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International Symposium on Mathematical Foundations of Computer Science

MFCS 1981: Mathematical Foundations of Computer Science 1981 pp 78–93Cite as

Time and space bounded complexity classes and bandwidth constrained problems

A survey

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 118))

Abstract

Bandwidth constraints on familiar natural computational problems are considered. It is seen that generally as the bandwidth of a problem decreases its space complexity decreases. More interestingly, for problems complete for a complexity class \(\mathbb{K}\), often as one decreases the bandwidth one obtains complete problems for space restricted subclasses of \(\mathbb{K}\). For example, (1) the \(\mathbb{N}\)SPACE(log n) complete graph accessibility problem (GAP), when restricted to graphs of bandwidth fk(n), for some k≥1, forms a complete family of problems for \(\mathbb{N}\)SPACE(log f(n)), (2) the \(\mathbb{P}\)complete and/or graph accessibility problem (AGAP), when restricted to graphs of bandwidth f(n), is complete for the simultaneous time-space complexity class \(\mathbb{D}\)TISP(poly,f(n)), (3) the \(\mathbb{N}\)P complete graph problems 3COLOR, SIMPLE MAX CUT, INDEPENDENT SET, VERTEX COVER, DOMINATING SET, and several others, when restricted to graphs of bandwidth f(n) are complete for the simultaneous time-space complexity class \(\mathbb{N}\)TISP(poly,f(n)), and (4) the \(\mathbb{P}\)-Space complete PEBBLE problem, when restricted to graphs of bandwidth f(n), can be solved in space f(n)×log2n. These results are used to show, for example, that:

  1. (1)

    \(\mathbb{N}\)SPACE( f(n)) (\(\mathbb{D}\)SPACE( f(n)×max(f(n),log n) ), for all functions f,

  2. (2)

    The class SC, called Steve's class in honor of Stephen Cook who showed, for example, that all DCFL's are in SC2 = \(\mathbb{D}\)TISP(poly,log2n), is identical to the log space closure of the class of sets accepted by one-way alternating Turing machines within loglog n space. ( Note: SC = Uk≥1 \(\mathbb{D}\)TISP(poly,logkn) )

  3. (3)

    The graph problems 3COLOR, SIMPLE MAX CUT, INDEPENDENT SET, VERTEX COVER, DOMINATING SET, and several other \(\mathbb{N}\)P complete problems, when restricted to graphs of bandwidth f(n), can be solved in polynomial time if and only if \(\mathbb{N}\)TISP(poly,f(n)) ( \(\mathbb{P}\).

The work of this author was partially supported by NSF grant #MCS-79-08919

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik/Informatik, Universität-Gesamthochschule Paderborn, 4790, Paderborn, West Germany

    Burkhard Monien

  2. Department of Electrical Engineering and Computer Science, Northwestern University, 60201, Evanston, Illinois, U.S.A.

    Ivan Hal Sudborough

Authors
  1. Burkhard Monien
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  2. Ivan Hal Sudborough
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Jozef Gruska Michal Chytil

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© 1981 Springer-Verlag Berlin Heidelberg

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Monien, B., Sudborough, I.H. (1981). Time and space bounded complexity classes and bandwidth constrained problems. In: Gruska, J., Chytil, M. (eds) Mathematical Foundations of Computer Science 1981. MFCS 1981. Lecture Notes in Computer Science, vol 118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-10856-4_75

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  • DOI: https://doi.org/10.1007/3-540-10856-4_75

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  • Print ISBN: 978-3-540-10856-6

  • Online ISBN: 978-3-540-38769-5

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