Alon, N. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6, 3 (1986), 207-219.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....two distinct elements a; a 0 2 GF (q) with N(2A) a = a 02 . Thus G q;t contains q 1 loops, so the trace of M is q 1. Hence 1 = T rM = q 1) i=1 i = q 1 2 q (2m (q 1) q 2) which implies that the multiplicity m is (q 1) q 2) 2. Remark. Alon [1] proved that the C 4 free Erd os R enyi graphs on n vertices have independence number at most 2n (which can actually be improved to n (1 o(1) 2] This represents the best known constructive lower bound for the Ramsey number r(C 4 ; K n ) A referee pointed out that via Theorem 1 the ....

N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6 (1986), 207-219.

....over nite elds to construct graphs. He used the eigenvalue methods of [21] to prove his graphs had the relevant properties. Pippenger used a variation of Alon s graphs to obtain csort(2;n) log n) ######## ### If q is a prime power than F # is the nite eld on q elements. ######### ### [2] Let d; q # and let q be a prime power. The number d will be referred to as the dimension. The Geometric Expander over F # of dimension d is the bipartite graph that we construct below: 1. Create a set of d 1 tuples of the following form (1;a # ;a # ; a ### ) a # ; a ### (0; 1;a # ....

....which represent u and v are orthogonal to one another. #### ### Note that the number of vertices in the graph in De nition 7. 2 is (q ) and the number of edges is (q ) If we denote the number of vertices by n then the number of edges is ) The following de nition is implicit in [2]. ######### ### An ( n ) expander is a bipartite graph G = U; V; E) such that #U # = #V # = n and the following two properties hold. 1. #Z V ) #Z## n ###x#U:#N(x)#Z## n n 2. #Y V ) #Y ## n ##N(Y)#n#n Alon proved the following theorem using the eigenvalues ....

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N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica, 6, 1986.

....hence O( n 3 a 2 ) expanding. Theorem 4.8 csort(k; n) O(n 1 2 (k 1) log n) 2 4 (k 1) Proof sketch: This proof is similar to that of Theorem 3.13 except that we use Lemma 4.7 with a = n 1 1= 2 k 1) log n) 2= 2 k 1) 4. 4 Two Simple Constructive Algorithm Alon [2] showed that csort(2; n; 2) O(n 7=4 ) His algorithm is simpler than that of Theorem 4.8. Since Alon s result is about limited closure sorting we will discuss it in Section 7; however by combining it with the recurrence in Theorem 4.3 he obtained improvements over Theorem 4.3. We give the rst ....

....is about limited closure sorting we will discuss it in Section 7; however by combining it with the recurrence in Theorem 4.3 he obtained improvements over Theorem 4.3. We give the rst few improvements. More numbers can be generated; however, the asymptotic values do not improve. Theorem 4. 9 ([2]) 1. csort(2; n) O(n 7=4 ) 2. csort(3; n) O(n 8=5 ) 3. csort(4; n) O(n 26=17 ) 4. csort(5; n) O(n 22=15 ) Pippenger [18] noticed that a variant of Alon s algorithm actually yields csort(2; n) O(n 5=3 log n) We will discuss this algorithm when discussing Alon s ....

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N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica, 6, 1986.

....that we will use (aside from the fact that, given x and w, Hadw (x) can be computed in time poly(n) Lemma 5.2 Let X be any distribution on f0; 1g n of density and let 0. Then # fw : Hadw (X) has bias at least g 1 2 : Forms of Lemma 5. 2 have been proven by Lindsey, Alon [Alo86] and Chor and Goldreich [CG88] In the full version of this paper, we give a direct proof. Although Lemma 5.2 does not explicitly give an efficient decoding algorithm, we can easily obtain one using nondeterminism: Lemma 5.3 For every fixed i, there is a probabilistic i 2 algorithm ....

N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica, 6(3):207--219, 1986.

....a specific set of k vertices has less than h neighbors. 5 The last column (i.e. error prob. states the probability that a random construction (with given n and ) does not achieve the stated expansion. Actually, we only provide upper bounds on these probabilities. Alon s Geometric Expanders [2]. These constructions do not allow = O(log n) but rather is polynomially related to n. Still for our small numbers we get meaningful results, when using = q 1 and n = q 2 q 1, where q is a prime power. Specific values that may be used are tabulated below. 2 degree (i.e. ....

N. Alon. Eigenvalues, Geometric Expanders, Sorting in Rounds, and Ramsey Theory. Combinatorica, Vol. 6, pages 207--219, 1986.

....by n q (r, s, d) the number of s flats in PG(d, q) which contain a fixed r flat. Clearly n q (r, s, d) N q (s r 1, d r 1) q (s r) d s) 1 o(1) An r flat F is spanned by X, if dim(X # F ) r and we denote by F (r) X) the collection of r flats spanned by X. Alon [1] has shown that if X # PG(2, q) is of size X = 1 #) q 1) then F (1) X) # # 2 2 (q 2 q 1) For the case of spanned lines Alon s method easily extends to higher dimensions. Here we show the following generalization of Alon s result # Research supported in part by the ....

N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6 (1986), 207-219.

....can be used to realise a 1 relation in O(log p) steps on a randomly wired, bounded degree network known as a multibutterfly. An important feature of multibutterflies is that they have powerful expansion properties. In addition to permitting fast deterministic routing, such expander graphs [13, 14, 16, 141, 177, 255] also have very strong fault tolerance properties. The techniques and results that we have described for various types of randomised routing show convincingly that for the problem of routing h relations at least, there are a variety of theoretically and practically efficient methods which can be ....

N Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey Theory. Combinatorica, 6:207--219, 1986.

....of issues. First, our constants, in particular, the degree d, are rather large, and hence the results are bene cial for very large systems only. We are looking for graph constructions facilitating our methods for smaller system sizes. One such family of candidates is nite projective geometries [Bat97, A86]. Second, our adversarial assumption is rather strong, namely, fully adaptive malicious adversary, and it might be possible to improve eciency if we adopt a weaker adversarial model. In particular, one might accept in practice a non adaptive adversarial model, that is, one that gives the ....

N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6(3):207-219, 1986.

....of issues. First, our constants, in particular, the degree d, are rather large, and hence the results are bene cial for very large systems only. We are looking for graph constructions facilitating our methods for smaller system sizes. One such family of candidates is nite projective geometries [Bat97, A86]. Second, our adversarial assumption is rather strong, namely, fully adaptive malicious adversary, and it might be possible to improve eciency if we adopt a weaker adversarial model. In particular, one might accept in practice a non adaptive adversarial model, that is, one that gives the ....

N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6(3):207-219, 1986.

....that supplies a deterministic algorithm to construct a graph with the desired properties in time polynomial in the size of the graph. It is worth noting that for the case r = 2, corresponding to the usual Ramsey numbers, there are several known explicit constructions; see [10] 13] 9] [1], 2] 3] Despite a considerable amount of e ort, all these constructions supply bounds that are inferior to those proved by applying probabilistic arguments. The problem of nding explicit constructions matching the best known bounds is of great interest, and may have algorithmic applications ....

....O(log n) where n is the number of vertices. In the rest of this note we describe these two constructions and prove their properties. 2 2 The rst construction The rst construction we present applies nite geometries and the proof of its properties is based on the spectral technique used in [1] for a similar purpose, together with some additional ideas. Graphs considered in this section may have loops. Each loop contributes one to the degree of a vertex incident to it and contributes 1=2 when we count the number of edges spanned by a set of vertices. We need the following lemma. Lemma ....

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N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6 (1986), 207-219.

....of issues. First, our constants, in particular, the degree d, are rather large, and hence the results are bene cial for very large systems only. We are looking for graph constructions facilitating our methods for smaller system sizes. One such family of candidates are nite projective geometries [A86]. Second, our adversarial assumption is rather strong, namely, fully adaptive malicious adversary, and it might be possible to improve eciency if we adopt a weaker adversarial model. In particular, one might accept in practice a nonadaptive adversarial model, that is, one that gives the adversary ....

N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6(3):207-219, 1986.

....that supplies a deterministic algorithm to construct a graph with the desired properties in time polynomial in the size of the graph. It is worth noting that for the case r = 2, corresponding to the usual Ramsey numbers, there are several known explicit constructions; see [10] 13] 9] [1], 2] 3] Despite a considerable amount of effort, all these constructions supply bounds that are inferior to those proved by applying probabilistic arguments. The problem of finding explicit constructions matching the best known bounds is of great interest, and may have algorithmic applications ....

....O(log n) where n is the number of vertices. In the rest of this note we describe these two constructions and prove their properties. 2 The first construction The first construction we present applies finite geometries and the proof of its properties is based on the spectral technique used in [1] for a similar purpose, together with some additional ideas. Graphs considered in this section may have loops. Each loop contributes one to the degree of a vertex incident to it and contributes 1=2 when we count the number of edges spanned by a set of vertices. We need the following lemma. Lemma ....

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N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6 (1986), 207--219.

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Alon, N. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6, 3 (1986), 207-219.

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Alon, N. (1986) Eigenvalues, geometric expanders, sorting in rounds, and Ramsey Theory, Combinatorica 6 207219.

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, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinatorica 6 (1986), no. 3, 207-219. 1

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