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 peer-to-peer 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, simple and straightforward algorithms can be used to construct a skip graph, insert new nodes into it, search it, and detect and repair errors in a skip graph introduced due to node failures.

Abstract. For words of length n, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, q–analogues for these parameters from the usual permutation model are obtained. 1.

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. 1.

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.

Abstract. We study the asymptotic probability that a random composition of an integer n is gap-free, 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 gap-free. 1. introduction A composition of a natural number n is said to be gap-free 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 gap-free compositions arising from permuting the order of the parts of the partitions

We present a new type of search trees, called Skip trees, which are a generalization of Skip lists. To be precise, there is a one-to-one 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...

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.

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 ...

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

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. inria-00547935, version 1- 17 Dec 2010 1