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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
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
6
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
make many strings, so every demand is met!
9
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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)
15
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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.
. . .
? ? ??? ?
.
.
.
16
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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).).
17
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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.
19
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
20
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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]
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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
...
23
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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..
24
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Deterministic Construction!Deterministic Construction!
25
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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-..)
27
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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!
32
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
N
hash family
ConcatenationConcatenationm
m’
m’
inefficient solution
N
33
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Concatenation Works For Concatenation Works For Permutations Invariant DemandsPermutations Invariant Demands
m
k
m’
m’
34
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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))..
35
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Dividing Into Dividing Into BLOCKSBLOCKSm
36
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
37
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
38
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Necklace Splitting [A87]Necklace Splitting [A87]• b thieves
• t types
How many splits?
39
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Necklace Splitting [A87]Necklace Splitting [A87]
40
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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
42
Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
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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Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions
Further ResearchFurther Research
Applications: Applications: complexitycomplexity,, algorithmsalgorithms, , combinatoricscombinatorics, , cryptographycryptography……
Better constructions? Better constructions? different techniques?different techniques?