k-median [GMMO’00] : approximate k-

median of approximate weighted k-medians is an approximate k-median

Result: Constant depth tree Space: kn , >0 O(1) -approximation Works for any metric

space

k=3

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Use the solution, ctd. -Approximations: find a subset SP ,

such that for any rectangle/halfspace/etc R,

|RS|/|S| = |RP|/|P| [Matousek] : approximation of a union of

approximations is an approximation [BCEG’04] : convert it into streaming

algorithm, applications 1/2 space

[STZ’04] : better/optimal bounds for rectangles and halfspaces

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Idea 2: Core-Sets [AHP’01] Assume we want to

minimize CP(o) SP is an -core-set

for P, if for any o, and a set T:

CPT (o) = (1 ± ) CST (o) Note: this must hold

for all o, not just the optimal one

o

P

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Example: Core-set for MEB Compute extremal points:

Choose “densely” spaced direction v1 …vk

I.e., for any u there is vi such that u*vi ≥ ||u||2 / (1+)

For each direction maintain extremal point

k=O(1/)(d-1)/2 suffice

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Stream Algorithms via Core-sets Diameter/MEB/width: O(1/)(d-1)/2 log n

space [AHP’01] k-center: O(k/d) log n [HP’01] k-median:

O(k/d) log n [HPM’04] O(k2/d) [HPK’05] O(k2d log6 n/) [Chen’05] O(d3/7), k=1 [Indyk’05]

Faster algorithms and other results

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Limitations Small core-sets might not exist Do not support deletions

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Insertions and Deletions

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Insertions and Deletions Technique:

Reduction of geometric problems to vector problems

Use of randomized linear embeddings Problems:

Maintaining histograms of the data Classic geometric problems

(matching, MST, clustering etc)

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Streaming Algorithms for Vector Problems Norm estimation:

Stream elements: (i,b) , i=1…m Interpretation: xi=xi+b Want to maintain ||x||p

Why ? Examples: ||x||p

p =Σi xip = #non-zero coordinates in x, as

p0 …

How ?

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Dimensionality reduction x is an m-dimensional vector A is a “random” m times k matrix, k “small” Store Ax Recover (1±)||x||2 from Ax (with prob. 1-1/N )

[Alon-Matias-Szegedy’96] Estimator: median[ (A1x)2+..+ (Ac x)2, (Ac+1x)2+..+ (A2cx)2,..]1/2 ,

c=1/2 , k=c log N A: constructed from 4-wise independent random variables

[Johnson-Lindenstrauss’85] Estimator: ||Ax||2 A: each entry independently drawn from e.g. Gaussian distribution constructed using Nisan’s PRG [Indyk’00]

[Indyk’00] Estimator: median[ (A1x),…, (Ak x) ] A: as above Works for ||x||p any p(0,2] (using p-stable distributions)

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What it means To know ||x||2, suffices to know Ax Can maintain Ax when the coordinates are

incremented:A(x+ bei)=Ax+ bA ei

A xAx

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Applications of Vector Approach Histograms/wavelet approximation Classic geometric problems

(matching, MST, clustering etc)

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Histograms View x as a function x:[1…n] [1…M] Approximate it using piecewise constant

function h, with B pieces (buckets) Problem can be formulated in 2D as well

(buckets become rectangular tiles)

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Results: 1D [Gilbert-Guha-Indyk-Kotidis-Muthukrishnan-

Strauss’02] : Maintains h with B pieces such that

||x-h||2 ≤ (1+)||x-hOPT||2 Under increments/decrements of x Space: poly(B,1/,log n) Time: poly(B,1/,log n)

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Results: 2D [Thaper-Guha-Indyk-Koudas’02] :

Maintains h with B log (nM) tiles such that ||x-h||2 ≤ (1+)||x-hOPT||2

Under increments/decrements of x Space/Update time: poly(B,1/,log n) Histogram reconstruction time: poly(B,1/, n)

[Muthukrishnan-Strauss’03] : Maintains h with 4B tiles Time: poly(B,1/, log(nM))

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Minimum Weight Bi-chromatic Matching

• Estimate the cost of MWBM

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Minimum Weight Matching

• Estimate the cost of MWM

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Minimum Spanning Tree

• Estimate the cost of MST

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Facility Location

• Goal: choose a set F of facilities to minimize the sum of the distances to nearest facility plus the number of facilities times f• Again, report the cost

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Approach [Indyk’04] Assume P{1…}2

Reduce to vector problems Impose square grids G0…Gk, with

side lengths 20,21, …, 2k , shifted at random.

For each square cell c in Gi, let nP(c) be the number of points from P in c.

The algorithms will maintain certain statistics over nP(.), which will allow it to approximately solve the problems

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Estimators MST:∑i 2i ∑c Gi [nP(c)>0] MWM: ∑i 2i ∑c Gi [nP(c) is odd] MWBM: ∑i 2i ∑c Gi |nG(c)-nB(c)| Fac. Loc.: ∑i 2i ∑c Gi min[nP(c), Ti] K-median:∑i 2i ∑c Gi - B(Q, 2^i) nP(c) (given medians Q)

Maintain #non-zero entries in nP [FM’85]Maintain L1 difference [I’00]

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ResultsProblem Appr.MST log → 1+MWM log MWBM log Fac.Loc. log2 K-median XYZ → 1+

Space: (log +log n + K )O(1)

[Frahling-Indyk-Sohler’05]

[Frahling-Sohler’05]

[…, Charikar’02, …]

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XYZ

Space: (K+log + log n)O(1)

Computation Time ApproximationO(k) poly(log n+1/) 1+2 poly(log n+log +k) O(1)poly(log n+log +k) [ 1+ , log n log ]

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Probabilistic embeddings into HST’s

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Known [Bartal’96, Charikar-Chekuri-Goel-Guha-Plotkin’98]: • ||p-q|| ≤ Dtree (p,q) • E[ Dtree(p,q) ] ≤ ||p-q|| * O(log )

T

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MST

E[Cost(MST in T)] ≤ O(log ) Cost(MST) Cost(MST in T) Cost(T) How to compute Cost(T) ?

Sum over all levels i, of the #nodes at i, times 2i

Node c exists iff ni(c)>0

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Matching Algorithm:

Match what you can at the current level

Odd leftovers wait for the next level

Repeat Optimal on the HST Cost=∑i 2i ∑c Gi [nP(c) is odd]

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Conclusions Algorithms for geometric data

streams Insertions-only: merge and reduce,

coresets Insertions and deletions: randomized

linear embeddings

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Open Problems Matchings, Facility Location, etc:

Replace log by O(1) or even 1+ Possible for:

MST [Frahling-Indyk-Sohler’05] k-median [Frahling-Sohler’05]

Related to computing bi-chromatic matching [Agarwal-Varadarajan’04]

Min-sum clustering ?

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Open Problems High dimensions:

Diameter: 21/2-approx, O(d2 n1/2 ) space, follows from

[Goel-Indyk-Varadarajan’01] c-approx, O( dn1/(c2 - 1) ) [Indyk’03] Conjecture: 21/2-approx, O(d polylog n)

space Min-width cylinder: 18-approx, O(d)

space [Chan’04]

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Open Problems Range queries:

General lower bounds ? (Not just for - approximations)

(1/2) -bit bound for general queries follows from LB for dot product [Indyk-Woodruff’03] and is tight (for randomized algorithms)

What about e.g., half-space queries ? O(1/4/3) is known [STZ’04]

Other problems [STZ’04]