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 Error locating: degree constraints

2021 ◽  
Author(s):  
Christopher Dennis
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Mathematical Tool ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Special Cases ◽  
Locating Array ◽  
General Error

Error graphs are a useful mathematical tool for representing failing interactions in a system. This representation is used as the basis for constructing an error locating array (ELA). However, if too many errors are present in a given error graph, it may not be possible to locate all interactions. We say that a graph is locatable if an ELA can be built. Bounds on the total size of an error graph are known, bounds on the degree an error graph can have have not been considered. In this thesis we explore the maximum degree an error graph may have while still guaranteeing its locatability. We consider special cases for 3 and 4 partite error graphs as well as developing bounds on the degree of a general error graph. We describe a linear time algorithm which can be used to generate tests which have at most one failing interaction.

scholarly journals Comparing Degenerate Strings

2020 ◽  
Author(s):  
Mai Alzamel ◽  
Lorraine A.K. Ayad ◽  
Giulia Bernardini ◽  
Roberto Grossi ◽  
Costas S. Iliopoulos ◽  
...  
Keyword(s):  
Lower Bound ◽  
Linear Time ◽  
Simple Algorithm ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Empty Intersection ◽  
Compact Representations ◽  
String Comparison ◽  
Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, …, kn-1. Our main result is an O(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in O(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

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 Error locating: degree constraints

2021 ◽  
Author(s):  
Christopher Dennis
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Mathematical Tool ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Special Cases ◽  
Locating Array ◽  
General Error

Error graphs are a useful mathematical tool for representing failing interactions in a system. This representation is used as the basis for constructing an error locating array (ELA). However, if too many errors are present in a given error graph, it may not be possible to locate all interactions. We say that a graph is locatable if an ELA can be built. Bounds on the total size of an error graph are known, bounds on the degree an error graph can have have not been considered. In this thesis we explore the maximum degree an error graph may have while still guaranteeing its locatability. We consider special cases for 3 and 4 partite error graphs as well as developing bounds on the degree of a general error graph. We describe a linear time algorithm which can be used to generate tests which have at most one failing interaction.

scholarly journals Comparing Degenerate Strings

2020 ◽  
Vol 175 (1-4) ◽  
pp. 41-58
Author(s):  
Mai Alzamel ◽  
Lorraine A.K. Ayad ◽  
Giulia Bernardini ◽  
Roberto Grossi ◽  
Costas S. Iliopoulos ◽  
...  
Keyword(s):  
Lower Bound ◽  
Linear Time ◽  
Simple Algorithm ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Empty Intersection ◽  
Compact Representations ◽  
String Comparison ◽  
Combinatorial Result

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

A Linear time algorithm for deciding security

1976 ◽  
Author(s):  
A. K. Jones ◽  
R. J. Lipton ◽  
L. Snyder
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm
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