G.A.Margulis . Explicit constructions of concentrators. Problemy peredaci informacii 9 (1973) no.4,71-80  Home/Search   Document Not in Database   Summary   Related Articles   Check

....have multiple edges. The definition above is for the number of edges from S V to vertices in V having the di#erent index numbers from S while an expander defined in [3] is for the number of neighbors. But we can also obtain for our definition the same theorem as [3] According to [3] Margulis [2] first found a way to explicitly construct a family of expanders for k = 5, although the constant value # was not known. On the other hand, Gabber and Galil [3] proposed similar constructions for some specified value of #. Here we explain one of their constructions, which we will use for our ....

G. A. Margulis, "Explicit constructions of concentrators", Problemy Peredachi Informatsii, 71-80, 1973.

....that are very well connected in the sense that for every set of vertices S of size at most V there are at least # S vertices in V S that are adjacent to some vertex in S. In this case we say the expansion of the graph is #. An explicit construction for expanders was given by Margulis [18] and later by Gabber and Galil [7] In this construction a continuous graph G is defined over I by the following two transformations: f(x, y) x y, y) mod 1 , g(x, y) x, x y) mod 1. Theorem 12 ( 7] For every set A of points in I such that (A) it holds that ( f(A)#g(A) A) A) ....

G. A. Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, (9(4)):325--332, 1973.

....that are very well connected in the sense that for every set of vertices S of size at most 2 jV j there are at least jSj vertices in V nS that are adjacent to some vertex in S. In this case we say the expansion of the graph is . An explicit construction for expanders was given by Margulis [18] and later by Gabber and Galil [7] In this construction a continuous graph G is de ned over I by the following two transformations: f(x; y) x y; y) mod 1 , g(x; y) x; x y) mod 1. Theorem 12 ( 7] For every set A of points in I such that (A) 2 , it holds that ( f(A) g(A) nA) ....

G. A. Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, (9(4)):325-332, 1973.

....online job load balancing in a network is presented in [1] The efficiency of the algorithm depends only on the expansion of the network. Our framework could be used to guarantee that the topology of the network is indeed an expander. An explicit construction for expanders was given by Margulis [15] and later by Gabber and Galil [7] In this construction a continuous graph G is defined over I by the following two transformations: f(x; y) x y; y) mod 1 , g(x; y) x; x y) mod 1. Theorem 5.1 ( 7] For every set A of points in I such that (A) it holds that ( f(A) g(A) nA) ....

G. A. Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, (9(4)), October - December 1973.

....test results. Section 5 treats the nonadaptive or oblivious model [5, 9, 11, 13] in which the tests conducted in a round may not depend on previous results. 2 Preliminaries Some of our algorithms rely on expander graphs. Constructions, existence proofs, and analysis of expanders can be found in [1, 3, 4, 8, 10]. We require the following definitions and theorems: Definition 2.1 If G(V; E) is a graph and D V , then Gamma(D) fx 2 D j (9d 2 D) x; d) 2 E]g: The elements in Gamma(D) are called neighbors of D. Definition 2.2 An n by m expander graph with parameters ( ff; r) is a bipartite graph ....

G. A. Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, 9(4):71--80, October-December 1973. English translation available (pp. 325-332).

....A mathematical analysis of iterative decoding of LDC codes will be given in the third paper. This is the most complicated part of the theory of LDC codes. A corresponding investigation, of iterative decoding of low density block codes, has been done by Zyablov and Pinsker [9] and by Margulis [10]. In the first one, it was proved that there exist low density block codes, such that, with iterative decoding, after transmission over a binary symmetric channel, the number of errors that can be corrected is increasing linearly with the block length. We can understand the importance of this ....

G. A. Margulis, "Explicit Construction of a Concentrator", Probl. Inform. Transm., vol. 9, N4, pp. 71-80, 1973.

....a certain probability space of graphs, and then one shows that the probability of such graphs is non zero. In fact it is usually shown that such probability tends to 1. Thus not only such graphs exist, but they exist in abundance. The weakness of such a proof is that it is not explicit. Margulis [12] was the first to give an explicit construction of a sequence of graphs Gn . This major achievement uses group representation theory. However, while his construction is explicit, the constant of expansion was not explicitly known. Gabber and Galil [10] in a beautiful paper gave an explicit ....

G. A. Margulis, Explicit construction of concentrators. Problems Inform. Transmission 9, 1973, 325--332.

....an assignment of minimal size satisfying all the pebbled clauses at stage s, there are at least ex(G) Gamma b d 2 c pebbled clauses at this stage and ex(G) Gamma b d 2 c 1 pebbled clauses at stage s 1. Xi There exist expander graphs G with n nodes constant degree d and with ex(G) n [12]. In [17] it is shown that the degree for such expander graphs can be reduced to d = 8. For an odd marking of such a graph the formula (G; m) has at most dn 2 variables and n2 d Gamma1 clauses. By the above result, the space needed in a resolution refutation of (G; m) is at least n Gamma 3 ....

Margulis, A.: Explicit construction of concentrators. Problems Information Transmission 9 (1973) 71--80.

....of areas of computer science (e.g. AKS1, AKS2, FFP, GIL , Tom, Val2] It is not hard to show that a random graph is an expander. Yet the problem of deterministically constructing expanders has proved to be difficult; the construction of constant degree expanders was considered a breakthrough [Mar, GG]. The eigenvalue method has proved particularly useful in designing expander graphs. This method works by looking at the adjacency matrix A of an undirected graph G = V; E) To simplify matters for the moment, suppose that G is d regular, so A has d as its largest eigenvalue. Then G is an ....

G.A. Margulis, "Explicit Construction of Concentrators," Problems of Inform. Transmission, pp. 325-332.

....N o(1) if K N :49 . Yet for most applications random graphs are not useful; instead, explicit, deterministic constructions are required. Historically, constructing explicit expanders has been quite dicult. The explicit construction of constant degree expander graphs was a major breakthrough [12, 7]. These explicit constructions relied on showing an upper bound on the second largest eigenvalue of the adjacency matrix corresponding to the graph. Kahale [11] showed that such methods cannot achieve c T=2. Yet some applications, such as [4, 24, 5] need c = 1=2 433 T , as then the expander ....

G.A. Margulis. Explicit construction of concentrators. Problems of Inform. Transmission, 9:325-332, 1973.

....with a small modification to simplify the proof. The expander graph Hm is a simple bipartite graph in which each vertex has degree at most 5 and each side contains m 2 vertices (for brevity we write n = m 2 ) The particular family of expander graphs used here was first defined by Margulis [44]. The exact definition of the graphs is not needed; for the lower bound all that is needed is the expanding property proved by Margulis and stated in the next lemma. Lemma 5.8. There is a constant d 0 such that if V 1 is contained in one side of Hm , V 1 # n 2, and V 2 consists of all ....

G. A. Margulis, Explicit construction of concentrators, Problems of Information Transmission, vol. 9 (1973), pp. 325--332.

....a compressor as soon as we have a family of multigraphs of sizes m = 1; 2; with the property that the ratio of the second largest 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 ....

Grigorii A. Margulis. Explicit construction of concentrators. Problems of Inform. Transm., 9:71-80, 1974.

....with a small modification to simplify the proof. The expander graph H m is a simple bipartite graph in which each vertex has degree at most 5 and each side contains m 2 vertices (for brevity we write n = m 2 ) The particular family of expander graphs used here was first defined by Margulis [44]. The exact definition of the graphs is not needed; for the lower bound all that is needed is the expanding property proved by Margulis and stated in the next lemma. Lemma 5.8. There is a constant d 0 such that if V 1 is contained in one side of H m , jV 1 j n 2, and V 2 consists of all the ....

G. A. Margulis, Explicit construction of concentrators, Problems of Information Transmission, vol. 9 (1973), pp. 325--332.

....and fi 1. This property is most interesting when r l, for when r AE l, it is easy to construct graphs with expansion. As it happens, a random k regular l Theta l bipartite graph is likely to be an expander for any k 3, Pin73] Explicit constructions were first discovered by Margulis [Mar73, Mar75] and have since been greatly improved. So far, however, the expansion achieved by the explicit constructions is still about a factor of two smaller than the expected expansion of a random graph. A nice summary of the state of the art in expander graphs can be found in [Kah95] One drawback to ....

G. A. Margulis. Explicit constructions of concentrators. Problems of Information Transmission, 9:71--80, 1975. 39

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G.A.Margulis . Explicit constructions of concentrators. Problemy peredaci informacii 9 (1973) no.4,71-80

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G. A. Margulis, "Explicit constructions of concentrators", Problemy Peredachi Informatsii, 71-80, 1973.

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G. A. Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, pages 71--80, 1973.

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G.A. Margulis, Explicit constructions of concentrators, Prob. Per. Infor. 9 (1973) 71 - 80; English transl. in Problems of Information Transmission, (1975) 325 - 332.

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G.A. Margulis. Explicit construction of concentrators. Problems of Inform. Transmission, 9:325-332, 1973.

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Margulis, A.: Explicit construction of concentrators. Problems Information Transmission 9 (1973) 71--80.

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G.A. Margulis. Explicit construction of concentrators. Problems of Inform. Transmission, 9:325--332, 1973.

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G. A. Margulis. Explicit construction of concentrators. Problems of Information Transmission, 1975.

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G.A. Margulis, "Explicit Construction of Concentrators", Prob. Per. Infor. 9 (4) (

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G. A. Margulis. Explicit construction of concentrators. Problemy Peredaci Informacii, 9(4):71--80, 1973.

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G.A. Margulis. Explicit construction of a concentrator. Problems of Information Transmission, 9:71--80, 1973.

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