Jon Louis Bentley

Multidimensional binary search trees used for associative searching

This paper develops the multidimensional binary search tree (or <italic>k</italic>-d tree, where <italic>k</italic> is the dimensionality of the search space) as a data structure for storage of information to be retrieved by associative searches. The <italic>k</italic>-d tree is defined and examples are given. It is shown to be quite efficient in its storage requirements. A significant advantage of this structure is that a single data structure can handle many types of queries very efficiently. Various utility algorithms are developed; their proven average running times in an <italic>n</italic> record file are: insertion, <italic>O</italic>(log <italic>n</italic>); deletion of the root, <italic>O</italic>(<italic>n</italic><supscrpt>(<italic>k</italic>-1)/<italic>k</italic></supscrpt>); deletion of a random node, <italic>O</italic>(log <italic>n</italic>); and optimization (guarantees logarithmic performance of searches), <italic>O</italic>(<italic>n</italic> log <italic>n</italic>). Search algorithms are given for partial match queries with <italic>t</italic> keys specified [proven maximum running time of <italic>O</italic>(<italic>n</italic><supscrpt>(<italic>k</italic>-<italic>t</italic>)/<italic>k</italic></supscrpt>)] and for nearest neighbor queries [empirically observed average running time of <italic>O</italic>(log <italic>n</italic>).] These performances far surpass the best currently known algorithms for these tasks. An algorithm is presented to handle any general intersection query. The main focus of this paper is theoretical. It is felt, however, that <italic>k</italic>-d trees could be quite useful in many applications, and examples of potential uses are given.

An Algorithm for Finding Best Matches in Logarithmic Expected Time

An algorithm and data structure are presented for searching a file containing N records, each described by k real valued keys, for the m closest matches or nearest neighbors to a given query record. The computation required to organize the file is proportional to kNlogN. The expected number of records examined in each search is independent of the file size. The expected computation to perform each search is proportional to logN. Empirical evidence suggests that except for very small files, this algorithm is considerably faster than other methods.

Quad trees a data structure for retrieval on composite keys

SummaryThe quad tree is a data structure appropriate for storing information to be retrieved on composite keys. We discuss the specific case of two-dimensional retrieval, although the structure is easily generalised to arbitrary dimensions. Algorithms are given both for staightforward insertion and for a type of balanced insertion into quad trees. Empirical analyses show that the average time for insertion is logarithmic with the tree size. An algorithm for retrieval within regions is presented along with data from empirical studies which imply that searching is reasonably efficient. We define an optimized tree and present an algorithm to accomplish optimization in n log n time. Searching is guaranteed to be fast in optimized trees. Remaining problems include those of deletion from quad trees and merging of quad trees, which seem to be inherently difficult operations.

Multidimensional divide-and-conquer

Most results in the field of algorithm design are single algorithms that solve single problems. In this paper we discuss multidimensional divide-and-conquer, an algorithmic paradigm that can be instantiated in many different ways to yield a number of algorithms and data structures for multidimensional problems. We use this paradigm to give best-known solutions to such problems as the ECDF, maxima, range searching, closest pair, and all nearest neighbor problems. The contributions of the paper are on two levels. On the first level are the particular algorithms and data structures given by applying the paradigm. On the second level is the more novel contribution of this paper: a detailed study of an algorithmic paradigm that is specific enough to be described precisely yet general enough to solve a wide variety of problems.

Scaling and related techniques for geometry problems

Three techniques in computational geometry are explored: <italic>Scaling</italic> solves a problem by viewing it at increasing levels of numerical precision; <italic>activation</italic> is a restricted type of update operation, useful in sweep algorithms; the <italic>Cartesian tree</italic> is a data structure for problems involving maximums and minimums. These techniques solve the minimum spanning tree problem in R<supscrpt>k</supscrpt><subscrpt>1</subscrpt> and R<supscrpt>k</supscrpt><subscrpt>@@@@</subscrpt> in O(<italic>n</italic>(<italic>lg n</italic>)<supscrpt>r</supscrpt><italic>lg lg n</italic>) time and O(<italic>n</italic>) space, where for R<supscrpt>k</supscrpt><subscrpt>@@@@</subscrpt> and k ≥ 3, r &equil; k−2; for R<supscrpt>k</supscrpt><subscrpt>1</subscrpt>, r &equil; 1, 2, 4 for k &equil; 3, 4, 5 and r &equil; k for k > 5. Other problems solved include R<supscrpt>k</supscrpt><subscrpt>1</subscrpt>and R<supscrpt>k</supscrpt> all nearest neighbors, post office and maximum spanning tree; R<supscrpt>k</supscrpt> maxima, R<supscrpt>k</supscrpt> rectangle searching problems, and Z<supscrpt>k</supscrpt><subscrpt>p</subscrpt> all nearest neighbors (1 ≤ <italic>p</italic> ≤ @@@@).

On the Average Number of Maxima in a Set of Vectors and Applications

A maximal vector of a set ~s one which is not less than any other vector m all components We derive a recurrence relation for computing the average number of maxunal vectors m a set of n vectors m d-space under the assumpUon that all (nl) a relative ordermgs are equally probable. Solving the recurrence shows that the average number of maxmaa is O((ln n) a-~) for fixed d We use this result to construct an algorithm for finding all the maxima that have expected running tmae hnear m n (for sets of vectors drawn under our assumptions) We then use the result to find an upper bound on the expected number of convex hull points m a random point set

Optimal Expected-Time Algorithms for Closest Point Problems

Abstract : Geometric closest-point problems deal with the proximity relationships in k-dimensional point sets. Examples of closest-point problems include building minimum spanning trees, nearest neighbor searching, and triangulation construction. Shamos and Hoey (1975) have shown how the Voronoi diagram can be used to solve a number of planar closest-point problems in optimal worst-case time. In this paper we extend their work by giving optimal expected-time algorithms for solving a number of closest-point problems in k- space, including nearest neighbor searching, finding all nearest neighbors, and computing planar minimum spanning trees. In addition to establishing theoretical bounds, the algorithms in this paper can be implemented to solve practical problems very efficiently.

Decomposable searching problems I. Static-to-dynamic transformation ☆

Abstract Transformations that serve as tools in the design of new data structures are investigated. Specifically, general methods for converting static structures (in which all elements are known before any searches are performed) to dynamic structures (in which insertions of new elements can be mixed with searches) are studied. Three classes of such transformations are exhibited, each based on a different counting scheme for representing the integers, and a combinatorial model is used to show the optimality of many of the transformations. Issues such as online data structures and deletion of elements are also examined. To demonstrate the applicability of these tools, several new data structures that have been developed by applying the transformations are studied.

Efficient worst-case data structures for range searching

In this paper we investigate the worst-case complexity of range searching: preprocess N points in k-space such that range queries can be answered quickly. A range query asks for all points with each coordinate in some range of values, and arises in many problems in statistics and data bases. We develop three different structures for range searching in this paper. The first structure has absolutely optimal query time (which we prove), but has very high preprocessing and storage costs. The second structure we present has logarithmic query time and O(N1+2) preprocessing and storage costs, for any fixed ɛ>0. Finally we give a structure with linear storage, O(N ln N) preprocessing and O(Nɛ) query time.

Divide-and-conquer in multidimensional space

We investigate a divide-and-conquer technique in multidimensional space which decomposes a geometric problem on N points in k dimensions into two problems on N/2 points in k dimensions plus a single problem on N points in k−1 dimension. Special structure of the subproblems is exploited to obtain an algorithm for finding the two closest of N points in 0(N log N) time in any dimension. Related results are discussed, along with some conjectures and unsolved geometric problems.