An e-cient transformation for klee's measure problem in the streaming model

Document Type

Conference Proceeding

Publication Date

12-1-2012

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 can be seen as the discrete version of the two-dimensional Klee's measure problem for streaming inputs. We pro- vide an (ε δ)-Approximation for fat rectangles. For the case of arbitrary rectangles, we provide an O( p log U)- Approximation, where U is the total number of discrete points in the two-dimensional space. The time to pro- cess each rectangle, the total required space, and the time to answer a query for the total area are polylog- Arithmic in U. Our approximations are based on an eficient transformation technique which projects rect- Angle areas to one-dimensional ranges, and then uses a streaming algorithm for the Klee's measure problem in the one-dimensional space. The projection is determin- istic and to our knowledge it is the first approach of this kind which provides eficiency and accuracy trade-offs in the streaming model.

Publication Source (Journal or Book title)

Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG 2012

First Page

83

Last Page

88

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