N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42:1091-1106, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....log d n) for any 2d regular graph. Therefore, no family of 2d regular graphs can have smaller asymptotic ) bounds than Theorem 2. As a consequence of Theorem 2, H graphs are expanders with O(log d n) diameter with high probability because of the relation between eigenvalues and expanders [13] [14]. Corollary 3 is a direct consequence of Theorem 2. be a graph chosen from H n,2d uniformly at random. Then 2 # 2d with probability p ) where p depends on d. IV. PERFECT SAMPLING In this section, we will discuss several implementations for procedure SAMPLE of Section III. A. Global ....

N. Kahale, "Eigenvalues and expansion of regular graphs," vol. 42, no. 5, pp. 1091--1106, Sept. 1995.

.... expander graph of constant degree d and second eigenvalue 2 d (cf. LPS] The constant d will be determined so that d 2 4 ffl (and d 2 6 ffl ) It is well known that a random walk of length t in an expander avoids a set of density ae with probability at most (ae (cf. [AKS, Kah]) Thus, as a preparation step, we reduce the error probability of the pcp system to (24) This is done using the trivial reduction of Proposition 11.1. We derive a proof system with error probability p, randomness complexity = r Delta log 2 (1=p) r Delta log 2 ( d=2) O(r) 25) and ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, Vol. 42, No. 5, 1995, pp. 1091--1106.

....University, partially supported by DIMACS. 1 The problem with this equivalence is that it is not tight. For a random d regular graph, small sets S have roughly (d Gamma 1)jSj neighbors, yet bounding the second eigenvalue can only be used to show the existence of roughly (d=2)jSj neighbors [Kah]. The situation gets much worse for larger degree and stronger expansion. A definition that captures such strong expansion is: Definition 1.1 [Pip3] An undirected graph is a expanding if any two disjoint sets of vertices, each containing at least a vertices, are joined by an edge. Equivalently, ....

N. Kahale, "Eigenvalues and expansion of regular graphs," Journal of the ACM, 42 (1995), pp. 1091-1106.

....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 has the unique neighbors property. This means that for any subset A of vertices, there are jAj) vertices that are neighbors of exactly one vertex in A. ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42:1091-1106, 1995.

....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 the AKS network is that the big O notation hides large constant factors. In contrast, the depth of the bitonic sorting network is (log 2 n) 2 (log n) 2 [CLR90, p. 650] Some progress has been made in simplifying the AKS network and in improving the constant factors in its ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091--1106, September 1995.

....by Margulis [24, 25] 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 [17]. One drawback to the AKS network is that the big O notation hides large constant factors. In contrast, the depth of the bitonic sorting network is (log 2 n) 2 (log n) 2 [11, p. 650] Some progress has been made in simplifying the AKS network and in improving the constant factors in its depth ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091--1106, September 1995.

....nodes [20] Recently Youssef proved: Theorem 2. 7 Any Omega or Inverse Omega permutation can be off line routed on the MIMD G Theta H network in T (G) T (H) steps [330] Expander graphs became increasingly important in search for asymptotically optimal permutation routing and sorting networks [127]. The AKS sorting network [3, 4] and the multibutterfly are the only two known bounded degree networks which can sort in O(log N) bit steps with deterministic algorithms [200, 248] Recent proof that multibutterflies can sort is based on a constant load, congestion and dilation embedding of an ....

Kahale, N. Eigenvalues and expansion of regular graphs. J. ACM. 42 (5), 1995, pp. 1091--1106.

....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 the AKS network is that the big O notation hides large constant factors. In contrast, the depth of the bitonic sorting network is (log 2 n) 2 (logn) 2 [CLR90, p. 650] Some progress has been made in simplifying the AKS network and in improving the constant factors in its depth ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091--1106, September 1995.

.... Delta n log 2 n and depth ( 1 2 o(1) Delta log 2 n, and the AKS network of Ajtai, Koml os, and Szemer edi [2, 3] with depth O(log n) and a large constant behind the O notation. 2 Comparator networks for the related problems of merging and selection have also been widely studied [5, 8, 9, 10, 12, 13, 18, 20, 21, 26, 27]. A selection network is a comparator network that classifies the input values into two groups such that all values in the first group are smaller than all those in the other group. 3 A merging network is a comparator network 1 We often refer to a comparator network as a network, unless ....

N. Kahale. Eigenvalues and expansion of regular graphs. Technical Report 93-70R, DIMACS, Rutgers University, 1993.

....by Margulis [24, 25] 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 [17]. One drawback to the AKS network is that the big O notation hides large constant factors. In contrast, the depth of the bitonic sorting network is (log 2 n) 2 (log n) 2 [11, p. 650] Some progress has been made in simplifying the AKS network and in improving the constant factors in its depth ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091--1106, September 1995.

....by Margulis [34, 35] 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 [23]. One drawback to the AKS network is that the big O notation hides large constant factors. In contrast, the depth of the bitonic sorting network is (log 2 n) 2 (log n) 2 [14, p. 650] Some progress has been made in simplifying the AKS network and in improving the constant factors in its depth ....

N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091-- 1106, September 1995.

....ff = 1 a Gamma Delta=2 and fi = a Gamma Delta=2)b Gamma ffl (for any ffl 0) 2. If a Delta regular graph G is (ff; fi) immune, then it is an (a; b) expander for a = 1=ff and b = fffi. Based on Part 1 of the lemma and known results concerning the existence of strong expanders (cf. K95] we get Corollary 3.8 [P96b] For every sufficiently large constant integer Delta and for every ff 1 Delta=2 Gamma1 , there exists some fi 0 such that a random Delta regular graph is (ff; fi) immune with high probability. Definition 3.9 A graph G(V; E) is near Delta regular if ....

N. Kahale. Eigenvalues and expansion of regular graphs. J. of the ACM, 42:1091--1106, 1995.

....graph on n vertices [5, p. 238] is almost surely (1 o(1) log k01 n. We also prove that our bounds on the diameter in terms of ffi i are asymptotically tight, for any fixed i. Our proofs are based on elementary linear algebra, the use of Chebychev polynomials, and some techniques developed in [10, 11, 12]. Section 3 is based on [9] and a longer version of the paper appears in [11] 2 Notation and background Let G = V; E) be an undirected k regular graph on n vertices. Denote by L 2 (V ) the set of real valued functions on V and L 2 0 (V ) ff 2 L 2 (V ) P v2V f(v) 0g. As usual, we ....

....at least bD(G) ic from each other. By applying Theorem 3 to the set fu 0 ; u 1 ; u i g, we get the first bound on D(G) d(u; v) The second bound can be established similarly by applying Theorem 1 to the subsets N r (fu j g) 8 5 Tightness of bounds We use techniques similar to [10, 11, 12] to prove that the inequality Eq. 12 is asymptotically tight, for any fixed i. We start with the case i = 1. Theorem 5 For any integer k such that k 0 1 is prime congruent to 1 modulo 4, there exists an infinite family of k regular graphs Gn on n vertices with (Gn ) 2 o(1) p k 0 1 and ....

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N. Kahale. Eigenvalues and expansion of regular graphs. submitted, 1993.

....The second bound can be established similarly by applying Theorem 1 to the subsets N r (fu j g) 5 Tightness of bounds We show that, for any fixed i, Eq. 13 is asymptotically tight for certain families of k regular graphs having asymptotically optimal jffi i j. We use techniques similar to [11]. We start with the case i = 1. Theorem 4 For any integer k such that k Gamma 1 is prime congruent to 1 modulo 4, there exists an infinite explicit family of k regular graphs Gn on n vertices with (Gn ) 2 o(1) p k Gamma 1 and diameter (2 o(1) log k Gamma1 n. Proof Let H be a ....

....1 1) X v2V (T ) V (T 0 ) g(v) 2 : The third equation follows from the fact that the largest eigenvalue of T is at most 2 p k Gamma 1. Since g 2 L 2 0 (V (G) X w2V (H) f(w) Gamma X w2(V (T ) V (T 0 ) GammaV (H) g(w) We need the following lemma, whose proof can be found in [11]. Lemma 1 If H = V; E) is k regular on n vertices, then for any f 2 L 2 (V ) we have f Delta Af 1 (H)jjf jj 2 k Gamma 1 (H) n Gamma X v2V f(v) Delta 2 : Using Lemma 1 and the Cauchy Schwartz inequality, we get f Delta Af 1 (H)jjf jj 2 k n 0 X w2(V (T ) V (T 0 ....

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N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42(5):1091--1106, 1995.

....n) in [23] While this is the best result known so far, it gives only a lower order improvement over the trivial informationtheoretic bound of n log n Gamma n log e. In related work, a number of bounds have been established for the size of comparator networks for merging and selection (see [3, 8, 9, 11, 16, 20, 21, 24, 25], and see Theorem F on page 230 of [10] for a result due to Floyd) For the depth of sorting networks, Yao [24] has shown a lower bound of approximately (2:41 Gamma o(1) log n. This should be compared with the (2 Gamma o(1) log n trivial information theoretic bound. Note that each level of ....

N. Kahale. Eigenvalues and expansion of regular graphs. Technical Report 93-70R, DIMACS, Rutgers University, 1993.

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N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42:1091-1106, 1995.

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N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42:1091--1106, 1995.

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N. Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM, 42:1091-1106, 1995.

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N. Kahale, "Eigenvalues and Expansion of Regular Graphs", Journal of the ACM, 42(5):1091--1106, September 1995.

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N. Kahale, "Eigenvalues and Expansion of Regular Graphs", Journal of the ACM, 42(5):1091--1106, September 1995.

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