Two-Dimensional Range Diameter Queries

Davoodi, Pooya; Smid, Michiel; Walderveen, Freek van

doi:10.1007/978-3-642-29344-3_19

Cited by 10 publications

(13 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Overall, the combination of existing and new ideas is nontrivial (and interesting, in our opinion). Our conditional lower bound proof for three-dimensional orthogonal RCP is similar to some previous work (for example, Davoodi et al's conditional lower bound for two-dimensional range diameter [8]), and along the way, we introduce a new variant of colored range searching, color uniqueness query, which may be of independent interest.…”

mentioning

confidence: 56%

Range Closest-Pair Search in Higher Dimensions

Chan

Rahul

Xue

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Range closest-pair (RCP) search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S ∩ Q can be reported quickly. RCP search has received attention over years, but the primary focus was only on R 2 . In this paper, we study RCP search in higher dimensions. We give the first nontrivial RCP data structures for orthogonal, simplex, halfspace, and ball queries in R d for any constant d. Furthermore, we prove a conditional lower bound for orthogonal RCP search for d ≥ 3. IntroductionThe closest-pair problem is one of the most fundamental problems in computational geometry and finds numerous applications in various areas, such as collision detection, traffic control, etc. In many scenarios, instead of finding the global closest-pair, people want to know the closest pair contained in some specified ranges. This results in the notion of range closest-pair (RCP) search. RCP search is a range-search variant of the classical closest-pair problem, which aims to store a given set S of points into some space-efficient data structure such that when a query range Q is specified, the closest pair in S ∩ Q can be reported quickly. RCP search has received considerable attention over the years [1,4,9,10,16,17,20,19,21,22].Unlike most traditional range-search problems, RCP search is non-decomposable. That is, if we partition the dataset S into S 1 and S 2 , given a query range Q, the closest pair in S ∩ Q cannot be obtained efficiently from the closest pairs in S 1 ∩ Q and S 2 ∩ Q. Due to the non-decomposability, many traditional range-search techniques are inapplicable to RCP search, which makes the problem quite challenging. As such, despite of much effort made on this topic, most known results are restricted to the plane case, i.e., RCP search in R 2 . Beyond R 2 , only very specific query types have been studied, such as 2-sided box queries.In this paper, we investigate RCP search in higher dimensions. We consider four widely-studied query types: orthogonal queries, simplex queries, halfspace queries, and ball queries. We are interested in designing efficient RCP data structures (in terms of space cost, query time, and preprocessing time) for these kinds of query ranges, and proving conditional lower bounds for these problems.Related work. The closest-pair problem and range search are both well-studied problems in computational geometry; see [2,18] for surveys of these two topics.RCP search was for the first time introduced by Shan et al. [16] and subsequently studied in [1,4,9,10,17,20,19,21,22]. In R 2 , the query types studied include quadrants, strips, rectangles, and halfplanes. RCP search with these query ranges can be solved using near-linear space with poly-logarithmic query time. The best known data structures were given by Xue et al. [21], and we summarize the bounds in Table 1. For fat rectangles queries (i.e., rectangles of constant aspect ratio), Bae and S...

show abstract

mentioning

confidence: 56%

Range Closest-Pair Search in Higher Dimensions

Chan

Rahul

Xue

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…For this range version of CTRQ problem, we are able to show that it is unlikely that a solution exists that simultaneously achieves low space (close to linear) and low query time (polylogarithmic). This is via a reduction from the well-known Set Intersection problem [7]. However, it would still be useful to explore algorithms that are highly efficient in terms of space or query time or demonstrate a good trade-off between the two.…”

Section: Resultsmentioning

confidence: 98%

Range search on tuples of points

Agrawal¹,

Rahul²,

Li³

et al. 2015

Journal of Discrete Algorithms

View full text Add to dashboard Cite

Range search is a fundamental query-retrieval problem, where the goal is to preprocess a given set of points so that the points lying inside a query object (e.g., a rectangle, or a ball, or a halfspace) can be reported efficiently. This paper considers a new version of the range search problem: Several disjoint sets of points are given along with an ordering of the sets. The goal is to preprocess the sets so that for any query point q and a distance δ, all the ordered sequences (i.e., tuples) of points can be reported (one per set and consistent with the given ordering of the sets) such that, for each tuple, the total distance traveled starting from q and visiting the points of the tuple in the specified order is no more than δ. The problem has applications in trip planning, where the objective is to visit venues of multiple types starting from a given location such that the total distance traveled satisfies a distance constraint. Efficient solutions are given for the fixed distance and variable distance versions of this problem, where δ is known beforehand or is specified as part of the query, respectively.

show abstract

“…Specifically, in the so-called cell-probe model without the floor function and where the maximum cardinality of the sets is polylogarithmic in m, any algorithm to answer set intersection queries inÕ(α) time requiresΩ((n/α) 2 ) space, for 1 ≤ α ≤ n [3]. (The "tilde" notation is used to suppress polylogarithmic factors.…”

Section: Hardness Of Range-skyline Count-ingmentioning

confidence: 99%