The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes

Bodini, Olivier; Dien, Matthieu; Genitrini, Antoine; Peschanski, Frédéric

doi:10.1137/1.9781611974775.2

Cited by 5 publications

(12 citation statements)

References 10 publications

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“…Analysis of the functional equation -heuristics. Now we turn to the actual enumeration problem which amounts to the analysis of the function given in (3). First, we read off coefficients in (3) and get a recurrence relation for B n = [z n ]B(z), which is, of course, in compliance with (1): starting with B 2 = 1, we then get…”

Section: 3mentioning

confidence: 99%

“…Finally, by merging the nodes with the same label we obtain directed acyclic graphs whose nodes are increasingly labeled (without repetitions). Such an approach introduces a new model of concurrent processes with synchronization that induces processes whose description is more expressive than the classical series-parallel model that we have studied in [3,2].…”

Section: Introductionmentioning

confidence: 99%

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On the number of increasing trees with label repetitions

Bodini

Genitrini

Gittenberger

et al. 2020

Discrete Mathematics

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We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the same label can appear in distinct branches of the tree.In the course of the analysis we develop a method to extract asymptotic information on the coefficients of purely formal power series. The method is based on an approximate Borel transform (or, more generally, Mittag-Leffler transform) which enables us to quickly guess the exponential growth rate. With this guess the sequence is then rescaled and a singularity analysis of the generating function of the scaled counting sequence yields accurate asymptotics. The actual analysis is based on differential equations and a Tauberian argument.The counting problem for trees of size n exhibits interesting asymptotics involving powers of n with irrational exponents.

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the number of increasing trees with label repetitions

Bodini

Genitrini

Gittenberger

et al. 2020

Discrete Mathematics

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“…Combinatorially, the fact that nullable(P ) holds corresponds to the fact that the class of its executions contains one object of size 0: the empty execution. It is in fact important that we can express this in terms of disjoint unions because they fit in the framework of analytic combinatorics whereas arbitrary unions are more difficult to handle 6 .…”

Section: Combinatorial Interpretationmentioning

confidence: 99%

“…The rules of the symbolic method for computing a generating function from a combinatorial specification. In the case of the labelled product A B, the corresponding operation on the series is called the coloured product and is defined in [6] by…”

Section: Combinatorial Interpretationmentioning

confidence: 99%

“…The coloured product used in the parallel composition case can be implemented using the formula A(z) B(z) = Lap(Bor(A) • Bor(B)) where Bor and Lap are respectively the combinatorial Borel and Laplace transforms (see [6] for insights on the coloured product). This approach has the advantage of benefiting from the efficient polynomial multiplication algorithms from the literature at the cost of three linear transformations.…”

Section: Preprocessing: the Generating Function Of Executionsmentioning

confidence: 99%

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Statistical Analysis of Non-deterministic Fork-Join Processes

Genitrini¹,

Pépin²,

Peschanski³

2020

Lecture Notes in Computer Science

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We study the combinatorial structure of concurrent programs with non-deterministic choice and a fork-join style of coordination. As a first step we establish a link between these concurrent programs and a class of combinatorial structures. Based on this combinatorial interpretation, we develop and experiment algorithms aimed at the statistical exploration of the state-space of programs. The first algorithm is a uniform random sampler of bounded executions, providing a suitable default exploration strategy. The second algorithm is a random sampler of execution prefixes that allows to control the exploration with respect to the uniform distribution. The fundamental characteristic of these algorithms is that they work on the control graph of the programs and not directly on their state-space, thus providing a way to tackle the state explosion problem.

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Asymptotic enumeration of compacted binary trees of bounded right height

Genitrini

Gittenberger

Kauers

et al. 2020

Journal of Combinatorial Theory, Series A

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A compacted tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf.We solve the asymptotic counting problem for this class as well as a closely related, further simplified class.For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations.Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity. Proposition 4.3. From any binary tree of size n, we can build a compacted tree of size n, with the following operations:1. Add a leaf as left child of the leftmost node of the binary tree.2. Add pointers to every node such that every node except the leaf has out-degree 2.

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The Ordered and Colored Products in Analytic Combinatorics: Application to the Quantitative Study of Synchronizations in Concurrent Processes

Cited by 5 publications

References 10 publications

On the number of increasing trees with label repetitions

On the number of increasing trees with label repetitions

Statistical Analysis of Non-deterministic Fork-Join Processes

Asymptotic enumeration of compacted binary trees of bounded right height

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