Explicit dispersers with polylog degree

An (N, M, ‘T)-disperser is a duected bipartite Multigraph G = (V, W,E) with IV[ = N, IW[ = M and all edges directed from V to W, having the following expansion property: any subset of V having at least T vertices has a neighbor set of sise at least M/2. For any pair of constants (, ~, 1 z ~ > ~ z 0, ~y suffiaently large N, and for any T ~ 2(106@, M < 2(I%N)A , we give an explicit elementary construction ~f an (N, M, T)-disperser such that the out-degree of any vertex in V is at most polylogarithmic in N. Using this with known applications of dispersers yields several results. First, our construction implies that the complexity class Strong-RP defined by Sipser, equals RP. Second, for arty fixed q > 0, we give the first polynomial-time simulation of RP algorithms using the output of any “minimally randomn source. For any integral R >0, such a source accepts a single request for an R-bit string and generates the string according to a distribution that assigns probahiity at most 2-R’ to any string. It is minimally random in the sense that any weaker source is insufficient to do a blackbox polynomial-time simulation of RP algorithms. Third, we show improvements on the expander construction and the consequent applications given by Wlgderson and Zuck“The full version of thk work will be available at the DIMACS www site soon (URIJ http:ildirnacs. r-utgers. edu). The first and third authors were supported in part by NSF grant CCR-92 15293. The secoud author was supported in part by grant 93-6-6 of the Alfred P. Sloan Foundation to the Institute for Advanced Study, and in part by ESPRIT Bcsic Research Action Programme of the EC under contract No. 7141 (project ALCOM II). All three authors were also supported in part by DIMACS (Center for Discrete Mathematics & Theoretical Computer Science), through NSF grant NSF-STC91-19999 and by the New Jersey Commission on Science and Technology. *Dep~ment of Mathematics, Rutgem University, New Brunswick, NJ 08854, USA. Email: saka@math.ntgers. edn. $ school of Mathematics, Institute for Advanced Study, prince ton, NJ 08540, USA. Parts of this work were done while at DIMACS, while visiting the Department of Applied Mathematics and Computer Science, \Veizmann Institute of Science, Rehovot 76100, Israel, and while visiting the Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. Email: aravind@dca.warwick. ac.uk. SrJep~ment of Computer Science, Rutgers University, New Brunswick, NJ 08854, USA. Email: szhou@cs. rutgers. edu. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyri ht notice and the 3 title of the publication and. tis date appear,. an notice is given that copym IS by pernws~n of the Asscqatlon of Computing Machinery. o cop otherwe, or to repubkh, requires ‘{ a fee ancVor spec m permissmn. STOC’ 95, Las Vegas, Nevada, USA O 1995 ACM 0-89791 -718-9195/0005 ..$3.50 erman. Applying our disperser to a reduction due to Zuckerman, we show that approximating log log u(G) (where w(G) denotes the maximum clique sise of a given graph G) to within any constant tkctor cannot be done in deterministic quaei-polynomial time, assuming that rion-deterministic quasbpolynomial time does not equal deterministic quaaipolynomial time. Finally, we improve knolwn results due to Fiat and Naor and Zuckerman, for a problem in data structures: implicit 0(1 ) probe search.