Efficient transformations for Klee's measure problem in the streaming model

Gokarna Sharma, Costas Busch, Ramachandran Vaidyanathan, Suresh Rai, Jerry L. Trahan

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Abstract Given a stream of rectangles over a discrete space, we consider the problem of computing the total number of distinct points covered by the rectangles. This is the discrete version of the two-dimensional Klee's measure problem for streaming inputs. Given 0 < ∈, δ < 1, we provide (∈, δ)-approximations for bounded side length rectangles and for bounded aspect ratio rectangles. For the case of arbitrary rectangles, we provide an O(Formula presented.)-approximation, where U is the total number of discrete points in the two-dimensional space. The time to process each rectangle and the total required space are polylogarithmic in U. The time to answer a query for the total area is constant. We construct efficient transformation techniques that project rectangle areas to one-dimensional ranges and then use a streaming algorithm for the one-dimensional Klee's measure problem to obtain these approximations. The projections are deterministic, and to our knowledge, these are the first approaches of this kind that provide efficiency and accuracy trade-offs in the streaming model.

Original languageEnglish (US)
Article number1396
Pages (from-to)688-702
Number of pages15
JournalComputational Geometry: Theory and Applications
Volume48
Issue number9
DOIs
StatePublished - Oct 1 2015
Externally publishedYes

Keywords

  • Bounded aspect ratio rectangles
  • Bounded side length rectangles
  • Klee's measure problem
  • Streaming model
  • Transformations

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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