scholarly journals Symmetric Distributed Termination

1985 ◽  
Vol 14 (189) ◽  
Author(s):  
Ole Eriksen ◽  
Sven Skyum
Keyword(s):  
Upper Bound ◽  
Simple Algorithm ◽  
Distributed Computation ◽  
Directed Networks ◽  
Strongly Connected ◽  
Termination Problem ◽  
Special Master

In this paper we present a simple algorithm for deciding when to terminate a distributed computation. Former solutions to the termination problem are restricted to rings and undirected connected networks, and they rely on the existence of a special master processor present in the network. The class of configurations for which our algorithm is applicable is the most general possible, namely the class of configurations, where the underlying (directed) networks are strongly connected. If the network is not strongly connected then there exists a group of processors which cannot send messages to the rest and it becomes impossible to detect when to terminate. Furthermore, the algorithm does not require a special master processor acting differently from the others. The only information needed is an upper bound on the diameter of the network (the number of processors, for example). The protocol for all processors is in other words identical.

scholarly journals Randomized Construction of Complexes with Large Diameter

2020 ◽  
Author(s):  
Francisco Criado ◽  
Andrew Newman
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Large Diameter ◽  
Simplicial Complexes ◽  
Similar Improvement ◽  
Strongly Connected ◽  
Probabilistic Construction ◽  
Combinatorial Diameter

Abstract We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected $$(d-1)$$ ( d - 1 ) -dimensional simplicial complexes on n vertices, denoted $$H_s(n, d)$$ H s ( n , d ) . Using a probabilistic construction we give a new lower bound on $$H_s(n, d)$$ H s ( n , d ) that is within an $$O(d^2)$$ O ( d 2 ) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of $$e^{\varTheta (d)}$$ e Θ ( d ) of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

Inflation of strongly connected networks

2011 ◽  
Vol 150 (2) ◽  
pp. 367-384 ◽  
Author(s):  
NIKITA AGARWAL
Keyword(s):  
Sufficient Conditions ◽  
Simple Algorithm ◽  
Necessary And Sufficient Conditions ◽  
Connected Network ◽  
Cell Network ◽  
Strongly Connected ◽  
Necessary And Sufficient ◽  
Strongly Connected Network

AbstractA coupled cell network is an inflation of if the dynamics of is embedded in as a quotient network. We give necessary and sufficient conditions for the existence of a strongly connected inflation of a strongly connected network. We provide a simple algorithm for the construction of a strongly connected inflation as a sequence of simple inflations.

scholarly journals Upper bound for the energy of strongly connected digraphs

2011 ◽  
Vol 5 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Singaraj Ayyaswamy ◽  
Selvaraj Balachandran ◽  
Ivan Gutman
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Strongly Regular Graph ◽  
Simple Graphs ◽  
Strongly Connected Digraph ◽  
Complex Eigenvalues ◽  
Connected Digraph ◽  
Strongly Connected ◽  
Strongly Regular ◽  
Appl Math

The energy of a digraph D is defined as E(D) = ?n,i=1 ?Re(zi)?, where z1, z2, ..., zn are the (possibly complex) eigenvalues of D . We show that if D is a strongly connected digraph on n vertices, a arcs, and c2 closed walks of length two, such that Re(z1) ? (a + c2)=(2n) ? 1 , then E(D) ? n(1 + ?n)=2. Equality holds if and only if D is a directed strongly regular graph with parameters (n, n+?n/2, 3n+2?n/8, n+2?n/8, n+2?n/8). This bound extends to digraphs an earlier result [J. H. Koolen, V. Moulton:, Maximal energy graphs. Adv. Appl. Math., 26 (2001), 47-52], obtained for simple graphs.

scholarly journals Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

2016 ◽  
Vol 27 (02) ◽  
pp. 127-145 ◽  
Author(s):  
Jorge Almeida ◽  
Emanuele Rodaro
Keyword(s):  
Upper Bound ◽  
Broad Class ◽  
Regular Languages ◽  
Theoretic Approach ◽  
Synchronizing Automata ◽  
Natural Notion ◽  
Strongly Connected ◽  
Synchronizing Automaton ◽  
Synchronizing Word ◽  
Černý’S Conjecture

We present a ring theoretic approach to Černý's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý's series. Semisimplicity gives also the advantage of “factorizing” the problem of finding a synchronizing word into the sub-problems of finding “short” words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most [Formula: see text] where n is the number of states of the automaton. We show that the solution of this “Radical Conjecture” would give an upper bound [Formula: see text] for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and Černý's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

scholarly journals A modified GADIA-based upper-bound to the capacity of Gaussian general N-relay networks

2021 ◽  
Author(s):  
Zekeriya Uykan ◽  
Riku Jäntti
Keyword(s):  
Upper Bound ◽  
Simple Algorithm ◽  
Relay Network ◽  
Maximization Problem ◽  
Interference Avoidance ◽  
Gain Matrix ◽  
Power Domain ◽  
Information Maximization ◽  
Capacity Bound ◽  
Min Cut

AbstractIn this paper, we present a general Gaussian N-relay network by allowing relays to communicate to each other and allowing a direct channel between source and destination as compared to the standard diamond network in Nazaroğlu et al. (IEEE Trans Inf Theory 60:6329–6341, 2014) at the cost of extra channel uses. Our main focus is to examine the min-cut bound capacities of the relay network. Very recently, the results in Uykan (IEEE Trans Neural Netw Learn Syst 31:3294–3304, 2020) imply that the GADIA in Babadi and Tarokh (IEEE Trans Inf Theory 56:6228–6252, 2010), a pioneering algorithm in the interference avoidance literature, actually performs max-cut of a given power-domain (nonnegative) link gain matrix in the 2-channel case. Using the results of the diamond network in Nazaroğlu et al. (2014) and the results in Uykan (2020), in this paper, we (i) turn the mutual information maximization problem in the Gaussian N-relay network into an upper bound minimization problem, (ii) propose a modified GADIA-based algorithm to find the min-cut capacity bound and (iii) present an upper and a lower bound to its min-cut capacity bound using the modified GADIA as applied to the defined “squared channel gain matrix/graph”. Some advantages of the proposed modified GADIA-based simple algorithm are as follows: (1) The Gaussian N-relay network can determine the relay clusters in a distributed fashion and (2) the presented upper bound gives an insight into whether allowing the relays to communicate to each other pays off the extra channel uses or not as far as the min-cut capacity bound is concerned. The simulation results confirm the findings. Furthermore, the min-cut upper bound found by the proposed modified-GADIA is verified by the cut-set bounds found by the spectral clustering based solutions as well.

scholarly journals Giant strongly connected component of directed networks

2001 ◽  
Vol 64 (2) ◽  
Author(s):  
S. N. Dorogovtsev ◽  
J. F. F. Mendes ◽  
A. N. Samukhin
Keyword(s):  
Connected Component ◽  
Directed Networks ◽  
Strongly Connected Component ◽  
Strongly Connected

Exponential Consensus in Directed Networks of Multi-Agents with Saturated Protocols

2016 ◽  
Vol 16 (02) ◽  
pp. 1650003
Author(s):  
QINGLING WANG ◽  
YUANDA WANG
Keyword(s):  
Lyapunov Function ◽  
Directed Graph ◽  
Spanning Tree ◽  
Directed Graphs ◽  
Consensus Problem ◽  
Directed Networks ◽  
Strongly Connected ◽  
Theoretical Results ◽  
Multi Agents

This paper addresses the exponential consensus problem of single-integrator agents with saturated protocols on directed graphs. By employing an integral Lyapunov function, the exponential consensus problem of single-integrator agents is solved under the directed graph with strongly connected or a spanning tree. The main contribution is that under the directed graph, some conditions for exponential consensus with saturated protocols are first obtained. Finally, two examples are used to illustrate the effectiveness of the theoretical results.

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