Rectangle-efficient aggregation in spatial data streams

Abstract

We consider the estimation of aggregates over a data stream of multidimensional axis-aligned rectangles. Rectangles are a basic primitive object in spatial databases, and efficient aggregation of rectangles is a fundamental task. The data stream model has emerged as a de facto model for processing massive databases in which the data resides in external memory or the cloud and is streamed through main memory. For a point p, let n(p) denote the sum of the weights of all rectangles in the stream that contain p. We give nearoptimal solutions for basic problems, including (1) the k-th frequency moment F k = points p |n(p)| k , (2) the counting version of stabbing queries, which seeks an estimate of n(p) given p, and (3) identification of heavy-hitters, i.e., points p for which n(p) is large. An important special case of F k is F0, which corresponds to the volume of the union of the rectangles. This is a celebrated problem in computational geometry known as "Klee's measure problem", and our work yields the first solution in the streaming model for dimensions greater than one.

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

“…Similarly, as the algorithm of Tirthapura and Woodruff [13] has space as well as processing time per rectangle d log( mU…”

“…We call these as bounded aspect ratio rectangles. This includes the special case of "fat rectangles" (that have constant aspect ratios) [13,22]. We prove the following theorem for our algorithm for this special case of χ = 1 (refer Section 7.1.1 for the Side_Length_Map transformation proof and Section 8.1.1 for the Aspect_Ratio_Map transformation proof).…”

“…Bringmann and Friedrich [17] gave a (non-streaming) ( , δ)-approximation algorithm for KMP that runs in O m 2 log 1 δ time and has (m) words workspace, for any 0 < < 1 and δ ≥ 3/4. Recently, Tirthapura and Woodruff [13] gave a (streaming) randomized ( , δ)-approximation algorithm for KMP_SD with workspace and processing time per rectangle upper bounded by 1 …”

Sketching Earth-Mover Distance on Graph Metrics

“…By combining recent results on rangesummable random variables by Tirthapura and Woodruff [16] with a natural path-decomposition, we show how such a sketch can be applied in the datastream setting with O(polylog n) update time whereas, even in the cycle case, the existing sketch has Ω(n) update time. See Section 4.…”

“…However, this can be reduced to O(polylog n) time using the range-efficient 1 sketching algorithm of Tirthapura and Woodruff [16]. This allows contiguous segments of the vector z to be updated in O(polylog n) time rather than O(w polylog n) time where w is the length of the segment.…”