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18
Lower bounds for UnionSplitFind related problems on random access machines
, 1994
"... We prove \Omega\Gamma p log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the unionsplitfind problem, dynamic prefix problems and onedimensional range query problems. The proof techniques include a ..."
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Cited by 49 (3 self)
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We prove \Omega\Gamma p log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the unionsplitfind problem, dynamic prefix problems and onedimensional range query problems. The proof techniques include a general technique using perfect hashing for reducing static data structure problems (with a restriction of the size of the structure) into partially dynamic data structure problems (with no such restriction), thus providing a way to transfer lower bounds. We use a generalization of a method due to Ajtai for proving the lower bounds on the static problems, but describe the proof in terms of communication complexity, revealing a striking similarity to the proof used by Karchmer and Wigderson for proving lower bounds on the monotone circuit depth of connectivity. 1 Introduction and summary of results In this paper we give lower bounds for the complexity of implementing several dynamic and sta...
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...
Ordered Theta Graphs
"... Let V be a set of n points in R2. The `graph of V is a geometric graph with vertex set V that has been studied extensively and which has several nice properties. We introduce a new variant of `graphs which we call ordered `graphs. These are graphs that are built incrementally by inserting the ve ..."
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Cited by 20 (5 self)
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Let V be a set of n points in R2. The `graph of V is a geometric graph with vertex set V that has been studied extensively and which has several nice properties. We introduce a new variant of `graphs which we call ordered `graphs. These are graphs that are built incrementally by inserting the vertices one by one so that the resulting graph depends on the insertion order. We show that careful insertion orders can produce graphs with desirable properties.
New Results on Binary Space Partitions in the Plane
 COMPUT. GEOM. THEORY APPL
, 1994
"... We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ra ..."
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Cited by 19 (6 self)
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We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.
Efficient Algorithms for Constructing FaultTolerant Geometric Spanners
, 1997
"... Let S be a set of n points in IR d , and k an integer such that 1 k n 2. Algorithms are given that construct faulttolerant spanners for S. If in such a spanner at most k edges or vertices are removed, then each pair of points in the remaining graph is still connected by a \short" path. Our resu ..."
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Cited by 15 (2 self)
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Let S be a set of n points in IR d , and k an integer such that 1 k n 2. Algorithms are given that construct faulttolerant spanners for S. If in such a spanner at most k edges or vertices are removed, then each pair of points in the remaining graph is still connected by a \short" path. Our results include (i) an algorithm with running time O(n log d 1 n + kn log log n + k 2 n) that constructs a spanner with O(k 2 n) edges, that is resilient to k edge faults, (ii) an algorithm with running time O(n log n + k 2 n) that constructs a spanner with O(k 2 n) edges, that is resilient to k vertex faults, and (iii) an algorithm with running time O(n log n + c k n) that constructs a spanner of degree O(c k ), whose total edge length is bounded by O(c k ) times the weight of a minimum spanning tree of S, and that is resilient to k edge or vertex faults. Here, c is a constant that is independent of n and k. Our algorithms are based on wellseparated pairs, and approximate n...
Maintaining the Approximate Width of a Set of Points in the Plane (Extended Abstract)
, 1993
"... The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes lin ..."
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Cited by 12 (1 self)
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The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes linear space and allows for reporting the approximation with relative accuracy ffl in O( p 1=ffl log n) time; and the update time is O(log² n). The method uses the tentative pruneandsearch strategy of Kirkpatrick and Snoeyink.
Range Searching and Point Location among Fat Objects
 Journal of Algorithms
, 1994
"... We present a data structure that can store a set of disjoint fat objects in dspace such that point location and boundedsize range searching with arbitrarilyshaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or nonconvex polytopes. The m ..."
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Cited by 10 (0 self)
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We present a data structure that can store a set of disjoint fat objects in dspace such that point location and boundedsize range searching with arbitrarilyshaped ranges can be performed efficiently. The structure can deal with either arbitrary (fat) convex objects or nonconvex polytopes. The multipurpose data structure supports point location and range searching queries in time O(log d\Gamma1 n) and requires O(n log d\Gamma1 n) storage, after O(n log d\Gamma1 n log log n) preprocessing. The data structure and query algorithm are rather simple. 1 Introduction Fatness turns out to be an interesting phenomenon in computational geometry. Several papers present surprising combinatorial complexity reductions [3, 15, 22, 26, 32] and efficiency gains for algorithms [1, 4, 19, 28, 33] if the objects under consideration have a certain fatness. Fat objects are compact to some extent, rather than long and thin. Fatness is a realistic assumption, since in many practical instances of ...
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.
Approximating Energy Efficient Paths in Wireless MultiHop Networks
, 2003
"... Given the positions of n sites in a radio network we consider the problem of finding routes between any pair of sites that minimize energy consumption and do not use more than some constant number k of hops. Known exact algorithms for this problem required O(n log n) per query pair (p,q). In this pa ..."
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Cited by 6 (3 self)
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Given the positions of n sites in a radio network we consider the problem of finding routes between any pair of sites that minimize energy consumption and do not use more than some constant number k of hops. Known exact algorithms for this problem required O(n log n) per query pair (p,q). In this paper we relax the exactness requirement and only compute approximate (1+ε) solutions which allows us to guarantee constant query time using linear space and O(n log n) preprocessing time. The dependence on ε is polynomial in1/ε. One tool...
The Temporal Precedence Problem
 Algorithmica
, 1998
"... In this paper we analyze the complexity of the Temporal Precedence Problem on pointer machines. Simply stated, the problem is to efficiently support two operations: insert and precedes. The operation insert(a) introduces a new element a, while precedes(a; b) returns true iff element a was inserted ..."
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Cited by 5 (4 self)
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In this paper we analyze the complexity of the Temporal Precedence Problem on pointer machines. Simply stated, the problem is to efficiently support two operations: insert and precedes. The operation insert(a) introduces a new element a, while precedes(a; b) returns true iff element a was inserted before element b temporally. We provide a solution to the problem with worstcase time complexity O(lg lg n) per operation, where n is the number of elements inserted. We also demonstrate that the problem has a lower bound of \Omega\Gammaf/ lg n) on pointer machines. Thus the proposed scheme is optimal on pointer machines. Keywords: Algorithms, Dynamic Data Structures, Complexity. 1 Introduction In this paper we study the complexity of what we call the Temporal Precedence (T P) Problem on pointer machines. Informally, the problem is to manage the dynamic insertion of elements, with the ability of determining, given two elements, which one was inserted first. The problem is related to ...