Simple and semi-dynamic structures for cache-oblivious planar orthogonal range searching

Abstract

In this paper, we develop improved cache-oblivious data structures for two-and three-sided planar orthogonal range searching. Our main result is an optimal static structure for two-sided range searching that uses linear space and supports queries in O(log B N + T /B) memory transfers, where B is the block size of any level in a multi-level memory hierarchy and T is the number of reported points. Our structure is the first linear-space cache-oblivious structure for a planar range searching problem with the optimal O(log B N + T /B) query bound. The structure is very simple, and we believe it to be of practical interest.We also show that our two-sided range search structure can be constructed cache-obliviously in O(N log B N ) memory transfers. Using the logarithmic method and fractional cascading, this leads to a semi-dynamic linear-space structure that supports two-sided range queries in O(log 2 N + T /B) memory transfers and insertions in O(log 2 N · log B N ) memory transfers amortized. This structure is the first (semi-)dynamic structure for any planar range searching problem with a query bound that is logarithmic in the number of elements in the structure and linear in the output size.Finally, using a simple standard construction, we also obtain a static O(N log 2 N )-space structure for three-sided range searching that supports queries in the optimal bound of O(log B N +T /B) memory transfers. These bounds match the bounds of the best previously known structure for this *

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.…”

Cache-Oblivious Range Reporting with Optimal Queries Requires Superlinear Space

“…The first one, by Agarwal et al [5], works only if log log B is an integer. The second and third data structures, by Arge et al [10] and Arge and Zeh [7], remove this restriction. The data structure by Arge and Zeh is obtained using standard techniques from a linear-space data structure for optimal 2-d dominance reporting queries presented in the same paper.…”

“…The data structure by Arge and Zeh is obtained using standard techniques from a linear-space data structure for optimal 2-d dominance reporting queries presented in the same paper. By applying the same techniques to the O(N log N)-space three-sided range reporting data structures of [5,7,10], it is easy to obtain cache-oblivious 2-d orthogonal range reporting data structures that use O(N log 2 N) space and achieve the optimal query bound of O(log B N + K/B) block transfers. For 3-d dominance reporting, Vengroff and Vitter [30] presented a data structure with a query bound of O((log log log B N) log(N/B) + K/B) block transfers and using O(N log(N/B)) space in the I/O model.…”

Cache-oblivious range reporting with optimal queries requires superlinear space

“…The first one, by Agarwal et al [4], requires that log log B is an integer. The second and third structures, by Arge et al [7] and Arge and Zeh [11], remove this restriction. The structure by Arge and Zeh was obtained by combining standard techniques with a linear-space structure for optimal 2-d dominance queries proposed in the same paper.…”

“…All computation has to happen on data in internal memory. The transfer of data between internal and external memory happens in blocks of B consecutive data [3,4,7,11] 2-d dominance N log B N + K/B cache-oblivious, [11] 3-d dominance N log N + K internal memory, [1,20] N log B N + K/B I/O model, [1] N log N log B N + K/B cache-oblivious, [3] 3-d halfspace N log N + K internal memory, [2] N log * N log B N + K/B I/O model, [2] N log N log B N + K/B cache-oblivious, [3] Table 1: A summary of related work on three-sided range reporting, 2-d and 3-d dominance reporting, and 3-d halfspace range reporting with the optimal query bound.…”