algorithmic construction of sets for k-restrictions dana moshkovitz joint work with noga alon and...

Algorithmic Construction Algorithmic Construction of Sets for of Sets for kk-Restrictions-Restrictions

Dana MoshkovitzDana Moshkovitz

Joint work with Joint work with Noga AlonNoga Alon and and Muli SafraMuli Safra

Tel-Aviv UniversityTel-Aviv University

2

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions

Problem definition:Problem definition: k k-restrictions-restrictions Applications:Applications: … …

group testinggroup testing generealized hashinggenerealized hashing Set-Cover HardnessSet-Cover Hardness

BackgroundBackground Techniques and ResultsTechniques and Results

Talk PlanTalk Plan

3


TechniquesTechniques

GreedineGreedine$$$$

kk-wise approximating distributions-wise approximating distributions ConcatConcatenationenation multi-way splittersmulti-way splitters via the topologicalvia the topological

NeNeccklacklacee Spli Splittttiing Theong Theorremem

Problem DefinitionProblem Definition

5


One day the hot-tempered pirate asks the goldsmith to prepare him

a nice string in m.

On Forgetful Hot-Tempered Pirates and On Forgetful Hot-Tempered Pirates and Helpless Goldsmiths Helpless Goldsmiths

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But the capricious pirate has various contradicting local

demands he may pose when he comes to collect it…

this pattern!should

differ!

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What will the goldsmith do?

8


make many strings, so every demand is met!

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Formal Definition [~NSS95]Formal Definition [~NSS95]

Input:Input: alphabet alphabet , length , length mm. . demands demands ff11,…,f,…,fss::kk{0,1}{0,1}, ,

Solution:Solution: A Amm s.t s.t for every for every 11ii11<…<i<…<ikkmm, ,

11jjss, , there is there is aaAA s.t. s.t. ffjj(a(i(a(i11),…,a(i),…,a(ikk))=1))=1..

Measure:Measure: how small how small |A| |A| isis

m

k

ApplicationsApplications

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Goldsmith-Pirate Games Capture Goldsmith-Pirate Games Capture Many Known ProblemsMany Known Problems

universal setsuniversal sets hashing and its hashing and its

generalizationsgeneralizations group testinggroup testing set-cover gadgetset-cover gadget separating codesseparating codes superimposed codessuperimposed codes color codingcolor coding

……

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Application IApplication IUniversal SetUniversal Set

every every kk configuration is tried. configuration is tried.

circuit.. .

000

.

.

.00

001

.

.

.10

110

.

.

.01

010

.

.

.11

. . . m

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Application IIApplication IIHashingHashing

Goal:Goal: small set of small set of functions functions [m][m][q][q]

For every For every kkqq in in [m][m], , some function maps some function maps them to them to kk different different elementselements

small set of

functions

u1

u2

u3

u4...um

r1

r2

.

.

.rq

k

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Generalized Hashing Generalized Hashing TheoremTheorem

Definition Definition (t,u)-hash families(t,u)-hash families [ACKL][ACKL]: for all : for all TTUU, , |T|=t|T|=t, , |U|=u|U|=u, some function , some function ff satisfies satisfies f(i)≠f(j)f(i)≠f(j) for every for every iiTT, , jjU-{i}U-{i}..

Theorem:Theorem: For any fixed For any fixed 2≤t<u2≤t<u, for any , for any

>0>0, one can construct efficiently a , one can construct efficiently a (t,u)-(t,u)-hash familyhash family over alphabet of size over alphabet of size t+1t+1, , whose whose

rate (i.e rate (i.e loglogqqm/nm/n) ≥ ) ≥ (1-(1-)t!(u-t))t!(u-t)u-tu-t/u/uu+1u+1ln(t+1)ln(t+1)

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Application IIIApplication IIIGroup Testing Group Testing [DH,ND…][DH,ND…]

mm people people at most at most k-1k-1 are ill are ill can test a group: can test a group:

contains illness?contains illness? Goal:Goal: identify the identify the

ill people by few ill people by few tests.tests.

. . .

? ? ??? ?

.

.

.

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Group-Tests TheoremGroup-Tests Theorem

Theorem:Theorem: For every For every >0>0, there exists , there exists d(d()), s.t for any number of ill people , s.t for any number of ill people d>d(d>d()), there exists an algorithm that , there exists an algorithm that outputs a set of at most outputs a set of at most (1+(1+))eded22lnmlnm group-tests in time polynomial in the group-tests in time polynomial in the population’s size (population’s size (mm).).

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Application IVApplication IVOrientations [AYZ94]Orientations [AYZ94]

Input:Input: directed graph directed graph GG

Question:Question: simplesimple kk-path?-path? if if GG were DAG… were DAG…

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Application IVApplication IV Orientations [AYZ94] Orientations [AYZ94]

Pick an orientationPick an orientation

Delete ‘bad’ edgesDelete ‘bad’ edges Now Now GG is a DAG… is a DAG…

1 3 542

1

2

35

4

Need several orientations, s.t

wherever the path is, one reflects it.

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Application VApplication VSet-Cover GadgetSet-Cover Gadget

ele

ments

sets

Gadget:Gadget: a succinct set- a succinct set-cover instance so that: cover instance so that:

a small, illegal sub-a small, illegal sub-collection is not a collection is not a covercover..legal cover: set and its complement

small: its total weight ≤ …sets and complements differ in weight

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Approximability of Set-Approximability of Set-CoverCover

ln n

known app. algorithms

[Lov75,Sla95,Sri99]

approximation ratio (upto low-

order terms)

if NPDTIME(nloglogn) [Feige96]

if NPP [RS97]

BackgroundBackground

Random and Pseudo-Random Random and Pseudo-Random SolutionsSolutions

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DensityDensity

D:D:mm[0,1][0,1] - - probability probability distribution.distribution.

densitydensity w.r.t w.r.t DD is: is:

= = minminI,jI,j PrPraaDD[ [ ffjj(a(I))=1(a(I))=1 ]]

m

k

m

...

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Probabilistic StrategyProbabilistic Strategy

Claim:Claim: t=t=-1-1(klnm+lns+1)(klnm+lns+1) random random strings from strings from DD form a form a solutionsolution, ,

with probabilitywith probability≥½≥½..

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Deterministic Construction!Deterministic Construction!

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First ObservationFirst Observation

support(support(DD)) is a solution is a solution if density positive w.r.t if density positive w.r.t

DD..

m

k

every demand

is satisfied w.p ≥

|support(uniform)|=qm

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Second ObservationSecond Observation

A A kk-wise, -wise, O(O())-close to -close to DD is a solution.is a solution.

Theorem [EGLNV98]Theorem [EGLNV98]: : Product dist. are Product dist. are

efficiently (efficiently (poly(qpoly(qkk,m,,m,-1-1))) ) approximatableapproximatable

m

k

every demand is satisfied

w.p (1-..)

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So What’s the Problem?So What’s the Problem?

It’s much more costly than a random solution! It’s much more costly than a random solution!

Random solution: Random solution: ~ klogm/~ klogm/ for all for all distributions!distributions!

kk-wise -wise -close to uniform: -close to uniform: O(2O(2kkkk2 2 loglog22m /m /22) ) [AGHP90][AGHP90]

for other distributions, the state of affairs is

usually much worse…

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Background Sum-UpBackground Sum-Up

RandomRandom strings are good solutions strings are good solutions for for kk-restriction problems-restriction problems if one picks the ‘right’ distribution…if one picks the ‘right’ distribution…

kk-wise approximating-wise approximating distributions distributions are are deterministicdeterministic solutions solutions of larger size…of larger size…

Our goal:Our goal: simulate deterministically simulate deterministically the probabilistic boundthe probabilistic bound

Our ResultsOur Results

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OutlineOutline

Greedy on approximationk=O(1)

+

+

multi-way splitterslarger k’s

Concatenationk=O(logm/

loglogm)works for some problems

assumes invariance under permutations

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Greedine$$ Greedine$$

Claim:Claim: Can find a solution Can find a solution of size of size ---1-1(klnm+lns) (klnm+lns) in in timetime poly(poly(C(m,k), s, |support|C(m,k), s, |support|))

Proof:Proof: FormulateFormulate as as Set-CoverSet-Cover::

elements: elements: <position,constraint><position,constraint>

sets: sets: <support vector><support vector>

Apply greedy strategy.Apply greedy strategy.

m

k

same as random solution!

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N

hash family

ConcatenationConcatenationm

m’

m’

inefficient solution

N

33


Concatenation Works For Concatenation Works For Permutations Invariant DemandsPermutations Invariant Demands

m

k

m’

m’

34


TheoremTheorem

Theorem:Theorem: Fix some eff. approx. dist. Fix some eff. approx. dist. DD..Given a Given a kk-rest. prob. with density -rest. prob. with density w.r.t w.r.t DD, , obtain a solution of size arbitrarily close to obtain a solution of size arbitrarily close to

(2klnk+lns)/(2klnk+lns)/ ×× kk44logmlogm in time in time poly(m,s,kpoly(m,s,kkk,q,qkk,,-1-1))..

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Dividing Into Dividing Into BLOCKSBLOCKSm

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Splitters, [NSS95]Splitters, [NSS95]

What are they?What are they? several block divisionsseveral block divisions any any kk are splat by one are splat by one k-restriction problem!k-restriction problem!

How to construct?How to construct? needs only needs only (b-1)(b-1) cuts cuts use concatenationuse concatenation

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Multi-Way SplittersMulti-Way Splitters

For any For any II11⊎⊎……⊎⊎IItt[m][m], , ||⊎⊎IIjj||kk, , some partition some partition to to bb blocks is a split. blocks is a split.

k-restriction problemk-restriction problem!!

m

k

b

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Necklace Splitting [A87]Necklace Splitting [A87]• b thieves

• t types

How many splits?

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Necklace Splitting [A87]Necklace Splitting [A87]

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Necklace Splitting TheoremNecklace Splitting Theorem

Theorem (Alon, 1987):Theorem (Alon, 1987): Every necklace Every necklace with with babaii beads of color beads of color ii, , 11iitt, has a , has a bb-splitting of size at most -splitting of size at most (b-1)t(b-1)t.. tight!

Corollary:Corollary: A A multi-way splittermulti-way splitter of size of size

bb(b-1)t+1 (b-1)t+1 C(m, (b-1)t)C(m, (b-1)t)

is efficiently constructible.is efficiently constructible.C(k2

,·|Hashm,k2,k|

concatenation

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The b=t=2 CaseThe b=t=2 Case

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Sum-UpSum-Up

BeatBeat k-wise approximationsk-wise approximations for for kk--restriction problems.restriction problems.

Multi-way splittersMulti-way splitters via Necklace via Necklace Splitting.Splitting.

Substantial improvements for:Substantial improvements for: Group TestingGroup Testing Generalized HashingGeneralized Hashing Set-CoverSet-Cover

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Further ResearchFurther Research

Applications: Applications: complexitycomplexity,, algorithmsalgorithms, , combinatoricscombinatorics, , cryptographycryptography……

Better constructions? Better constructions? different techniques?different techniques?

algorithmic construction of sets for k-restrictions dana moshkovitz joint work with noga alon and...

Documents

theorem slide

dag slide

restrictions applications

weight slide

plan slide

problem definition slide

total weight sets

np p rs97 slide