DOCSLIB.ORG

Geometric Range Searching*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Range Searching* Geometric range searching Jir Matousek Department of Applied Mathematics Charles University Malostranskenam Praha Czech Republic Abstract In geometric range searching algorithmic problems of the following typ e are considered Given an np oint set P in the plane build a data structure so that given a query triangle R the number of points of P lying in R can b e determined quickly Problems of this typ e are of crucial imp ortance in computational geometry as they can b e used as subroutines in many seemingly unrelated algorithms We presenta survey of results and main techniques in this area Contents Intro duction Geometric range searching Intuition and lower b ounds Algorithms with approximately linear space Algorithms with logarithmic query time Extensions and examples of applications Halfspace range rep orting and other sp ecial situations Dynamization Multilevel data structures Searching with more general ranges Ray sho oting and linear optimization More on applications Last paragraph Part of the work on this pap er was done while the author was visiting the Computer Science Institute Free University Berlin The work has b een supp orted by the GermanIsraeli Foundation of Scientic Research and Development GIF Intro duction This pap er considers an algorithmic problem called range searching We de scrib e the problem and outline current theoretical knowledge ab out it including the main ideas of several pro ofs and constructions Together with the main material we also include various extensions historical notes etc Many of them are somewhat sub jective Several notions and results of computational geometry app earing in our discussion but not directly related to range searching are explained in separate b oxes Table on page summarizes the currentbestknown complexity b ounds for simplex and halfspace range searching Computational geometry general remarks The considered prob lems b elong into the area of computational geometry In the rest of this section we briey intro duce this eld and mention some features and conventions which seem particular to it reader somewhat familiar with computational geometry may probably skip the rest of section safely Older computational geometry monographs of a wide scop e are PS Ede Meh a recent one is Mul Useful survey pap ers can b e found in Pac GPS The sub ject of computational geometry is the design and analysis of ef cient algorithms for computing various prop erties and parameters of nite congurations of geometric ob jects The last sentence may sound rather in comprehensible and the reader can get a better picture from few examples of simple problems studied in computational geometry P Given an np oint set P in the plane or in a Euclidean space of dimension d nd a pair of p oints of P with a smallest distance P Given n segments in the plane compute the numb er of their intersections P Given a convex p olytop e P in ddimensional space sp ecied as an inter section of n halfspaces determine a p ointofP minimizing a given linear function P Given a polygon P in the plane not necessarily convex and two points a b in its interior nd a shortest path from a to b inside P or another version a path from a to b consisting of a minimum p ossible number of segments And so on In computational geometry it is deceptively easy to formu late problems Many problems have practical or theoretical motivations and for almost any problem one can makeup an acceptable lo oking ap plication ex post In the pioneer era of the eld say around the year and few years later it was not to o dicult to nd an unsolved elemen tary and interesting problem Today more complicated and more p owerful metho ds are known and nding a really remarkable new problem b ecame quite hard This is probably the most frequentpreamble of a problem in computational geometry The sub ject of computational geometry is similar to the areas of interest of other elds such as geometric asp ects of combinatorial optimization etc For instance a careful reader might have recognized Problem P as a linear programming problem The approach of computational geometry is usually distinguished by twotypical features The problems are considered in a xed dimension and an innite precision mo del of computation is used Dimension and hidden constants The original computational geometry problems were formulated in the plane or in dimensional space Generaliza tions to an arbitrary dimension are also studied but almost always we imagine that the dimension is quite small say at most while the number of objects p oints segments etc app earing in the problem is large Formally the dimension is usually considered as a constant The eciency of typical computational geometry algorithms decreases with the dimension quite rapidly for example the multiplicative constants in the asymptotic b ounds for the complexity of the algorithms are usually exp onential in the dimension this is also the case for range searching This xed dimension assumption contrasts eg with the theory of linear programming algorithms where the dimension is comparable with the numb er of constraints In theoretical research the eciency of algorithms is compared almost exclusivelyby the asymptotic order of growth of their complexity as a function of the input size One can raise various ob jections against this criterion mainly from the practical p oint of view but no b etter theoretical criterion seems to b e available For simpler algorithms mainly older ones the eectivity expressed in this way agrees with the intuitive notion how fast the algorithm computes and even with the sp eed of actual implementations for real life problems For more complicated algorithms this touch with reality is sometimes lost Sometimes it is even apparent that an asymptotic improvementhas nothing to do with the sp eed of computation for any physically feasible inputs Attempts to improve an algorithm with O n log log n complexity to an O n log n algorithm mightlooklike some p eculiar kind of sp ort In defense of such activities one can say that the actual goal is or at least should be a b etter understanding of the studied problem A discovery of a complicated and quite impractical algorithm with a good asymptotic complexity indicates the p ossibility of an improvement and often a simpler and easily implementable algorithm of a similar complexity is found so on afterwards Mo del of computation rounding problems Algorithms in computa tional geometry are designed for an ideal computer mo del of computation called Real RAM This is an analog of the usual RAM ie of an abstraction of an actual computer programmed in a machine co de However a Real RAM can store an arbitrary real number in one register and p erform arithmetic op era tions with real numb ers in unit time while the usual RAM works with integers whose size is b ounded by a p olynomial in the input size In the sequel we will also use the Real RAM mo del of computation The innite precision computation in the Real RAM mo del is very conve nient theoretically but it often behaves quite dierently than an actual com putation with limited precision Not only that the result of a calculation will b e imprecise a carelessly programmed algorithm may give completely wrong results with a limited precision as the combinatorial information derived from comparisons of imprecise numbers may be erroneous or inconsistent Dicul ties with rounding errors are a very serious p otential obstacle p erhaps the most serious one to successful implementation of geometric algorithms Several promising attempts have app eared at developing program pack ages for a suciently precise and relatively quick arithmetic eg FvW However probably wehave to exp ect that if a more complicated geomet ric algorithm is implemented reliably ie in such a waythat it cannot give an erroneous result b ecause of rounding errors it causes a signicant slowdown compared to a straightforward implementation ignoring these asp ects General p osition assumption In many computational geometry algo rithms one has to deal separately with congurations of input ob jects that are degenerate in some sense for instance when three input p oints lie on a common line Such exceptions complicate both the description and implementation of algorithms Luckily for theoreticians it is known howtoavoid such degenerate cases systematically although at the cost of a slowdown by a constant factor at least in versions published until now Conceptually one p erturbs the input ob jects by innitesimal amounts which brings them into a general p ositions Such metho ds are called simulation of simplicity The idea of innitesimal p er turbations can be realized in various ways see eg EM EC EC and the b est metho d is still b eing sought Simulation of simplicity has also theoretical drawbacks Since we are in eect replacing a given input problem by a dierent problem although an innitesimally close one an algorithm run on this dierent problem may sometimes yield a dierent answer than the correct one for the original problem and it may actually be quite complicated to recover a correct answer to the original problem Of course we may say that the input numbers for a real world problem are inaccurate anyway and so the answer for the perturbed problem is equally appropriate as the one for the original problem However this is not suitable for many applications where degeneracies in the input are not mere coincidences
Load more

Recommended publications
  • Parameter-Free Locality Sensitive Hashing for Spherical Range Reporting˚
    Parameter-free Locality Sensitive Hashing for Spherical Range Reporting˚ Thomas D. Ahle, Martin Aumüller, and Rasmus Pagh IT University of Copenhagen, Denmark, {thdy, maau, pagh}@itu.dk July 21, 2016 Abstract We present a data structure for spherical range reporting on a point set S, i.e., reporting all points in S that lie within radius r of a given query point q (with a small probability of error). Our solution builds upon the Locality-Sensitive Hashing (LSH) framework of Indyk and Motwani, which represents the asymptotically best solutions to near neighbor problems in high dimensions. While traditional LSH data structures have several parameters whose optimal values depend on the distance distribution from q to the points of S (and in particular on the number of points to report), our data structure is essentially parameter-free and only takes as parameter the space the user is willing to allocate. Nevertheless, its expected query time basically matches that of an LSH data structure whose parameters have been optimally chosen for the data and query in question under the given space constraints. In particular, our data structure provides a smooth trade-off between hard queries (typically addressed by standard LSH parameter settings) and easy queries such as those where the number of points to report is a constant fraction of S, or where almost all points in S are far away from the query point. In contrast, known data structures fix LSH parameters based on certain parameters of the input alone. The algorithm has expected query time bounded by Optpn{tqρq, where t is the number of points to report and ρ P p0; 1q depends on the data distribution and the strength of the LSH family used.
    [Show full text]
  • On Range Searching with Semialgebraic Sets*
    Discrete Comput Geom 11:393~,18 (1994) GeometryDiscrete & Computational 1994 Springer-Verlag New York Inc. On Range Searching with Semialgebraic Sets* P. K. Agarwal 1 and J. Matou~ek 2 1 Computer Science Department, Duke University, Durham, NC 27706, USA 2 Katedra Aplikovan6 Matamatiky, Universita Karlova, 118 00 Praka 1, Czech Republic and Institut fiir Informatik, Freie Universit~it Berlin, Arnirnallee 2-6, D-14195 Berlin, Germany matou~ek@cspguk 11.bitnet Abstract. Let P be a set of n points in ~d (where d is a small fixed positive integer), and let F be a collection of subsets of ~d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following F-range searching problem: Given P, build a data structure for efficient answering of queries of the form, "Given a 7 ~ F, count (or report) the points of P lying in 7." Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1- x/b+~) query time, where d < b < 2d - 3 and ~ > 0 is an arbitrarily small constant. The acutal value of b is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of F-range searching problem, including improved ray shooting among triangles in ~3 1. Introduction Let F be a family of subsets of the d-dimensional space ~d (d is a small constant) such that each y e F can be described by some fixed number of real parameters (for example, F can be the set of balls, or the set of all intersections of two ellipsoids, * Part of the work by P.
    [Show full text]
  • New Data Structures for Orthogonal Range Searching
    Alcom-FT Technical Report Series ALCOMFT-TR-01-35 New Data Structures for Orthogonal Range Searching Stephen Alstrup∗ The IT University of Copenhagen Glentevej 67, DK-2400, Denmark [email protected] Gerth Stølting Brodal† BRICS ‡ Department of Computer Science University of Aarhus Ny Munkegade, DK-8000 Arhus˚ C, Denmark [email protected] Theis Rauhe§ The IT University of Copenhagen Glentevej 67, DK-2400, Denmark [email protected] 5th April 2001 Abstract We present new general techniques for static orthogonal range searching problems in two and higher dimensions. For the general range reporting problem in R3, we achieve ε query time O(log n + k) using space O(n log1+ n), where n denotes the number of stored points and k the number of points to be reported. For the range reporting problem on ε an n n grid, we achieve query time O(log log n + k) using space O(n log n). For the two-dimensional× semi-group range sum problem we achieve query time O(log n) using space O(n log n). ∗Partially supported by a grant from Direktør Ib Henriksens Fond. †Partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM- FT). ‡Basic Research in Computer Science, Center of the Danish National Research Foundation. §Partially supported by a grant from Direktør Ib Henriksens Fond. Part of the work was done while the author was at BRICS. 1 1 Introduction Let P be a finite set of points in Rd and Q a query range in Rd. Range searching is the problem of answering various types of queries about the set of points which are contained within the query range, i.e., the point set P Q.
    [Show full text]
  • Range Searching
    Range Searching ² Data structure for a set of objects (points, rectangles, polygons) for efficient range queries. Y Q X ² Depends on type of objects and queries. Consider basic data structures with broad applicability. ² Time-Space tradeoff: the more we preprocess and store, the faster we can solve a query. ² Consider data structures with (nearly) linear space. Subhash Suri UC Santa Barbara Orthogonal Range Searching ² Fix a n-point set P . It has 2n subsets. How many are possible answers to geometric range queries? Y 5 Some impossible rectangular ranges 6 (1,2,3), (1,4), (2,5,6). 1 4 Range (1,5,6) is possible. 3 2 X ² Efficiency comes from the fact that only a small fraction of subsets can be formed. ² Orthogonal range searching deals with point sets and axis-aligned rectangle queries. ² These generalize 1-dimensional sorting and searching, and the data structures are based on compositions of 1-dim structures. Subhash Suri UC Santa Barbara 1-Dimensional Search ² Points in 1D P = fp1; p2; : : : ; png. ² Queries are intervals. 15 71 3 7 9 21 23 25 45 70 72 100 120 ² If the range contains k points, we want to solve the problem in O(log n + k) time. ² Does hashing work? Why not? ² A sorted array achieves this bound. But it doesn’t extend to higher dimensions. ² Instead, we use a balanced binary tree. Subhash Suri UC Santa Barbara Tree Search 15 7 24 3 12 20 27 1 4 9 14 17 22 25 29 1 3 4 7 9 12 14 15 17 20 22 24 25 27 29 31 u v xlo =2 x hi =23 ² Build a balanced binary tree on the sorted list of points (keys).
    [Show full text]
  • Further Results on Colored Range Searching
    28:2 Further Results on Colored Range Searching colored range reporting: report all the distinct colors in the query range. colored “type-1” range counting: find the number of distinct colors in the query range. colored “type-2” range counting: report the number of points of color χ in the query range, for every color χ in the range. In this paper, we focus on colored range reporting and type-2 colored range counting. Note that the output size in both instances is equal to the number k of distinct colors in the range, and we aim for query time bounds that depend linearly on k, of the form O(f(n) + kg(n)). Naively using an uncolored range reporting data structure and looping through all points in the range would be too costly, since the number of points in the range can be significantly larger than k. 1.1 Colored orthogonal range reporting The most basic version of the problem is perhaps colored orthogonal range reporting: report the k distinct colors inside an orthogonal range (an axis-aligned box). It is not difficult to obtain an O(n polylog n)-space data structure with O(k polylog n) query time [22] for any constant dimension d: one approach is to directly modify the d-dimensional range tree [15, 43], and another approach is to reduce colored range reporting to uncolored range emptiness [26] (by building a one-dimensional range tree over the colors and storing a range emptiness structure at each node). Both approaches require O(k polylog n) query time rather than O(polylog n + k) as in traditional (uncolored) orthogonal range searching: the reason is that in the first approach, each color may be discovered polylogarithmically many times, whereas in the second approach, each discovered color costs us O(log n) range emptiness queries, each of which requires polylogarithmic time.
    [Show full text]
  • Orthogonal Range Searching in Moderate Dimensions: K-D Trees and Range Trees Strike Back
    Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back Timothy M. Chan∗ March 20, 2017 Abstract We revisit the orthogonal range searching problem and the exact `1 nearest neighbor search- ing 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(n1+δ) for any constant δ > 0, and the expected query time is n1−1=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 al- gorithms that work only in the case of offline queries [Impagliazzo, Lovett, Paturi, and Schneider (2014) and Chan (SODA'15)]. 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 n2−1=O(log c)-time non-combinatorial algorithm [Abboud, Williams, and Yu (SODA'15)]. 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 [FOCS'98] for approximate `1 nearest neighbor search when d = c log n. 1 Introduction In this paper, we revisit some classical problems in computational geometry: d • In orthogonal range searching, we want to preprocess n data points in R so that we can detect if there is a data point inside any query axis-aligned box, or report or count all such points.
    [Show full text]
  • Dynamic Orthogonal Range Searching on the RAM, Revisited∗
    Dynamic Orthogonal Range Searching on the RAM, Revisited∗ Timothy M. Chan1 and Konstantinos Tsakalidis2 1 Department of Computer Science, University of Illinois at Urbana-Champaign [email protected] 2 Cheriton School of Computer Science, University of Waterloo, Canada [email protected] Abstract We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range log n reporting. We present a new data structure achieving O 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+ε n update time for an arbitrarily small constant ε. In the case of 3-sided queries, our update time reduces to O log1/2+ε n , improving Wilkin- son’s previous bound [ESA 2014] of O log2/3+ε n . 1998 ACM Subject Classification “F.2.2 Nonnumerical Algorithms and Problems” Keywords and phrases dynamic data structures, range searching, computational geometry Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.28 1 Introduction Orthogonal range searching is one of the most well-studied and fundamental problems in computational geometry: the goal is to design a data structure to store a set of n points so that we can quickly report all points inside a query axis-aligned rectangle. In the “emptiness” version of the problem, we just want to decide if the rectangle contains any point.
    [Show full text]
  • Solving Database Queries Using Orthogonal Range Searching
    SOLVING DATABASE QUERIES USING ORTHOGONAL RANGE SEARCHING CPS234: Computational Geometry | Prof. Pankaj Agarwal Prepared by: Khalid Al Issa ([email protected]) Fall 2005 BACKGROUND : RANGE SEARCHING, a database-oriented view Range searching is the side of geometry science in which a set of items is analyzed to determine the subset that belong to a given “range”. In other words, we are given as an input a set of elements with certain properties, and we are also given a “range” of properties. So, our output is expected to be “the set of elements whose properties exist in the given range”. In mathematical notation, we have: INPUT: P : {P0, P1, P2,.. , Pn} a set of elements with given properties (ex. weights..etc) Q : a range of properties (ex. Weights between x and y) OUTPUT: The set of elements that belong in the “intersection” of P and Q. Database queries happen to be a major application in which theories of range searching are applied. Let’s take an example of a database query, and interpret it in geometrical form: Assume we have a database for students’ records, in which we keep every student’s ID, name, number of completed hours, and the cumulative GPA. We now would like to apply the following SQL query to the database: SQL> SELECT ID,NAME FROM STUDENTS WHERE HOURS BETWEEN 75 AND 90 AND GPA BETWEEN 2.5 AND 3.0 Figure.1: Range searching for a 2-D query 1 So this query above will give us the ID and name of every student whose GPA lies in the range [2.5,3.0] AND who has successfully completed hours in the range [75,90].
    [Show full text]
  • Further Results on Colored Range Searching Timothy M
    Further Results on Colored Range Searching Timothy M. Chan Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA [email protected] Qizheng He Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA [email protected] Yakov Nekrich Department of Computer Science, Michigan Technological University, Houghton, MI, USA [email protected] Abstract We present a number of new results about range searching for colored (or “categorical”) data: 1. For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1, . , n}. Previous data structures require log n O( log log n + k) query time. Our result also implies improvements in higher constant dimensions. 2. Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log2 n) query time. 3. For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored “type-2” range counting queries: report the number of occurrences log n of every distinct color in a query orthogonal range. The query time is O( log log n + k log log n), where k is the number of distinct colors in the range.
    [Show full text]
  • Orthogonal Range Searching on the RAM, Revisited
    Orthogonal Range Searching on the RAM, Revisited Timothy M. Chan∗ Kasper Green Larsen† Mihai Pˇatras¸cu‡ March 30, 2011 Abstract We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2-d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves ε the previous results by Alstrup, Brodal, and Rauhe (FOCS’00), with O(n lg n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg2 lg n) query time. Our second data structure uses ε O(n) space and answers queries in O(lg n) time. The best previous O(n)-space data structure, due to Nekrich (WADS’07), answers queries in O(lg n/ lg lg n) time. ε 2. We give a data structure for 3-d orthogonal range reporting with O(n lg1+ n) space and O(lg lg n+ k) query time for points in rank space, for any constant ε> 0. This improves the previous results by Afshani (ESA’08), Karpinski and Nekrich (COCOON’09), and Chan (SODA’11), with O(n lg3 n) ε space and O(lg lg n + k) query time, or with O(n lg1+ n) space and O(lg2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant di- mensions above 3.
    [Show full text]
  • Lecture 8: Orthogonal Range Searching 8.1 Range Searching 8.2
    CPS234 Computational Geometry September 22nd, 2005 Lecture 8: Orthogonal Range Searching Lecturer: Pankaj K. Agarwal Scribe: Mason F. Matthews 8.1 Range Searching The general problem of range searching is as follows: Given input S, a set of n points in Rd, preprocess the data into a data structure (if necessary) and answer a series of queries about the data. Queries include a d-dimensional region r and may be any of the following types: • Emptiness: Does r ∩ S = ∅? The output will be boolean. • Reporting: Output r ∩ S. The output will be a subset of S. • Counting: Compute |r ∩ S|. The output will be an integer. Since solving the reporting problem provides an answer to the other two queries, this will be our focus. In the broadest sense, these regions can have any form, but if it is assumed that region inclusion can be computed in constant time, the naive algorithm of checking every point can be run in O(n) time for each query. In the database community, these problems are known as “indexing” problems, and any preprocessing of the input into a data structure is referred to as “building an index.” 8.2 Orthogonal Range Searching Suppose instead that our queries are limited to d-dimensional axis-aligned rectangles, input as the cross- product of 1-D intervals (i.e. r = [a1, b1] × [a2, b2] × ... × [ad, bd]). Queries of this form are referred to as rectangular range queries, or more commonly in computational geometry as orthogonal range queries. With this restriction in place, the naive query time is still O(n), but now a wide range of data structures can be used to speed the process.
    [Show full text]
  • R-Trees: Introduction
    1. Introduction The paper entitled “The ubiquitous B-tree” by Comer was published in ACM Computing Surveys in 1979 [49]. Actually, the keyword “B-tree” was standing as a generic term for a whole family of variations, namely the B∗-tree, the B+-tree and several other variants [111]. The title of the paper might have seemed provocative at that time. However, it represented a big truth, which is still valid now, because all textbooks on databases or data structures devote a considerable number of pages to explain definitions, characteristics, and basic procedures for searches, inserts, and deletes on B-trees. Moreover, B+-trees are not just a theoretical notion. On the contrary, for years they have been the de facto standard access method in all prototype and commercial relational systems for typical transaction processing applications, although one could observe that some quite more elegant and efficient structures have appeared in the literature. The 1980s were a period of wide acceptance of relational systems in the market, but at the same time it became apparent that the relational model was not adequate to host new emerging applications. Multimedia, CAD/CAM, geo- graphical, medical and scientific applications are just some examples, in which the relational model had been proven to behave poorly. Thus, the object- oriented model and the object-relational model were proposed in the sequel. One of the reasons for the shortcoming of the relational systems was their inability to handle the new kinds of data with B-trees. More specifically, B- trees were designed to handle alphanumeric (i.e., one-dimensional) data, like integers, characters, and strings, where an ordering relation can be defined.
    [Show full text]