53,796 research outputs found
Trellis decoding complexity of linear block codes
In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of âuniformâ optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths
One-way permutations, computational asymmetry and distortion
Computational asymmetry, i.e., the discrepancy between the complexity of
transformations and the complexity of their inverses, is at the core of one-way
transformations. We introduce a computational asymmetry function that measures
the amount of one-wayness of permutations. We also introduce the word-length
asymmetry function for groups, which is an algebraic analogue of computational
asymmetry. We relate boolean circuits to words in a Thompson monoid, over a
fixed generating set, in such a way that circuit size is equal to word-length.
Moreover, boolean circuits have a representation in terms of elements of a
Thompson group, in such a way that circuit size is polynomially equivalent to
word-length. We show that circuits built with gates that are not constrained to
have fixed-length inputs and outputs, are at most quadratically more compact
than circuits built from traditional gates (with fixed-length inputs and
outputs). Finally, we show that the computational asymmetry function is closely
related to certain distortion functions: The computational asymmetry function
is polynomially equivalent to the distortion of the path length in Schreier
graphs of certain Thompson groups, compared to the path length in Cayley graphs
of certain Thompson monoids. We also show that the results of Razborov and
others on monotone circuit complexity lead to exponential lower bounds on
certain distortions.Comment: 33 page
Permutation Complexity and Coupling Measures in Hidden Markov Models
In [Haruna, T. and Nakajima, K., 2011. Physica D 240, 1370-1377], the authors
introduced the duality between values (words) and orderings (permutations) as a
basis to discuss the relationship between information theoretic measures for
finite-alphabet stationary stochastic processes and their permutation
analogues. It has been used to give a simple proof of the equality between the
entropy rate and the permutation entropy rate for any finite-alphabet
stationary stochastic process and show some results on the excess entropy and
the transfer entropy for finite-alphabet stationary ergodic Markov processes.
In this paper, we extend our previous results to hidden Markov models and show
the equalities between various information theoretic complexity and coupling
measures and their permutation analogues. In particular, we show the following
two results within the realm of hidden Markov models with ergodic internal
processes: the two permutation analogues of the transfer entropy, the symbolic
transfer entropy and the transfer entropy on rank vectors, are both equivalent
to the transfer entropy if they are considered as the rates, and the directed
information theory can be captured by the permutation entropy approach.Comment: 26 page
Generating All Permutations by Context-Free Grammars in Greibach Normal Form
We consider context-free grammars in Greibach normal form and, particularly, in Greibach -form () which generates the finite language of all strings that are permutations of different symbols (). These grammars are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols and the number of production rules of as functions of . As in the case of Chomsky normal form these descriptional complexity measures grow faster than any polynomial function
Computational Complexity Results for Genetic Programming and the Sorting Problem
Genetic Programming (GP) has found various applications. Understanding this
type of algorithm from a theoretical point of view is a challenging task. The
first results on the computational complexity of GP have been obtained for
problems with isolated program semantics. With this paper, we push forward the
computational complexity analysis of GP on a problem with dependent program
semantics. We study the well-known sorting problem in this context and analyze
rigorously how GP can deal with different measures of sortedness.Comment: 12 page
Generating all permutations by context-free grammars in Chomsky normal form
Let Ln be the finite language of all n! strings that are permutations of n different symbols (n1). We consider context-free grammars Gn in Chomsky normal form that generate Ln. In particular we study a few families {Gn}n1, satisfying L(Gn)=Ln for n1, with respect to their descriptional complexity, i.e. we determine the number of nonterminal symbols and the number of production rules of Gn as functions of n
Compressed Representations of Permutations, and Applications
We explore various techniques to compress a permutation over n
integers, taking advantage of ordered subsequences in , while supporting
its application (i) and the application of its inverse in
small time. Our compression schemes yield several interesting byproducts, in
many cases matching, improving or extending the best existing results on
applications such as the encoding of a permutation in order to support iterated
applications of it, of integer functions, and of inverted lists and
suffix arrays
Analysis of Algorithms for Permutations Biased by Their Number of Records
The topic of the article is the parametric study of the complexity of
algorithms on arrays of pairwise distinct integers. We introduce a model that
takes into account the non-uniformness of data, which we call the Ewens-like
distribution of parameter for records on permutations: the weight
of a permutation depends on its number of records. We show that
this model is meaningful for the notion of presortedness, while still being
mathematically tractable. Our results describe the expected value of several
classical permutation statistics in this model, and give the expected running
time of three algorithms: the Insertion Sort, and two variants of the Min-Max
search
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