Widgerson and Zuckerman .Expanders that beat eigen value bounds: Explicit construction and applications. STOC93, ACM Press, pp. 245-251  Home/Search   Document Details and Download   Summary   Related Articles   Check

....idea can be used to extract all of the randomness from the weak random source. This is accomplished by replacing the combinatorial designs underlying the Nisan Wigderson generator and Trevisan s construction with a weaker (and more suitable) notion. Applying a result of Wigderson and Zuckerman [WZ99] to these extractors, we also obtain improved constructions of highly expanding graphs and superconcentrators. The second idea improves Trevisan s construction in the case where the output bits are required to be of a relatively small statistical di erence from uniform distribution. The two ideas ....

.... with weak random sources [Zuc96] constructing oblivious samplers [Zuc97] constructive leader election [Zuc97, RZ98] randomness ecient error reduction in randomized algorithms and interactive proofs [Zuc97] explicit constructions of expander graphs, superconcentrators, and sorting networks [WZ99]; hardness of approximation [Zuc96, Uma99] pseudorandom generators for space bounded computation [NZ96, RR99] derandomizing BPP under circuit complexity assumptions [ACR97, STV99] and other problems in complexity theory [Sip88, GZ97] For a detailed survey of previous work on extractors and ....

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Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

....few improvements. More numbers can be generated; however, the asymptotic values do not improve. ####### #### log n) use Pippenger s modi cation of Alon) use Theorem 4.9 or Theorem 4. 3) 10 ### # ############ ######### ### ############# ########## Wigderson and Zuckerman [23] present a constructive proof that sort(n; k) O(n ) Their algorithm is based upon Pippenger s non constructive sorting algorithm (See Section 3.3) Recall that Pippenger showed that small a expander graphs were useful for sorting, and then showed that small a expander graphs exist. The value ....

....then state that such can be used to construct n expanding graphs for an appropriate . ##### #### [17] Let ; be functions of n such that 1=n 1=2 and 2 ## 1=n. Let t = O( # ### ) There exists polynomial time linear space computable (n; m; t; extractors. ##### #### [23] If there is an (n; m; t; 1=4) extractor computable in linear space then there is an N expanding graph on N = 2 nodes with maximum degree N2 constructible in DSPACE(log n) hence in P. ##### ####### Let E : be the extractor. First de ne a bipartite graph H ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19:125-138, 1999. Earlier version appeared in FOCS93.

.... decreasing the error probability of randomized algorithms (deterministic amplification, oblivious sampling) and the construction of graphs with random properties (superconcentrators, expanders) More information about applications of extractors can be found in the work of Zuckerman [Zuc91, SZ94, WZ95, NZ95, Zuc96a, Zuc96b] and Nisan [Nis96] 4.4 Spoiling Knowledge We now turn to further characterizations of smooth entropy and to lower bounds in terms of R enyi entropy. Corollary 4.1 shows that R enyi entropy of order 2 is a lower bound for the smooth entropy of a distribution. As ....

Avi Wigderson and David Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, Preprint available from the authors, preliminary version presented at 25th STOC (1993), 1995.

....the randomness from the weakly random source using fewer truly random bits than any previous construction. This is accomplished by improving the combinatorial construction underlying the Nisan Wigderson generator used in Trevisan s construction. Applying a construction of Wigderson and Zuckerman [WZ95], we also obtain improved expanders. Extractors. A distribution X on f0; 1g is said to have min entropy k if for all x 2 f0; 1g , Pr [X = x] Think of this as saying that X has k bits of randomness. A function Ext: f0; 1g is called an (k; extractor if for every distribution ....

.... [NZ96] Much of the motivation for research on extractors comes from work done on somewhat random sources [SV86, CG88, Vaz87b, VV85, Vaz84, Vaz87a] There have been a number of papers giving explicit constructions of dispersers and extractors, with a steady improvement in the parameters [Zuc96, NZ96, WZ95, GW97, SZ98, SSZ98, NT98, TS98, Tre98]. Most of the work on extractors is based on techniques such as k wise independence, the Leftover hash lemma [ILL89] and various forms of composition. A new approach to constructing extractors was recently initiated by Trevisan [Tre98] who discovered that the Nisan Wigderson pseudorandom ....

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Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Technical Report CS-TR-95-21, University of Texas Department of Computer Sciences, 1995. To appear in Combinatorica.

....with jSjjT j n =36 we have e(S; T ) C dn ; where C 0:00009 is a constant. Before proving this lemma, we need some background. For a graph G, denote by (G) the second largest eigenvalue of its adjacency matrix. The following well known inequality (see for example [3, pp. 119 125] and [24]) relates (G) to the cut property we are interested in. Lemma 2 Let G = V; E) be a d regular n vertex graph with second largest eigenvalue (G) Then, for all disjoint sets S; T V we have (G) This lemma suggest to look for graphs G with small (G) Fortunately, such graphs (called ....

A. Widgerson and D. Zuckerman, Expanders that beat the eigenvalue bound: explicit construction and applications, in Proc. 25th Annual ACM Symp. on Theory of Computing (STOC'93), pp. 245--251, ACM, 1993.

....of truly random bits. These extractors have been used to show that randomized scee(S( using only lol(S( random bits can be simulated deterministically in slcee(S( for S(t) lgt [25] In addition, the extractor function has been used to construct high degree expanders in polynomial time [29]. Using techniques similar to those found in [29] we construct regular, high degree bipartite graphs with the expansion properties necessary to obtain an O(lg t) depth counting network. In essence, this bipartite expander graph allows us to find the desired network in A deterministically in ....

.... used to show that randomized scee(S( using only lol(S( random bits can be simulated deterministically in slcee(S( for S(t) lgt [25] In addition, the extractor function has been used to construct high degree expanders in polynomial time [29] Using techniques similar to those found in [29], we construct regular, high degree bipartite graphs with the expansion properties necessary to obtain an O(lg t) depth counting network. In essence, this bipartite expander graph allows us to find the desired network in A deterministically in polynornial tirne. We believe that the bipartite ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Pz'oceecligs of sium o Theor'y of Computig, pages 245 251, May 1993. 92

....constructions and our construction of (k, #) extractors Ext : In the expressions, # is fixed and arbitrarily small, and # 0 is an arbitrarily small constant. O( notations hide dependencies on # and #. They also yield expander graphs, as discovered by Wigderson and Zuckerman [WZ93] that in turn have applications to superconcentrators, sorting in rounds, and routing in optical networks. Constructions of expanders via constructions of extractors and the Wigderson Zuckerman connection appeared in [NZ93, SZ94, TS96] among others. Extractors can also be used to give simple ....

.... k and the output is required to be of length m, then the additional randomness is O(m 1 log(k 2m) log n) log(k 2m) In [RRV99] the dependency is O( log n) log(k m) Raz et al. RRV99] also show how to recursively compose their construction with itself (along the lines of [WZ93] and they obtain in this way another construction where k = m and the additional randomness is O(log n) Constructions of extractors with parameters k = m have applications to the explicit construction of expander graphs [WZ93] In particular, Raz et al. RRV99] present constructions of ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th ACM Symposium on Theory of Computing, pages 245--251, 1993.

....such a graph with b = p n and k = p(n) First, we define a concentrator. Definition: An (n; m; concentrator with expansion ff is a network with n inputs and m outputs such that every set of t inputs expands to at least ct outputs. We use the following result from Wigderson and Zuckerman [WZ93]. Theorem 3: For all n, there are explicitly constructible (n; 2ff p n; p n) concentrators with expansion ff, depth one and size ffn Delta p(n) Application of Theorem 3 For our case we set ff to be one, and get that there exists a bipartite graph H 0 (S 0 ; B 0 ; F 0 ) where jS ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. In Proc. of the 25th ACM Symp. on Theory of Computing, San Diego, CA, May 1993. 22

....with x inputs and y outputs such that every set of t inputs expands to at least t outputs. The size of the network is the number of edges and the depth of the network is the length of the longest path from an input to an output. We use the following result from Wigderson and Zuckerman [WZ93]. Theorem 4 (WZ93) For all x, there are explicitly constructible (x; 2 p x; p x) concentrators with expansion , depth 1 and size x (x) We now show how to apply Theorem 4. For our case, we set x = 2n and = 1 and get that there exists a bipartite graph H 0 (S 0 ; B 0 ; F 0 ) ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. In Proc. of the 25th ACM Symp. on Theory of Computing, San Diego, CA, May 1993. 30

....s = O(log n) 2.1. 2 Previous work and Applications Extractors were first defined and constructed by Nisan and Zuckerman [NZ96] This followed a large body of work in the late 1980 s, regarding weak notions of randomness [SV86, VV85, CG88, Zuc90] Improved constructions of extractors appeared in [WZ95, GW97, SZ98, Zuc97, NT98]. A new approach to constructing extractors was recently initiated by Trevisan [Tre99] and then [RRV99a] who uses the Nisan Wigderson pseudo random generator [NW94] for constructing extractors (these constructions will be discussed in detail in subsection 2.1.3) Besides the natural ....

.... a wide variety of applications, including constructing oblivious samplers [Zuc97] constructive leader election [Zuc97] randomness efficient error reduction in randomized algorithms and interactive proofs [Zuc97] explicit constructions of expander graphs, superconcentrators, and sorting networks [WZ95]; hardness of approximation [Zuc96] pseudorandom generators for space bounded computation [NZ96, RR99] derandomizing BPP under circuit complexity assumptions [STV99] and other problems in complexity theory [GZ97] For a detailed survey on previous work on extractors and their applications, see ....

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Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Technical Report CS-TR-95-21, University of Texas Department of Computer Sciences, 1995. To appear in Combinatorica.

.... (see [Zuc96b, GZ97] and references therein) They yield oblivious samplers (as defined in [BR94] that have applications to interactive proofs and leader election in anonymous networks (see [Zuc96b] and references therein) They also yield expander graphs, as discovered by Wigderson and Zuckerman [WZ93], that in turn have applications to superconcentrators, sorting in rounds, and routing in optical networks. Constructions of expanders via constuction of extractors and the Wigderson Zuckerman connection appeared in [NZ93, SZ94, TS96] among others. Extractors can also be used to give simple ....

.... k and the output is required to be of length m, then the additional randomness is O(m 1 log(k 2m) log n) 2 log(k 2m) In [RRV99] the dependency is O( log n) 2 log(k m) Raz et al. RRV99] also show how to recursively compose their construction with itself (along the lines of [WZ93]) and they obtain in this way another construction where k = m and the additional randomness is O(log 3 n) Constructions of extractors with parameters k = m have applications to the explicit construction of expander graphs [WZ93] In particular, Raz et al. RRV99] present constructions of ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th ACM Symposium on Theory of Computing, pages 245--251, 1993.

.... of error in randomized algorithms (see [Zuc96b, GZ97] and references therein) They yield oblivious samplers (as de ned in [BR94] that have applications to interactive proofs (see [Zuc96b] and references therein) They also yield expander graphs, as discovered by Wigderson and Zuckerman [WZ93] that in turn have applications to superconcentrators, sorting in rounds, and routing in optical networks. Constructions of expanders via constructions of extractors and the Wigderson Zuckerman connection appeared in [NZ93, SZ94, TS96] among others. Extractors can also be used to give simple ....

....is bounded by O(log 2 n) even when k = m=3. Furthermore, the running of the extractors of [RRV99] is poly(n; t) rather than poly(n; 2 t ) as in the construction presented in this paper. Raz et al. RRV99] also show how to recursively compose their construction with itself (along the lines of [WZ93] and obtain another construction where k = m and the additional randomness is O(log 3 n) Constructions of extractors with parameters k = m have applications to the explicit construction of expander graphs [WZ93] In particular, Raz et al. RRV99] present constructions of expander graphs and ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th ACM Symposium on Theory of Computing, pages 245-251, 1993.

....4.10 1. csort(2; n) O(n 5=3 log n) use Pippenger s modi cation of Alon) 2. csort(3; n) O(n 8=5 ) use Theorem 4.9 or Theorem 4.3) 3. csort(4; n) O(n 3=2 ) 4. csort(5; n) O(n 23=16 ) 10 4. 5 A Constructive Algorithm via Pseudo Random Generators Wigderson and Zuckerman [23] present a constructive proof that sort(n; k) O(n 1 1=k o(1) Their algorithm is based upon Pippenger s non constructive sorting algorithm (See Section 3.3) Recall that Pippenger showed that small a expander graphs were useful for sorting, and then showed that small a expander graphs exist. ....

....can be used to construct n expanding graphs for an appropriate . Lemma 4.13 [17] Let ; be functions of n such that 1=n 1=2 and 2 n 1=n. Let t = O( log 1 log 2 n log 1 ) There exists polynomial time linear space computable (n; m; t; extractors. Lemma 4. 14 [23] If there is an (n; m; t; 1=4) extractor computable in linear space then there is an N expanding graph on N = 2 n nodes with maximum degree N2 1 2t m constructible in DSPACE(log n) hence in P. Proof sketch: Let E : f0; 1g n f0; 1g t f0; 1g m be the extractor. First de ne a ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19:125-138, 1999. Earlier version appeared in FOCS93.

....can be applied to designing randomized algorithms that achieve maximal robustness with respect to imperfections in the random source. There are other consequences of our disperser construction. For example, it gives improvements on the expander construction and the consequent applications given in [WZ93], on the the hardness results of approximating the clique function [Zuc93, SZ94] and on the results for a problem in 46 data structures: implicit O(1) probe search [FN93, Zuc91] These consequences were each observed by previous researchers, and provided much of the motivation for the search for ....

A. Wigderson and D. Zuckerman. Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications. In Proc. of ACM Symposium on Theory of Computing, pp. 245-251, 1993.

....with x inputs and y outputs such that every set of t inputs expands to at least fft outputs. The size of the network is the number of edges and the depth of the network is the length of the longest path from an input to an output. We use the following result from Wigderson and Zuckerman [WZ93]. Theorem 4 (WZ93) For all x, there are explicitly constructible (x; 2ff p x; p x) concentrators with expansion ff, depth 1 and size ffx Delta (x) We now show how to apply Theorem 4. For our case, we set x = 2n and ff = 1 and get that there exists a bipartite graph H 0 (S 0 ; B 0 ; ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. In Proc. of the 25th ACM Symp. on Theory of Computing, San Diego, CA, May 1993. 24

....m 0 = m; ffl) extractor E. That is, this extractor works for any min entropy, small or large, and extracts all the randomness present in the given source. These properties turn out to be very important for some applications, most notably the following two corollaries: Corollary: improving [WZ93] For any N and 1 a N there is an explicitly constructible a expanding graph with N vertices, and maximum degree O( N a 2 polyloglog(N) 5 . Another important corollary, that solves a problem similar to the network problem presented in section 1.1, is: Corollary: improving [WZ93] ....

....[WZ93] For any N and 1 a N there is an explicitly constructible a expanding graph with N vertices, and maximum degree O( N a 2 polyloglog(N) 5 . Another important corollary, that solves a problem similar to the network problem presented in section 1. 1, is: Corollary: improving [WZ93] For any N there is an explicitly constructible superconcentrator over N vertices, with linear size and polyloglog(N) depth 6 . See section 2.5 for more details on these and other applications. Simulating random classes with sources having high min entropy Our second extractor is motivated ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM, pages 245--251, 1993.

....random source. In particular, the idea can be used to extract all of the randomness from the weak random source. This is accomplished by improving the combinatorial construction underlying the Nisan Wigderson generator used in Trevisan s construction. Applying a result of Wigderson and Zuckerman [WZ95] to these extractors, we also obtain improved constructions of highly expanding graphs and superconcentrators. The second idea improves Trevisan s construction in the case where the output bits are required to be of a relatively small statistical difference from uniform distribution. The two ideas ....

.... with weak random sources [Zuc96] constructing oblivious samplers [Zuc97] constructive leader election [Zuc97] randomness efficient error reduction in randomized algorithms and interactive proofs [Zuc97] explicit constructions of expander graphs, superconcentrators, and sorting networks [WZ95]; hardness of approximation [Zuc96] pseudorandom generators for space bounded computation [NZ96, RR99] derandomizing BPP under circuit complexity assumptions [ACR97, STV99] and other problems in complexity theory [Sip88, GZ97] For a detailed survey of previous work on extractors and their ....

[Article contains additional citation context not shown here]

Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Technical Report CS-TR-95-21, University of Texas Department of Computer Sciences, 1995. To appear in Combinatorica.

.... have been the focus of much research in recent years (see the survey in [16] and have found applications in a wide range of areas, including simulating randomized algorithms with weak random sources [37, 27] explicit constructions of expanders, superconcentrators and sorting networks [35, 30], constructive leader election [38, 23] and several diverse applications in complexity theory [18, 2, 29, 26, 10] The dispersers we use in this paper are ones that work on sources with very small min entropy; constructions for this parameter range are in [27, 9] logarithmic degree) and [16, 32, ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedingsof the 25th AnnualACM Symposiumon the Theory of Computing (STOC 93), pages 245--251, San Diego, California, 16--18 May 1993.

.... and Goldreich, and others [SV86, Vaz87a, Vaz86, Vaz87b, VV85, CG88] The direct development of the constructions and applications of extractors and dispersers came first in papers written by Zuckerman in the early 1990 s [Zuc90, Zuc91] and then in a sequence of papers by various authors [NZ93, WZ93, SZ94, SSZ95, Zuc93, Zuc] The first explicit construction of extractors came in [NZ93] and relied on techniques developed in [Zuc90, Zuc91] This construction had d = polylog(n) for k n=polylog(n) An efficient extractor working for small k s, k = Theta(log(n) was obtained by [GW94, SZ94] ....

....or in explicit constructions. Our new constructions improve some of them. In the following we state the results we improve. Formal definitions and proofs are given in Section 6. The first application is constructing explicit a expanding graphs, obtained by plugging our first extractor into the [WZ93] construction: Corollary 1.1 For any N and 1 a N there is an explicitly constructible a expanding graph with N vertices, and maximum degree O( N a 2 polyloglog(N) 1 . This corollary has applications on sorting [Pip87a, WZ93] and selecting [AKSS89, WZ93] in k rounds. Corollary 1.2 ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM, pages 245--251, 1993.

....understood for all depths except two. In this paper, we close this gap. Let size(N ) denote the size of the smallest depthtwo N superconcentrator. Theorem 1 (Main result) size(N ) Theta(N Delta log 2 N log log N ) For the upper bound, we use the method of Wigderson and Zuckerman [WZ93], who showed how supercon centrators can be constructed using a type of expander graphs called dispersers. Definition 1.1 (Sipser [Sip88] A bipartite graph G = V 1 = N ] V 2 = M ] E) is a (K; ffl) disperser, if for every X V 1 of cardinality K, j Gamma(X )j (1 Gamma ffl)M (i.e. ....

....)j (1 Gamma ffl)M (i.e. every large enough set in V 1 misses less than an ffl fraction of the vertices of V 2 ) The size of the disperser G is jE(G)j. Following a hint given by Nisan and Wigderson in their course, Research Pearls in Theory of Computation [NW] we refine the construction in [WZ93] and get the upper bound by putting together a small number of dispersers. These dispersers are obtained by probabilistic arguments; the best explicit construction known uses dispersers constructed by TaShma [T96a] and gives N superconcentrators of size O(N(log N ) poly log log n ) see [T96b, ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM, pages 245--251, 1993.

....results were d = O(log 2 (n= and d = O(log 2 n Delta log(1= proved in [RRV99] 3. Extracting all k random bits (i.e. m = k) In this case we achieve d = O( log 2 n log(1= Delta log k) This is obtained by iterative application of the previous result O(log k) times (as in [WZ95]) The best previous results were d = O(log 2 (n= Delta log k) and d = O(log 2 n Delta log(1= Delta log k) proved in [RRV99] Strong Extractors. The original definition of extractors [NZ96] is somewhat stronger than the definition given above (which is due to [NT99] Such a strong ....

Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Technical Report CS-TR-95-21, University of Texas Department of Computer Sciences, 1995. To appear in Combinatorica.

....and remarks. Construction of quasi random objects, mainly graphs, is discussed in Alon and Spencer [AS93] This subject is very wide and has many often unexpected connections; another long survey would be needed to cover just the basics. A recent work in this area is Wigderson and Zuckerman [WZ93] where also more references can be found. Explicit quasi random graph (expanders) were applied in geometric algorithms by Ajtai and Megiddo [AM92] and by Katz and Sharir [KS93a] KS93b] Small k wise independent probability spaces were constructed by Joffe [Jof74] an asymptotically optimal ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proc. 25th ACM Symposium on Theory of Computing, pages 245--51, 1993.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

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A. Wigderson, D. Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, Combinatorica 19 (1) (1999) 125-- 138.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

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A. Wigderson, D. Zuckerman, " Expanders that Beat the Eigenvalue Bound, Explicit Construction and Applications", Proc. of the 25th STOC, pp. 245--251, 1993. To appear in Combinatorica.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

....by Nisan and Zuckerman [14] extractors have played a fundamental and unifying role in the theory of pseudorandomness. In particular, it has been discovered that they are intimately related to a number of other important and widely studied objects, such as hash functions [9] expander graphs [14, 28, 18, 23, 4], samplers [7, 30] pseudorandom generators [26] and error correcting codes [26, 25, 24] In addition, extractors have been found to have a vast and ever growing collection of applications in diverse aspects of computational complexity, combinatorics, and cryptography. See the excellent surveys ....

....merger in Corollary 12 with k # = # m) # k) using a seed of length O(log(n k) log(1 #) The result is a distribution that is 2# close to an # k) source of length m k, on which we can apply Zuckerman s extractor [30] to extract# k) bits. We can iterate the extraction using the technique of [28] to extract all but a small constant fraction of randomness. Then we have the following. Lemma 31. For any constant # n, and every # (0, 1) where # exp( k 2 there is an explicit (k, #) extractor O(log n log(n k)log(1 #) For constant #, this gives our main extractor. ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

....jf Gamma1 (0)j jf Gamma1 (1)j, so jf Gamma1 (0)j 2 n Gamma1 . Then any ffi source outputting only values in f Gamma1 (0) contradicts the claim about f . Given this, is there any hope in using a ffi source Idea: add a small number of truly random bits. Building on earlier work [NZ96, SZ94, WZ95, Zuc96], in [Zuc97] I constructed an efficient extractor E : f0; 1g n Theta f0; 1g O(logn) f0; 1g :99ffin : This has the property that if x is output according to any ffi source and y is uniformly random, then E(x; y) is close to random. The small amount of randomness may be eliminated by ....

....may be eliminated by trying all possible O(log n) bit strings. 2 Extractors have had many applications to seemingly unrelated areas: ffl random sampling using few random bits [Zuc97] ffl pseudo random generators for space bounded computation [NZ96] ffl construction of highly expanding graphs [WZ95] ffl construction of non blocking networks [WZ95] ffl sorting in rounds [WZ95] ffl unapproximability results [Zuc96] ffl leader election protocols [RZ98, Zuc97] ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica. To appear. Revised version appears as Technical Report TR-95-21, Department of Computer Sciences, The University of Texas at Austin, June

.... n fl ) fl 1 Gamma 1=k O( log n) log (k) n) 2 n ; 2 n kfl Gammak 1 = log 2k Gamma1 n ; n O(log (k) n) 2 n fl ) fl 1 Gamma 1= 2 log n) O(log n) 2 n ; 2 p n ; n O(1) 2 n fl ) Just as extractors and dispersers for constant ffi had important applications [NZ, WZ], so too do our results for subconstant ffi. The first application is to a relationship between the RP = P question and time space tradeoffs. Sipser [Sip] showed that if certain expander graphs can be constructed efficiently, then for some ffl 0 and any time bound t(n) either RP = P or all ....

.... take time n O(log (k) n) we instead show unconditionally that either RP T k DT IME(n log (k) n ) or all unary languages in DTIME(t(n) are accepted infinitely often in T k SPACE(t(n) 1 Gamma1= log (k) n ) Our second application is to improve the expanders constructed in [WZ], and hence all of the applications given there. In [WZ] graphs on n nodes were constructed such that for every pair of disjoint subsets S 1 and S 2 of the vertices with jS 1 j n ffi and jS 2 j n ffi , there is an edge joining S 1 and S 2 ; the graphs so constructed had essentially optimal ....

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A. Wigderson and D. Zuckerman, "Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications," Combinatorica, to appear. Also see Technical Report CS-TR-95-21, Computer Science Dept., The University of Texas at Austin. Preliminary version appeared in Proc. 25th ACM Symposium on Theory of Computing, 1993, pp. 245-251.

....k, there are at least k vertex disjoint paths from X to Y . Size Reference O(N 2 poly(log log N) 21] O(N polylogN) This paper O(N log 2 N log log N ) Lower bound, 17] Table 3: Explicit depth two super concentrators We obtain this result by plugging our disperser into [31]. A long line of papers has tried to solve this problem. The previous best result and our result is summarized in Table 3. We also get an explicit a expanding graph with N vertices that has degree O( N a polylogN ) and we improve a hardness result of Umans [29] See Section 6 for more details ....

.... the following results from [19] Theorem 7 ( 19] For every n, k n, and 0, there exists an explicit (k; extractor E : f0; 1g n f0; 1g t f0; 1g k t , where t = O(log 2 n log k log 1 ) and = 2 log 1 O(1) The following is a slight strengthening of a lemma of [31] Lemma 4.0.3. 19] Suppose E1 : f0; 1g n f0; 1g t 1 f0; 1g m 1 is a (k; 1 ) extractor with entropy loss 1 and E2 : f0; 1g n t 1 f0; 1g t 2 f0; 1g m 2 is a ( 1 1; 2 ) extractor with entropy loss 2 . Then E(x; y1 y2 ) E1(x; y1)E2 (x y1 ; y2) is a (k; 2 1 2 ) extractor ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

....to simulating randomized algorithms using weak sources, extractors have had applications to many areas in derandomization that are seemingly unrelated to weak sources, and below we list some of them. Extractors have been used to construct expanders that beat the second eigenvalue method [WZ99], superconcentrators and non blocking networks [WZ99] sorting and selecting in rounds [WZ99] pseudorandom generators for space bounded computation [NZ96] unapproximability of clique [Zuc96] and certain S P 2 minimization problems [Uma99] time versus space complexities [Sip88] leader ....

....weak sources, extractors have had applications to many areas in derandomization that are seemingly unrelated to weak sources, and below we list some of them. Extractors have been used to construct expanders that beat the second eigenvalue method [WZ99] superconcentrators and non blocking networks [WZ99], sorting and selecting in rounds [WZ99] pseudorandom generators for space bounded computation [NZ96] unapproximability of clique [Zuc96] and certain S P 2 minimization problems [Uma99] time versus space complexities [Sip88] leader election [Zuc97, RZ98] another proof that BPP PH [GZ97] ....

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

....randomness from a defective random source, using a small additional number of truly random bits. Extractors were first defined and constructed in [NZ96] and have since been improved by several authors. Extractors have found numerous applications to pseudo randomness and explicit constructions [NZ96, WZ99, Zuc97, STV99]. This paper gives yet another application. We need a weaker variant of extractors called slice extractors . The output of an extractor needs to be almost random with respect to any statistical test; the output of a slice extractor just needs to be almost random with respect to tests of a certain ....

A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

....n 30 , and later extended for any k in [ISW00] 2. Maximizing the output size: Explicit extractors can extract large fractions of the randomness only at the cost of enlarging the seed length. A general method to increase the fraction extracted at the cost of enlarging the seed length was given in [WZ99]. The best extractors which extract all the randomness out of random sources are constructed this way from extractors which extract a constant fraction of the randomness. In light of this, we focus our attention to extractors which extract a constant fraction. The best such explicit extractor is ....

....) This improves the best previous such result by [RRV99b] which uses a seed of length O(log 2 n) Theorem 1 For every n; k and constant , there are explicit (k; extractors Ext : f0; 1g n f0; 1g O(log n (log log n) 2 ) f0; 1g (1 )k , where 0 is an arbitrary constant. Using [WZ99], we get the following corollary 3 , which also improves the previous best construction which extract all the randomness by [RRV99b] 1 [RRV99a] gave a general explicit transformation which transforms an extractor with constant error into an extractor with arbitrary small error while harming ....

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Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

....which they defined addresses all of the above three problems. Moreover, it turned out to be fundamental derandomization tool and found other applications, such as sampling [Zuc97] and various combinatorial constructions (including certain kinds of expander graphs and other networks) WZ99] For the purpose of the introduction, we will use a simplified definition. Intuitively, an extractor is a function that converts the k bits of entropy hidden in every k source on n bits into an (almost) uniform distribution on k bits It is not hard to see that this task is impossible if the ....

....constructions have failed to match this bound. An application of the best known relation between eigenvalues and vertex expansion [Tan84] shows that Ramanujan graphs (e.g. as given by [LPS88, Mar88, Mor94] of degree (A 2 ) suffice. To beat this eigenvalue bound, Wigderson and Zuckerman [WZ99] suggested to build such graphs from extractors and obtained degree A N o(1) which was important for many applications where A is a fixed power of N . However, for very small A, even much better dependence on N obtained in subsequent work (e.g. NT99] does not beat the eigenvalue bound. ....

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Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica, 19(1):125--138, 1999.

.... comparison reference min entropy k output length m additional randomness d [Tre99] any k m = k 1 d = O( log 2 n log k ) Theorem 2 any k m = k 1 d = O(log n) RRV99b] any k m = 1 )k d = O(log 2 n) Theorem 3 any k m = k log log log n ) d = O(log n) O(log 2 k) Theorem 3 [WZ93] k 2 p log n m = 1 )k d = O(log n log log log n) Ultimate goal any k m = k d = O(log n) The results are given for constant . is an arbitrary small constant. Theorem 3 For every n; k and there exists an explicit (k; 1=m) extractor Ext : f0; 1g n f0; 1g O(log n log 2 ....

....n) f0; 1g m , where m = k 1 for arbitrary 0) Theorem 11 For every n; k there exists an explicit (k; 1=m) extractor Ext : f0; 1g n f0; 1g O(log n log 2 k) f0; 1g m , for m = k= log log log n) Note that this extractor uses optimal seed length for k 2 p log n . In [WZ93] it was shown how to transform an extractor which extract m = k=r bits into one which extracts m = 1 )k, for arbitrary constant ) while multiplying d by O(r) Using this with theorem 11 gives: Theorem 12 For every n and k 2 p log n there exists an explicit (k; 1=m) extractor Ext : f0; 1g ....

Avi Wigderson and David Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the Twenty-Fifth Annual ACM Symposium on the Theory of Computing, pages 245-251, San Diego, California, 16-18 May 1993.

....small and shallow superconcentrators has been a major object of study [Pi78, Pi82, DDPW] We shall need the following two upper bounds on the size of superconcentrators of depth 2, the first being nonconstructive and the second explicit. Theorem 3 [Pi82] c 2 (n) O(n(log n) 2 ) Theorem 4 [WZ93] There is a polynomial time algorithm which for every n outputs an n superconcentrator of depth 2 and size n 1 o(1) The best bound on the o(1) term is actually (log n) Gamma1=2 [SZ94] In any depth 2 graph G, the vertices decompose to three sets, V = I [ R [ O, which are resp. the ....

A. Wigderson, D. Zuckerman, "Expanders that Beat the Eigenvalue Bound, Explicit Construction and Applications", Proc. of the 25th STOC, pp. 245--251, 1993. 12

....f0; 1g m , where x 2 f0; 1g n is connected to z 2 f0; 1g m if there exists y 2 f0; 1g t such that E(x; y) z. As in the constructions of [20, 21] this graph has good expansion properties, which are better than what can be obtained using eigenvalue methods. These ideas are further used in [19]. Weak random sources and deterministic amplification: Given an extractor and the first parameter x to it, an algorithm may go over all the possible values of y. It is not difficult to see that this can be used to simulate BPP using a ffi source [20, 21] or to do deterministic amplification ....

A. Wigderson and D. Zuckerman, Expanders that Beat the Eigenvalue Bound: Explicit Construction and Applications, Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993, pp. 245-251.

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Widgerson and Zuckerman .Expanders that beat eigen value bounds: Explicit construction and applications. STOC93, ACM Press, pp. 245-251

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A. Wigderson, and D. Zuckerman, Expanders that beat the eigenvalue bound: explicit construction and applications, Combinatorica, 19 (1999) 125--138. 17

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A. Wigderson and D. Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, in Proceedings ACM Symposium on the Theory of Computing, ACM, New York, 1993, pp. 245--251.

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A. Widgerson and D. Zuckerman. Expanders that beat the eigenvalue bound: explicit construction and applications. In Proc. 25th Annual ACM Symp. on Theory of Computing (STOC '93), pages 245--251, 1993.

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A. Wigderson, D. Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, in: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993, pp. 245--251.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, ACM, pages 245--251, 1993.

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A. Wigderson and D. Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, in Proceedings, 25th Annual ACM Symposium on the Theory of Computing, ACM, 1993," pp. 245#251.

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A. Wigderson and D. Zuckerman. Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica, 19(1):125-138, 1999.

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A. Wigderson and D. Zuckerman, Expanders that beat the eigenvalue bound: Explicit construction and applications, Combinatorica 19(1) (1999), 125--138.

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