12,931 research outputs found

    Fringe analysis for parallel MacroSplit insertion algorithms in 2--3 trees

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    We extend the fringe analysis (used to study the expected behavior of balanced search trees under sequential insertions) to deal with synchronous parallel insertions on 2--3 trees. Given an insertion of k keys in a tree with n nodes, the fringe evolves following a transition matrix whose coefficients take care of the precise form of the algorithm but does not depend on k or n. The derivation of this matrix uses the binomial transform recently developed by P. Poblete, J. Munro and Th. Papadakis. Due to the complexity of the preceding exact analysis, we develop also two approximations. A first one based on a simplified parallel model, and a second one based on the sequential model. These two approximated analysis prove that the parallel insertions case does not differ significantly from the sequential case, namely on the terms O(1/n^2).Postprint (published version

    Finger Search in Grammar-Compressed Strings

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    Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures many popular compression schemes. Given a grammar, the random access problem is to compactly represent the grammar while supporting random access, that is, given a position in the original uncompressed string report the character at that position. In this paper we study the random access problem with the finger search property, that is, the time for a random access query should depend on the distance between a specified index ff, called the \emph{finger}, and the query index ii. We consider both a static variant, where we first place a finger and subsequently access indices near the finger efficiently, and a dynamic variant where also moving the finger such that the time depends on the distance moved is supported. Let nn be the size the grammar, and let NN be the size of the string. For the static variant we give a linear space representation that supports placing the finger in O(logN)O(\log N) time and subsequently accessing in O(logD)O(\log D) time, where DD is the distance between the finger and the accessed index. For the dynamic variant we give a linear space representation that supports placing the finger in O(logN)O(\log N) time and accessing and moving the finger in O(logD+loglogN)O(\log D + \log \log N) time. Compared to the best linear space solution to random access, we improve a O(logN)O(\log N) query bound to O(logD)O(\log D) for the static variant and to O(logD+loglogN)O(\log D + \log \log N) for the dynamic variant, while maintaining linear space. As an application of our results we obtain an improved solution to the longest common extension problem in grammar compressed strings. To obtain our results, we introduce several new techniques of independent interest, including a novel van Emde Boas style decomposition of grammars

    The fluctuations of the giant cluster for percolation on random split trees

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    A split tree of cardinality nn is constructed by distributing nn "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as mm-ary search trees, quad trees, median-of-(2k+1)(2k+1) trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality nn. We show for appropriate percolation regimes that depend on the cardinality nn of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as nn \rightarrow \infty are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random mm-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.Comment: 43 page

    Optimal prefix codes for pairs of geometrically-distributed random variables

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    Optimal prefix codes are studied for pairs of independent, integer-valued symbols emitted by a source with a geometric probability distribution of parameter qq, 0<q<10{<}q{<}1. By encoding pairs of symbols, it is possible to reduce the redundancy penalty of symbol-by-symbol encoding, while preserving the simplicity of the encoding and decoding procedures typical of Golomb codes and their variants. It is shown that optimal codes for these so-called two-dimensional geometric distributions are \emph{singular}, in the sense that a prefix code that is optimal for one value of the parameter qq cannot be optimal for any other value of qq. This is in sharp contrast to the one-dimensional case, where codes are optimal for positive-length intervals of the parameter qq. Thus, in the two-dimensional case, it is infeasible to give a compact characterization of optimal codes for all values of the parameter qq, as was done in the one-dimensional case. Instead, optimal codes are characterized for a discrete sequence of values of qq that provide good coverage of the unit interval. Specifically, optimal prefix codes are described for q=21/kq=2^{-1/k} (k1k\ge 1), covering the range q1/2q\ge 1/2, and q=2kq=2^{-k} (k>1k>1), covering the range q<1/2q<1/2. The described codes produce the expected reduction in redundancy with respect to the one-dimensional case, while maintaining low complexity coding operations.Comment: To appear in IEEE Transactions on Information Theor

    B-urns

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    The fringe of a B-tree with parameter mm is considered as a particular P\'olya urn with mm colors. More precisely, the asymptotic behaviour of this fringe, when the number of stored keys tends to infinity, is studied through the composition vector of the fringe nodes. We establish its typical behaviour together with the fluctuations around it. The well known phase transition in P\'olya urns has the following effect on B-trees: for m59m\leq 59, the fluctuations are asymptotically Gaussian, though for m60m\geq 60, the composition vector is oscillating; after scaling, the fluctuations of such an urn strongly converge to a random variable WW. This limit is C\mathbb C-valued and it does not seem to follow any classical law. Several properties of WW are shown: existence of exponential moments, characterization of its distribution as the solution of a smoothing equation, existence of a density relatively to the Lebesgue measure on C\mathbb C, support of WW. Moreover, a few representations of the composition vector for various values of mm illustrate the different kinds of convergence

    Rebalancing operations for deletions in AVL-trees

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