Results 1  10
of
21
New data structures for orthogonal range searching
 In Proc. 41st IEEE Symposium on Foundations of Computer Science
, 2000
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Dynamic Ordered Sets with Exponential Search Trees
 Combination of results presented in FOCS 1996, STOC 2000 and SODA
, 2001
"... We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. This leads to an optimal bound of O ( √ log n/log log n) for searching and updating a dynamic set of n integer keys i ..."
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Cited by 26 (1 self)
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We introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fullydynamic linear space data structures. This leads to an optimal bound of O ( √ log n/log log n) for searching and updating a dynamic set of n integer keys in linear space. Here searching an integer y means finding the maximum key in the set which is smaller than or equal to y. This problem is equivalent to the standard text book problem of maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein: Introduction to Algorithms, 2nd ed., MIT Press, 2001). The best previous deterministic linear space bound was O(log n/log log n) due Fredman and Willard from STOC 1990. No better deterministic search bound was known using polynomial space.
Range searching over tree cross products
 In Proc. 8th European Symposium on Algorithms (ESA
, 2000
"... Abstract. We introduce the tree crossproduct problem, which abstracts a data structure common to applications in graph visualization, string matching, and software analysis. We design solutions with a variety of tradeoffs, yielding improvements and new results for these applications. 1 ..."
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Cited by 19 (0 self)
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Abstract. We introduce the tree crossproduct problem, which abstracts a data structure common to applications in graph visualization, string matching, and software analysis. We design solutions with a variety of tradeoffs, yielding improvements and new results for these applications. 1
Fast algorithms for 3d dominance reporting and counting
 International Journal of Foundations of Computer Science
, 2003
"... We present in this paper fast algorithms for the 3D dominance reporting and counting problems, and generalize the results to the ddimensional case. Our 3D dominance reporting algorithm achieves O(log n = log log n + f) 1 query time using O(n log n) space, where f is the number of points satisfyin ..."
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Cited by 8 (5 self)
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We present in this paper fast algorithms for the 3D dominance reporting and counting problems, and generalize the results to the ddimensional case. Our 3D dominance reporting algorithm achieves O(log n = log log n + f) 1 query time using O(n log n) space, where f is the number of points satisfying the query and>0 is an arbitrarily small constant. For the 3D dominance counting problem (which isequivalent to the 3D range counting problem), our algorithm runs in O((log n = log log n) 2)timeusingO(nlog 1+ n = log log n) space. 1
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 ..."
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Cited by 8 (3 self)
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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.
Fullydynamic orthogonal range reporting on RAM
, 2003
"... In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of p ..."
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Cited by 7 (2 self)
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In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of points stored and r is the number of points reported. The space usage is n). In the standard comparison model the result holds for d 3.
A TradeOff For WorstCase Efficient Dictionaries
"... We consider dynamic dictionaries over the universe U = {0, 1}^w on a unitcost RAM with word size w and a standard instruction set, and present a linear space deterministic dictionary accommodating membership queries in time (log log n)^O(1) and updates in time (log n)^O(1), where n is the size of t ..."
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Cited by 7 (2 self)
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We consider dynamic dictionaries over the universe U = {0, 1}^w on a unitcost RAM with word size w and a standard instruction set, and present a linear space deterministic dictionary accommodating membership queries in time (log log n)^O(1) and updates in time (log n)^O(1), where n is the size of the set stored. Previous solutions either had query time (log n) 18 or update time 2 !( p log n) in the worst case.
Fast fractional cascading and its applications
, 2003
"... Using the notions of Qheaps and fusion trees developed by Fredman and Willard, we develop a faster version of the fractional cascading technique while maintaining the linear space structure. The new version enables sublogarithmic iterative search in the case when we have a search tree and the degre ..."
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Cited by 4 (2 self)
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Using the notions of Qheaps and fusion trees developed by Fredman and Willard, we develop a faster version of the fractional cascading technique while maintaining the linear space structure. The new version enables sublogarithmic iterative search in the case when we have a search tree and the degree of each node is bounded by O(log n), for some constant> 0, where n is the total size of all the lists stored in the tree. The fast fractional cascading technique is used in combination with other techniques to derive sublogarithmic time algorithms for the geometric retrieval problems: orthogonal segment intersection and rectangular point enclosure. The new algorithms use O(n) space and achieve a query time of O(log n = log log n + f), where f is the number of objects satisfying the query. All our algorithms assume the version of the RAM model used by Fredman and Willard. 1
Hashing, Randomness and Dictionaries
, 2002
"... This thesis is centered around one of the most basic information retrieval problems, namely that of storing and accessing the elements of a set. Each element in the set has some associated information that is returned along with it. The problem is referred to as the dictionary problem, due to the si ..."
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Cited by 3 (0 self)
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This thesis is centered around one of the most basic information retrieval problems, namely that of storing and accessing the elements of a set. Each element in the set has some associated information that is returned along with it. The problem is referred to as the dictionary problem, due to the similarity to a bookshelf dictionary, which contains a set of words and has an explanation associated with each word. In the static version of the problem the set is fixed, whereas in the dynamic version, insertions and deletions of elements are possible. The approach
A new framework for addressing temporal range queries and some preliminary results
, 2003
"... Given a set of n objects, each characterized by d attributes speci ed at m xed time instances, we are interested in the problem of designing space e cient indexing structures such that arbitrary temporal range search queries can be handled e ciently. Whenm =1, our problem reduces to the ddimensiona ..."
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Cited by 2 (1 self)
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Given a set of n objects, each characterized by d attributes speci ed at m xed time instances, we are interested in the problem of designing space e cient indexing structures such that arbitrary temporal range search queries can be handled e ciently. Whenm =1, our problem reduces to the ddimensional orthogonal search problem. We establish e cient data structures to handle several classes of the general problem. Our results include a linear size data structure that enables a query time of O(log n log m = log log n + f) for onesided queries when d = 1, where f is the number of objects satisfying the query. A similar result is shown for counting queries. We alsoshow that the most general problem can be solved with a polylogarithmic query time using nonlinear space data structures. 1