N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. Journal of Algorithms, 8:337--347, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....recursion. It is worth noting that spectral methods have not been limited to graph partitioning; work has been done using the spectrum of the adjacency matrix in graph coloring [4] and using the Laplacian spectrum to prove theorems about expander graph and superconcentrator properties [3] 1] [2]. The work on expanders has explored the relationship of # 2 to the isoperimetric number; Mohar has given an upper bound on the isoperimetric number using a strong discrete version of the Cheeger inequality [22] Reference [8] is a book length treatment of graph spectra, and it predates many of ....

N. Alon, Z. Galil, and V. D. Milman, Better expanders and superconcentrators, Journal of Algorithms, 8 (1987), pp. 337--347.

....expected probing for unsuccessful search. It should be noted that the [16] result only needs O(log n) wise independent hash functions that map, say, 0; n 4 ] 7 [0; n Gamma 1] Randomized routing schemes and PRAM emulation have had a substantial and fruitful recent literature [21] 17] [2], 12] 18] 19] 5] 13] In particular, 5] and [13] show formally (and perhaps plausibly) how n log n processor Omega like networks 22 On universal classes of extremely random constant time hash functions and their time space tradeoff can, with very high probability, emulate an n log ....

N. Alon, Z. Galil, and V.D. Milman. Better Expanders and Superconcentrators. Journal of Algorithms, 8, 1987, pp. 337--347.

....on vertices I (inputs) and O (outputs) where jIj = jOj = n, such that every subset A ae I of size at most ffn is joined by edges to at least jAj i 1 c(1 Gamma jAj n ) j different outputs. The constant c is called the expansion factor of the graph. Theorem2 (Alon, Galil, and Milman [2], Cor. 2.3) If n = m 2 for some integer m, then we can construct an (n; 9; 1 2 ; 0:41) expander in O(n) time. Corollary 3. Let 0 ffi 1 2 and fl ffi = 3 Gamma5ffi 5ffi(1 Gammaffi) Then there exists an integer q ffi such that for any n, where n = m 2 for some integer m, we can ....

N. Alon, Z. Galil and V.D. Milman. Better expanders and superconcentrators. Journal of Algorithms 8 (1987), pp. 337--347.

....by Tanner [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 ....

....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 our approach are O(1=D 1=3 ) 2.3 Squaring and Tensoring ....

N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. J. Algorithms, 8(3):337--347, 1987.

....are its crosspoint and depth complexity. It was established in [NM82] that any bipartite (n, m) concentrator requires at least m(n m 1) crosspoints, and it was recently shown in [OH94] that this bound is tight. In the case of multipartite concentrators, it was shown in a series of e#orts [Pin73, Mar73, Pip77, Chu78, Bas81, GG81, Alo86, GAM87, JM87, Lee92] that a multipartite (n, m) concentrator can be constructed with O(n) crosspoints and O(log n) depth. The cited concentrator constructions are obtained in one of two ways. In the first approach, one employs counting arguments to prove the existence of a bounded capacity concentrator and then uses ....

....(expanders) 1 . In the second approach, one relies on an explicit construction of a bounded capacity concentrator and any of the recursive constructions used in the first approach. An explicit bounded capacity concentrator construction was first reported in [Mar73] and subsequently extended in [GG81, Alo86, GAM87, JM87, Lee92]. 1 An (a, b, c, d) expander is a bipartite graph with a inputs, b outputs in which any k # c inputs are connected to at least k outputs. As in a concentrator, c denotes the capacity of the expander, and the new parameter d is called its expansion coe#cient. 2 While explicit constructions of ....

Z.Galil, N. Alon, and V.D. Milman. Better expanders and superconcentrators. Journal of Algorithms, 8:337--347, 1987.

.... [Sim91] We note that spectral methods have not been limited to graph partitioning; work has been done using the spectrum of the adjacency matrix in graph coloring [AG84] and using the Laplacian spectrum to prove theorems about expander graph and superconcentrator properties [AM85] Alo86] [AGM87]. The work on expanders has explored the relationship of 2 to the isoperimetric number; Mohar has given an upper bound on the isoperimetric number using a strong discrete version of the Cheeger inequality [Moh89] Reference [CDS79] is a book length treatment of graph spectra, and it predates many ....

N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. Journal of Algorithms, 8:337--347, 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 = ....

....[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, 6, 12], but the bound they get, when applied to non bipartite Ramanujan graphs and for sufficiently large k, is no better than Tanner s bound. Other results about expanders are contained in [9, 22, 30] Some applications, such as the construction of non blocking networks in [7] required an expansion ....

N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. J. Algorithms, 8:337--347, 1987.

.... [Sim91] We note that spectral methods have not been limited to graph partitioning; work has been done using the spectrum of the adjacency matrix in graph coloring [AG84] and using the Laplacian spectrum to prove theorems about expander graph and superconcentrator properties [AM85] Alo86] [AGM87]. The work on expanders has explored the relationship of 2 to the isoperimetric number; Mohar has given an upper bound on the isoperimetric number using a strong discrete version of the Cheeger inequality [Moh89] Reference [CDS79] is a book length treatment of graph spectra, and it predates ....

N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. Journal of Algorithms, 8:337--347, 1987.

....(2ffll 1) the theorem follows. Proof of Theorem 1.3. Lubotsky et al. have given, for every prime p congruent to 1 modulo 4, an explicit construction of a (p 1) regular graph fGng, for n = q 1 and q prime congruent to 1 mod 4 and distinct from p. These graphs are Ramanujan Graphs. By [2] Gn is a (n; p 1; ff) expander, with ff = 4 c p 1 c 2 and c = 1 2 p 1 2(p 1) Gamma4 p p . Furthermore, it can be easily verified that a (n; d; ff) expander is a (1 ff 2 ) expanding graph. Therefore it is possible to construct for infinitely many ff 1:656 an expander with ....

N. Alon, Z. Galil, and V.D. Milman, "Better Expanders and Superconcentrators". Journal of Algorithms 8, pp. 337--347 (1987).

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N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. Journal of Algorithms, 8:337--347, 1987.

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N. Alon, Z. Galil, and V. D. Milman. Better expanders and superconcentrators. J. Algorithms, 8(3):337--347, 1987.

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N. Alon, Z. Galil, and V. Milman. Better expanders and superconcentrators. J. of Algorithms, 8:337-347, 1987.

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N. Alon, Z. Galil, and V.D. Milman, Better Expanders and Superconcentrators. Journal of Algorithms 8, pp. 337--347 (1987).

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