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

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

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

Problem Definition

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 6 But the capricious pirate has various contradicting local demands he may pose when he comes to collect it … this pattern! should differ!

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

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 9 Formal Definition [~NSS95] Input: alphabet , length m. demands f 1,…,f s :  k  {0,1}, Input: alphabet , length m. demands f 1,…,f s :  k  {0,1}, Solution: A  m s.t Solution: A  m s.t for every 1  i 1 <…<i k  m, 1  j  s, for every 1  i 1 <…<i k  m, 1  j  s, there is a  A s.t. f j (a(i 1 ),…,a(i k ))=1. there is a  A s.t. f j (a(i 1 ),…,a(i k ))=1. Measure: how small |A| is Measure: how small |A| is m k 

Applications

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 12 Application I Universal Set every k configuration is tried. every k configuration is tried. circuit m

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 13 Application II Hashing Goal: small set of functions [m]  [q] Goal: small set of functions [m]  [q] For every k  q in [m], some function maps them to k different elements For every k  q in [m], some function maps them to k different elements small set of functions u1u2u3u4...umu1u2u3u4...um r1r2...rqr1r2...rq k

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 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}. 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 0, one can construct efficiently a (t,u)-hash family over alphabet of size t+1, whose Theorem: For any fixed 2≤t 0, one can construct efficiently a (t,u)-hash family over alphabet of size t+1, whose rate (i.e log q m/n) ≥ (1-  )t!(u-t) u-t /u u+1 ln(t+1)

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 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+  ) ed 2 lnm group-tests in time polynomial in the population’s size (m).

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 17 Application IV Orientations [AYZ94] Input: directed graph G Input: directed graph G Question: simple k-path? Question: simple k-path? if G were DAG… if G were DAG…

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 18 Application IV Orientations [AYZ94] Pick an orientation Pick an orientation Delete ‘bad’ edges Delete ‘bad’ edges Now G is a DAG… Now G is a DAG… Need several orientations, s.t wherever the path is, one reflects it.

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 20 Approximability of Set-Cover ln n known app. algorithms [Lov75,Sla95,Sri99] approximation ratio (upto low- order terms) if NP  DTIME(n loglogn ) [Feige96] if NP  P [RS97]

Background Random and Pseudo-Random Solutions

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 22 Density D:  m  [0,1] - probability distribution. D:  m  [0,1] - probability distribution. density w.r.t D is: density w.r.t D is:  = min I,j Pr a  D [ f j (a(I))=1 ] m k mm    ......

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 24 Deterministic Construction!

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

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 27 So What’s the Problem? It’s much more costly than a random solution! Random solution: ~ klogm/  for all distributions! Random solution: ~ klogm/  for all distributions! k-wise  -close to uniform: O(2 k k 2 log 2 m /  2 ) [AGHP90] k-wise  -close to uniform: O(2 k k 2 log 2 m /  2 ) [AGHP90] for other distributions, the state of affairs is usually much worse…

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

Our Results

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

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

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

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 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)/  × k 4 logm in time poly(m,s,k k,q k,  -1 ).

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 36 Splitters, [NSS95] What are they? What are they? several block divisions several block divisions any k are splat by one any k are splat by one k-restriction problem! k-restriction problem! How to construct? How to construct? needs only (b-1) cuts needs only (b-1) cuts use concatenation use concatenation

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 37 Multi-Way Splitters For any I 1 ⊎ … ⊎ I t  [m], | ⊎ I j |  k, some partition to b blocks is a split. For any I 1 ⊎ … ⊎ I t  [m], | ⊎ I j |  k, some partition to b blocks is a split. k-restriction problem! k-restriction problem! m k b

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 38 Necklace Splitting [A87] b thieves t types How many splits?

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 39 Necklace Splitting [A87]

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

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

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

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

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 44 Context [NSS95] Universality/Hashing via split and “smart search”. [NSS95] Universality/Hashing via split and “smart search”. Our work: Generalizing [NSS95] techniques Generalizing [NSS95] techniques approximating distributions approximating distributions multi-way splitters via topological Necklace Splitting multi-way splitters via topological Necklace Splitting multi-way splitters multi-way splitters Many more applications: group testing, an improved hardness for SET-COVER under P≠NP… Many more applications: group testing, an improved hardness for SET-COVER under P≠NP…

Dana MoshkovitzAlgorithmic Construction of Sets for k-Restrictions 45 invariance under permutations Definition: We say C 1,…,C s  k are invariant under permutations, if for any permutation  :[k]  [k], {C 1,…,C s } = {  (C 1 ),…,  (C s )}