scholarly journals Enumeration and Random Generation of Concurrent Computations

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Olivier Bodini ◽  
Antoine Genitrini ◽  
Frédéric Peschanski
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Random Generation ◽  
Worst Case ◽  
Analytic Combinatorics ◽  
Mean Width ◽  
Concurrent Processes ◽  
International Audience ◽  
The Mean

International audience In this paper, we study the shuffle operator on concurrent processes (represented as trees) using analytic combinatorics tools. As a first result, we show that the mean width of shuffle trees is exponentially smaller than the worst case upper-bound. We also study the expected size (in total number of nodes) of shuffle trees. We notice, rather unexpectedly, that only a small ratio of all nodes do not belong to the last two levels. We also provide a precise characterization of what ``exponential growth'' means in the case of the shuffle on trees. Two practical outcomes of our quantitative study are presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random generation of concurrent runs.

scholarly journals A Quantitative Study of Pure Parallel Processes

10.37236/3977 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
O. Bodini ◽  
A. Genitrini ◽  
F. Peschanski
Keyword(s):  
Quantitative Study ◽  
Random Sampling ◽  
Linear Time ◽  
Time Algorithm ◽  
Point Of View ◽  
Total Size ◽  
Combinatorial Explosion ◽  
Analytic Combinatorics ◽  
Parallel Processes ◽  
Uniform Random Sampling

In this paper, we study the interleaving – or pure merge – operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes the analysis of process behaviours e.g. by model-checking, very hard – at least from the point of view of computational complexity. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.

scholarly journals Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

scholarly journals Characterization of the larval stages of Alphitobius diaperinus (Panzer) (Coleoptera: Tenebrionidae) using head capsule width

2001 ◽  
Vol 61 (1) ◽  
pp. 125-131 ◽  
Author(s):  
O. FRANCISCO ◽  
A. P. do PRADO
Keyword(s):  
Age Structure ◽  
Developmental Rate ◽  
Head Capsule ◽  
Larval Instars ◽  
Alphitobius Diaperinus ◽  
Larval Stages ◽  
Mean Width ◽  
The Mean ◽  
Coleoptera Tenebrionidae

The mean width (n = 5) of the cephalic capsule instar of Alphitobius diaperinus was determined. The larvae were reared at 27ºC (± 0.1ºC). The result showed that A. diaperinus has eight larval instars. The head capsule of the 1st instar larvae measured x = 0.228 (SD = 0.0192) and the last instar larval measured x = 1.339 (SD = 0.0436). The developmental rate, determined by Dyar-Hutchinson's rule, was 1.29. These data may be useful for studies on phenology and age structure of A. diaperinus in the field.

CONSTANT-MEMORY ITERATIVE GENERATION OF SPECIAL STRINGS REPRESENTING BINARY TREES

2012 ◽  
Vol 23 (02) ◽  
pp. 375-387
Author(s):  
SEBASTIAN SMYCZYŃSKI
Keyword(s):  
Linear Time ◽  
Special Property ◽  
Time Algorithm ◽  
Lexicographic Order ◽  
Binary Trees ◽  
Natural Order ◽  
Constant Amount ◽  
Worst Case ◽  
Average Case ◽  
Additional Memory

The shapes of binary trees can be encoded as permutations having a very special property. These permutations are tree permutations, or equivalently they avoid subwords of the type 231. The generation of binary trees in natural order corresponds to the generation of these special permutations in the lexicographic order. In this paper we use a stringologic approach to the generation of these special permutations: decompositions of essential parts into the subwords having staircase shapes. A given permutation differs from the next one with respect to its tail called here the working suffix. Some new properties of such working suffixes are discovered in the paper and used to design effective algorithms transforming one tree permutation into its successor or predecessor in the lexicographic order. The algorithms use a constant amount of additional memory and they look only at those elements of the permutation which belong to the working suffix. The best-case, average-case and worst-case time complexities of the algorithms are O(1), O(1), and O(n) respectively. The advantages of our stringologic approach are constant time and iterative generation, while other known algorithms are usually recursive or not constant-memory ones. In this paper we also present a new compact non-recursive linear time algorithm solving a related problem of decoding the shape of a binary tree from its corresponding tree permutation.

A LINEAR-TIME ALGORITHM FOR TESTING THE INSCRIBABILITY OF TRIVALENT POLYHEDRA

1995 ◽  
Vol 05 (01n02) ◽  
pp. 21-36 ◽  
Author(s):  
MICHAEL B. DILLENCOURT ◽  
WARREN D. SMITH
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Combinatorial Characterization ◽  
Integer Arithmetic

We present an algorithm for testing the inscribability of a trivalent polyhedron or, equivalently, testing the circumscribability of a simplicial polyhedron. Our algorithm, which runs in linear time and uses only low-precision integer arithmetic, is based on a purely combinatorial characterization of inscribable trivalent polyhedra.

scholarly journals Liveness in broadcast networks

Computing ◽  
2021 ◽  
Author(s):  
Peter Chini ◽  
Roland Meyer ◽  
Prakash Saivasan
Keyword(s):  
Model Checking ◽  
Polynomial Time ◽  
Message Passing ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Final State ◽  
Broadcast Network ◽  
Broadcast Networks

AbstractWe study liveness and model checking problems for broadcast networks, a system model of identical clients communicating via message passing. The first problem that we consider is Liveness Verification. It asks whether there is a computation such that one clients visits a final state infinitely often. The complexity of the problem has been open. It was shown to be $$\texttt {P}$$ P -hard but in $$\texttt {EXPSPACE}$$ EXPSPACE . We close the gap by a polynomial-time algorithm. The latter relies on a characterization of live computations in terms of paths in a suitable graph, combined with a fixed-point iteration to efficiently check the existence of such paths. The second problem is Fair Liveness Verification. It asks for a computation where all participating clients visit a final state infinitely often. We adjust the algorithm to also solve fair liveness in polynomial time. Both problems can be instrumented to answer model checking questions for broadcast networks against linear time temporal logic specifications. The first problem in this context is Fair Model Checking. It demands that for all computations of a broadcast network, all participating clients satisfy the specification. We solve the problem via the Vardi–Wolper construction and a reduction to Liveness Verification. The second problem is Sparse Model Checking. It asks whether each computation has a participating client that satisfies the specification. We reduce the problem to Fair Liveness Verification.

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

scholarly journals Packing Three-Vertex Paths in a Subcubic Graph

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Adrian Kosowski ◽  
Michal Malafiejski ◽  
Pawel Zyliński
Keyword(s):  
Linear Time ◽  
Packing Problem ◽  
Time Algorithm ◽  
Vertex Degree ◽  
Linear Time Algorithm ◽  
Cubic Graph ◽  
Subcubic Graph ◽  
International Audience ◽  
Subcubic Graphs

International audience In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.

scholarly journals The Cycles of the Multiway Perfect Shuffle Permutation

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
John Ellis ◽  
Hongbing Fan ◽  
Jeffrey Shallit
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Equal Size ◽  
Linear Time Algorithm ◽  
Theoretic Analysis ◽  
Cycle Structure ◽  
International Audience ◽  
Perfect Shuffle

International audience The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

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