Margulis, G.A.: Explicit Group-Theoretical Constructions of Combinatorial Schemes and their Application to the Design of Expanders and Concentrators. Problemy Peredachi Informatsii 24, 51--60 (1988)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....largest eigenvalue of the adjacency matrix of G, then G is an (n; d; c) expander with c = d Gamma ) 2d. Thus, if is much smaller than d (which is the largest eigenvalue of the adjacency matrix of G) then G is a good expander. Lubotzky, Phillips and Sarnak [26] and independently Margulis [27]) have given an explicit and very simple description of a d regular graph G with n vertices, for which 2 d Gamma 1, for any d = p 1 and n = q 1, where p and q are primes congruent to 1 modulo 4. These graphs actually have the stronger property that all their eigenvalues (except d) have ....

G.A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24 (1988), 51--60 (in Russian). English translation in Problems of Information Transmission 24, 39--46.

....A. Spielman. well under iterative decoding, only a few tenths of a decibel away from the Shannon limit. For further results, see [21 24] Since the early work of Gallager, the first significant work in constructing LDPC codes based on graphtheoretic algebraic approach was reported in [25] In [26, 27], explicit group theoretic constructions of graphs were proposed. These graphs have girth exceeding the Erdos Sachs bound [28] which is a non constructive lower bound on the girth of random graphs and has the same significance as the Gilbert Varshamov bound does in the context of minimum ....

G. A. Margulis, "Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators, " Problems of Information Transmission (USA), vol. 24, no. 1, pp. 39--46, Jan.-March 1988.

....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 Ramanujan graphs) have already been constructed. Lemma 3 ([19,20]) If p 6= q are primes, p 1 mod 4, q 1 mod 4, then there exists a simple n vertex graph G p;q = V; E) with the following properties: G p;q is d regular with d = p 1. q(q 1) 2 n q(q 1) G p;q ) 2 d 1. Remark. This second largest eigenvalue is known to be ....

A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and superconcentrators. Problems of Information Transmission 24 (1988), 39--64.

....order v i . Let g i denote the girth of G i . The family is called a family of graphs with large girth if g i # log k 1 (v i ) for some positive constant # and all i 1. The lower bound for v(k, g) shows that 2, but no infinite family has been found for which # = 2. F1. Margulis [16] and, independently, Lubotzky, Phillips and Sarnak [15] came up with similar examples of graphs with # 4 3 and arbitrary large valency (they turned out to be so called Ramanujan graphs) These are Cayley graphs of the group PGL 2 (Z q ) with respect to a set of p 1 generators, where p and q ....

....graphs) These are Cayley graphs of the group PGL 2 (Z q ) with respect to a set of p 1 generators, where p and q are distinct primes, each congruent to 1 mod 4, with the Legendre symbol p = 1. Denoted by X , they are (p 1) regular bipartite graphs of order q(q 1) Margulis [16] and, independently, Biggs and Boshier [3] showed that the asymptotic value of # for the graphs X is exactly 4 3. Moreover, in both papers an explicit formula for the girth g(X ) of X was found. To state their results (formulae (5) below) we first need the following definition. Call an ....

G. A. Margulis, Explicit group-theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators, Journal of Problems of Information Transmission (1988), 39-46.

....n = 0:22n. Furthermore, Kostochka and Melnikov show that almost every 3 regular graph has a bisection width of at least 1 9:9 n 0:101n [16] There are some (slightly weaker) results for explicitly constructible in nite graph classes with high bisection width. The Ramanujan Graphs (see e.g. [6, 17, 19, 22]) have a regular degree d and a bisection width of at least ( d 2 p d 1) n 2 . This value is derived by the use of the wellknown spectral lower bound 2 n 4 with 2 being the second smallest eigenvalue of the Laplacian of the graph (cf. 10] This implies lower bounds of 0:042n and ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes. Probl. Inf. Transm., 24(1):39-46, 1988.

.... regular graphs is 2 p 1 o(1) Amazingly enough, there are explicitly known constructions of an in nite family of regular graphs fG i g i 1 with lim sup i 1 (G i ) 2 p 1 2 p . These graphs, which are called Ramanujan graphs, were constructed independently in [7] and [8]. 4 Codes with rate 2 ) decodable up to a fraction (1=2 ) of errors Theorem 4 For any 0 there is an explicitly speci ed code family with rate 2 ) relative distance at least (1 ) and alphabet size 2 O(1= 2 ) such that a code of blocklength n from the family can be (a) ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and superconcentrators. Problems of Information Transmission, 24:39-46, 1988.

....be modified to construct several irregular graphs. We now briefly review various approaches to constructing LDPC codes. In his original monograph, Gallager [2, Appendix] gave an algorithm for the fixed rate construction. For various algebraic constructions, also in fixed rate setting, see Margulis[5] and Lubotzky, Phillips, and Sarnak [6] For various recent designs of graphs with girth 6 using mainly random constructions, see, for example, MacKay, Simon, and Davey [7] and MacKay and Davey [8] For designs of irregular graphs using random constructions and linear programming, see Luby et al. ....

....number of check nodes, the check degree, the bit degree, and the girth. Another interesting problem, first considered in [2] is given the number of check nodes m, the check degree b, and bitdegree a, maximize the girth g. For various algebraic constructions of fixed rate, high girth graphs, see [5], 6] In this case, the number of bit nodes is n = mb) a. Our algorithm can be adapted to this case as follows. We simply apply the algorithm with given parameters m, a, b, and some girth g. If the algorithm finds exactly (resp. less than) n bit nodes, then the girth g is achievable (resp. not ....

G. A. Margulis, "Explicit group-theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators," Problems of Inform. Transmission, vol. 24, no. 1, pp. 39--46, 1988.

.... of the second eigenvalue # 2 (G) of the graph G: ad(G U ) # U (d # 2 (G) V # 2 (G) This implies c E (r, G) # d # 1 r V # # 2 (G) Recall that a Ramanujan graph is a d regular graph G with # 2 (G) # 2 # d 1; explicit constructions of such graphs were given in [LPS88, Mar88]. Summing up the above, we have: Lemma 4.2 The incidence matrix of any d regular Ramanujan graph G on n vertices is an (r, d, d(1 r n) 2 # d 1) expander for any parameter r 0. 4.2 Tseitin tautologies: Boolean version A Tseitin tautology is an unsatisfiable CNF capturing the basic ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators. Problemy Peredachi Informatsii (in Russian), 24:51--60, 1988. English translation in Problems of Information Transmission, Vol. 24, pages 39-46.

....sets of nodes whose cardinalities form a dense sequence. To simplify the presentation we assume here that there are sufficiently many expanders in these families whose number of nodes is divisible by any desired constant. It is not difficult to show that this assumption can be omitted. By [11] [12] the sequence of integers m for which there is a (d; 2 p d Gamma 1) expander on m nodes is a dense sequence. We need the following from [6] Proposition 1 [6] The number of edges induced by any set of x nodes in a (d; graph on m nodes does not exceed 1 2 x(d x m (1 Gamma x m ....

G. A. Margulis, "Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators " Problemy Peredachi Informatsii, Vol. 24, 1988, pp. 51-60 (in Russian). (English translation in Problems of Information Transmission, Vol. 24, 1988, pp. 39-46).

....and 4regular graphs. We use them to establish tighter lower bounds for the bisection width of 3 and 4 regular Ramanujan graphs. Ramanujan graphs of degree d have 2 d 2 p d 1 and, thus, for a fixed d it holds bw = n) There are several methods of explicitly constructing Ramanujan graphs [5, 12, 13, 15]. Equation (1) leads to lower bounds of 0:043n and 0:134n for the bisection widths of 3 and 4 regular Ramanujan graphs. With the use of the level structure of a bisection we can prove that any 3 regular Ramanujan graph has a bisection width of at least 0:082n and any 4 regular Ramanujan graph a ....

.... for d regular graphs, since 2 d 2 p d 1 4 p d 1 log d 1 (n) O(1) 16] see also [1, 12] There are known construction of infinite families of d regular Ramanujan graphs for any d of the form d = p k 1, where p is any prime number, and k is an arbitrary positive integer (see [5, 12, 13, 15]) Theorem 3. The bisection width of any 1. 4 regular graph with 2 2 is at least minf n 2 ; 5 2 7 ( 2 1) 2 2n 2 g. 2. 3 regular graph with 2 2 is at least minf n 2 ; 4 2 4 2 2 2 2 2n 2 g. 3. 3 regular graph with 2 5 p 17 2 is at least minf n 6 ; ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Probl. Inf. Transm., 24(1):39--46, 1988.

....y Ecole Nationale Sup erieure des T el ecommunications, 75 634 Paris 13, France. Email : zemor infres.enst.fr 1 The first result of this kind (Sipser and Spielman [7] gives us a family of codes, based on a graph theoretic approach of Tanner [8] and constructions of Ramanujan graphs [5, 6]. The result is as follows: Theorem 1 [7] For any 0 there exists a polynomial time constructible family of codes with distance ffi Gamma and rate 1 Gamma 2H( p ffi ) for which any ff ffi=48 fraction of errors can be corrected by a circuit of size O(N log N) and depth O(log N) The ....

G. A. Margulis, "Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators," Problems Inform. Transm., vol. 24, no. 1, pp. 39--46, 1988.

....75 634 Paris 13, France. Email : zemor infres.enst.fr Submitted to IEEE Trans. on Information Theory, september 2000. 1 The rst result of this kind (Sipser and Spielman [7] gives us a family of codes, based on a graph theoretic approach of Tanner [8] and constructions of Ramanujan graphs [5, 6]. The result is as follows: Theorem 1 [7] For any 0 there exists a polynomial time constructible family of codes with distance and rate 1 2H( p ) for which any =48 fraction of errors can be corrected by a circuit of size O(N log N) and depth O(log N) The complexity of a sequential ....

G. A. Margulis, \Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators," Problems Inform. Transm., vol. 24, no. 1, pp. 39-46, 1988.

....eigenvalue of M T M to the largest one is smaller than c 1 where c does not depend on m. There are various constructions of such multigraphs, the rst one in [25] The proof that they have the desired property is never very simple. For a relatively recent construction, see . The papers [24] [26] contain a construction of graphs of degree k with smallest eigenvalue 2 p k 1. This bound is now known to be optimal. Exercise 3.1. We can de ne the notion of an (n; k; compressor as follows. This is a bipartite multigraph with n input nodes and n output nodes and the property that for every ....

Grigorii A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Inform. Transm., 24:39-46, 1988.

.... of the second eigenvalue # 2 (G) of the graph G: ad(G U ) # U (d # 2 (G) V # 2 (G) This implies c E (r, G) # d # 1 r V # # 2 (G) Recall that a Ramanujan graph is a d regular graph G with # 2 (G) # 2 # d 1; explicit constructions of such graphs were given in [LPS88, Mar88]. Summing up the above, we have: Lemma 4.2 The incidence matrix of any d regular Ramanujan graph G on n vertices is an (r, d, d(1 r n) 2 # d 1) expander for any parameter r 0. 17 4.2 Tseitin tautologies: Boolean version A Tseitin tautology is an unsatisfiable CNF capturing the basic ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators. Problemy Peredachi Informatsii (in Russian), 24:51--60, 1988. English translation in Problems of Information Transmission, Vol. 24, pages 39-46.

....that for any sequence Gn;k of k regular graphs on n vertices, lim inf (G n;k ) 2 p k 0 1 as n goes to infinity [2, 13, 15] A Ramanujan graph is a k regular graph where all eigenvalues not equal to 6k are at most 2 p k 0 1 in absolute value. Such graphs have been constructed explicitly in [13, 14]. Note that when = 2 p k 0 1, the right hand side of Eq. 3 is roughly 2 log k01 n. In this paper, we establish some isoperimetric bounds that are function of the subsequent eigenvalues and do not depend on the second eigenvalue. More precisely, Theorem 1 Let G = V; E) be an undirected ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and concentrators. Problemy Peredaci Informacii, 24(1):51--60, 1988.

.... systems (basically, they consist of binomials necessarily containing among them the polynomials X 2 i Gamma 1, 1 i n) They extend slightly Tseitin s tautologies [16] 9] 17] 18] We exploit the construction of the Tseitin s tautologies ( 9] 17] 18] based on expanders ( 1] 11] [12]) and give a somewhat simpler proof of a linear lower bound for the case of used in section 1 notion of refutations (lemma 4) Relying on it, we first prove a linear lower degree bound for Nullstellensatz refutations for the systems which include the polynomials X 2 i Gamma 1; 1 i n (theorem ....

G. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and concentrators. Problems Inform. Transm., 1988, 24, p. 39--46.

.... Kreuter and Steger showed in [11] for r=2 the following: for fixed k 2, it holds N(m; 2k; 2) Omega Gamma m 2k= 2k Gamma1) Delta (ln m) 1= 2k Gamma1) For this special case r=2, there are better constructions known from the work of Lubotzky, Phillips and Sarnak [15] and Margulis [16]. However, the intention in [11] was to study Tur an numbers in the random situation. The constructions from [15] and [16] use algebraic techniques, and yield the so called Ramanujan graphs, which are graphs on m vertices with at least Omega i m (3k 5) 3k 3) j edges which do not contain any ....

.... Delta (ln m) 1= 2k Gamma1) For this special case r=2, there are better constructions known from the work of Lubotzky, Phillips and Sarnak [15] and Margulis [16] However, the intention in [11] was to study Tur an numbers in the random situation. The constructions from [15] and [16] use algebraic techniques, and yield the so called Ramanujan graphs, which are graphs on m vertices with at least Omega i m (3k 5) 3k 3) j edges which do not contain any cycle of length smaller than 2k 1. i.e. N(m; 2k; 2) Omega i n (3k 5) 3k 3) j . Recently Lazebnik, Ustimenko and ....

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G. A. Margulis, Explicit Group Theoretical Construction of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators, J. Probl. Inform. Transmission 24, 1988, 39-46.

....reduced) By the pseudo degree of a monomial we mean the number of variables which occur in its reduction. Observe that the pseudo degree of a Laurent monomial does not exceed the double degree of this Laurent monomial (see section 1) From now on we assume that G = G k is an expander [LPS 88] M 88] with k nodes and being r regular (r will be a constant, one could take, say r = 6 [LPS 88] M 88] That means that for any subset S of the set of the nodes of G the number of adjacent to S nodes in G is at least (1 ffl(1 Gamma jSj=k) jSj for an appropriate constant ffl 0. The ....

.... Observe that the pseudo degree of a Laurent monomial does not exceed the double degree of this Laurent monomial (see section 1) From now on we assume that G = G k is an expander [LPS 88] M 88] with k nodes and being r regular (r will be a constant, one could take, say r = 6 [LPS 88] M 88] That means that for any subset S of the set of the nodes of G the number of adjacent to S nodes in G is at least (1 ffl(1 Gamma jSj=k) jSj for an appropriate constant ffl 0. The corresponding to G k Boolean Thue system we denote by TS k (2) Any Laurent monomial in fX 2 e g e ; fX(v)g ....

G. Margulis "Explicit group-theoretical construction of combinatorial schemes and their applications to the design of expanders and concentrators," Problems Inform. Transm., V. 24, 1988, 39--46. 13

....unweighted (i.e. unit weighted) graph of girth g, any proper subgraph of H distorts the distances by at least g Gamma 1. Now, for all integer g 3, there exist (explicitly constructible) graphs with n vertices and girth g, which have more than 1 2 n 1 4= 3g Gamma6) edges (see Margulis [9]) The lower bound follows. Let us return to our original question: what is the smallest possible distortion in representing a given finite metric space X by a graph G of the same size as X, but with a bounded number of edges More generally, what is the smallest possible distortion in ....

G.A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problems of Information Transmission 24 (1988), 39-46. (Translated from Russian, Problemy Peredachi Informatsii 24 (1988), 51-66.)

....(given that the degree is no object) there are other constructions that are both more economical from a practical point of view and essential for certain theoretical purposes. The most prominent of these are the Ramanujan graphs 11 introduced by Lubotzky, Phillips and Sarnak [12] and by Margulis [15]. Whether there are efficient time division multiplex implementations of these graphs remains an open question. 6. ....

G. A. Margulis, "Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators", Problems of Inform. Transm., 24 (1988) 39--46.

....depends only on d 0 and graphical parameters of G. More precisely, is a function of d 0 , and , the second largest eigenvalue of the adjacency matrix of G. Furthermore, is such that 0 when =d 0 0. This result becomes especially interesting when one brings in the Ramanujan graphs of [2, 4]. These families of graphs are constructive, have an arbitrarily large number of vertices for xed degrees , and satisfy 2 p 1. By choosing G to be Ramanujan and xing a large enough we can therefore make as small as we like and obtain constructions of asymptotically good (G; C 0 ....

....in reference to the expanding properties of Ramanujan graphs. For example, if C 0 is chosen to be a shortened extended BCH code of length = 224 dimension k 0 = 115 and minimum distance proved to satisfy d 0 30 [3] then it can be checked that Ramanujan graphs G of degree 224 (constructed in [4, 2]) yield asymptotically good (G; C 0 ) codes. Furthermore Sipser and Spielman exhibit a decoding algorithm of low complexity, namely time O(log N) for a circuit of size O(N log N ) that, for any xed 1, will always return the original codeword provided the error vector has weight less than ....

G. A. Margulis, \Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators," Problems Inform. Transm., vol. 24, no. 1, pp. 39-46, 1988.

....progressing through Gabber and Galil [8] and culminating with Jimbo and Maruoka [10] yields expanders of grades 0 and 1. These constructions are now based on elementary algebraic arguments. The introduction of Ramanujan graphs, in the works of Lubotzky, Phillips and Sarnak [12] and Margulis [14] brought explicit constructions for grade 2 expanders, but the mathematics required to establish the properties of these graphs lies much deeper. No explicit constructions have yet been found for expanders of grade 3. Our original construction for self routing superconcentrators required grade ....

....lightened, over the course of research, through grade 2 expanders to the present requirement for grade 0 expanders, and can thus be met with elementary constructions. The techniques we have used to achieve this lightening are applicable to at least three other problems that currently require grade 14 2 or 3 expanders, lightening their requirements to grade 0. Specifically, we can adapt the constructions of Leighton and Maggs [11] for fault tolerant packet routing networks and of Arora, Leighton and Maggs [6] for self routing non blocking networks to use any construction for regular ....

G. A. Margulis, "Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators", Problems of Inform. Transm., 24 (1988) 39--46.

....) edges, which have no cycles of lengths 3; 2k, are known only for k = 2; 3; 5. In particular, Benson [Be 66] gave an algebraic construction, while Wenger [We 90] used finite projective geometries. Using the so called Ramanujan Graphs, Lubotzky, Phillips and Sarnak [LPS 88] and Margulis [Ma 88] have improved the probabilistic lower bound to ex(m; fC 3 ; C 2k g) Omega Gamma m 3k 5 3k 3 ) hence, N(m; 2k; 2) c Delta n 3k 5 3k 3 = c Delta n 1 2 3k 3 : Recent constructive improvements by Lazebnik, Ustimenko and Woldar [LUW 95] on the lower bound on ex(m; fC 3 ; ....

G. A. Margulis, Explicit Group Theoretical Construction of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators, J. Probl. Inform. Transmission 24, 1988, 39-46.

....graph on n vertices and is the second largest eigenvalue of its adjacency matrix, then for any set U of m vertices of G, the number of edges between U and its complement is at least (d; m(n;m) n : It follows from Theorem 8 of Section 2.2 that if d ; 2 then G is honest. In [15] [16], for each prime p j 1 ( mod 4) an infinite explicit family of d regular graphs whose second largest eigenvalue is at most 2 p d ; 1 is constructed. Thus, for example, by packing two 2 such 5 regular graphs together we get explicitly infinitely many 10 regular honest graphs. Our first ....

G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators , ProblemyPeredachi Informatsii 24(1988), 51-60 (in Russian). English translation in Problems of Information Transmission 24(1988), 39-46.

....to define c(G) to be the lim inf of (log k 1 n i ) g i . Lubotsky, Phillips and Sarnak [32] constructed families L p 1 of degree p 1,wherep is a prime congruent to1modulo4,andshowedthatc(L p 1 ) # 3 4. The fact that the value of c(L p 1 )is exactly 3 4 was established independently by Margulis [34] and Biggs and Boshier [9] The basic idea of [32] is to use quaternion algebras, and this was extended to cubic graphs by Chiu [16] Recently, Lazebnik, Ustimenko and Woldar [29, 30] have constructed families G k such c(G k ) 3 4) log k 1 k for every k # 3 Unfortunately, their results are ....

G. A. Margulis, Explicit group-theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators. Journal of Problems of Information Transmission (1988) 39--46.

....walks is n O(1= 2 ) i.e. polynomial in n and therefore can be handled in NC. So, it remains to explain the following two points: first, how to construct in NC a constant degree expander, and second, how to simulate the random walk in NC. Both these points have been extensively treated in [10, 13, 9, 1], and we give a simple explanation as indicated in [12] A constant degree d expander on n nodes is an n node d regular graph in which the number of neighbors of any set of vertices S is larger than some positive constant multiple of the cardinality of S (see, e.g. 15, pages 108 112] It is ....

G. Margulis. Explicit Group-Theoretical Constructions of Combinatorial Schemes and their Application to the Design of Expanders and Superconcentrators. Problems of Information Transmission, 24:39--46, 1988.

....of a d regular expander graph G and the points of an ffl biased probability space with random variables y 1 ; y n . Instead of choosing O(logn) subsets uniformly at random, choose a random walk of length O(logn) on G as described in [4, 10, 16] By using the expander constructions of [21, 22] this produces an algorithm for deciding maximality with error probability O( 1 n c ) using [4] corollary 2.8) and decisional and randomness complexities O(logn) A similar procedure can be applied for testing membership in any W defined by a conjunction of polynomial inequalities (s.t. the ....

G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemesand their applications to the design of expanders and superconcentrators, Problems of Information Transmissions, 24:39--46, 1988.

....All these graphs will be constructed from a single expander. Let G = V; E) be a d regular graph on a set V = fv 1 ; v n g of n vertices in which the absolute value of all nontrivial eigenvalues is at most 2 p d Gamma 1. There are known explicit constructions of such graphs (see [10] [11]) for any d for which d Gamma 1 is a prime power congruent to 1 modulo 4, where for each such d there are constructions for an infinite set of values of n containing a number between x and cx for all large x, where c is some absolute constant. By adding dummy bolts and nuts, if necessary, we may ....

....completing the proof. 2 4 Conclusions We have presented an O(n log 4 n) time deterministic algorithm for the nuts and bolts matching problem. It is worth noting that since we applied expanders of polylogarithmic degrees there are simpler explicit constructions than those given in [10] and [11] and we can take appropriate Cayley graphs of the groups Z k 2 by applying some known constructions of Linear Error Correcting Codes, as described in [6] This gives somewhat simpler graphs at the cost of increasing the complexity by a polylogarithmic factor. We omit the details. It is ....

G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24, 1988, pp. 51-60 (in Russian). English translation in Problems of Information Transmission 24 (1988), pp. 39-46.

....graph H , and its proof is different from that of the lower bound discussed above. 2 Derandomized graph products Our graph H is based on an explicit construction of a constant degree expander graph. It is simplest to assume that H is a non bipartite d regular Ramanujan graph as in [21] [22], where d 16 (b Gammaa) 2 . If n is such that no respective H graph exists, then G can be slightly modified by adding dummy vertices until a desirable value of n is reached. We construct the graph DG k (D stands for derandomized ) in the following way. We consider all possible random walks ....

G. A. Margulis, "Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators", Problemy Peredachi Informatsii 24(1988), 51-60 (in Russian). English translation in Problems of Information Transmission 24(1988), 39-46.

....Ramanujan graphs have expansion approaching d=2, see [Kah93] ffl for each of these values d, the second largest eigenvalues of G n;d are bounded from above by constants d such that lim d 1 d =d = 0. The expander graphs constructed by Lubotzky, Phillips and Sarnak [LPS88] and by Margulis [Mar88] are such a family. Pippenger [Pip93] points out that one can also obtain such a family by exponentiating the expander graphs of Gabber and Galil [GG81] From a d regular graph G on n vertices, we will derive a (2; d) regular graph with dn=2 vertices on one side, and n vertices on the other. ....

....if n i 1 Gamma n i = o(n i ) One can observe that by exponentiating the expander graphs of Gabber and Galil, one obtains a dense family of expander graphs. The results of [Hei33] can be used to prove that the expander graphs constructed by Lubotzky, Phillips and Sarnak [LPS88] and by Margulis [Mar88] are dense. We can use a dense family of d regular good expander graphs to construct a family of good expander graphs of every sufficiently large number of vertices. Let G be a d regular graph on n vertices. Let S be an independent subset of the vertices of G such that no two vertices in S have ....

G. A. Margulis. Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24(1):39--46, July 1988.

....All these graphs will be constructed from a single expander. Let G = V; E) be a d regular graph on a set V = fv 1 ; v n g of n vertices in which the absolute value of all nontrivial eigenvalues is at most 2 p d Gamma 1. There are known explicit constructions of such graphs (see [10] [11]) for any d for which d Gamma 1 is a prime power congruent to 1 modulo 4, where for each such d there are constructions for an infinite set of values of n containing a number between x and cx for all large x, where c is some absolute constant. By adding dummy bolts and nuts, if necessary, we may ....

....completing the proof. 2 4 Conclusions We have presented an O(n log 4 n) time deterministic algorithm for the nuts and bolts matching problem. It is worth noting that since we applied expanders of polylogarithmic degrees there are simpler explicit constructions than those given in [10] and [11] and we can take appropriate Cayley graphs of the groups Z k 2 by applying some known constructions of Linear Error Correcting Codes, as described in [6] This gives somewhat simpler graphs at the cost of increasing the complexity by a polylogarithmic factor. We omit the details. It is ....

G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders Matching nuts and bolts 7 and superconcentrators, Problemy Peredachi Informatsii 24, 1988, pp. 51-60 (in Russian). English translation in Problems of Information Transmission 24 (1988), pp. 39-46.

....bits are used in the computation where random bits are needed. The method to obtain the pseudo random bits is based on random walks on explicitly given expanders. The idea of using expanders to reduce the error was proposed in [15, 1] There are different kinds of explicit expander constructions [12, 13, 7, 10], the first one given by Margulis [12] Because of its simplicity we shall use the expander from [7] A short description of this graph can be found at the end of Section 2.1. To generate the pseudo random sequence the algorithm of [8] performs a random walk on an expander with constant degree of ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Probl. Peredachi Informatsii(Problems of Information Transmission), 24/1:51--60(in Russian), 39--46(English translation), 1988.

....n;k of k regular graphs on n vertices, lim inf (G n;k ) 2 p k Gamma 1 as n goes to infinity [3, 23, 27] Therefore, the best expansion coefficient we can obtain by applying Tanner s result is approximately k=4. This bound is achieved by Ramanujan graphs, which have been explicitly constructed [23, 26] for many pairs (k; n) By definition, a Ramanujan graph is a connected k regular graph whose eigenvalues 6= Sigmak are at most 2 p k Gamma 1 in absolute value. The relationship between the eigenvalues of the adjacency matrix and the expansion coefficient has also been investigated in [3, 5, ....

....of k regular graphs Gn on n vertices whose linear expansion is k=2 and such that 1 (Gn ) 2 p k Gamma 1(1 2log 2 log n= log 2 k n) Proof We construct the family (Gn ) by altering the known constructions of explicit Ramanujan graphs, so that the expansion of (Gn ) is k=2. From [23] and [26], we know that we can explicitly construct an infinite family of bipartite Ramanujan graphs (Fn ) on n vertices whose girth c(Fn ) is (4=3 o(1) log k Gamma1 n. Let Fn = V; E) be an element of the family, u 2 V a vertex of Fn and l = bc(F n ) 2c Gamma 2. Let u 1 ; u k be the neighbors ....

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G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and concentrators. Problemy Peredaci Informacii, 24(1):51--60, 1988.

....The first explicit construction was given by G. A. Margulis, using representation theory of Lie groups [Mar1] A simplified and improved version, requiring only (commutative) Fourier analysis, was given in [GaG] The last word so far has come in simultaneous and nearly identical work by Margulis [Mar2] and Lubotzky, Phillips, Sarnak [LPS] greatly improving the isoperimetric ratio. Their bound is obtained through an eigenvalue estimate (linked to isoperimetry by [AlM] which rests on the theory of arithmetic and algebraic groups, including results on the Ramanujan conjecture on the number of ....

Margulis, G. A.: Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy Peredachi Informatsii (Problems of Information Transmission) 24/1 (1988), 51-60 (in Russian), English translation pp. 39-46.

....that (1 Gammaa) 1 Gammab) Theorem 10. There exists a constructive algorithm that will determine, for infinitely many values of n, which of n processors are good and which processors are faulty in 84 rounds of testing. Proof: Lubotzky, Phillips, and Sarnak [8] and independently Margulis [10] construct certain Cayley graphs which have second largest eigenvalue p 1 and degree p 1 whenever p is congruent to 1 modulo 4. The number of vertices is q(q Gamma 1) 2 where q is any prime that is congruent to 1 modulo 4 and p is a quadratic nonresidue mod q. For p = 37, a = 4 14 ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39--46, 1988.

....p ab p (1 Gammaa) 1 Gammab) Theorem 10. There exists a constructive algorithm that will determine, for infinitely many values of n, which of n processors are good and which processors are faulty in 84 rounds of testing. Proof: Lubotzky, Phillips, and Sarnak [8] and independently Margulis [10] construct certain Cayley graphs which have second largest eigenvalue 2 p p p 1 and degree p 1 whenever p is congruent to 1 modulo 4. The number of vertices is q(q 2 Gamma 1) 2 where q is any prime that is congruent to 1 modulo 4 and p is a quadratic nonresidue mod q. For p = 37, a = ....

G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39--46, 1988. Translated from Problemy Peredachi Informatsii, Vol. 24, No. 1, pp. 51--60, Jan-Mar, 1988.

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Margulis, G.A.: Explicit Group-Theoretical Constructions of Combinatorial Schemes and their Application to the Design of Expanders and Concentrators. Problemy Peredachi Informatsii 24, 51--60 (1988)

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G. A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24: 51-60 (Russian). English translation in Problems of Information Transmission 24, 1988, 39-46.

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G.A.Margulis . Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredaci informacii 24 (1988) no.1,51-60

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G.A. Margulis. Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information and Transmission, Vol. 24, No. 1, pp. 39-- 46, July, 1988.

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G.A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and concentrators, Problems of Information Transmission, 24 (1988) 39 - 46.

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G.A. Margulis. Explicit group--theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39--46, 1988.

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G.A. Margulis. Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators. Problems Inform. Transmission 11, pp. 39-46, 1988.

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G. Margulis. Explicit group--theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39--46, 1988.

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A. Margulis. Explicit group-theoretical constructions of combinatorial schemes and their applications to the design of expanders and superconcentrators. Problems of Information Transmission, 24:39--64, 1988.

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G.A. Margulis. Explicit group--theoretical construction of combinatorial schemes and their application to the design of expanders and concentrators. Problems of Information Transmission, 24:39--46, 1988.

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G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24(

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G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24(

No context found.

G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators, Problemy Peredachi Informatsii 24(

No context found.

G.A. Margulis. Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and superconcentrators. Problems Inform. Transmission 11, pp. 39-46, 1988.

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