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Procreating tiles of double commutative-step digraphs

Procreating tiles of double commutative-step digraphs Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguiló et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2, …, over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x, y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, …) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ |y − x| ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Procreating tiles of double commutative-step digraphs

Acta Mathematicae Applicatae Sinica , Volume 24 (2) – Aug 6, 2008

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-004-4102-y
Publisher site
See Article on Publisher Site

Abstract

Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguiló et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2, …, over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x, y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, …) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ |y − x| ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Aug 6, 2008

References