Using alternating de Bruijn sequences to construct de Bruijn tori

A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While some methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants de Bruijn sequences called `Alternating de Bruijn sequences' and `De Bruijn families'.Comment: 21 pages, comments welcom

Fault-tolerant meshes with minimal numbers of spares

This paper presents several techniques for adding fault-tolerance to distributed memory parallel computers. More formally, given a target graph with n nodes, we create a fault-tolerant graph with n + k nodes such that given any set of k or fewer faulty nodes, the remaining graph is guaranteed to contain the target graph as a fault-free subgraph. As a result, any algorithm designed for the target graph will run with no slowdown in the presence of k or fewer node faults, regardless of their distribution. We present fault-tolerant graphs for target graphs which are 2-dimensional meshes, tori, eight-connected meshes and hexagonal meshes. In all cases our fault-tolerant graphs have smaller degree than any previously known graphs with the same properties

The k-center Problem for Classes of Cyclic Words

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique

Growing perfect cubes

AbstractAn (n,a,b)-perfect double cube is a b×b×b sized n-ary periodic array containing all possible a×a×a sized n-ary array exactly once as subarray. A growing cube is an array whose cj×cj×cj sized prefix is an (nj,a,cj)-perfect double cube for j=1,2,…, where cj=njv/3,v=a3 and n1<n2<⋯. We construct the smallest possible perfect double cube (a 256×256×256 sized 8-ary array) and growing cubes for any a

On the Existence of de Bruijn Tori with Two by Two Windows

Necessary and sucient conditions for the existence of de Bruijn Tori (or Perfect Maps) with two by two windows over any alphabet are given. This is the rst two-dimensional window size for which the existence question has been completely answered for every alphabet. The techniques used to construct these arrays utilise existing results on Perfect Factors and Perfect Multi-Factors in one and two dimensions and involve new results on Perfect Factors with `puncturing capabilities&apos;. Finally, the existence question for two-dimensional Perfect Factors is considered and is settled for two by two windows and alphabets of prime-power size. 2

A powerful heuristic for telephone gossiping

A refined heuristic for computing schedules for gossiping in the telephone model is presented. The heuristic is fast: for a network with n nodes and m edges, requiring R rounds for gossiping, the running time is O(R n log(n) m) for all tested classes of graphs. This moderate time consumption allows to compute gossiping schedules for networks with more than 10,000 PUs and 100,000 connections. The heuristic is good: in practice the computed schedules never exceed the optimum by more than a few rounds. The heuristic is versatile: it can also be used for broadcasting and more general information dispersion patterns. It can handle both the unit-cost and the linear-cost model. Actually, the heuristic is so good, that for CCC, shuffle-exchange, butterfly de Bruijn, star and pancake networks the constructed gossiping schedules are better than the best theoretically derived ones. For example, for gossiping on a shuffle-exchange network with 2^{13} PUs, the former upper bound was 49 rounds, while our heuristic finds a schedule requiring 31 rounds. Also for broadcasting the heuristic improves on many formerly known results. A second heuristic, works even better for CCC, butterfly, star and pancake networks. For example, with this heuristic we found that gossiping on a pancake network with 7! PUs can be performed in 15 rounds, 2 fewer than achieved by the best theoretical construction. This second heuristic is less versatile than the first, but by refined search techniques it can tackle even larger problems, the main limitation being the storage capacity. Another advantage is that the constructed schedules can be represented concisely