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Zhi-Zhong CHEN, Takumi KASAI, "Parallel Algorithms for the Maximal Tree Cover Problems" in IEICE TRANSACTIONS on Information,
vol. E75-D, no. 1, pp. 30-34, January 1992, doi: .
Abstract: A maximal l-diameter tree cover of a graph G(V,E) is a spanning subgraph C(V,EC) of G such that each connected component of C is a tree, C contains no path with more than l edges, and adding any edge in EEC to C yields either a path of length l1 or a cycle. For every function f from positive integers to positive integers, the maximal f-diameter tree cover prolem (MDTC(f) problem for short) is to find a maximal f(n)-diameter tree cover of G, given an n-node graph G. In this paper, we give two parallel algorithms for the MDTC(f) problem. The first algorithm can be implemented in time O(TMSP(n,f(n))log2n) using polynomial number of processors on an EREW PRAM, where TMSP(n,f(n) is the time needed to find a maximal set of vertex disjoint paths of length f(n) in a given n-node graph using polynomial number of processors on an EREW PRAM. We then show that if suitable restrictions are imposed on the input graph and/or on the magnitude of f, then TMSP(n,f(n))O(logkn) for some constant k and thus, for such cases, we obtain an NC algorithm for the MDTC(f) problem. The second algorithm runs in time O(n log2n/{f(n)1}) using polynomial number of processors on an EREW PRAM. Thus if f(n)Ω(n/logkn) for some kO, we obtain an NC algorithm for the MDTC(f) problem.
URL: https://global.ieice.org/en_transactions/information/10.1587/e75-d_1_30/_p
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@ARTICLE{e75-d_1_30,
author={Zhi-Zhong CHEN, Takumi KASAI, },
journal={IEICE TRANSACTIONS on Information},
title={Parallel Algorithms for the Maximal Tree Cover Problems},
year={1992},
volume={E75-D},
number={1},
pages={30-34},
abstract={A maximal l-diameter tree cover of a graph G(V,E) is a spanning subgraph C(V,EC) of G such that each connected component of C is a tree, C contains no path with more than l edges, and adding any edge in EEC to C yields either a path of length l1 or a cycle. For every function f from positive integers to positive integers, the maximal f-diameter tree cover prolem (MDTC(f) problem for short) is to find a maximal f(n)-diameter tree cover of G, given an n-node graph G. In this paper, we give two parallel algorithms for the MDTC(f) problem. The first algorithm can be implemented in time O(TMSP(n,f(n))log2n) using polynomial number of processors on an EREW PRAM, where TMSP(n,f(n) is the time needed to find a maximal set of vertex disjoint paths of length f(n) in a given n-node graph using polynomial number of processors on an EREW PRAM. We then show that if suitable restrictions are imposed on the input graph and/or on the magnitude of f, then TMSP(n,f(n))O(logkn) for some constant k and thus, for such cases, we obtain an NC algorithm for the MDTC(f) problem. The second algorithm runs in time O(n log2n/{f(n)1}) using polynomial number of processors on an EREW PRAM. Thus if f(n)Ω(n/logkn) for some kO, we obtain an NC algorithm for the MDTC(f) problem.},
keywords={},
doi={},
ISSN={},
month={January},}
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TY - JOUR
TI - Parallel Algorithms for the Maximal Tree Cover Problems
T2 - IEICE TRANSACTIONS on Information
SP - 30
EP - 34
AU - Zhi-Zhong CHEN
AU - Takumi KASAI
PY - 1992
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E75-D
IS - 1
JA - IEICE TRANSACTIONS on Information
Y1 - January 1992
AB - A maximal l-diameter tree cover of a graph G(V,E) is a spanning subgraph C(V,EC) of G such that each connected component of C is a tree, C contains no path with more than l edges, and adding any edge in EEC to C yields either a path of length l1 or a cycle. For every function f from positive integers to positive integers, the maximal f-diameter tree cover prolem (MDTC(f) problem for short) is to find a maximal f(n)-diameter tree cover of G, given an n-node graph G. In this paper, we give two parallel algorithms for the MDTC(f) problem. The first algorithm can be implemented in time O(TMSP(n,f(n))log2n) using polynomial number of processors on an EREW PRAM, where TMSP(n,f(n) is the time needed to find a maximal set of vertex disjoint paths of length f(n) in a given n-node graph using polynomial number of processors on an EREW PRAM. We then show that if suitable restrictions are imposed on the input graph and/or on the magnitude of f, then TMSP(n,f(n))O(logkn) for some constant k and thus, for such cases, we obtain an NC algorithm for the MDTC(f) problem. The second algorithm runs in time O(n log2n/{f(n)1}) using polynomial number of processors on an EREW PRAM. Thus if f(n)Ω(n/logkn) for some kO, we obtain an NC algorithm for the MDTC(f) problem.
ER -
English
