Efficient processing of 3-sided range queries with probabilistic guarantees

Abstract

This work studies the problem of 2-dimensional searching for the 3-sided range query of the form [a, b] × (−∞, c] in both main and external memory, by considering a variety of input distributions. A dynamic linear main memory solution is proposed, which answers 3-sided queries in O(log n + t) worst case time and scales with O(log log n) expected with high probability update time, under continuous µ-random distributions of the x and y coordinates, where n is the current number of stored points and t is the size of the query output. Our expected update bound constitutes a considerable improvement over the O(log n) update time bound achieved by the classic Priority Search Tree of McCreight [23], as well as over the Fusion Priority Search Tree of Willard [30], which requires O( log n log log n ) time for all operations. Moreover, we externalize this solution, gaining O(log B n + t/B) worst case and O(logBlogn) amortized expected with high probability I/Os for query and update operations respectively, where B is the disk block size. Then, combining the Modified Priority Search Tree [27] with the Priority Search Tree [23], we achieve a query time of O(log log n + t) expected with high probability and an update time of O(log log n) expected with high probability, under the assumption that the x-coordinates are continuously drawn from a smooth distribution and the ycoordinates are continuously drawn from a more restricted class of distributions. The total space is linear. Finally, we externalize this solution, obtaining a dynamic data strucPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ICDT 2010, March 22-25, 2010, Lausanne, Switzerland. Copyright 2010 ture that answers 3-sided queries in O(log B log n + t/B) I/Os expected with high probability, and it can be updated in O(log B log n) I/Os amortized expected with high probability and consumes O(n/B) space, under the same assumptions.

Skyline Queries in O(1) time?

“…The proposed data structure borrows ideas from the Modified Priority Search Tree presented in [5] that supports simple 3-sided range reporting queries. However, the modifications to support skyline queries are novel and non-trivial.…”

“…In the following, we show that under assumptions on the generating distributions of the x and y coordinates of points we prove that during an epoch 3 only O(log N ) violations will happen. We provide a sketch of the structure since it is an easy adaptation of the probabilistic results of [20,5].…”

I/O-Efficient Data Structures for Colored Range and Prefix Reporting

“…[10,24] for the I/Omodel, [22,17] for the pointer machine, [5,9,11,3,4] for the cache-oblivious and [8,21,14] for the word-RAM model. One of the main reason why the problem has seen so much attention stems from the fact that range searching with more than three sides no longer admits linear space data structures with polylogarithmic query cost and a linear term in the output size.…”

Management of historical information and preference in massive datasets

“…This form of query resembles a 3-sided range reporting query with an additional skyline requirement and is handled by the ML-Tree for points drawn from specific distributions in O(log 2 N log log N) expected update time and O (t log log N) query time with high probability. The proposed data structure is inspired from the Modified Priority Search Tree presented in [64] that supports 3-sided range reporting queries. However, the modifications performed to support skyline queries are non-trivial.…”

“…After the end of the epoch, the new structure has no violating points stored explicitly. The following theorem from [64] guarantees the small number of violating points during an epoch: The theorem that describes the result attained in this chapter for 3-sided dynamic skyline queries follows: Theorem 9. Given a set of N 2-d points, whose x coordinates are generated by an (N/(log log N) 1+ , N 1−δ )-smooth distribution, where > 0 and δ ∈ (0, 1) are constants, and the y coordinates are drawn from the restricted class of distributions Y, we can store them in a dynamic main memory data structure that uses O (N) space and supports update operations in O(log 2 N log log N) expected time with high probability.…”