On the minimum total length of interval systems expressing all intervals, and range-restricted queries

  • Authors:

  • Hee-Kap Ahn;Peter Brass;Hyeon-Suk Na;Chan-Su Shin

  • Affiliations:

  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea;Department of Computer Science, City College, New York, USA;School of Computing, Soongsil University, Seoul, South Korea;School of Electrical and Information Engineering, Hankuk University of Foreign Studies, Yongin, South Korea

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2009

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Abstract

In this paper, we study the classical one-dimensional range-searching problem, i.e., expressing any interval {i,...,j}@?{1,...,n} as a disjoint union of at most k intervals in a system of intervals, though with a different lens: we are interested in the minimum total length of the intervals in such a system (and not their number, as is the concern traditionally). We show that the minimum total length of a system of intervals in {1,...,n} that allows to express any interval as a disjoint union of at most k intervals of the system is @Q(n^1^+^2^k) for any fixed k. We also prove that the minimum number of intervals k=k(n,c), for which there exists a system of intervals of total length cn with that property, satisfies k(n,c)=@Q(n^1^c) for any integer c=1. We also discuss the situation when k=@Q(logn).