Hybrid LSH: Faster Near Neighbors Reporting in High-dimensional Space

We study the $r$-near neighbors reporting problem ($r$-NN), i.e., reporting \emph{all} points in a high-dimensional point set $S$ that lie within a radius $r$ of a given query point $q$. Our approach builds upon on the locality-sensitive hashing (LSH) framework due to its appealing asymptotic sublinear query time for near neighbor search problems in high-dimensional space. A bottleneck of the traditional LSH scheme for solving $r$-NN is that its performance is sensitive to data and query-dependent parameters. On datasets whose data distributions have diverse local density patterns, LSH with inappropriate tuning parameters can sometimes be outperformed by a simple linear search. In this paper, we introduce a hybrid search strategy between LSH-based search and linear search for $r$-NN in high-dimensional space. By integrating an auxiliary data structure into LSH hash tables, we can efficiently estimate the computational cost of LSH-based search for a given query regardless of the data distribution. This means that we are able to choose the appropriate search strategy between LSH-based search and linear search to achieve better performance. Moreover, the integrated data structure is time efficient and fits well with many recent state-of-the-art LSH-based approaches. Our experiments on real-world datasets show that the hybrid search approach outperforms (or is comparable to) both LSH-based search and linear search for a wide range of search radii and data distributions in high-dimensional space.Comment: Accepted as a short paper in EDBT 201

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

Optimal Color Range Reporting in One Dimension

Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a \emph{color}. While the answer to an orthogonal point reporting query contains all points in the query range $Q$, the answer to a color reporting query contains only distinct colors of points in $Q$. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal $O(k+1)$ time, where $k$ is the number of colors in the answer and $N$ is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model