4.6 Article

Unpaired Many-to-Many Disjoint Path Covers in Nonbipartite Torus-Like Graphs With Faulty Elements

Journal

IEEE ACCESS
Volume 10, Issue -, Pages 127589-127600

Publisher

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/ACCESS.2022.3226687

Keywords

Disjoint path; path cover; path partition; torus; toroidal grid; interconnection network

Funding

  1. National Research Foundation of Korea (NRF) - Korean Government (MSIT) [2021R1F1A1048180]
  2. Catholic University of Korea
  3. National Research Foundation of Korea [2021R1F1A1048180] Funding Source: Korea Institute of Science & Technology Information (KISTI), National Science & Technology Information Service (NTIS)

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This paper discusses the problem of finding disjoint paths in the underlying graph of an interconnection network in parallel processing. It reveals that constructing a nonbipartite torus-like graph with good disjoint-path-cover properties from lower dimensional torus-like graphs can retain the good property. Consequently, nonbipartite tori with at most f vertex and/or edge faults can have unpaired many-to-many disjoint path covers joining arbitrary disjoint sets.
One of the key problems in parallel processing is finding disjoint paths in the underlying graph of an interconnection network. The disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. Given disjoint source and sink sets, S = {s(1), ... , s(k)} and T = {t(1), ... , t(k)}, in graph G, an unpaired many-to-many k-disjoint path cover joining S and T is a disjoint path cover {P-1, ... , P-k}, in which each path P-i runs from source s(i) to some sink t(j). In this paper, we reveal that a nonbipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties of the unpaired type, retains such a good property. As a result, an m-dimensional nonbipartite torus, m >= 2, with at most f vertex and/or edge faults has an unpaired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each, subject to k >= 1 and f + k <= 2m - 2. The bound of 2m - 2 on f + k is nearly optimal.

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