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101
Vertical decomposition of shallow levels in 3dimensional arrangements and its applications
 SIAM J. Comput
"... Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤klevel of the arrangement A(F) is O(k 3+ε ψ(n/k)), for any ε> 0, where ψ(r) is the maximum complexity of the lower envelope of a subse ..."
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Cited by 53 (13 self)
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Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the ≤klevel of the arrangement A(F) is O(k 3+ε ψ(n/k)), for any ε> 0, where ψ(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F. This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [52], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized threedimensional rangesearching problems; (ii) dynamic data structures for planar nearest and farthestneighbor searching under various fairly general distance functions; (iii) an improved (nearquadratic) algorithm for minimumweight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static and dynamic settings.
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint 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 be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint 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 be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 46 (11 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Complexity Models for Incremental Computation
, 1994
"... We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have sma ..."
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Cited by 42 (4 self)
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We present a new complexity theoretic approach to incremental computation. We define complexity classes that capture the intuitive notion of incremental efficiency and study their relation to existing complexity classes. We show that problems that have small sequential space complexity also have small incremental time complexity. We show that all common LOGSPACEcomplete problems for P are also incrPOLYLOGTIMEcomplete for P. We introduce a restricted notion of completeness called NRPcompleteness and show that problems which are NRPcomplete for P are also incrPOLYLOGTIMEcomplete for P. We also give incrementally complete problems for NLOGSPACE, LOGSPACE, and nonuniform NC¹. We show that under certain restrictions problems which have efficient dynamic solutions also have efficient parallel solutions. We also consider a nonuniform model of incremental computation and show that in this model most problems have almost linear complexity. In addition, we present some techniques f...
Efficient ExternalMemory Data Structures and Applications
, 1996
"... In this thesis we study the Input/Output (I/O) complexity of largescale problems arising e.g. in the areas of database systems, geographic information systems, VLSI design systems and computer graphics, and design I/Oefficient algorithms for them. A general theme in our work is to design I/Oeffic ..."
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Cited by 38 (12 self)
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In this thesis we study the Input/Output (I/O) complexity of largescale problems arising e.g. in the areas of database systems, geographic information systems, VLSI design systems and computer graphics, and design I/Oefficient algorithms for them. A general theme in our work is to design I/Oefficient algorithms through the design of I/Oefficient data structures. One of our philosophies is to try to isolate all the I/O specific parts of an algorithm in the data structures, that is, to try to design I/O algorithms from internal memory algorithms by exchanging the data structures used in internal memory with their external memory counterparts. The results in the thesis include a technique for transforming an internal memory tree data structure into an external data structure which can be used in a batched dynamic setting, that is, a setting where we for example do not require that the result of a search operation is returned immediately. Using this technique we develop batched dynamic external versions of the (onedimensional) rangetree and the segmenttree and we develop an external priority queue. Following our general philosophy we show how these structures can be used in standard internal memory sorting algorithms
SelfAdjusting Computation
 In ACM SIGPLAN Workshop on ML
, 2005
"... From the algorithmic perspective, we describe novel data structures for tracking the dependences ina computation and a changepropagation algorithm for adjusting computations to changes. We show that the overhead of our dependence tracking techniques is O(1). To determine the effectiveness of change ..."
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Cited by 35 (13 self)
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From the algorithmic perspective, we describe novel data structures for tracking the dependences ina computation and a changepropagation algorithm for adjusting computations to changes. We show that the overhead of our dependence tracking techniques is O(1). To determine the effectiveness of changepropagation, we present an analysis technique, called trace stability, and apply it to a number of applications.
Cacheoblivious algorithms and data structures
 IN LECTURE NOTES FROM THE EEF SUMMER SCHOOL ON MASSIVE DATA SETS
, 2002
"... A recent direction in the design of cacheefficient and diskefficient algorithms and data structures is the notion of cache obliviousness, introduced by Frigo, Leiserson, Prokop, and Ramachandran in 1999. Cacheoblivious algorithms perform well on a multilevel memory hierarchy without knowing any pa ..."
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Cited by 34 (3 self)
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A recent direction in the design of cacheefficient and diskefficient algorithms and data structures is the notion of cache obliviousness, introduced by Frigo, Leiserson, Prokop, and Ramachandran in 1999. Cacheoblivious algorithms perform well on a multilevel memory hierarchy without knowing any parameters of the hierarchy, only knowing the existence of a hierarchy. Equivalently, a single cacheoblivious algorithm is efficient on all memory hierarchies simultaneously. While such results might seem impossible, a recent body of work has developed cacheoblivious algorithms and data structures that perform as well or nearly as well as standard externalmemory structures which require knowledge of the cache/memory size and block transfer size. Here we describe several of these results with the intent of elucidating the techniques behind their design. Perhaps the most exciting of these results are the data structures, which form general building blocks immediately
Optimal External Memory Interval Management
, 2002
"... In this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central pro ..."
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Cited by 31 (6 self)
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In this paper we present the external interval tree, an optimal external memory data structure for answering stabbing queries on a set of dynamically maintained intervals. The external interval tree can be used in an optimal solution to the dynamic interval management problem, which is a central problem for objectoriented and temporal databases and for constraint logic programming. Part of the structure uses a novel weightbalancing technique for efficient worstcase manipulation of balanced trees of independent interest. The external interval tree, as well at our new balancing technique, have recently been used to develop several efficient external data structures.
I/OEfficient Dynamic Planar Point Location
"... We present the first provably I/Oefficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks to store a subdivision of size N , where B is the disk block size. Queries can be answered in ... I/Os in the worstcase, and insertions and de ..."
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Cited by 29 (17 self)
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We present the first provably I/Oefficient dynamic data structure for point location in a general planar subdivision. Our structure uses O(N/B) disk blocks to store a subdivision of size N , where B is the disk block size. Queries can be answered in ... I/Os in the worstcase, and insertions and deletions can be performed in ... and ... I/Os amortized, respectively. Previously, an I/Oefficient dynamic point location structure was only known for monotone subdivisions. Part of our data structure...
Cell probe complexity  a survey
 In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop
"... The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 ..."
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Cited by 28 (0 self)
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The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1