scholarly journals Minimizing the Alphabet Size in Codes with Restricted Error Sets

2021 ◽  
Author(s):  
Mira Gonen ◽  
Michael Langberg ◽  
Alex Sprintson
Keyword(s):  
Alphabet Size

Reducing Local Alphabet Size in Recognizable Picture Languages

2021 ◽  
pp. 103-116
Author(s):  
Stefano Crespi Reghizzi ◽  
Antonio Restivo ◽  
Pierluigi San Pietro
Keyword(s):  
Alphabet Size ◽  
Picture Languages

Concentration inequalities for the empirical distribution of discrete distributions: beyond the method of types

2019 ◽  
Vol 9 (4) ◽  
pp. 813-850 ◽  
Author(s):  
Jay Mardia ◽  
Jiantao Jiao ◽  
Ervin Tánczos ◽  
Robert D Nowak ◽  
Tsachy Weissman
Keyword(s):  
Sample Size ◽  
Empirical Distribution ◽  
Discrete Distributions ◽  
Concentration Inequalities ◽  
Sample Sizes ◽  
Alphabet Size ◽  
Large Sample ◽  
Kl Divergence ◽  
The Difference ◽  
True Distribution

Abstract We study concentration inequalities for the Kullback–Leibler (KL) divergence between the empirical distribution and the true distribution. Applying a recursion technique, we improve over the method of types bound uniformly in all regimes of sample size $n$ and alphabet size $k$, and the improvement becomes more significant when $k$ is large. We discuss the applications of our results in obtaining tighter concentration inequalities for $L_1$ deviations of the empirical distribution from the true distribution, and the difference between concentration around the expectation or zero. We also obtain asymptotically tight bounds on the variance of the KL divergence between the empirical and true distribution, and demonstrate their quantitatively different behaviours between small and large sample sizes compared to the alphabet size.

General Methodology to Identify the Minimum Alphabet Size for Heteropolymer Design

2019 ◽  
Vol 2 (7) ◽  
pp. 1900031 ◽  
Author(s):  
Chiara Cardelli ◽  
Francesca Nerattini ◽  
Luca Tubiana ◽  
Valentino Bianco ◽  
Christoph Dellago ◽  
...  
Keyword(s):  
Alphabet Size ◽  
General Methodology

scholarly journals ABELIAN PRIMITIVE WORDS

2012 ◽  
Vol 23 (05) ◽  
pp. 1021-1033 ◽  
Author(s):  
MICHAEL DOMARATZKI ◽  
NARAD RAMPERSAD
Keyword(s):  
Linear Time ◽  
Alphabet Size ◽  
Primitive Words ◽  
Context Free

We investigate Abelian primitive words, which are words that are not Abelian powers. We show the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time (for fixed alphabet size). Also differently from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration of Abelian primitive words.

AVERAGE VALUE OF SUM OF EXPONENTS OF RUNS IN A STRING

2009 ◽  
Vol 20 (06) ◽  
pp. 1135-1146 ◽  
Author(s):  
KAZUHIKO KUSANO ◽  
WATARU MATSUBARA ◽  
AKIRA ISHINO ◽  
AYUMI SHINOHARA
Keyword(s):  
Upper Bound ◽  
Unit Length ◽  
Closed Formula ◽  
Alphabet Size ◽  
Average Value ◽  
Limit Value ◽  
Value Of It ◽  
Binary Strings

A substring w[i.j] in w is called a repetition of period p if w[k] = w[k + p] for any i ≤ k ≤ j - p. Especially, a maximal repetition, which cannot be extended neither to left nor to right, is called a run. The ratio of the length of the run to its period, i.e. [Formula: see text], is called an exponent. The sum of exponents of runs in a string is of interest. The maximal value of the sum is still unknown, and the current upper bound is 2.9n given by Crochemore and Ilie, where n is the length of a string. In this paper we show a closed formula which exactly expresses the average value of it for any n and any alphabet size, and the limit of this value per unit length as n approaches infinity. For binary strings, the limit value is approximately 1.13103. We also show the average number of squares in a string of length n and its limit value.

ON THE AVERAGE SIZE OF GLUSHKOV AND PARTIAL DERIVATIVE AUTOMATA

2012 ◽  
Vol 23 (05) ◽  
pp. 969-984 ◽  
Author(s):  
SABINE BRODA ◽  
ANTÓNIO MACHIAVELO ◽  
NELMA MOREIRA ◽  
ROGÉRIO REIS
Keyword(s):  
Partial Derivative ◽  
Upper Bound ◽  
Regular Expression ◽  
Regular Expressions ◽  
Average Case ◽  
Alphabet Size ◽  
Large Alphabet ◽  
Exact Counting ◽  
Average Transition ◽  
Average Size

In this paper, the relation between the Glushkov automaton [Formula: see text] and the partial derivative automaton [Formula: see text] of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of [Formula: see text] was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of [Formula: see text]. Here we present a new quadratic construction of [Formula: see text] that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of [Formula: see text] to the number of states of [Formula: see text] which is about ½ for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in [Formula: see text], which we then use to get an average case approximation. In conclusion, assymptotically, and for large alphabets, the size of [Formula: see text] is half the size of the [Formula: see text]. This is corroborated by some experiments, even for small alphabets and small regular expressions.

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