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22
Skip Graphs
 Proc. of the 14th Annual ACMSIAM Symp. on Discrete Algorithms
, 2003
"... Skip graphs are a novel distributed data structure, based on skip lists, that provide the full functionality of a balanced tree in a distributed system where resources are stored in separate nodes that may fail at any time. They are designed for use in searching peertopeer systems, and by providin ..."
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Cited by 311 (9 self)
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Skip graphs are a novel distributed data structure, based on skip lists, that provide the full functionality of a balanced tree in a distributed system where resources are stored in separate nodes that may fail at any time. They are designed for use in searching peertopeer systems, and by providing the ability to perform queries based on key ordering, they improve on existing search tools that provide only hash table functionality. Unlike skip lists or other tree data structures, skip graphs are highly resilient, tolerating a large fraction of failed nodes without losing connectivity. In addition, constructing, inserting new nodes into, searching a skip graph, and detecting and repairing errors in the data structure introduced by node failures can be done using simple and straightforward algorithms. 1
Combinatorics of geometrically distributed random variables: New qtangent an qsecant numbers, Int
 Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A1090
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A design of a parallel dictionary using skip lists
, 1996
"... We present a topdown design of a parallel PRAM dictionary using skip lists. More precisely, we give detailed algorithms to search for, insert or delete k elements in a skip list of n elements in parallel. The algorithms are iterative and easy to implement on real machines. We discuss some implementa ..."
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Cited by 16 (9 self)
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We present a topdown design of a parallel PRAM dictionary using skip lists. More precisely, we give detailed algorithms to search for, insert or delete k elements in a skip list of n elements in parallel. The algorithms are iterative and easy to implement on real machines. We discuss some implementation issues and give concrete examples in C*. The algorithms run on an EREW PRAM in expected time O(logn + log k) using k processors. We also show an explicit protocol to avoid read conflicts thus obtaining an efficient EREW version of our algorithms. Although the asymptotic performance of the parallel skip list algorithms is not better compared to that of other parallel dictionaries, they are a practical alternative. Skip list algorithms are very simple and there is a small probability of large deviations from their expected performance.
Gapfree compositions and gapfree samples of geometric random variables
 Discrete Math
, 2005
"... Abstract. We study the asymptotic probability that a random composition of an integer n is gapfree, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distribute ..."
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Cited by 11 (3 self)
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Abstract. We study the asymptotic probability that a random composition of an integer n is gapfree, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gapfree. 1. introduction A composition of a natural number n is said to be gapfree if the part sizes occuring in it form an interval. In addition if the interval starts at 1, the composition is said to be complete. Example Of the 32 compositions of n = 6, there are 21 gapfree compositions arising from permuting the order of the parts of the partitions
Asymptotic normality of recursive algorithms via martingale difference arrays
 Discrete Mathematics and Theoretical Computer Science
, 2000
"... We propose martingale central limit theorems as an appropriate tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs LN, then L ..."
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Cited by 9 (0 self)
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We propose martingale central limit theorems as an appropriate tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs LN, then LN D = Ln + ¯LN−n + RN for N ≥ n0 ≥ 2, where n follows a certain distribution PN on the integers {0,...,N} and Lk D = ¯Lk for k ≥ 0. Ln, LN−n and RN are independent, conditional on n, and RN are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N − n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to ZN: = LN−IE LN √. Under certain Var LN compatibility assumptions on the sequence (PN)N≥0 we show that a pair of sufficient conditions (of Lyapunov type) for ZN D → N (0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (IE LN)N≥0. In the case that the PN are binomial distributions with the same parameter p, and for deterministic RN, we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (RN)N≥0 (and for the scale RN = N α a characterization of those α) leading to asymptotic normality of ZN.
The first descent in samples of geometric random variables and permutations
"... For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the fir ..."
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Cited by 5 (3 self)
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For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.
Skip trees, an alternative data structure to Skip lists in a concurrent approach
, 1997
"... We present a new type of search trees, called Skip trees, which are a generalization of Skip lists. To be precise, there is a onetoone mapping between the two data types which commutes with the sequential update algorithms. A Skip list is a data structure used to manage data bases which stores val ..."
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We present a new type of search trees, called Skip trees, which are a generalization of Skip lists. To be precise, there is a onetoone mapping between the two data types which commutes with the sequential update algorithms. A Skip list is a data structure used to manage data bases which stores values in a sorted way and in which it is insured that the form of the Skip list is independent of the order of updates by using randomization techniques. Skip trees inherit all the properties of Skip lists, including the time bounds of sequential algorithms. The algorithmic improvement of the Skip tree type is that a concurrent algorithm on the fly approach can be designed. Among other advantages, this algorithm is more compressive than the one designed by Pugh for Skip lists and accepts a higher degree of concurrence because it is based on a set of local updates. From a practical point of view, although the Skip list should be in the main memory, Skip trees can be registered into a secondary...
On the search path length of random binary skip graphs
, 2009
"... In this paper we consider the skip graph data structure, a load balancing alternative to skip lists, designed to perform better in a distributed environment. We extend previous results of Devroye on skip lists, and prove that the maximum length of a search path in a random binary skip graph of size ..."
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Cited by 2 (0 self)
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In this paper we consider the skip graph data structure, a load balancing alternative to skip lists, designed to perform better in a distributed environment. We extend previous results of Devroye on skip lists, and prove that the maximum length of a search path in a random binary skip graph of size n is of order log n with high probability. 1
A TopDown Design of a Parallel Dictionary using Skip Lists
, 1994
"... We present a top down design of a parallel PRAM dictionary using skip lists. More precisely, we give algorithms to search, insert and delete k ordered elements in a skip list of n elements in parallel. The algorithms are simple and easy to implement on real machines. All of them are iterative. Th ..."
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Cited by 1 (0 self)
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We present a top down design of a parallel PRAM dictionary using skip lists. More precisely, we give algorithms to search, insert and delete k ordered elements in a skip list of n elements in parallel. The algorithms are simple and easy to implement on real machines. All of them are iterative. They can be implemented in the EREW PRAM model using O(k) processors in expected time O(log n+ log k). The probability that there is a significant deviation from the expected time decreases as O(n \Gamma2 ) in the search and as O(n \Gamma2 + k \Gamma2 ) in the insertion and deletion. 1 Introduction Parallel dictionaries are important data structures widely studied. In a systolic framework, priority queues and search trees were designed by C.E. Leiserson in [13]. Later, M.J. Atallah and S.R. Kosaraju [1] developped a generalized dictionary where a sequence of operations can be pipelined at constant rate. In the PRAM context, W. Paul, U. Vishkin and H. Wagener developed and analyzed ...
The first descent in samples of geometric random variables and permutations
"... For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the fir ..."
Abstract
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For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first strict and weak descents in the word. Higher moments and limiting distributions are also derived. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.