scholarly journals The Sum-Free Process

10.37236/8095 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Patrick Bennett
Keyword(s):  
Lower Bound ◽  
General Result ◽  
High Probability ◽  
Independent Set ◽  
Final Size ◽  
Sharp Threshold ◽  
Free Process

$S \subseteq \mathbb{Z}_{2n}$ is said to be sum-free if $S$ has no solution to the equation $a+b=c$. The sum-free process on $\mathbb{Z}_{2n}$ starts with $S:=\varnothing$, and iteratively inserts elements of $\mathbb{Z}_{2n}$, where each inserted element is chosen uniformly at random from the set of all elements that could be inserted while maintaining that $S$ is sum-free. We prove a lower bound (which holds with high probability) on the final size of $S$, which matches a more general result of Bennett and Bohman, and also matches the order of a sharp threshold result proved by Balogh, Morris and Samotij. We also show that the set $S$ produced by the process has a particular non-pseudorandom property, which is in contrast with several known results about the random greedy independent set process on hypergraphs.

scholarly journals The Cyclic Triangle-Free Process

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 955
Author(s):  
Yu Jiang ◽  
Meilian Liang ◽  
Yanmei Teng ◽  
Xiaodong Xu
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Independent Set ◽  
Ramsey Number ◽  
Free Process ◽  
Vertex Transitive ◽  
Cyclic Graphs ◽  
Positive Integers ◽  
General Graphs ◽  
Independence Numbers

For positive integers s and t, the Ramsey number R ( s , t ) is the smallest positive integer n such that every graph of order n contains either a clique of order s or an independent set of order t. The triangle-free process begins with an empty graph of order n, and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It has been an important tool in studying the asymptotic lower bound for R ( 3 , t ) . Cyclic graphs are vertex-transitive. The symmetry of cyclic graphs makes it easier to compute their independent numbers than related general graphs. In this paper, the cyclic triangle-free process is studied. The sizes of the parameter sets and the independence numbers of the graphs obtained by the cyclic triangle-free process are studied. Lower bounds on R ( 3 , t ) for small t’s are computed, and R ( 3 , 35 ) ≥ 237 , R ( 3 , 36 ) ≥ 244 , R ( 3 , 37 ) ≥ 255 , R ( 3 , 38 ) ≥ 267 , etc. are obtained based on the graphs obtained by the cyclic triangle-free process. Finally, some problems on the cyclic triangle-free process and R ( 3 , t ) are proposed.

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

scholarly journals Leader Election Requires Logarithmic Time in Population Protocols

2020 ◽  
Vol 30 (01) ◽  
pp. 2050005 ◽  
Author(s):  
Yuichi Sudo ◽  
Toshimitsu Masuzawa
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Leader Election ◽  
Stabilization Time ◽  
Population Protocols ◽  
Population Protocol ◽  
Time Optimal ◽  
Exact Size

This paper shows that every leader election protocol requires logarithmic stabilization time both in expectation and with high probability in the population protocol model. This lower bound holds even if each agent has knowledge of the exact size of a population and is allowed to use an arbitrarily large number of agent states. This lower bound concludes that the protocol given in [Sudo et al., SSS 2019] is time-optimal in expectation.

New bounds on the independence number of connected graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850069
Author(s):  
Nader Jafari Rad ◽  
Elahe Sharifi
Keyword(s):  
Lower Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics  179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

Synchronizing Almost-Group Automata

2020 ◽  
pp. 1-22
Author(s):  
Mikhail V. Berlinkov ◽  
Cyril Nicaud
Keyword(s):  
Lower Bound ◽  
Efficient Algorithm ◽  
High Probability ◽  
Worst Case ◽  
Average Case ◽  
Model Of Computation ◽  
Letter Alphabet ◽  
Strongly Connected ◽  
Small Change ◽  
Random Automata

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on [Formula: see text] states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability. More precisely, we establish that the probability that a strongly-connected almost-group automaton is not synchronizing is [Formula: see text], for a [Formula: see text]-letter alphabet. We also present an efficient algorithm that decides whether a strongly-connected almost-group automaton is synchronizing. For a natural model of computation, we establish a [Formula: see text] worst-case lower bound for this problem ([Formula: see text] for the average case), which is almost matched by our algorithm.

A lower-bound limit pressure analysis for the oblique intersection of a flush cylindrical nozzle and the torus of a cylindrical vessel with a torispherical end

1974 ◽  
Vol 9 (4) ◽  
pp. 247-262 ◽  
Author(s):  
K S Dinno ◽  
S S Gill
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Independent Set ◽  
Cylindrical Vessel ◽  
Linear Programming Method ◽  
Non Linear Programming ◽  
Cylindrical Nozzle ◽  
Limit Pressure ◽  
Lower Bound Limit Analysis ◽  
Pressure Analysis

A lower-bound limit analysis is presented for the calculation of the limit pressure for the oblique intersection of a flush cylindrical nozzle and the torus of a cylindrical vessel with a torispherical end. A rotationally asymmetric formulation of stress resultants is specified in assumed plastic regions in terms of an independent set of variables and the limit pressure is computed for a limited number of geometric parameters by a non-linear programming method.

The Final Size of the $C_{\ell}$-free Process

2014 ◽  
Vol 28 (3) ◽  
pp. 1276-1305 ◽  
Author(s):  
Michael E. Picollelli
Keyword(s):  
Final Size ◽  
Free Process

Infection Spread in Random Geometric Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 164-181 ◽  
Author(s):  
Ghurumuruhan Ganesan
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Contact Process ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Long Time ◽  
Explicit Lower Bound ◽  
Infection Spread

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

scholarly journals Learning Immune-Defectives Graph through Group Tests

2015 ◽  
Author(s):  
Abhinav Ganesan ◽  
Sidharth Jaggi ◽  
Venkatesh Saligrama
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
High Probability ◽  
Chemical Compounds ◽  
Pathogen Identification ◽  
Pooling Design ◽  
Two Stage ◽  
Decoding Algorithms ◽  
Lead Compounds ◽  
Multiplicative Factor

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Here, the ``lead compounds'' are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of ``ineffective'' chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive iff it does not contain its ``associated'' inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their ``associations'' with high probability, or in other words, learn the Immune Defectives Graph (IDG). We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. For the non-adaptive pooling design, in the large inhibitor regime, we show that the upper bound exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of inhibitors.

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