Some observations on skip-lists
TL;DR: It is proved that the performance of the Skip-list data-structure can be further guaranteed in the following sense: the probability of the search time or space complexity exceeding their expected values, approaches 0 rapidly as the number of keys increases.
About: This article is published in Information Processing Letters.The article was published on 1991-09-01. It has received 12 citations till now. The article focuses on the topics: Skip list & Binary search tree.
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02 Jan 1993
TL;DR: This thesis investigates various forms of skip lists, and with probabilistic analyses of algorithms, and derives exact and asymptotic expressions for the average search cost of a fixed key and of an average key, and proposes several versions of deterministic skip lists.
Abstract: This thesis is concerned with various forms of skip lists, and with probabilistic analyses of algorithms. We investigate three topics; one topic from each of these two areas, and another topic common to both of them.
First, we consider Pugh's skip list. We derive exact and asymptotic expressions for the average search cost of a fixed key and of an average key. Our results improve previously known upper bounds of these two average search costs. We also derive exact and asymptotic expressions for the variance of the search cost for the largest key.
Next, we propose several versions of deterministic skip lists. They all have guaranteed logarithmic search and update costs per operation, they lead to an interesting "bridge" structure between the original skip list and standard search trees, they are simpler to implement than standard balanced search trees, and our experimental results suggest that they are also competitive in terms of space and time.
Finally, we consider the elastic-budget trie, a variant of the standard trie, in which each external node (bucket) has precisely as many key slots as the number of keys stored in it. We examine the number of buckets of each size, and we derive exact and asymptotic expressions for their average values, as well as asymptotic expressions for their variances and covariances under the closely related "Poisson model" of randomness. Our experimental results suggest that maintaining only two bucket sizes may be a very reasonable practical choice.
29 citations
TL;DR: There is a one-to-one mapping between the two data types which commutes with the sequential update algorithms, and the ability of Skip trees to manage data bases in comparison with B-trees is analysed.
Abstract: 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 proeprties 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 an 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 or external storage. Therefore we analyse the ability, of Skip trees to manage data bases in comparison with B-trees.
18 citations
TL;DR: A top-down design of a parallel PRAM dictionary using skip lists is presented and detailed algorithms to search for, insert or delete k elements in a skip list of n elements in parallel are given.
16 citations
Cites methods from "Some observations on skip-lists"
...Parallel search, insertion and deletion algorithms for k items in a 2-3 tree storing n items were shown to take time O(logn + log k) in the worst-case....
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25 Sep 1995
TL;DR: To any sequence of real numbers, the binomial transform is defined through the rule a_s = \mathcal{B}_s a_n =sum\limits_n {( - 1)^n}\left( {\begin{array}{*{20}c}s \
\right)a_n .
Abstract: To any sequence of real numbers 〈a n 〉n≥0, we can associate another sequence 〈â s 〉 s ≥0, called its binomial transform. This transform is defined through the rule
$$\hat a_s = \mathcal{B}_s a_n = \sum\limits_n {( - 1)^n \left( {\begin{array}{*{20}c}s \
\\\end{array} } \right)a_n .}$$
11 citations
TL;DR: These methods are used to perform a detailed analysis of skip lists, a probabilistic data structure introduced by Pugh as an alternative to balanced trees, and obtain the mean and variance for the cost of searching for the first or the last element in the list (confirming results obtained previously by other methods).
8 citations
Cites background from "Some observations on skip-lists"
...Sen [20] and also Devroye [2,3] have obtained results about the distribution of C n ....
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References
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TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.
3,760 citations
TL;DR: Skip lists as mentioned in this paper are data structures that use probabilistic balancing rather than strictly enforced balancing, and the algorithms for insertion and deletion in skip lists are much simpler and significantly faster than equivalent algorithms for balanced trees.
Abstract: Skip lists are data structures that use probabilistic balancing rather than strictly enforced balancing. As a result, the algorithms for insertion and deletion in skip lists are much simpler and significantly faster than equivalent algorithms for balanced trees.
1,113 citations
TL;DR: Three simple efficient algorithms with good probabilistic behaviour are described and an algorithm with a run time of O ( n log n ) which almost certainly finds a perfect matching in a random graph of at least cn log n edges is analyzed.
681 citations
30 Oct 1989
TL;DR: A randomized strategy for maintaining balance in dynamically changing search trees that has optimal expected behavior and generalizes naturally to weighted trees, where the expected time bounds for accesses and updates again match the worst case time bounds of the best deterministic methods.
Abstract: A randomized strategy for maintaining balance in dynamically changing search trees that has optimal expected behavior is presented. In particular, in the expected case an update takes logarithmic time and requires fewer than two rotations. Moreover, the update time remains logarithmic, even if the cost of a rotation is taken to be proportional to the size of the rotated subtree. The approach generalizes naturally to weighted trees, where the expected time bounds for accesses and updates again match the worst case time bounds of the best deterministic methods. The balancing strategy and algorithms are exceedingly simple and should be fast in practice. >
200 citations
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TL;DR: This paper describes and analyzes algorithms to use search fingers, merge, split and concatenate skip lists, and implement linear list operations using skip lists.
Abstract: Skip lists are a probabilistic data structure that seem likely to supplant balanced trees as the implementation method of choice for many applications. Skip list algorithms have the same asymptotic expected time bounds as balanced trees and are simpler, faster and use less space. The original paper on skip lists only presented algorithms for search, insertion and deletion. In this paper, we show that skip lists are as versatile as balanced trees. We describe and analyze algorithms to use search fingers, merge, split and concatenate skip lists, and implement linear list operations using skip lists. The skip list algorithms for these actions are faster and simpler than their balanced tree cousins. The merge algorithm for skip lists we describe has better asymptotic time complexity than any previously described merge algorithm for balanced trees.
85 citations