Results 1  10
of
14
Efficient Data Structures for Range Searching on a Grid
, 1987
"... We consider the 2dimensional range searching problem in the case where all point lie on an integer grid. A new data structure is preented that solves range queries on a U U grid in O(k + loglog U) time using O(n log n) storage, where n is the number of points and k the number of reported answers ..."
Abstract

Cited by 36 (0 self)
 Add to MetaCart
We consider the 2dimensional range searching problem in the case where all point lie on an integer grid. A new data structure is preented that solves range queries on a U U grid in O(k + loglog U) time using O(n log n) storage, where n is the number of points and k the number of reported answers. Although the query
Static Dictionaries on AC^0 RAMs: Query time Θ(,/log n / log log n) is necessary and sufficient
, 1996
"... In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©���������������� ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
In this paper we consider solutions to the static dictionary problem ���� � on RAMs, i.e. random access machines where the only restriction on the finite instruction set is that all computational instructions are ���� � in. Our main result is a tight upper and lower bound ���� � ���©��������������������� of on the time for answering membership queries in a set of � size when reasonable space is used for the data structure storing the set; the upper bound can be obtained using space ������ � �� � ���� �. Several variations of this result are also obtained. Among others, we show a tradeoff between time and circuit depth under the unitcost assumption: any RAM instruction set which permits a linear space, constant query time solution to the static dictionary problem must have an instruction of depth �������©���������������©���� � , where � is the word size of the machine (and ���© � the size of the universe). This matches the depth of multiplication and integer division, used in the perfect hashing scheme by Fredman, Komlós and Szemerédi.
Bounded Ordered Dictionaries in O(log log N) Time and O(n) Space
 Information Processing Letters
, 1990
"... In this paper we show how to implement bounded ordered dictionaries, also called bounded priority queues, in O(log log N) time per operation and O(n) space. Here n denotes the number of elements stored in the dictionary and N denotes the size of the universe. Previously, this time bound required O(N ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
In this paper we show how to implement bounded ordered dictionaries, also called bounded priority queues, in O(log log N) time per operation and O(n) space. Here n denotes the number of elements stored in the dictionary and N denotes the size of the universe. Previously, this time bound required O(N) space [E77].
Computational Geometry on a Grid  An Overview
, 1987
"... In this paper an overview is given of a number of algorithms solving problems in computational geometry on a grid, i.e., in the case where objects have integer coordinates in some bounded universe. The emphasis is on simple, yet efficient solutions. Especially problems with some relevance to comp ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
In this paper an overview is given of a number of algorithms solving problems in computational geometry on a grid, i.e., in the case where objects have integer coordinates in some bounded universe. The emphasis is on simple, yet efficient solutions. Especially problems with some relevance to computer graphics are studied. Simple and
Delaunay Triangulations in O(sort(n)) Time and More
"... We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We present several results about Delaunay triangulations (DTs) and convex hulls in transdichotomous and hereditary settings: (i) the DT of a planar point set can be computed in expected time O(sort(n)) on a word RAM, where sort(n) is the time to sort n numbers. We assume that the word RAM supports the shuffleoperation in constant time; (ii) if we know the ordering of a planar point set in x and in ydirection, its DT can be found by a randomized algebraic computation tree of expected linear depth; (iii) given a universe U of points in the plane, we construct a data structure D for Delaunay queries: for any P ⊆ U, D can find the DT of P in time O(P  log log U); (iv) given a universe U of points in 3space in general convex position, there is a data structure D for convex hull queries: for any P ⊆ U, D can find the convex hull of P in time O(P (log log U) 2); (v) given a convex polytope in 3space with n vertices which are colored with χ> 2 colors, we can split it into the convex hulls of the individual color classes in time O(n(log log n) 2). The results (i)–(iii) generalize to higher dimensions. We need a wide range of techniques. Most prominently, we describe a reduction from DTs to nearestneighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling.
Geometric searching over the rationals
 In Proceedings of the 7th European Symposium on Algorithms, Lecture Notes in Computer Science
, 1999
"... We revisit classical geometric search problems under the assumption of rational coordinates. Our main result is a tight bound for point separation, ie, to determine whether n given points lie on one side of a query line. We show that with polynomial storage the query time is Θ(logb / log log b), whe ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We revisit classical geometric search problems under the assumption of rational coordinates. Our main result is a tight bound for point separation, ie, to determine whether n given points lie on one side of a query line. We show that with polynomial storage the query time is Θ(logb / log log b), where b is the bit length of the rationals used in specifying the line and the points. The lower bound holds in Yao’s cell probe model with storage in n O(1) and word size in b O(1). By duality, this provides a tight lower bound on the complexity on the polygon point enclosure problem: given a polygon in the plane, is a query point in it? 1
Neighbours on a Grid
, 1996
"... . We address the problem of a succinct static data structure representing points on an M \Theta M grid (M = 2 m where m is size of a word) that permits to answer the question of finding the closest point to a query point under the L1 or L1 norm in constant time. Our data structure takes essential ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
. We address the problem of a succinct static data structure representing points on an M \Theta M grid (M = 2 m where m is size of a word) that permits to answer the question of finding the closest point to a query point under the L1 or L1 norm in constant time. Our data structure takes essentially minimum space. These results are extended to d dimensions under L1 . 1 Introduction Given a set of points, a query point, and a distance metric, the closest neighbour problem is that of determining the point of the set whose distance from the query point is minimal. If the query point is a member of the given set then it will be the solution, and if two or more points are of equal distance from the query point we choose one of them arbitrarily. This problem arises in many areas such as modeling of robot arm movements and integrated circuits layouts (cf. [26]). The problem has been heavily studied in the IR 2 contiguous domain where it is solved using Voronoi diagrams (cf. [29]). Furthe...
On the Longest Upsequence Problem for Permutations
, 1999
"... Given a permutation of n numbers, its longest upsequence can be found in time O(n log log n). Finding the longest upsequence (resp. longest downsequence) of a permutation solves the maximum independent set problem (resp. the clique problem) for the corresponding permutation graph. Moreover, we discu ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Given a permutation of n numbers, its longest upsequence can be found in time O(n log log n). Finding the longest upsequence (resp. longest downsequence) of a permutation solves the maximum independent set problem (resp. the clique problem) for the corresponding permutation graph. Moreover, we discuss the problem of effeciently constructing the Young tableau for a given permutation.
LowEntropy Computational Geometry
, 2010
"... The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional informa ..."
Abstract
 Add to MetaCart
The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a threedimensional convex polytope and a twocoloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offlineproblem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the selfimproving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a boundeddegree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.
Point Location in Ó ÐÓ � Ò Time, Voronoi Diagrams in Ó Ò ÐÓ � Ò Time, and Other Transdichotomous Results in Computational Geometry
"... Given Ò points in the plane with integer coordinates bounded by Í � Û, we show that the Voronoi diagram can be constructed in Ç Ñ�Ò�Ò ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ò Ô ÐÓ � Í� expected time by a randomized algorithm on the unitcost RAM with word size Û. Similar results are also obtained for many other fun ..."
Abstract
 Add to MetaCart
Given Ò points in the plane with integer coordinates bounded by Í � Û, we show that the Voronoi diagram can be constructed in Ç Ñ�Ò�Ò ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ò Ô ÐÓ � Í� expected time by a randomized algorithm on the unitcost RAM with word size Û. Similar results are also obtained for many other fundamental problems in computational geometry, such as constructing the convex hull of adimensional point set, computing the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. These are the first results to beat the ª Ò ÐÓ � Ò algebraicdecisiontree lower bounds known for these problems. The results are all derived from a new twodimensional version of fusion trees that can answer point location queries in Ç Ñ�Ò�ÐÓ � Ò � ÐÓ � ÐÓ � Ò � Ô ÐÓ � Í � time with linear space. Higherdimensional extensions and applications are also mentioned in the paper. 1.