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22
Optimal Bounds for the Predecessor Problem and Related Problems
 Journal of Computer and System Sciences
, 2001
"... We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved ..."
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Cited by 72 (0 self)
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We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.
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 53 (4 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...
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 27 (8 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 20 (7 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 ..."
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Cited by 19 (1 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 11 (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 ...
Lower bounds for intersection searching and fractional cascading in higher dimension
, 2003
"... Given an nedge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in Oðk þ polylogðnÞÞ time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve Oðk þ polylogðnÞÞ time with subcubic sto ..."
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Cited by 11 (0 self)
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Given an nedge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in Oðk þ polylogðnÞÞ time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve Oðk þ polylogðnÞÞ time with subcubic storage? Does any convex polytope have a boundary dominant Dobkin–Kirkpatrick hierarchy? Can fractional cascading be generalized to planar maps instead of linear lists? We prove that the answer to all of these questions is no, and we derive nearoptimal solutions to these classical problems.
Threedimensional layers of maxima
 Algorithmica
"... Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima problem, an improvement over the prior O(n logn log log n)time solution. A previous claimed O(n log n)time solution due to Atallah, Goodrich, and Ramaiyer [SCG’94] has technical flaws. Our algorithm ..."
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Cited by 7 (0 self)
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Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima problem, an improvement over the prior O(n logn log log n)time solution. A previous claimed O(n log n)time solution due to Atallah, Goodrich, and Ramaiyer [SCG’94] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph. 1
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 7 (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+&epsilon;) solutions which allows us to guarantee constant query time using linear space and O(n log n) preprocessing time. The dependence on &epsilon; is polynomial in1/&epsilon;. One tool...