S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. Combinatorica, 7(4):343--355, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... that if G is a d regular graph such that 2ff 0 2 then the probability that at least one of the l samples is a member of S is at least 1 Gamma p 2j j 0 l : Constructions for regular graphs of constant degree such that the second eigenvalue is bounded away from 0 are known [27, 29, 37]. For degree regular graphs of degree d, 0 = d and the value of can be almost 2 p d. For a given expander G, let ff G = 1 2 0 2 and let l = log 1 ffl log ip 2j j 0 j : In our case, fi, the precentage of good vectors, is too small to apply the method directly and we ....

S. Jimbo and A. Marouka, Expanders obtained from affine transformations, Proceedings of the 17th annual ACM Symposium on Theory of Computing, (1985) pp. 88-97. 25

....[Tan84] see also [AM85] the second largest eigenvalue of the adjacency matrix should be strictly smaller than the degree. This naturally lead researchers to consider algebraic constructions, where this eigenvalue can be estimated. The celebrated sequence of papers [Mar73, GG81, AM85, AGM87, JM87, LPS88, Mar88, Mor94] provided such constant degree expanders. All these graphs are extremely simple to describe: given the name of a vertex (in binary) its neighbors can be computed in polynomial time (or even logarithmic space) This level of explicitness is essential for many of the ....

....arguments provide strong existential results, applications of expanders in computer science often require explicit families of constant degree expanders. The first such construction was given by Margulis [Mar73] with improvements and simplifications by Gabber and Galil [GG81] Jimbo and Maruoka [JM87] Alon and Milman [AM85] and Alon, Galil, and Milman [AGM87] Explicit families of Ramanujan graphs were first constructed by Lubotzky, Philips, and Sarnak [LPS88] and Margulis [Mar88] with more recent constructions given by Morgenstern [Mor94] The best eigenvalues we know how to achieve using ....

Shuji Jimbo and Akira Maruoka. Expanders obtained from affine transformations. Combinatorica, 7(4):343--355, 1987.

....for graphs with a prescribed eigenvalue separation ratio ) This construction was first proposed by Margulis [14] in 1974. Quantitative estimates essential for its application were provided by Gabber and Galil [8] in 1981, and improvements to these estimates have been given by Jimbo and Maruoka [10], whose version of the spacedivision multiplex construction we follow. In Section 5 we present some open problems prompted by this work. 2. Connectors Imagine a juggler who can with complete reliability throw balls to a fixed height, so that they always return a fixed amount of time after they ....

....4. Expanders This section is devoted to the time division multiplex implementation of expanding graphs. Our implementation will be based upon a particular explicit construction for expanding graphs, originated by Margulis [14] with improvements due to Gabber and Galil [8] and Jimbo and Maruoka [10]. We shall construct a basic expanding graph, which is a regular bipartite multigraph G = A; B;E) in which every vertex (in A[B) has degree 8 (meets 8 edges in E) and in which A and B each contain n vertices, where n = m 2 is a perfect square, and m = 2 is a perfect power of 2 (so that n = ....

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S. Jimbo and A. Maruoka, "Expanders Obtained from Affine Transformations", Combinatorica, 7 (1987) 343--355.

.... 2 d 2 is absolute value. We shall assume that we have available an explicit construction (say, one that can be carried out in logarithmic space) for regular bipartite multigraphs of various sizes n, but fixed degree d and fixed eigenvalue separation 1. For example, Jimbo and Maruoka [10] give a construction for n any perfect square, d = 8 and = p 3=2. In fact, a result of Alon [1] ensures that any explicit construction for expanders with fixed degree and expansion also yields a fixed eigenvalue separation. For k 0, let us write G k for the regular bipartite multigraph ....

....separation k . Thus we can obtain graphs with fixed (though perhaps very large) degree and eigenvalue separation as small as we please. We may also assume that we can obtain such graphs with size n any integral power of 2. To see this, we note that the construction of Jimbo and Maruoka [10] like those of Margulis [13] and of Gabber and Galil [8] before it) works for n any integral power of the perfect square 4; this gives us graphs for the even integral powers of 2. If we have a graph G with size n, degree d and eigenvalue separation 1, we can obtain from 4 copies of G a graph ....

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S. Jimbo and A. Maruoka, "Expanders Obtained from Affine Transformations", Combinatorica, 7 (1987) 343--355.

....y Delta and A 2 Gamma x y Delta . GN is an 8 regular graph. Margulis proved that its second eigenvalue is less than 8, hinting it may be a good expander. However, he did not achieve a bound on the ratio d . Gabber and Galil [4] were the first to present such a bound. Jimbo and Maruoka [7] obtained the best estimation known for the second eigenvalue of this graph: 6 Actually, the standard definition requires the neighborhood set to be of size at least (1 fl(1 Gamma 2jSj jGj ) jSj. Theorem 4.2 (Margulis, Gabber Galil, Jimbo Maruoka) The graph GN with T 1 = 1 1 0 1 ....

S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. Combinatorica, 7(4):343--355, 1987.

....all the subsets of a random k regular graph of size at most ffn have expansion at least fi. The explicit construction of expander graphs is much more difficult, however. The first explicit construction of an infinite family of expanders was discovered by Margulis [25] in 1973, and improved in [15, 5, 17]. The best currently known method to calculate lower bounds on the expansion in polynomial time relies on analyzing the second eigenvalue of the graph. Since the adjacency matrix A is symmetric, all its eigenvalues are real and will be denoted by 0 1 : n Gamma1 . We have 0 = k, and = ....

S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. Combinatorica, 7(4):343--355, 1987.

....families, using nontrivial group theoretic notions, and with cardinalities defined in terms of primes but it is not clear that these construction are efficient. All the other constructions of constant degree separable families that have appeared in the literature (e.x. Mar75, GG81, Buc86, AM85, JM85] are far from optimal, yielding separation exponents quite larger than 1 2 . However the difficulty of efficiently constructing 1 2 separable digraphs can be obviated if the degree of the digraphs is not required to be bounded. We shall now describe an efficient construction of digraphs with ....

S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. In STOC, page 88, 1985.

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S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. Combinatorica, 7(4):343--355, 1987.

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