''Efficient Transformations for Klee's Measure Problem in the Streaming Model''

Gokarna Sharma

Costas Busch

Ramachandran Vaidyanathan

Suresh Rai

Jerry L. Trahan

Computational Geometry: Theory and Applications, vol. 48, no. 9, pp. 688-702, 2015


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 epsilon, delta between 0 and 1, we provide (epsilon, delta)-approximations for bounded side length rectangles and for bounded aspect ratio rectangles. For the case of arbitrary rectangles, we provide an O(sqrt(logU))-approximation, where U is the total number of discrete points in the two-dimensional space. The time to process each rectangle, the total required space, and the time to answer a query for the total area are polylogarithmic in U. 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.