Triple-loop networks with arbitrarily many minimum distance diagrams
TL;DR: This paper shows that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams, and builds-up on the relations between minimumdistance diagrams and monomial ideals.
About: This article is published in Discrete Mathematics.The article was published on 2009-04-01 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Monomial.
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TL;DR: An algorithm to compute the set of primitive elements for an embedding dimension three numerical semigroups is given and it is shown how this procedure is used in the study of the construction of L-shapes and the tame degree of the semigroup.
4 citations
TL;DR: A polynomial‐time algorithm for deciding whether a given distance graph with a finite number of vertices is connected and is conjectured to be NP‐hard is described.
Abstract: We describe a polynomial-time algorithm for deciding whether a given distance graph with a finite number of vertices is connected. This problem was conjectured to be NP-hard in Draque Penso et al. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012 © 2012 Wiley Periodicals, Inc.
2 citations
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...These diagrams and their extensions to higher dimensions (graphs with more than two jumps) have aroused interest both from the pure combinatorial point of view and because of its applications in optimal network design [6, 7, 9, 12]....
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TL;DR: In this paper, a geometric method for obtaining an infinite family of Cayley digraphs of constant density on finite Abelian groups is presented, which works for any given degree and it is based on consecutive dilates of a minimum distance diagram associated with a given initial Cayley detector.
Abstract: A geometric method for obtaining an infinite family of Cayley digraphs of constant density on finite Abelian groups is presented. The method works for any given degree and it is based on consecutive dilates of a minimum distance diagram associated with a given initial Cayley digraph. The method is used to obtain infinite families of dense or asymptotically dense Cayley digraphs. In particular, for degree $d=3$, an infinite family of maximum known density is proposed.
2 citations
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TL;DR: Two procedures, E and Q, which generate a particular type of extensions and quotients of 2{Cayley digraphs, respectively, are used to obtain optimal quotients and extensions and are proved to be also optimal.
Abstract: Given a nite Abelian group G and a generator subset A G of cardinality two, we consider the Cayley digraph = Cay(G ;A). This digraph is called 2{Cayley digraph. An extension of is a 2{Cayley digraph, 0 = Cay(G 0 ;A) with G < G 0 , such that there is some subgroup H < G 0 satisfying the digraph isomorphism Cay(G 0 =H;A) = Cay(G;A). We also call the digraph a quotient of 0 . Notice that the generator set does not change. A 2{Cayley digraph is called optimal when its diameter is optimal with respect to its order. In this work we dene two procedures, E and Q, which generate a particular type of extensions and quotients of 2{Cayley digraphs, respectively. These procedures are used to obtain optimal quotients and extensions. Quotients obtained by procedure Q of optimal 2{Cayley digraphs are proved to be also optimal. The number of tight extensions, generated by procedure E from a given tight digraph, is characterized. Tight digraphs for which procedure E gives innite tight extensions are also characterized. Finally, these two procedures allow the obtention of new optimal families of 2{Cayley digraphs and also the improvement of the diameter of many proposals in the literature.
Cites background from "Triple-loop networks with arbitrari..."
...Sabariego and Santos [11] gave the algebraic definition of MDD in the general case (for any cardinality of A)....
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References
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TL;DR: In the Groebner package, the most commonly used commands are NormalForm, for doing the division algorithm, and Basis, for computing a Groebners basis as mentioned in this paper. But these commands require a large number of variables.
Abstract: (here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In Maple, a monomial ordering is called a monomial order. The monomial orderings lex, grlex, and grevlex from Chapter 2 are easy to use in Maple. Lex order is called plex (for “pure lexicographic”), grlex order is called grlex, and grevlex order is called tdeg (for “total degree”). Be careful not to confuse tdeg with grlex. Since a monomial order depends also on how the variables are ordered, Maple needs to know both the monomial order you want (plex, grlex or tdeg) and a list of variables. For example, to tell Maple to use lex order with variables x > y > z, you would need to input plex(x,y,z). The Groebner package also knows some elimination orders, as defined in Exercise 5 of Chapter 3, §1. To eliminate the first k variables from x1, . . . , xn, one can use the monomial order lexdeg([x 1,. . .,x k],[x {k+1},. . . ,x n]) (remember that Maple encloses a list inside brackets [. . .]). This order is the elimination order of Bayer and Stillman described in Exercise 6 of Chapter 3, §1. The Maple documentation for the Groebner package also describes how to use certain weighted orders, and we will explain below how matrix orders give us many more monomial orderings. The most commonly used commands in the Groebner package are NormalForm, for doing the division algorithm, and Basis, for computing a Groebner basis. NormalForm has the following syntax:
3,332 citations
Book•
14 Dec 1995TL;DR: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry as mentioned in this paper.
Abstract: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry Some specific Grobner bases Bibliography Index.
1,809 citations
"Triple-loop networks with arbitrari..." refers background or methods in this paper
...Following the terminology used in the theory of toric ideals [15] we call the MDD’s that can be obtained as initial ideals coherent....
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...Mimicking the literature on A-graded ideals and toric Hilbert schemes (see Chapter 10 in [15]) we call MDD’s coherent or non-coherent depending on whether they can be obtained from monomial orderings or not....
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TL;DR: This paper deals with a combinatorial minimization problem arising from studies on multimodule memory organizations with a particular solution proposed and it is demonstrated that it is close to optimum.
Abstract: This paper deals with a combinatorial minimization problem arising from studies on multimodule memory organizations. Instead of searching for an optimum solution, a particular solution is proposed and it is demonstrated that it is close to optimum. Lower bounds for the objective functions are obtained and compared with the corresponding values of the particular solution. The maximum percentage deviation of this solution from optimum is also established.
263 citations
"Triple-loop networks with arbitrari..." refers background in this paper
...It is known [ 18 , 13] that the MDD’s of double-loop networks have a very precise form for which they are called L-shapes (see Figure 1). Aguil´o and Miralles [4] have shown that for each double-loop network there are at most two such L-shapes that are MDD’s for it. We give a new proof of this in Lemma 2.4....
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...Their mathematical study was initiated in [ 18 ], where the problem of finding the network parameters that minimize the diameter (and/or the average distance) for networks of given size and degree was posed....
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TL;DR: This paper presents the solution of the following optimization problem that appears in the design of double-loop structures for local networks and also in data memory, allocation and data alignment in SIMD processors.
Abstract: This paper presents the solution of the following optimization problem that appears in the design of double-loop structures for local networks and also in data memory, allocation and data alignment in SIMD processors.
132 citations
TL;DR: In this paper, a standard basis with respect to a nonstrict order is defined, and the standard basis can be improved to a strict order by breaking any ties with a fixed strict order > 2, which is then a compatible order.
Abstract: Let k be an infinite field of any characteristic, and let S = k[xl9..., xn] be a graded polynomial ring, where each xf has degree one. Let / c S be a homogeneous ideal. Let Sd denote the finite vector space of all homogeneous, degree d polynomials in S, so S = SQ © #! © • • • © Sd © • • • . Writing / in the same manner as / = 70 © /a © • • • ©7rf © • • •, we have Id c Sd for each d. An order > on the monomials of Sd for each d is compatible with the monoid structure on the monomials of S if whenever X > X for two monomials X, X, then XX > XX for all monomials x. We shall only consider orders satisfying this compatibility condition. If an order > is a strict order on the monomials of each degree, one can use > in applying the division algorithm to constructing a standard (Grobner) basis for /. The standard basis for /, and its properties, will vary in a crucial way with the choice of order > . The subject of computing standard or Grobner bases has a long history; see [Bay85] for a recent survey. One can generalize the necessary definitions to nonstrict orders > , which fail to distinguish between all monomials of a given degree: For each polynomial / e S9 define in(/) to be the sum of those terms CX A of / which are greatest with respect to the order > . Define in(7) to be the ideal generated by {in(/)|/ e /}. Define fl9..., fr to be a standard basis for / with respect to the order > if in(/1),..., in(/r) generate the ideal in(7). If > is a strict order, in(J) will be a monomial ideal; if > is not strict, in(/) may fail to be a monomial ideal. A nonstrict order >l can be refined to a strict order by breaking any ties with a fixed strict order >2 ; the resulting order >3 is then a compatible order, so the usual division algorithm can be applied to compute standard bases with respect to >3 . Let ml9in29m3 correspond to >x, >2 , >3 . We shall see that in3(/) = in2(in1(/)), so a standard basis with respect to >3 is already a standard basis with respect to >x. Call >3 the refinement of >x by >2 . Thus, refinements provide a mechanism for computing with nonstrict orders. This has been observed for example in [MoM683], where in the affine setting, homogenizing bases (in the above sense, standard bases with respect to the total degree order) are computed via standard bases with respect to a strict order. We recall two frequently used strict orders: The lexicographic order is defined by X > X if the first nonzero entry in A-B is positive. The reverse lexico-
131 citations