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Algorithmic Construction of Sets for k -Restrictions

Algorithmic Construction of Sets for k -Restrictions. Dana Moshkovitz Joint work with Noga Alon and Muli Safra Tel-Aviv University. Talk Plan. Problem definition: k -restrictions Applications: … group testing generealized hashing Set-Cover Hardness Background

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Algorithmic Construction of Sets for k -Restrictions

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  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 splittersvia the topologicalNecklace 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!

  9. m Formal Definition [~NSS95] • Input: alphabet , length m. demands f1,…,fs:k{0,1}, • Solution: Am s.t • for every 1i1<…<ikm, 1js, • there is aA s.t. fj(a(i1),…,a(ik))=1. • Measure: how small |A| is  k

  10. Applications

  11. 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 1 0 . . . 1 1 1 1 0 . . . 0 1 0 0 1 . . . 1 0 0 0 0 . . . 0 0 . . . . . . m

  13. k 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

  14. Generalized Hashing Theorem • Definition (t,u)-hash families[ACKL]: for all TU, |T|=t, |U|=u, some function f satisfies f(i)≠f(j) for every iT, jU-{i}. • Theorem: For any fixed 2≤t<u, for any >0, one can construct efficiently a (t,u)-hash family over alphabet of size t+1, whose rate (i.e logqm/n) ≥ (1-)t!(u-t)u-t/uu+1ln(t+1)

  15. . . . 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. . . . ? ? ? ? ? ?

  16. Group-Tests Theorem Theorem: For every >0, there exists d(), s.t for any number of ill people d>d(), there exists an algorithm that outputs a set of at most (1+)ed2lnm group-tests in time polynomial in the population’s size (m).

  17. Application IVOrientations [AYZ94] • Input: directed graph G • Question:simplek-path? • if G were DAG…

  18. 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 2 3 4 5

  19. 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    

  20. 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]

  21. Background Random and Pseudo-Random Solutions

  22. m Density • D:m[0,1] - probability distribution. • density w.r.t D is:  = minI,j PraD[ fj(a(I))=1]    m    . . . k

  23. Probabilistic Strategy Claim:t=-1(klnm+lns+1) random strings from D form a solution, with probability≥½.

  24. Deterministic Construction!

  25. m k First Observation support(D)is a solution if density positive w.r.t D.  every demand is satisfied w.p ≥ |support(uniform)|=qm

  26. m  every demand is satisfied w.p  (1-..) k Second Observation A k-wise, O()-close to D is a solution. Theorem [EGLNV98]: Product dist. are efficiently (poly(qk,m,-1)) approximatable

  27. So What’s the Problem? It’s 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…

  28. 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

  29. Our Results

  30. Outline Greedy on approximation k=O(1) assumes invariance under permutations + k=O(logm/loglogm) Concatenation works for some problems + multi-way splitters larger k’s

  31. m Greedine$$ same as random solution! Claim: Can find a solution of size --1(klnm+lns) in timepoly(C(m,k), s, |support|) Proof: • Formulateas Set-Cover: • elements: <position,constraint> • sets: <support vector> • Apply greedy strategy. k

  32. m m’ N N Concatenation m’ hash family inefficient solution

  33. m m’ k Concatenation Works For Permutations Invariant Demands m’

  34. 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 (2klnk+lns)/×k4logm in time poly(m,s,kk,qk,-1).

  35. m Dividing Into BLOCKS

  36. 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

  37. m k Multi-Way Splitters • For any I1⊎…⊎It[m], |⊎Ij|k, some partition to b blocks is a split. • k-restriction problem! b

  38. Necklace Splitting [A87] • b thieves • t types • How many splits?

  39. Necklace Splitting [A87]

  40. 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)t+1 C(m, (b-1)t) is efficiently constructible. C(k2, ·|Hashm,k2,k| concatenation

  41. The b=t=2 Case

  42. Sum-Up • Beatk-wise approximations for k-restriction problems. • Multi-way splitters via Necklace Splitting. •  Substantial improvements for: • Group Testing • Generalized Hashing • Set-Cover

  43. Further Research • Applications: complexity, algorithms, combinatorics, cryptography… • Better constructions? different techniques?

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