Algorithmic Construction of Sets for k-Restrictions

1
Algorithmic Construction of Sets for
k-Restrictions

• Dana Moshkovitz
• Joint work with Noga Alon and Muli Safra
• Tel-Aviv University

2
Talk Plan

• Problem definition k-restrictions
• Applications
• group testing
• generealized hashing
• Set-Cover Hardness
• Background
• Techniques and Results

3
Techniques

• Greedine
• k-wise approximating distributions
• Concatenation
• multi-way splitters via the topological Necklace
  Splitting Theorem

4
Problem Definition
5
On Forgetful Hot-Tempered Pirates and Helpless
Goldsmiths
One day the hot-tempered pirate asks the
goldsmith to prepare him a nice string in ?m.

6
But the capricious pirate has various
contradicting local demands he may pose when he
comes to collect it

this pattern!
should differ!

7
What will the goldsmith do?
8
make many strings, so every demand is met!








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Formal Definition NSS95

• Input alphabet ?, length m. demands
  f1,,fs?k?0,1,
• Solution A??m s.t
• for every 1?i1ltltik?m, 1?j?s,
• there is a?A s.t. fj(a(i1),,a(ik))1.
• Measure how small A is

?
k
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Applications
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Goldsmith-Pirate Games Capture Many Known Problems

• universal sets
• hashing and its generalizations
• group testing
• set-cover gadget
• separating codes
• superimposed codes
• color coding

12
Application IUniversal Set

• every k configuration is tried.

circuit
0 0 0 . . . 0 0
0 0 1 . . . 1 0
1 1 0 . . . 0 1
0 1 0 . . . 1 1
. . .
. . .
m
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Application IIHashing

• Goal small set of functions m?q
• For every k?q in m, some function maps them to
  k different elements

small set of functions
u1 u2 u3 u4 . . . um
r1 r2 . . . rq
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Generalized Hashing Theorem

• Definition (t,u)-hash families ACKL for all
  T?U, Tt, Uu, some function f satisfies
  f(i)?f(j) for every i?T, j?U-i.
• Theorem For any fixed 2tltu, for any ?gt0, one
  can construct efficiently a (t,u)-hash family
  over alphabet of size t1, whose
• rate (i.e logqm/n) (1-?)t!(u-t)u-t/uu1ln(t1)

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Application IIIGroup Testing DH,ND
. . .

• m people
• at most k-1 are ill
• can test a group contains illness?
• Goal identify the ill people by few tests.

?
?
?
?
?
?
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Group-Tests Theorem

• Theorem For every ?gt0, there exists d(?), s.t
  for any number of ill people dgtd(?), there exists
  an algorithm that outputs a set of at most
  (1?)ed2lnm group-tests in time polynomial in the
  populations size (m).

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Application IVOrientations AYZ94

• Input directed graph G
• Question simple k-path?
• if G were DAG

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Application IV Orientations AYZ94
Need several orientations, s.t wherever the path
is, one reflects it.

• Pick an orientation
• Delete bad edges
• Now G is a DAG

3
5
1
4
2
1
3
5
4
2
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Application VSet-Cover Gadget
sets
Gadget a succinct set-cover instance so that a
small, illegal sub-collection is not a cover.
?
elements
legal cover set and its complement
small its total weight sets and complements
differ in weight
?
?
?
?
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Approximability of Set-Cover
approximation ratio (upto low-order terms)
known app. algorithms Lov75,Sla95,Sri99
ln n
if NP?DTIME(nloglogn) Feige96
if NP?P RS97
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Background

• Random and Pseudo-Random Solutions

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Density

• D?m?0,1 - probability distribution.
• density w.r.t D is
• ? minI,j Pra?D fj(a(I))1

?
?
?
?m
?
?
?
. . .
k
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Probabilistic Strategy

• Claim t?-1(klnmlns1) random strings from D
  form a solution,
• with probability½.

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• Deterministic Construction!

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First Observation

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

?
every demand is satisfied w.p ?
support(uniform)qm
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Second Observation

• A k-wise, O(?)-close to D is a solution.
• Theorem EGLNV98 Product dist. are efficiently
  (poly(qk,m,?-1)) approximatable

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So Whats the Problem?

• Its much more costly than a random solution!
• Random solution klogm/? for all distributions!
• k-wise ?-close to uniform O(2kk2 log2m /?2)
  AGHP90

for other distributions, the state of affairs is
usually much worse
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Background Sum-Up

• Random strings are good solutions for
  k-restriction problems
• if one picks the right distribution
• k-wise approximating distributions are
  deterministic solutions
• of larger size
• Our goal simulate deterministically the
  probabilistic bound

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Our Results
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Outline

• Greedy
• on approximation

kO(1)
assumes invariance under permutations

kO(logm/loglogm)
Concatenation
works for some problems

multi-way splitters
larger ks
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Greedine
same as random solution!

• Claim Can find a solution of size -?-1(klnmlns)
  in time poly(C(m,k), s, support)
• Proof
• Formulate as Set-Cover
• elements ltposition,constraintgt
• sets ltsupport vectorgt
• Apply greedy strategy.

k
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Concatenation
m
















hash family
inefficient solution
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Concatenation Works For Permutations Invariant
Demands
m
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Theorem
Theorem Fix some eff. approx. dist. D. Given a
k-rest. prob. with density ? w.r.t D, obtain a
solution of size arbitrarily close to
(2klnklns)/? k4logm in time
poly(m,s,kk,qk,?-1).
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Dividing Into BLOCKS
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Splitters, NSS95

• What are they?
• several block divisions
• any k are splat by one
• k-restriction problem!
• How to construct?
• needs only (b-1) cuts
• use concatenation

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Multi-Way Splitters

• For any I1??It?m, ?Ij?k, some partition to b
  blocks is a split.
• k-restriction problem!

b
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Necklace Splitting A87

• b thieves
• t types
• How many splits?

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Necklace Splitting A87
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Necklace Splitting Theorem

• Theorem (Alon, 1987) Every necklace with bai
  beads of color i, 1?i?t, has a b-splitting of
  size at most (b-1)t.

tight!
Corollary A multi-way splitter of size
b(b-1)t1 C(m, (b-1)t) is efficiently
constructible.
C(k2,
Hashm,k2,k
concatenation
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The bt2 Case
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Sum-Up

• Beat k-wise approximations for k-restriction
  problems.
• Multi-way splitters via Necklace Splitting.
• ? Substantial improvements for
• Group Testing
• Generalized Hashing
• Set-Cover

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Further Research

• Applications complexity, algorithms,
  combinatorics, cryptography
• Better constructions? different techniques?

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Context

• NSS95 Universality/Hashing via split and smart
  search.
• Our work
• Generalizing NSS95 techniques
• approximating distributions
• multi-way splitters via topological Necklace
  Splitting
• Many more applications group testing, an
  improved hardness for SET-COVER under P?NP

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invariance under permutations

• Definition We say C1,,Cs ??k are invariant
  under permutations,
• if for any permutation ?k?k,
• C1,,Cs ?(C1),,?(Cs)