Efficient processing of aggregated range queries on two-dimensional grids is a common requirement in information retrieval and data mining systems, for example in Geographic Information Systems and OLAP cubes. We introduce a technique to represent grids supporting aggregated range queries that requires little space when the data points in the grid are clustered, which is common in practice. We show how this general technique can be used to support two important types of aggregated queries, which are ranked range queries and counting range queries. Our experimental evaluation shows that this technique can speed up aggregated queries up to more than an order of magnitude, with a small space overhead. HighlightsSpace-efficient representation for two-dimensional grids.Efficient support for aggregated range queries.Proved performance in main memory.Results competitive with the state of the art.Applications to several domains: Geographic Information Systems, OLAP cubes, etc.
A better result, using multi-ary wavelet trees, was introduced by @cite_15 . They match the optimal @math time using just @math bits on an @math grid. @cite_23 extended the results to @math grids. This query time is optimal within space @math @cite_0 .
@cite_0 @cite_15 @cite_23
2065240689 1495731517 2116258248
Proving lower bounds for range queries has been an active topic of research since the late 70s, but so far nearly all results have been limited to the (rather restrictive) semigroup model. We consider one of the most basic range problem, orthogonal range counting in two dimensions, and show almost optimal bounds in the group model and the (holy grail) cell-probe model. Specifically, we show the following bounds, which were known in the semigroup model, but are major improvements in the more general models:* In the group and cell-probe models, a static data structure of size n lgO(1) n requires Omega(lg n lglg n) time per query. This is an exponential improvement over previous bounds, and matches known upper bounds.* In the group model, a dynamic data structure takes time Omega((lg n lglg n)2) per operation. This is close to the O(lg2 n) upper bound, where as the previous lower bound was Omega(lg n). Proving such (static and dynamic) bounds in the group model has been regarded as an important challenge at least since [Fredman, JACM 1982] and [Chazelle, FOCS 1986]. We present a succinct representation of a set of n points on an n ×n grid using @math bits to support orthogonal range counting in @math time, and range reporting in @math time, where k is the size of the output. This achieves an improvement on query time by a factor of @math upon the previous result of Makinen and Navarro [1], while using essentially the information-theoretic minimum space. Our data structure not only can be used as a key component in solutions to the general orthogonal range search problem to save storage cost, but also has applications in text indexing. In particular, we apply it to improve two previous space-efficient text indexes that support substring search [2] and position-restricted substring search [1]. We also use it to extend previous results on succinct representations of sequences of small integers, and to design succinct data structures supporting certain types of orthogonal range query in the plane. Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify many of those operations arising in applications and generalize them into a wide set of desirable queries for a binary relation representation. We also identify reductions among those operations. We then introduce several novel binary relation representations, some simple and some quite sophisticated, that not only are space-efficient but also efficiently support a large subset of the desired queries.