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Instanceoptimal geometric algorithms
"... ... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of ..."
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Cited by 7 (1 self)
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... in 2d and 3d, and offline point location in 2d. We prove the existence of an algorithm A for computing 2d or 3d convex hulls that is optimal for every point set in the following sense: for every set S of n points and for every algorithm A ′ in a certain class A, the maximum running time of A on input 〈s1,..., sn〉 is at most a constant factor times the maximum running time of A ′ on 〈s1,..., sn〉, where the maximum is taken over all permutations 〈s1,..., sn 〉 of S. In fact, we can establish a stronger property: for every S and A ′ , the maximum running time of A is at most a constant factor times the average running time of A ′ over all permutations of S. We call algorithms satisfying these properties instanceoptimal in the orderoblivious and randomorder setting. Such instanceoptimal algorithms simultaneously subsume outputsensitive algorithms and distributiondependent averagecase algorithms, and all algorithms that do not take advantage of the order of the input or that assume the input is given in a random order. The class A under consideration consists of all algorithms in a decision tree model where the tests involve only multilinear functions with a constant number of arguments. To establish an instancespecific lower bound, we deviate from traditional Ben–Orstyle proofs and adopt an interesting adversary argument. For 2d convex hulls, we prove that a version of the well known algorithm by Kirkpatrick and Seidel (1986) or Chan, Snoeyink, and Yap (1995) already attains this lower bound. For 3d convex hulls, we propose a new algorithm. To demonstrate the potential of the concept, we further obtain instanceoptimal results for a few other standard problems in computational geometry, such as maxima in 2d and 3d, orthogonal line segment intersection in 2d, finding bichromatic L∞close pairs in 2d, offline orthogonal range searching in 2d, offline dominance reporting in 2d and 3d, offline halfspace range reporting 1.
Distributionsensitive set multipartitioning
 1st International Conference on the Analysis of Algorithms
, 2005
"... Given a set S with realvalued members, associated with each member one of two possible types; a multipartitioning of S is a sequence of the members of S such that if x, y ∈ S have different types and x < y, x precedes y in the multipartitioning of S. We give two distributionsensitive algorithms ..."
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Cited by 3 (2 self)
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Given a set S with realvalued members, associated with each member one of two possible types; a multipartitioning of S is a sequence of the members of S such that if x, y ∈ S have different types and x < y, x precedes y in the multipartitioning of S. We give two distributionsensitive algorithms for the set multipartitioning problem and a matching lower bound in the algebraic decisiontree model. One of the two algorithms can be made stable and can be implemented in place. We also give an outputsensitive algorithm for the problem.
DistributionSensitive Construction of MinimumRedundancy Prefix Codes
, 2005
"... Abstract. A new method for constructing minimumredundancy prefix codes is described. This method does not build a Huffman tree; instead it uses a property of optimal codes to find the codeword length of each weight. The running time of the algorithm is shown to be O(nk), where n is the number of we ..."
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Cited by 1 (0 self)
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Abstract. A new method for constructing minimumredundancy prefix codes is described. This method does not build a Huffman tree; instead it uses a property of optimal codes to find the codeword length of each weight. The running time of the algorithm is shown to be O(nk), where n is the number of weights and k is the number of different codeword lengths. When the given sequence of weights is already sorted, it is shown that the codes can be constructed using O(log 2k−1 n) comparisons, which is sublinear if the value of k is small.
Exploring Active Networks and BTrees Using Bun
"... In recent years, much research has been devoted to the simulation of SMPs; on the other hand, few have constructed the synthesis of IPv4. In fact, few information theorists would disagree with the refinement of DHTs. In this paper we show that while XML and semaphores can collude to address this que ..."
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In recent years, much research has been devoted to the simulation of SMPs; on the other hand, few have constructed the synthesis of IPv4. In fact, few information theorists would disagree with the refinement of DHTs. In this paper we show that while XML and semaphores can collude to address this question, Internet QoS can be made compact, probabilistic, and certifiable. 1
PRIMITIVE ROOTS IN QUADRATIC FIELDS II
, 2005
"... Abstract. This paper is continuation of the paper ”Primitive roots in quadratic field”. We consider an analogue of Artin’s primitive root conjecture for algebraic numbers which is not a unit in real quadratic fields. Given such an algebraic number, for a rational prime p which is inert in the field ..."
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Abstract. This paper is continuation of the paper ”Primitive roots in quadratic field”. We consider an analogue of Artin’s primitive root conjecture for algebraic numbers which is not a unit in real quadratic fields. Given such an algebraic number, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p 2 −1. An extension of Artin’s conjecture is that there are infinitely many such inert primes for which this order is maximal. we show that for any choice of 85 algebraic numbers satisfying a certain simple restriction, there is at least one of the algebraic numbers which satisfies the above version of Artin’s conjecture. 1.
PRIMITIVE ROOTS IN QUADRATIC FIELDS
, 2003
"... Abstract. We consider an analogue of Artin’s primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of Artin’s conjecture is that there are infinitely m ..."
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Abstract. We consider an analogue of Artin’s primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of Artin’s conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin’s conjecture. 1.
unknown title
, 812
"... Using a computer algebra system to simplify expressions for TitchmarshWeyl mfunctions ..."
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Using a computer algebra system to simplify expressions for TitchmarshWeyl mfunctions