Algorithmic Construction of Sets for kRestrictions Dana Moshkovitz

Algorithmic Construction of Sets for k-Restrictions Dana Moshkovitz Joint work with Noga Alon and Muli Safra Tel-Aviv University

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Talk Plan n n Problem definition: k-restrictions Applications: … n n n group testing generealized hashing Set-Cover Hardness Background Techniques and Results 2

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Techniques Greedine$$ n k-wise approximating distributions n Concatenation n multi-way splitters via the topological Necklace Splitting Theorem n 3

Problem Definition

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions On Forgetful Hot-Tempered Pirates and Helpless Goldsmiths One day the hot-tempered pirate asks the m goldsmith to prepare him a nice string in . 5

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… should differ! this pattern! 6

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions What will the goldsmith do? 7

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions make many strings, so every demand is met! 8

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Formal Definition [~NSS 95] n n Input: alphabet , length m. demands f 1, …, fs: k {0, 1}, Solution: A m s. t n for every 1 i 1<…<ik m, 1 j s, n n m there is a A s. t. fj(a(i 1), …, a(ik))=1. Measure: how small |A| is k 9

Applications

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Goldsmith-Pirate Games Capture Many Known Problems n n n n … universal sets hashing and its generalizations group testing set-cover gadget separating codes superimposed codes color coding 11

Dana Moshkovitz n Algorithmic Construction of Sets for k-Restrictions Application I Universal Set every k configuration is tried. . 0 1 1 0 0 0 . . . 1 1 . . . 0 1 . . . 1 0 . . . 0 0 . . . circuit m 12

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Application II Hashing Goal: small set of functions [m] [q] n For every k q in [m], some function maps them to k different elements n k small set of functions u 1 u 2 u 3 u 4. . . um r 1 r 2. . . rq 13

Dana Moshkovitz n n Algorithmic Construction of Sets for k-Restrictions 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+1 ln(t+1) 14

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Application III Group Testing [DH, ND…]. . . n n m people at most k-1 are ill can test a group: contains illness? Goal: identify the ill people by few tests. . ? ? 15

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Group-Tests 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+ )ed 2 lnm group-tests in time polynomial in the population’s size (m). 16

Dana Moshkovitz n Algorithmic Construction of Sets for k-Restrictions Application IV Orientations [AYZ 94] Input: directed graph G Question: simple k-path? n if G were DAG… n 17

Dana Moshkovitz n Algorithmic Construction of Sets for k-Restrictions Application IV Orientations [AYZ 94] Pick an orientation 3 Need several orientations, s. t wherever the path is, one reflects it. 5 1 4 2 Delete ‘bad’ edges n Now G is a DAG… n 1 2 3 4 5 18

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Application V Set-Cover Gadget sets Gadget: a succinct setcover instance so that: a small, illegal subcollection is not a cover. elements legal cover: set and its complement small: its total weight ≤ … sets and complements differ in weight 19

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Approximability of Set-Cover approximation ratio (upto loworder terms) known app. algorithms [Lov 75, Sla 95, Sri 99] ln n if NP DTIME(nloglogn) [Feige 96] if NP P [RS 97] 20

Background Random and Pseudo-Random Solutions

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Density D: m [0, 1] - probability distribution. n density w. r. t D is: = min. I, j Pra D[ fj(a(I))=1 ] n m . . . k 22

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Probabilistic Strategy Claim: t= -1(klnm+lns+1) random strings from D form a solution, with probability≥½. 23

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Deterministic Construction! 24

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions First Observation support(D) is a solution if density positive w. r. t D. |support(uniform)|=qm m every demand is satisfied w. p ≥ k 25

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Second Observation A k-wise, O( )-close to D is a solution. Theorem [EGLNV 98]: Product dist. are efficiently (poly(qk, m, -1)) approximatable m every demand is satisfied w. p (1 -. . ) k 26

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions So What’s the Problem? It’s much more costly than a random solution! n n Random solution: ~ klogm/ for all distributions! k-wise -close to uniform: O(2 kk 2 log 2 m / 2) [AGHP 90] for other distributions, the state of affairs is usually much worse… 27

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Background Sum-Up n Random strings are good solutions for krestriction problems n n k-wise approximating distributions are deterministic solutions n n if one picks the ‘right’ distribution… of larger size… Our goal: simulate deterministically the probabilistic bound 28

Our Results

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Outline Greedy k=O(1) on approximation + assumes invariance under permutations Concatenation k=O(logm/loglogm) works for some problems + multi-way largersplitters k’s 30

Dana Moshkovitz same as random solution! Algorithmic Construction of Sets for k-Restrictions Greedine$$ Claim: Can find a solution of size - -1(klnm+lns) in time poly(C(m, k), s, |support|) Proof: n Formulate as Set-Cover: m elements: <position, constraint> n sets: <support vector> n n Apply greedy strategy. k 31

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Concatenation m m’ m’ N hash family N inefficient solution 32

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Concatenation Works For Permutations Invariant Demands m m’ m’ k 33

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions 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 (2 klnk+lns)/ × k 4 logm in time poly(m, s, kk, qk, -1). 34

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Dividing Into BLOCKS m 35

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Splitters, [NSS 95] n What are they? several block divisions n any k are splat by one n k-restriction problem! n n How to construct? needs only (b-1) cuts n use concatenation n 36

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Multi-Way Splitters For any I 1⊎…⊎It [m], |⊎Ij| k, some partition to b blocks is a split. n k-restriction problem! m n b k 37

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Necklace Splitting [A 87] • b thieves • t types How many splits? 38

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Necklace Splitting [A 87] 39

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Necklace Splitting Theorem (Alon, 1987): Every necklace with bai beads of color i, 1 i t, has a bsplitting of size at most (b-1)t. tight! Corollary: A multi-way splitter of size b(b-1)t+1 C(k C(m 2, , (b-1)t)·|Hashm, k , k| is efficiently constructible. concatenation 2 40

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions The b=t=2 Case 41

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Sum-Up Beat k-wise approximations for krestriction problems. n Multi-way splitters via Necklace Splitting. n Substantial improvements for: n Group Testing n Generalized Hashing n Set-Cover n 42

Dana Moshkovitz Algorithmic Construction of Sets for k-Restrictions Further Research n n Applications: complexity, algorithms, combinatorics, cryptography… Better constructions? different techniques? 43