449 research outputs found

    Long path and cycle decompositions of even hypercubes

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    We consider edge decompositions of the nn-dimensional hypercube QnQ_n into isomorphic copies of a given graph HH. While a number of results are known about decomposing QnQ_n into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if nn is even, β„“<2n\ell < 2^n and β„“\ell divides the number of edges of QnQ_n, then the path of length β„“\ell decomposes QnQ_n. Tapadia et al.\ proved that any path of length 2mn2^mn, where 2m<n2^m<n, satisfying these conditions decomposes QnQ_n. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to 2n+1/n2^{n+1}/n decompose QnQ_n. As a consequence, we show that QnQ_n can be decomposed into copies of any path of length at most 2n/n2^{n}/n dividing the number of edges of QnQ_n, thereby settling Erde's conjecture up to a linear factor

    Decomposing the cube into paths

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    We consider the question of when the nn-dimensional hypercube can be decomposed into paths of length kk. Mollard and Ramras \cite{MR2013} noted that for odd nn it is necessary that kk divides n2nβˆ’1n2^{n-1} and that k≀nk\leq n. Later, Anick and Ramras \cite{AR2013} showed that these two conditions are also sufficient for odd n≀232n \leq 2^{32} and conjectured that this was true for all odd nn. In this note we prove the conjecture.Comment: 7 pages, 2 figure

    Compact Oblivious Routing

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    Oblivious routing is an attractive paradigm for large distributed systems in which centralized control and frequent reconfigurations are infeasible or undesired (e.g., costly). Over the last almost 20 years, much progress has been made in devising oblivious routing schemes that guarantee close to optimal load and also algorithms for constructing such schemes efficiently have been designed. However, a common drawback of existing oblivious routing schemes is that they are not compact: they require large routing tables (of polynomial size), which does not scale. This paper presents the first oblivious routing scheme which guarantees close to optimal load and is compact at the same time - requiring routing tables of polylogarithmic size. Our algorithm maintains the polylogarithmic competitive ratio of existing algorithms, and is hence particularly well-suited for emerging large-scale networks
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