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

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( log U )-approximation, where U is the total number of discrete points in the twodimensional 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 onedimensional 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.

A Dyadic Simulation Approach to Efficient Range-Summability

“…We illustrate a DST using a "small universe" example (with U = 16) shown in Figure 1. Sitting at the root of the tree is the S[0, 16), which has distribution X * 16 by initialization. Its two children are the two half-range-sums S[0, 8) and S [8,16) resulting from splitting S[0, 16), its four grandchildren are the four quarter-range-sums S[0, 4), S [4,8), S [8,12) and S [12,16) resulting from splitting S[0, 8) and S [8,16) respectively, and so on.…”

“…Existing solutions to this problem, such as range-efficient sampling [12,18], are sampling-based in the sense they maintain a select subset of sampled data items instead of a sketch (e.g., accumulators like in [8]). The range-efficient F 0 problem has been generalized to high-dimensional spaces, where it is called the Klee's measure problem in computational geometry [20,16]. Existing solutions to Klee's measure problem are also sampling based.…”