Discrete Mathematics & Theoretical Computer Science 

About: Discrete Mathematics & Theoretical Computer Science is an academic journal published by French association. The journal publishes majorly in the area(s): Bijection & Permutation. It has an ISSN identifier of 1365-8050. It is also open access. Over the lifetime, 1834 publications have been published receiving 16712 citations. The journal is also known as: Discrete mathematics and theoretical computer science & DMTCS.

HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm

Abstract: This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of \emphdistinct elements (the cardinality) of very large data ensembles. Using an auxiliary memory of m units (typically, "short bytes''), HYPERLOGLOG performs a single pass over the data and produces an estimate of the cardinality such that the relative accuracy (the standard error) is typically about $1.04/\sqrt{m}$. This improves on the best previously known cardinality estimator, LOGLOG, whose accuracy can be matched by consuming only 64% of the original memory. For instance, the new algorithm makes it possible to estimate cardinalities well beyond $10^9$ with a typical accuracy of 2% while using a memory of only 1.5 kilobytes. The algorithm parallelizes optimally and adapts to the sliding window model.

An Approximate L^p Difference Algorithm for Massive Data Streams

Abstract: Several recent papers have shown how to approximate the difference ∑ _i|a_i-b_i| or ∑ |a_i-b_i|^2 between two functions, when the function values a_i and b_i are given in a data stream, and their order is chosen by an adversary. These algorithms use little space (much less than would be needed to store the entire stream) and little time to process each item in the stream. They approximate with small relative error. Using different techniques, we show how to approximate the L^p-difference ∑ _i|a_i-b_i|^p for any rational-valued p∈(0,2], with comparable efficiency and error. We also show how to approximate ∑ _i|a_i-b_i|^p for larger values of p but with a worse error guarantee. Our results fill in gaps left by recent work, by providing an algorithm that is precisely tunable for the application at hand. These results can be used to assess the difference between two chronologically or physically separated massive data sets, making one quick pass over each data set, without buffering the data or requiring the data source to pause. For example, one can use our techniques to judge whether the traffic on two remote network routers are similar without requiring either router to transmit a copy of its traffic. A web search engine could use such algorithms to construct a library of small ''sketches,'' one for each distinct page on the web; one can approximate the extent to which new web pages duplicate old ones by comparing the sketches of the web pages. Such techniques will become increasingly important as the enormous scale, distributional nature, and one-pass processing requirements of data sets become more commonplace.

On linear layouts of graphs

Abstract: In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \par

Pseudo-BCK Algebras: An Extension of BCK Algebras

Abstract: We extend BCK algebras to pseudo-BCK algebras, as MV algebras and BL algebras were extended to pseudo-MV algebras and pseudo-BL algebras, respectively. We make the connection with pseudo-MV algebras and with pseudo-BL algebras.

Some exactly solvable models of urn process theory

Abstract: We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Polya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Polya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.