Orthogonal Range Search Research Articles

Succinct data structure for path graphs

We consider the problem of designing a succinct data structure for path graphs, that generalizes interval graphs, on n vertices while efficiently supporting degree, adjacency, and neighbourhood queries. We provide the following two solutions for this problem:1.an nlog⁡n+o(nlog⁡n)-bit succinct data structure that supports adjacency query in O(log⁡n) time, neighbourhood query in O(dlog⁡n) time and finally, degree query in min⁡{O(log2⁡n),O(dlog⁡n)} time where d is the degree of the queried vertex.2.an O(nlog2⁡n)-bit space-efficient data structure that supports adjacency, neighborhood, and degree queries optimally. Central to our data structures is the usage of the heavy path decomposition, followed by careful bookkeeping using an orthogonal range search data structure using wavelet trees among others, which may be of independent interest for designing succinct data structures for other graph classes.

Fully Dynamic No-Back-Edge-Traversal Forest via 2D-Range Queries

Orthogonal range search is ubiquitous nowadays, with natural applications in databases, data mining, and text indexing. Very recently, yet another application was discovered, which is to maintain a DFS forest in a dynamic graph. In this paper, we want to extend the above recent study, by applying orthogonal range search to efficient maintenance of a BFS-like forest, called no-back-edge-traversal (NBET) forest, which refers to a spanning forest obtained from a traversal that does not create any back edge. The study of such a problem is motivated by the fact that NBET forest can be used as a strong certificate of 2-connectivity of an undirected graph, which is more general than a spanning forest obtained from a scan-first search traversal.

The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points. Focusing on constant dimension d, we show that any such k-quantifier, d-dimensional problem is solvable in $$O(n^{k-1} \log ^{d-1} n)$$ time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, $$(3k-3)$$ -dimensional problem in this class that requires time $$\Omega (n^{k-1-o(1)})$$ . Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time $$O(n\log ^{d-1} n)$$ , and k-quantifier problems with $$k>3$$ reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination $$\mathsf {VCND}_d$$ (Given three sets of vectors X, Y and Z of dimension d, d and 2d, respectively, is there an $$x \in X$$ and a $$y \in Y$$ so that their concatenation $$x \circ y$$ is not dominated by any $$z \in Z$$ , where vector u is dominated by vector v if $$u_i \le v_i$$ for each coordinate $$1 \le i \le d$$ ), and determine it as the “unique” candidate to be complete for this class (under fine-grained assumptions).

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(ncdot left( {begin{array}{c}k+lceil log nrceil kend{array}}right) cdot 2^k log n), where k is linear in the treewidth of the graph. For every epsilon >0, this bound is n^{1+epsilon }exp O(k), which matches a hardness result of Abboud et al. (in: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2016. https://doi.org/10.1137/1.9781611974331.ch28) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comput Geom 42:815–824, 2009. https://doi.org/10.1016/j.comgeo.2009.02.001) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log ^d n to left( {begin{array}{c}d+lceil log nrceil dend{array}}right), as originally observed by Monier (J Algorithms 1:60–74, 1980. https://doi.org/10.1016/0196-6774(80)90005-X). We also investigate the parameterization by vertex cover number.

Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back

We revisit the orthogonal range searching problem and the exact \(\ell _\infty \) nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of \(\log n\). Specifically, the preprocessing time and space are \(O(n^{1+\delta })\) for any constant \(\delta >0\), and the expected query time is \(n^{1-1/O(c\log c)}\) for \(d=c\log n\). The data structure is simple and is based on a new “augmented, randomized, lopsided” variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [(Impagliazzo et al. in arXiv:1401.5512, 2014) and (Chan in Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication. In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an \(n^{2-1/O(\log c)}\)-time non-combinatorial algorithm [Abboud et al. in Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015]. This reduction is also simple and is based on range trees. Finally, we use a similar approach to obtain a small improvement to Indyk’s data structure (J Comput Syst Sci 63(4):627–638, 2001) for approximate\(\ell _\infty \) nearest neighbor search when \(d=c\log n\).

Dynamic Orthogonal Range Searching on the RAM, Revisited

© Timothy M. Chan and Konstantinos Tsakalidis. We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n/log log n+k) optimal query time and O(log2/3+o(1)n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen's previous result [SIAM J. Comput., 2006], which has O (log7/8/ϵn) update time for an arbitrarily small constant ϵ. In the case of 3-sided queries, our update time reduces to O (log1/2+ϵn), improving Wilkinson's previous bound [ESA 2014] of O(log2/3+ϵ).

A fault tolerant peer-to-peer spatial data structure

In recent years, many layered indexing techniques over distributed hash table (DHT)-based peer-to-peer (P2P) systems have been proposed to realize distributed range search. In this paper, we present a fault tolerant constant degree dynamic Distributed Spatial Data Structure called DSDS that supports orthogonal range search on a set of N d-dimensional points published on n nodes. We describe a total order binary relation algorithm to publish points among supernodes and determine supernode keys. A non-redundant rainbow skip graph is used to coordinate message passing among nodes. The worst case orthogonal range search cost in a d-dimensional DSDS with n nodes is \(O\left (\log n+m+\frac {K}{B}\right )\) messages, where m is the number of nodes intersecting the query, K is the number of points reported in range, and B is the number of points that can fit in one message. A complete backup copy of data points stored in other nodes provides redundancy for our DSDS. This redundancy permits answering a range search query in the case of failure of a single node. For single node failure, the DSDS routing system can be recovered to a fully functional state at a cost of O(log n) messages. Backup sets in DSDS nodes are used to first process a query in the most efficient dimension, and then used to process a query containing the data in a failed node in d-dimensional space. The DSDS search algorithm can process queries in d-dimensional space and still tolerate failure of one node. Search cost in the worst case with a failed node increases to \(O\left (d\log n+dm+\frac {K}{B}\right )\) messages for d dimensions.

Querying simplexes in quasi-triangulation

Given a quasi-triangulation, the dual structure of the Voronoi diagram of a molecule, querying its simplexes of a particular bounding state for the spherical probe of a given radius occurs frequently. The β -complex and the β -shape are such examples with various applications for reasoning the spatial structure of molecules. While such simplexes can be found by linearly scanning all simplexes in the quasi-triangulation, it is desirable to do the query faster. This paper introduces the Q2P-transformation that transforms the simplex query problem in the three-dimensional space to the orthogonal range search problem of points in another three-dimensional space. Based on this observation, we show that the k d -tree data structure suits very well for developing efficient query algorithms for the quasi-triangulation of molecules using the set of sixteen primitive and the set of ten high-level query operators. In particular, the proposed method shows an extremely efficient performance for incremental queries. A subset of the presented observations facilitates efficient simplex queries for any simplicial complexes including the Delaunay and regular triangulations.

COMPUTING MULTISCALE ENTROPY WITH ORTHOGONAL RANGE SEARCH

Multi-scale entropy (MSE) is a measurement of a system's complexity. It has received a great deal of attention in recent years, and its effectiveness has been verified, and applied in a number of different fields. However, the algorithms proposed in past studies required O (N^2), which represented a degree of execution time insufficient for on-line applications, or for applications with long term correlations. In this study, we showed that the probability function in the entropy term could be transformed into an orthogonal range search in the field of computational geometry. We then developed an efficient new algorithm for computing multi-scale entropy. The execution time in the results of our experiments with electrocardiogram (ECG), electroencephalography (EEG), interbeat interval (RR), and mechanical and ecological signals showed a significant improvement from 10 to 70 times over that of conventional methods for N=80, 000. Because the execution time has been significant reduced, the new algorithm could be applied to online diagnosis, in the computation of MSE for long-term correlation of signal.

Untangled monotonic chains and adaptive range search

We present the first adaptive data structure for two-dimensional orthogonal range search. Our data structure is adaptive in the sense that it gives improved search performance for data that is better than the worst case (Demaine et al., 2000) [8]; in this case, data with more inherent sortedness. Given n points on the plane, the linear space data structure can answer range queries in O ( log n + k + m ) time, where m is the number of points in the output and k is the minimum number of monotonic chains into which the point set can be decomposed, which is O ( n ) in the worst case. Our result matches the worst-case performance of other optimal-time linear space data structures, or surpasses them when k = o ( n ) . Our data structure can be made implicit, requiring no extra space beyond that of the data points themselves (Munro and Suwanda, 1980) [16], in which case the query time becomes O ( k log n + m ) . We also present a novel algorithm of independent interest to decompose a point set into a minimum number of untangled, similarly directed monotonic chains in O ( k 2 n + n log n ) time.

On k-d Range Search with Patricia Tries

Patricia tries are explored for indexing combined text and spatial data. A combined text and spatial data range search algorithm is presented for reporting all data from a set of size n intersecting a query hyperrectangle. We also use Patricia tries to answer $\epsilon$-approximate orthogonal range search on a set of n random points and hyperrectangles in k-dimensional data space. $\epsilon$-approximate orthogonal range counting queries can be answered in $O(k {\rm log} n \epsilon^{k-1})$ time, and the number of nodes visited for orthogonal range counting queries is shown to be $O({\rm log} n + k(1+2n^{1/k}\Delta)^{k-1})$ for cubical range of side length $\Delta$. Patricia tries are evaluated experimentally for both orthogonal range search and $\epsilon$-approximate orthogonal range search (for $2 \leq k \leq 14$ and n up to 1,000,000) using uniformly distributed random data. The expected range search time is determined theoretically and found to agree with experimental results.

Improving the performance of multidimensional search using fingers

We propose two variants of K -d trees where fingers are used to improve the performance of orthogonal range search and nearest neighbor queries when they exhibit locality of reference. The experiments show that the second alternative yields significant savings. Although it yields more modest improvements, the first variant does it with much less memory requirements and great simplicity, which makes it more attractive on practical grounds.

Personalized Trust Management in Decision Making: A Dynamic Clustering Approach

This paper presents a personalized approach for distributed trust management by employing the k-means range algorithm, a combination of the partitional k-means clustering algorithm with orthogonal range search concepts. The aim of this approach is to aid the human or computer agent in organizing information from multiple sources into clusters according to its “trust features”. Thus the agent can perform complicated trust assessments in real-time situations and cooperate with decision-making software to assist in purchasing activities. We conclude by discussing the implications and advantages of this approach in trust management in traditional and mobile e-commerce applications.

On the average performance of orthogonal range search in multidimensional data structures

In this work we present the average-case analysis of orthogonal range search for several multidimensional data structures. We first consider random relaxed K-d trees as a prototypical example. Later we extend these results to many different multidimensional data structures. We show that the performance of range searches is related to the performance of a variant of partial matches using a mixture of geometric and combinatorial arguments. This reduction simplifies the analysis and allows us to give exact upper and lower bounds for the performance of range searches (Theorems 3 and 4) and a useful characterization of the cost of range search as a sum of the costs of partial match-like operations (Theorem 5). Using these results, we can get very precise asymptotic estimates for the expected cost of range searches (Theorem 6).

Average case analysis of dynamic geometric optimization

We maintain the maximum spanning tree of a planar point set, as points are inserted or deleted, in O(log 3 n) expected time per update in Mulmuley's average-case model of dynamic geometric computation. We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest and the rotating caliper graph related to an algorithm for static computation of point set widths and diameters. We maintain the former graph in expected time O(log 2 n) per update and the latter in expected time O(log n) per update. We also use the rotating caliper graph to maintain the diameter, width, and minimum enclosing rectangle of a point set in expected time O(log n) per update. A subproblem uses a technique for average-case orthogonal range search that may also be of interest.

Fractional cascading: II. Applications

This paper presents several applications offractional cascading, a new searching technique which has been described in a companion paper. The applications center around a variety of geometric query problems. Examples include intersecting a polygonal path with a line, slanted range search, orthogonal range search, computing locus functions, and others. Some results on the optimality of fractional cascading, and certain extensions of the technique for retrieving additional information are also included.