scholarly journals On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time

Information ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 254
Author(s):  
Matthew Connor ◽  
Othon Michail ◽  
Paul Spirakis
Keyword(s):  
Polynomial Time ◽  
High Probability ◽  
Network Construction ◽  
Parallel Time ◽  
Finite State ◽  
Partial Characterisation ◽  
Regular Networks ◽  
State Assumption ◽  
Stable Network ◽  
Anonymous Agents

We study the class of networks, which can be created in polylogarithmic parallel time by network constructors: groups of anonymous agents that interact randomly under a uniform random scheduler with the ability to form connections between each other. Starting from an empty network, the goal is to construct a stable network that belongs to a given family. We prove that the class of trees where each node has any k≥2 children can be constructed in O(logn) parallel time with high probability. We show that constructing networks that are k-regular is Ω(n) time, but a minimal relaxation to (l,k)-regular networks, where l=k−1, can be constructed in polylogarithmic parallel time for any fixed k, where k>2. We further demonstrate that when the finite-state assumption is relaxed and k is allowed to grow with n, then k=loglogn acts as a threshold above which network construction is, again, polynomial time. We use this to provide a partial characterisation of the class of polylogarithmic time network constructors.

scholarly journals Copyful Streaming String Transducers

2021 ◽  
Vol 178 (1-2) ◽  
pp. 59-76
Author(s):  
Emmanuel Filiot ◽  
Pierre-Alain Reynier
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Expressive Power ◽  
The Other ◽  
Finite State Automata ◽  
Finite State ◽  
Variable Content ◽  
Deterministic Automata ◽  
Intermediate Output ◽  
Linear Manner

Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Černý in 2010 as a one-way deterministic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i)HDT0L systems and total deterministic copyful SST have the same expressive power, (ii)the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in quadratic time. As a consequence, equivalence of deterministic SST is decidable, (iii)the functionality of non-deterministic copyful SST is decidable, (iv)determining whether a non-deterministic copyful SST can be transformed into an equivalent non-deterministic copyless SST is decidable in polynomial time.

scholarly journals Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

2021 ◽  
Author(s):  
Thomas Bläsius ◽  
Philipp Fischbeck ◽  
Tobias Friedrich ◽  
Maximilian Katzmann
Keyword(s):  
Computational Complexity ◽  
Structural Properties ◽  
Random Graphs ◽  
Polynomial Time ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Vertex Cover ◽  
Optimal Solution ◽  
Np Complete

AbstractThe computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.

INFIX-FREE REGULAR EXPRESSIONS AND LANGUAGES

2006 ◽  
Vol 17 (02) ◽  
pp. 379-393 ◽  
Author(s):  
YO-SUB HAN ◽  
YAJUN WANG ◽  
DERICK WOOD
Keyword(s):  
Polynomial Time ◽  
Regular Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Regular Languages ◽  
Finite State Automaton ◽  
Primality Test ◽  
Finite State ◽  
State Pair ◽  
Free Decomposition

We study infix-free regular languages. We observe the structural properties of finite-state automata for infix-free languages and develop a polynomial-time algorithm to determine infix-freeness of a regular language using state-pair graphs. We consider two cases: 1) A language is specified by a nondeterministic finite-state automaton and 2) a language is specified by a regular expression. Furthermore, we examine the prime infix-free decomposition of infix-free regular languages and design an algorithm for the infix-free primality test of an infix-free regular language. Moreover, we show that we can compute the prime infix-free decomposition in polynomial time. We also demonstrate that the prime infix-free decomposition is not unique.

scholarly journals Computing the Expected Edit Distance from a String to a Probabilistic Finite-State Automaton

2017 ◽  
Vol 28 (05) ◽  
pp. 603-621 ◽  
Author(s):  
Jorge Calvo-Zaragoza ◽  
Jose Oncina ◽  
Colin de la Higuera
Keyword(s):  
Polynomial Time ◽  
Edit Distance ◽  
Input String ◽  
Finite State Automaton ◽  
Global View ◽  
Finite State ◽  
Median String

In a number of fields, it is necessary to compare a witness string with a distribution. One possibility is to compute the probability of the string for that distribution. Another, giving a more global view, is to compute the expected edit distance from a string randomly drawn to the witness string. This number is often used to measure the performance of a prediction, the goal then being to return the median string, or the string with smallest expected distance. To be able to measure this, computing the distance between a hypothesis and that distribution is necessary. This paper proposes two solutions for computing this value, when the distribution is defined with a probabilistic finite state automaton. The first is exact but has a cost which can be exponential in the length of the input string, whereas the second is a fully polynomial-time randomized schema.

LISAS — Simulation tool for regular networks of finite state machines

1991 ◽  
Vol 32 (1-5) ◽  
pp. 645-650
Author(s):  
T. Müller-Wipperfürth ◽  
H. Hellwagner ◽  
F. Pichler
Keyword(s):  
Finite State Machines ◽  
Simulation Tool ◽  
State Machines ◽  
Finite State ◽  
Regular Networks

scholarly journals On the Complexity of Deciding Behavioural Equivalences and Preorders. A Survey

1996 ◽  
Vol 3 (39) ◽  
Author(s):  
Hans Hüttel ◽  
Sandeep Shukla
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Branching Time ◽  
Complexity Results ◽  
Special Cases ◽  
Finite State ◽  
Bisimulation Equivalence ◽  
Infinite State ◽  
Context Free

This paper gives an overview of the computational complexity of all the equivalences in the linear/branching time hierarchy [vG90a] and the preorders<br />in the corresponding hierarchy of preorders. We consider finite state or regular processes as well as infinite-state BPA [BK84b] processes. <br />A distinction, which turns out to be important in the finite-state processes, is that of simulation-like equivalences/preorders vs. trace-like equivalences<br />and preorders. Here we survey various known complexity results for these relations. For regular processes, all simulation-like equivalences and preorders are decidable in polynomial time whereas all trace-like equivalences and preorders are PSPACE-Complete. We also consider interesting special<br />classes of regular processes such as deterministic, determinate, unary, locally unary, and tree-like processes and survey the known complexity results in<br />these special cases. For infinite-state processes the results are quite different. For the class of context-free processes or BPA processes any preorder or equivalence beyond bisimulation is undecidable but bisimulation equivalence is polynomial time<br />decidable for normed BPA processes and is known to be elementarily decidable in the general case. For the class of BPP processes, all preorders and equivalences apart from bisimilarity are undecidable. However, bisimilarity<br />is decidable in this case and is known to be decidable in polynomial time for normed BPP processes.

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