the information lost in erasures we consider sources and channels with memory observed through erasure channels in particular we examine the impact of sporadic erasures on the fundamental limits of lossless data compression lossy data compression channel coding and denoising we define the erasure entropy of a collection of random variables as the sum of entropies of the individual variables conditioned on all the rest the erasure entropy measures the information content carried by each symbol knowing its context the erasure entropy rate is shown to be the minimal amount of bits per erasure required to recover the lost information in the limit of small erasure probability when we allow recovery of the erased symbols within a prescribed degree of distortion the fundamental tradeoff is described by the erasure rate distortion function which we characterize we show that in the regime of sporadic erasures knowledge at the encoder of the erasure locations does not lower the rate required to achieve a given distortion when no additional encoded information is available the erased information is reconstructed solely on the basis of its context by a denoiser connections between erasure entropy and discrete denoising are developed the decrease of the capacity of channels with memory due to sporadic memoryless erasures is also characterized in wide generality

the reliability functions of the general source with fixed length coding the reliability function problems with fixed length source coding for the general source are studied for all rates r our fundamental philosophy in doing so is to convert all of the reliability function problems to the pertinent computation problems in the large derivation probability theory it turns out that this kind of new methodology which was previously developed by han see ibid vol 43 p 1145 64 1997 enables us to establish quite compact general formulas of the reliability function for general sources including all nonstationary and or nonergodic sources with countably infinite alphabet such general formulas are presented from the information spectrum point of view

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

attractors for iterated function schemes on 0 1 n are exceptional abstract let x 0 1 n with s s 1 s m a finite set of contraction maps from x to itself a nonempty compact subset e of x is an attractor for the iterated function scheme s if e i 1 m s i e s e we show that the typical closed set in 0 1 n is not an attractor of any iterated function scheme

fuzzy attractors appearing from gifzs abstract cabrelli forte molter and vrscay in 1992 considered a fuzzy version of the theory of iterated function systems ifss in short and their fractals which now is quite rich and important part of the fractals theory on the other hand miculescu and mihail in 2008 introduced another generalization of the ifss theory instead of selfmaps of a metric space x they considered mappings defined on the finite cartesian product x m in this paper we show that the fuzzification ideas of cabrelli et al can be naturally adjusted to the case of mappings defined on finite cartesian product in particular we define the notion of a generalized iterated fuzzy function system gifzs in short and prove that it generates a unique fuzzy fractal set we also study some basic properties of gifzss and their fractals and consider the question whether our setting gives us some new fuzzy fractal sets

nonasymptotic noisy lossy source coding this paper shows new general nonasymptotic achievability and converse bounds and performs their dispersion analysis for the lossy compression problem in which the compressor observes the source through a noisy channel while this problem is asymptotically equivalent to a noiseless lossy source coding problem with a modified distortion function nonasymptotically there is a noticeable gap in how fast their minimum achievable coding rates approach the common rate distortion function as evidenced both by the refined asymptotic analysis dispersion and the numerical results the size of the gap between the dispersions of the noisy problem and the asymptotically equivalent noiseless problem depends on the stochastic variability of the channel through which the compressor observes the source

coding of sources with two sided geometric distributions and unknown parameters lossless compression is studied for a countably infinite alphabet source with an unknown off centered two sided geometric tsg distribution which is a commonly used statistical model for image prediction residuals we demonstrate that arithmetic coding based on a simple strategy of model adaptation essentially attains the theoretical lower bound to the universal coding redundancy associated with this model we then focus on more practical codes for the tsg model that operate on a symbol by symbol basis and study the problem of adaptively selecting a code from a given discrete family by taking advantage of the structure of the optimum huffman tree for a known tsg distribution which enables simple calculation of the codeword of every given source symbol an efficient adaptive strategy is derived

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

analyticity of entropy rate of hidden markov chains with continuous alphabet we first prove that under certain mild assumptions the entropy rate of a hidden markov chain observed when passing a finite state stationary markov chain through a discrete time continuous output channel is analytic with respect to the input markov chain parameters we then further prove under strengthened assumptions on the channel that the entropy rate is jointly analytic as a function of both the input markov chain parameters and the channel parameters in particular the main theorems establish the analyticity of the entropy rate for two representative channels 1 cauchy and 2 gaussian the analyticity results obtained are expected to be helpful in computation estimation of entropy rate of hidden markov chains and capacity of finite state channels with continuous output alphabet

epsilon entropy and critical distortion of random fields the link between epsilon entropy for binary random fields and the lee yang theorem in statistical mechanics makes it possible to prove the existence of positive critical distortion for most binary random fields with pair interactions and some with many body interactions lower bounds for critical distortion of random fields on two dimensional 2 d lattices have been obtained in particular the new lower bounds for some 2 d ising models improve upon previously known bounds

asymptotic stability of the optimal filter with respect toits initial condition consider the problem of estimation of a diffusion signal observed in additive white noise if the solution to the filtering equations initialized with an incorrect prior distribution approaches the true conditional distribution asymptotically in time then the filter is said to be asymptotically stable with respect to perturbations of the initial condition this paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior for the linear case stability of the riccati equation of kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non gaussian initial conditions in the nonlinear case asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution

generalized iterated function systems on the space l x abstract in the last decades there has been a current effort to extend the classical hutchinson theory of iterated function systems composed by contractions on a metric space x into itself to more general spaces and infinitely many mappings in this paper we consider the countable iterated function systems consisting of some generalized contractions on the product space x i into x where i is an arbitrary set of natural numbers some approximations of the attractors of the respective iterated function systems are given

denoising and filtering under the probability of excess loss criterion subclasses of finite alphabet denoising and filtering causal denoising schemes are compared performance is measured by the normalized cumulative loss a k a distortion as measured by a single letter loss function we aim to minimize the probability that the normalized cumulative loss exceeds a given threshold we call this quantity the probability of excess loss specifically we consider a scheme to be optimal if it attains the maximal exponential decay rate of the probability of excess loss this provides another way of comparing schemes that complements and contrasts previous work which considered the expected value of the normalized cumulative loss in particular the question of whether the optimal denoiser is symbol by symbol for an independent and identically distributed i i d source and a discrete memoryless channel dmc is investigated for hamming loss the optimal denoiser is proven to be symbol by symbol perhaps somewhat counterintuitively for a general single letter loss function the optimal scheme need not be symbol by symbol the optimal denoiser requires unbounded delay and unbounded look ahead while symbol by symbol schemes mandate zero delay and look ahead it is natural to wonder about the effect of limited delay and limited look ahead consequently finite sliding window denoisers and finite block denoisers are defined they are shown to perform no better than symbol by symbol denoisers finally the effect of causality is investigated while it is difficult to characterize the performance of filters with unbounded memory explicitly it is shown that finite memory filters perform no better than symbol by symbol filters

on the entropy of a hidden markov process we study the entropy rate of a hidden markov process hmp defined by observing the output of a binary symmetric channel whose input is a first order binary markov process despite the simplicity of the models involved the characterization of this entropy is a long standing open problem by presenting the probability of a sequence under the model as a product of random matrices one can see that the entropy rate sought is equal to a top lyapunov exponent of the product this offers an explanation for the elusiveness of explicit expressions for the hmp entropy rate as lyapunov exponents are notoriously difficult to compute consequently we focus on asymptotic estimates and apply the same product of random matrices to derive an explicit expression for a taylor approximation of the entropy rate with respect to the parameter of the binary symmetric channel the accuracy of the approximation is validated against empirical simulation results we also extend our results to higher order markov processes and to renyi entropies of any order

an optimal algorithm for finding a maximum independent set of a circular arc graph a new algorithm is presented for finding a maximum independent set of a circular arc graph when the graph is given in the form of a family of n arcs our algorithm requires only o n cdot log n time and o n space furthermore if the endpoints of the arcs are already sorted it runs in o n time this algorithm is time and space optimal to within a constant factor

jump number maximization for proper interval graphs and series parallel graphs in this paper the order theoretic problem of finding linear extensions of a partially ordered set poset with certain properties is mapped to the corresponding graphtheorectic problem given a poset a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the poset we want to find a topological sequence that minimizes or maximizes the total number of jumps in graph theoretic terms define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized jump number minimization problem or minimized jump number maximization problem the complexities of both problems with respect to some special graphs are investigated first in this paper we then study the jump maximization problem for proper interval graphs and series parallel graphs from this study we hope that more researches can be derived from these interdisciplinary knowledge domains

estimating distortion in a binary symmetric channel consistently the problem of estimating the distortion in a binary symmetric channel is examined a consistent estimator is described and its performance is assessed numerically the estimator is based solely on the information contained in the channel output the channel input is viewed as an unobservable nuisance parameter that satisfies a weak basic assumption that 1 makes the problem of estimation viable and 2 seems swell suited for use in an imaging context under which the input is viewed as the true image and the output is viewed as the observed image

stationary measures for projective transformations the blackwell and furstenberg measures in this paper we study the blackwell and furstenberg measures which play an important role in information theory and the study of lyapunov exponents for the blackwell measure we determine parameter domains of singularity and give upper bounds for the hausdorff dimension for the furstenberg measure we establish absolute continuity for some parameter values our method is to analyze linear fractional iterated function schemes which are contracting on average have no separation properties that is we do not assume that the open set condition holds see hutchinson in indiana univ math j 30 713 747 1981 and in the case of the blackwell measure have place dependent probabilities in such a general setting even an effective upper bound on the dimension of the measure is difficult to achieve

certifying algorithms for recognizing interval graphs and permutation graphs a certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces the certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation we give linear time certifying algorithms for recognition of interval graphs and permutation graphs and for a few other related problems previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class we show that our certificates of nonmembership can be authenticated in o v time

o m n algorithms for the recognition and isomorphism problems on circular arc graphs circular arc graphs have a rich combinatorial structure the circular endpoint sequence of arcs in a model for a circular arc graph is usually far from unique we present a natural restriction on these models to make it meaningful to define the unique representations for circular arc graphs we characterize those circular arc graphs which have unique restricted models and give an o m cdot n algorithm for recognizing circular arc graphs we think a more careful implementation could reduce the complexity to o n 2 our approach is to reduce the recognition problem of circular arc graphs to that of circle graphs this approach has the following advantages it is conceptually simpler than tucker s o n 3 recognition algorithm it exploits the similarity between circle graphs and circular arc graphs in a natural fashion it yields an isomorphism algorithm a main contribution of this result is an illustration of the transformed decomposition technique the decomposition tree developed for circular arc graphs generalizes the concept of the pq tree which is a data structure that keeps track of all possible interval representations of a given interval graph as a consequence our approach also yields an o m cdot n isomorphism algorithm for circle graphs

on a question of a kameyama concerning self similar metrics abstract a kameyama introduced the concept of self similar topological system and asked the following fundamental question given a topological self similar system k f i i 1 2 n does there exist a metric on k comparable to the topology such that all the functions f i are contractions we modify kameyama s question which has a negative answer for an arbitrary topological self similar system by weakening the requirement that the functions in the topological self similar system be contractions to requiring that they be contractions more precisely we give an affirmative answer to the following question given a topological self similar system k f i i 1 2 n does there exist a metric on k which is compatible with the original topology and a comparison function 0 0 such that all the functions f i k k are contractions consequently the iterated function system k d f i i 1 2 n where d is a metric on k compatible with the original topology on k is hyperbolic

a decomposition theorem for binary markov random fields consider a binary markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree under a minimal assumption we prove a decomposition theorem to the effect that such a markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields one of which is white binary noise of positive weight said decomposition provides the information theorist with an exact expression for the per site rate distortion function of the random field over an interval of distortions not exceeding this weight we mention possible implications for communication theory probability theory and statistical physics

universal zero delay joint source channel coding we consider zero delay joint source channel coding of individual source sequences for a general known channel given an arbitrary finite set of schemes with finite memory not necessarily time invariant decoders a scheme is devised that does essentially as well as the best in the set on all individual source sequences using this scheme we construct a universal zero delay joint source channel coding scheme that is guaranteed to achieve asymptotically the performance of the best zero delay encoding decoding scheme with a finite state encoder and a markov decoder on all individual sequences for the case where the channel is a discrete memoryless channel dmc we construct an implementable zero delay joint source channel coding scheme that is based on the follow the perturbed leader scheme of gyoumlrgy for lossy source coding of individual sequences our scheme is guaranteed to attain asymptotically the performance of the best in the set of all encoding decoding schemes with a symbol by symbol decoder and arbitrary encoder on all individual sequences

approximation and limit results for nonlinear filters over an infinite time interval this paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion or just a process that is approximated by a jump diffusion the observation noise can be either white or of wide bandwidth the observations can be taken in either discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval it is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used analogous results are obtained with appropriate scaling if the observations are taken in discrete time and the sampling interval also goes to zero for these cases the approximating filter is a numerical approximation to the optimal filter for the presumed limit signal observation noise problem

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

markov property and ergodicity of the nonlinear filter in this paper we first prove under quite general conditions that the nonlinear filter and the pair signal filter are feller markov processes the state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded our proofs in contrast to those of kunita j multivariate anal 1 1971 pp 365 393 spatial stochastic processes birkhauser 1991 pp 233 256 and stettner stochastic differential equations springer verlag 1989 pp 279 292 do not depend upon the uniqueness of the solutions to the filtering equations we then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process these results extend those of kunita and stettner which hold for locally compact state space and bounded h to our general framework finally we show that the recent results of ocone and pardoux siam j control optim 34 1996 pp 226 243 on asymptotic stability of the nonlinear filter which use the kunita stettner setup hold for the general situation considered in this paper

polynomial algorithms for hamiltonian cycle in cocomparability graphs finding a hamiltonian cycle in a graph is one of the classical np complete problems complexity of the hamiltonian problem in permutation graphs has been a well known open problem in this paper the authors settle the complexity of the hamiltonian problem in the more general class of cocomparability graphs it is shown that the hamiltonian cycle existence problem for cocomparability graphs is in p a polynomial time algorithm for constructing a hamiltonian path and cycle is also presented the approach is based on exploiting the relationship between the hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph

refinements of stochastic domination in a recent paper by two of the authors the concepts of upwards and downwards epsilon movability were introduced mainly as a technical tool for studying dynamical percolation of interacting particle systems in this paper we further explore these concepts which can be seen as refinements or quantifications of stochastic domination and we relate them to previously studied concepts such as uniform insertion tolerance and extractability

the interval merging problem abstract a closed interval is an ordered pair of real numbers x y with x y the interval x y represents the set i r x i y given a set of closed intervals i a 1 b 1 a 2 b 2 a k b k the interval merging problem is to find a minimum cardinality set of intervals m i x 1 y 1 x 2 y 2 x j y j j k such that the real numbers represented by i i 1 k a i b i equal those represented by m i i 1 j x i y i in this paper we show the problem can be solved in o d log d sequential time and in o log d parallel time using o d processors on an erew pram where d is the number of the endpoints of i moreover if the input is given as a set of sorted endpoints then the problem can be solved in o d sequential time and in o log d parallel time using o d log d processors on an erew pram

weighted connected k domination and weighted k dominating clique in distance hereditary graphs a graph is distance hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph this paper presents efficient algorithms for solving the weighted connected k domination and the weighted k dominating clique problems in distance hereditary graphs

the efficiency of investment information we investigate how the description of a correlated information v improves the investment in the stock market x the objective is to maximize the growth rate of wealth in repeated investments we find a single letter characterization of the incremental growth rate spl delta r the maximum increase in growth rate when v is described to the investor at rate r the incremental growth rate specialized to the horse race market is related to source coding with side information of wyner and ahlswede kormer we provide two horse race examples jointly binary and jointly gaussian the initial efficiency spl delta 0 is the maximum possible increase in the growth rate per bit of description we show that the initial efficiency is related to the dependency between v and the market in particular for the horse race market the initial efficiency is the square of the hirschfeld gebelein renyi maximal correlation between v and x this provides a connection with the hypercontraction of the markov operator of ahlswede and gacs for the general market the initial efficiency is 1 when the side information v is equal to the stock market outcome x

analyticity of entropy rates of continuous state hidden markov models the analyticity of the entropy and relative entropy rates of continuous state hidden markov models is studied here using the analytic continuation principle and the stability properties of the optimal filter the analyticity of these rates is established for analytically parameterized models the obtained results hold under relatively mild conditions and cover several useful classes of hidden markov models these results are relevant for several theoretically and practically important problems arising in statistical inference system identification and information theory

attractors of generalized ifss that are not attractors of ifss abstract mihail and miculescu introduced the notion of a generalized iterated function system gifs in short and proved that every gifs generates an attractor in our previous paper we gave this notion a more general setting in this paper we show that for any m 2 there exists a cantor subset of the plane which is an attractor of some gifs of order m but is not an attractor of a gifs of order m 1 in particular this result shows that there is a subset of the plane which is an attractor of some gifs but is not an attractor of an ifs we also give an example of a cantor set which is not an attractor of a gifs

optimal prefix codes for sources with two sided geometric distributions a complete characterization of optimal prefix codes for off centered two sided geometric distributions of the integers is presented these distributions are often encountered in lossless image compression applications as probabilistic models for image prediction residuals the family of optimal codes described is an extension of the golomb codes which are optimal for one sided geometric distributions the new family of codes allows for encoding of prediction residuals at a complexity similar to that of golomb codes without recourse to the heuristic approximations frequently used when modifying a code designed for nonnegative integers so as to apply to the encoding of any integer optimal decision rules for choosing among a lower complexity subset of the optimal codes given the distribution parameters are also investigated and the relative redundancy of the subset with respect to the full family of optimal codes is bounded

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

on estimation for a binary symmetric channel we consider a binary symmetric channel where the input modeled as an infinite sequence of bits is distorted by bernoulli noise in a paper by simons 1991 a consistent estimator of the distortion i e of the probability that a single bit is changed is described under the basic assumption that the complexity of the input is finite here we deal with two shortcomings 1 the simons estimator requires some information on the value of the complexity we describe a modified approach through the generalized bernoulli components in which determination of the complexity is not needed 2 the simons estimator exhibits a serious negative bias when the distortion is great our modification largely overcomes this deficiency while reducing the variance the asymptotics for the proposed estimators are studied and their properties are illustrated through numerical studies

asymptotics of entropy rate in special families of hidden markov chains we derive an asymptotic formula for entropy rate of a hidden markov chain under certain parameterizations we also discuss applications of the asymptotic formula to the asymptotic behaviors of entropy rate of hidden markov chains as outputs of certain channels such as binary symmetric channel binary erasure channel and some special gilbert elliot channel

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

on state estimation of a two state hidden markov model with quantization we consider quantization from the perspective of minimizing filtering error when quantized instead of continuous measurements are used as inputs to a nonlinear filter specializing to discrete time two state hidden markov models hmms with continuous range output an explicit expression for the filtering error when continuous measurements are used is presented we also propose a quantization scheme based on maximizing the mutual information between quantized observations and the hidden states of the hmm

universal denoising for the finite input general output channel we consider the problem of reconstructing a finite alphabet signal corrupted by a known memoryless channel with a general output alphabet the goodness of the reconstruction is measured by a given loss function we constructively establish the existence of a universal sequence of denoiser s attaining asymptotically the optimum distribution dependent performance for any stationary source that may be generating the noiseless signal we show in fact that there is a whole family of denoiser sequences with this property these schemes are shown to be universal also in a semistochastic setting where the only randomness assumed is that associated with the channel noise the scheme is practical requiring o n sup 1 spl epsiv operations for any spl epsiv 0 and working storage size sublinear in the input data length this extends recent work that presented a discrete universal denoiser for recovering a discrete source corrupted by a discrete memoryless channel dmc

linear time algorithms for the hamiltonian problems on distance hereditary graphs a hamiltonian path of a graph g is a simple path that contains each vertex of g exactly once a hamiltonian cycle of a graph is a simple cycle with the same property the hamiltonian path resp cycle problem involves testing whether a hamiltonian path resp cycle exists in a graph the 1hp resp 2hp problem is to determine whether a graph has a hamiltonian path starting from a specified vertex resp starting from a specified vertex and ending at the other specified vertex the hamiltonian problems include the hamiltonian path hamiltonian cycle 1hp and 2hp problems a graph is a distance hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them in this paper we present a unified approach to solving the hamiltonian problems on distance hereditary graphs in linear time

analyticity convergence and convergence rate of recursive maximum likelihood estimation in hidden markov models this paper considers the asymptotic properties of the recursive maximum likelihood estimator for hidden markov models the paper is focused on the analytic properties of the asymptotic log likelihood and on the point convergence and convergence rate of the recursive maximum likelihood estimator using the principle of analytic continuation the analyticity of the asymptotic log likelihood is shown for analytically parameterized hidden markov models relying on this fact and some results from differential geometry lojasiewicz inequality the almost sure point convergence of the recursive maximum likelihood algorithm is demonstrated and relatively tight bounds on the convergence rate are derived as opposed to the existing result on the asymptotic behavior of maximum likelihood estimation in hidden markov models the results of this paper are obtained without assuming that the log likelihood function has an isolated maximum at which the hessian is strictly negative definite

a polynomial solution to the k fixed endpoint path cover problem on proper interval graphs we study a variant of the path cover problem namely the k fixed endpoint path cover problem or kpc for short given a graph g and a subset t of k vertices of v g a k fixed endpoint path cover of g with respect to t is a set of vertex disjoint paths p that covers the vertices of g such that the k vertices of t are all endpoints of the paths in p the kpc problem is to find a k fixed endpoint path cover of g of minimum cardinality note that if t is empty or equivalently k 0 the stated problem coincides with the classical path cover problem the kpc problem generalizes some path cover related problems such as the 1hp and 2hp problems which have been proved to be np complete note that the complexity status for both 1hp and 2hp problems on interval graphs remains an open question damaschke 1993 9 in this paper we show that the kpc problem can be solved in linear time on the class of proper interval graphs that is in o n m time on a proper interval graph on n vertices and m edges the proposed algorithm is simple requires linear space and also enables us to solve the 1hp and 2hp problems on proper interval graphs within the same time and space complexity

optimal covering of cacti by vertex disjoint paths moran s and y wolfstahl optimal covering of cacti by vertex disjoint paths theoretical computer science 84 1991 179 197 a path couer or in short cover of a graph g is a set of vertex disjoint paths which cover all the vertices of g an optimal couer of g is a cover of the smallest possible cardinality notable applications of graph covering are code optimization and mapping parallel programs to parallel architectures the optimal covering problem is known to be np complete even for cubic 3connected planar graphs motivated by the intractability of this problem we develop an efficient optimal covering algorithm for cacti i e graphs where no edge lies on more than one cycle in doing so we generalize the results of boesch gimpel mchugh 1974 and of pinter wolfstahl 1987 where optimal covering algorithms for trees and graphs where no two cycles share a vertex were presented

on a typical compact set as the attractor of generalized iterated function systems of infinite order abstract in 2013 balka and mathe showed that in uncountable polish spaces the typical compact set is not a fractal of any ifs in 2008 miculescu and mihail introduced a concept of a generalized iterated function system gifs in short a particular extension of classical ifs in which they considered families of mappings defined on finite cartesian product x m with values in x recently secelean extended these considerations to mappings defined on the space l x of all bounded sequences of elements of x endowed with supremum metric in the paper we show that in euclidean spaces a typical compact set is an attractor in sense of secelean and that in general in the polish spaces it can be perceived as selfsimilar in such sense

analyticity of entropy rate of hidden markov chains we prove that under mild positivity assumptions the entropy rate of a hidden markov chain varies analytically as a function of the underlying markov chain parameters a general principle to determine the domain of analyticity is stated an example is given to estimate the radius of convergence for the entropy rate we then show that the positivity assumptions can be relaxed and examples are given for the relaxed conditions we study a special class of hidden markov chains in more detail binary hidden markov chains with an unambiguous symbol and we give necessary and sufficient conditions for analyticity of the entropy rate for this case finally we show that under the positivity assumptions the hidden markov chain itself varies analytically in a strong sense as a function of the underlying markov chain parameters

noisy constrained capacity for bsc channels we study the classical problem of noisy constrained capacity in the case of the binary symmetric channel bsc namely the capacity of a bsc whose input is a sequence from a constrained set as stated by fan while calculation of the noise free capacity of constrained sequences is well known the computation of the capacity of a constraint in the presence of noise has been an unsolved problem in the half century since shannon s landmark paper we first express the constrained capacity of a binary symmetric channel with d k constrained input as a limit of the top lyapunov exponents of certain matrix random processes then we compute asymptotic approximations of the noisy constrained capacity for cases where the noise parameter e is small in particular we show that when k 2d the error term excess of capacity beyond the noise free capacity is o e whereas it is o eloge when k 2d in both cases we compute the coefficient of the error term in the course of establishing these findings we also extend our previous results on the entropy of a hidden markov process to higher order finite memory processes these conclusions are proved by a combination of analytic and combinatorial methods

hamilton paths in grid graphs a grid graph is a node induced finite subgraph of the infinite grid it is rectangular if its set of nodes is the product of two intervals given a rectangular grid graph and two of its nodes we give necessary and sufficient conditions for the graph to have a hamilton path between these two nodes in contrast the hamilton path and circuit problem for general grid graphs is shown to be np complete this provides a new relatively simple proof of the result that the euclidean traveling salesman problem is np complete

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

non self similar sets in 0 1 n of arbitrary dimension abstract we consider 0 1 n the unit cube of r n n 1 let s s 1 s m be a finite set of contraction maps from x to itself a non empty subset e of x is an attractor or an invariant set for the iterated function system ifs s if e i 1 m s i e we construct for each s 0 n a nowhere dense perfect set e contained in 0 1 n with hausdorff dimension s which is not an attractor for any iterated function system composed of weak contractions from 0 1 n to itself

derivatives of entropy rate in special families of hidden markov chains consider a hidden markov chain obtained as the observation process of an ordinary markov chain corrupted by noise recently zuk et al showed how in principle one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise namely they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time we generalize this result to a natural class of hidden markov chains called black holes we also discuss in depth special cases of binary markov chains observed in binary symmetric noise and give an abstract formula for the first derivative in terms of a measure on the simplex due to blackwell

an efficient certifying algorithm for the hamiltonian cycle problem on circular arc graphs abstract a certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces the certificate is an evidence that can be used to authenticate the correctness of the answer a hamiltonian cycle in a graph is a simple cycle in which each vertex of the graph appears exactly once the hamiltonian cycle problem is to determine whether or not a graph contains a hamiltonian cycle the best result for the hamiltonian cycle problem on circular arc graphs is an o n 2 log n time algorithm where n is the number of vertices of the input graph in fact the o n 2 log n time algorithm can be modified as a certifying algorithm although it was published before the term certifying algorithms appeared in the literature however whether there exists an algorithm whose time complexity is better than o n 2 log n for solving the hamiltonian cycle problem on circular arc graphs has been opened for two decades in this paper we present an o n time certifying algorithm to solve this problem where represents the maximum degree of the input graph the certificates provided by our algorithm can be authenticated in o n time

a new method to estimate the critical distortion of random fields it was shown by c m newman and g a baker 1991 that the critical distortion d sub c of the ising model on z sup k is related to the radius of convergence r of the classical mayer series by d sub c r 1 r an approach from statistical mechanics to estimate r as well as d sub c by using the matrix representation of mayer series is found

on the optimality of symbol by symbol filtering and denoising we consider the problem of optimally recovering a finite alphabet discrete time stochastic process x sub t from its noise corrupted observation process z sub t in general the optimal estimate of x sub t will depend on all the components of z sub t on which it can be based we characterize nontrivial situations i e beyond the case where x sub t z sub t are independent for which optimum performance is attained using symbol by symbol operations a k a singlet decoding meaning that the optimum estimate of x sub t depends solely on z sub t for the case where x sub t is a stationary binary markov process corrupted by a memoryless channel we characterize the necessary and sufficient condition for optimality of symbol by symbol operations both for the filtering problem where the estimate of x sub t is allowed to depend only on z sub t sub t spl les t and the denoising problem where the estimate of x sub t is allowed dependence on the entire noisy process it is then illustrated how our approach which consists of characterizing the support of the conditional distribution of the noise free symbol given the observations can be used for characterizing the entropy rate of the binary markov process corrupted by the binary symmetric channel bsc in various asymptotic regimes for general noise free processes not necessarily markov general noise processes not necessarily memoryless and general index sets random fields we obtain an easily verifiable sufficient condition for the optimality of symbol by symbol operations and illustrate its use in a few special cases for example for binary processes corrupted by a bsc we establish under mild conditions the existence of a spl delta sup 0 such that the say what you see scheme is optimal provided the channel crossover probability is less than spl delta sup finally we show how for the case of a memoryless channel the large deviations ld performance of a symbol by symbol filter is easy to obtain thus characterizing the ld behavior of the optimal schemes when these are singlet decoders and constituting the only known cases where such explicit characterization is available

universal coding with minimum probability of codeword length overflow lossless block to variable length source coding is studied for finite state finite alphabet sources the aim is to minimize the probability that the normalized length of the codeword will exceed a given threshold b subject to the kraft inequality it is shown that the lempel ziv algorithm 1978 asymptotically attains the optimal performance in the sense just defined independently of the source and the value of b for the subclass of unifilar markov sources faster convergence to the asymptotic optimum performance can be accomplished by using the minimum description length universal code for this subclass it is demonstrated that these universal codes are also nearly optimal in the sense of minimizing buffer overflow probability and asymptotically optimal in a competitive sense

on finite block length quantization distortion we investigate the upper and lower bounds on the quantization distortions for independent and identically distributed continuous sources in the finite block length regime we derive a lower bound on the quantization distortion as well as an upper bound on the quantization distortion based on random quantization codebooks moreover we apply the obtained bounds to the continuous gaussian source for the gaussian source we propose a computationally tractable method to numerically compute the upper and lower bounds for both bounded and unbounded quantization codebooks numerical results show that the gap between the upper and lower bounds is small for the block length of several hundreds and the upper and lower bounds are tighter than those proposed in the existing works

scanning and sequential decision making for multidimensional data part i the noiseless case we investigate the problem of scanning and prediction ldquoscandiction rdquo for short of multidimensional data arrays this problem arises in several aspects of image and video processing such as predictive coding for example where an image is compressed by coding the error sequence resulting from scandicting it thus it is natural to ask what is the optimal method to scan and predict a given image what is the resulting minimum prediction loss and whether there exist specific scandiction schemes which are universal in some sense specifically we investigate the following problems first modeling the data array as a random field we wish to examine whether there exists a scandiction scheme which is independent of the field s distribution yet asymptotically achieves the same performance as if this distribution were known this question is answered in the affirmative for the set of all spatially stationary random fields and under mild conditions on the loss function we then discuss the scenario where a nonoptimal scanning order is used yet accompanied by an optimal predictor and derive bounds on the excess loss compared to optimal scanning and prediction this paper is the first part of a two part paper on sequential decision making for multidimensional data it deals with clean noiseless data arrays the second part deals with noisy data arrays namely with the case where the decision maker observes only a noisy version of the data yet it is judged with respect to the original clean data

on universal hypotheses testing via large deviations a prototype problem in hypotheses testing is discussed the problem of deciding whether an i i d sequence of random variables has originated from a known source p sub 1 or an unknown source p sub 2 is considered the exponential rate of decrease in type ii probability of error under a constraint on the minimal rate of decrease in type i probability of error is chosen for a criterion of optimality using large deviations estimates a decision rule that is based on the relative entropy of the empirical measure with respect to p sub 1 is proposed in the case of discrete random variables this approach yields weaker results than the combinatorial approach used by hoeffding 1965 however it enables the analysis to be extended to the general case of r sup n valued random variables finally the results are extended to the case where p sub 1 is an unknown parameter dependent distribution that is known to belong to a set of distributions p sup 0 sub 1 theta in theta

a randomized algorithm for the capacity of finite state channels inspired by ideas from the field of stochastic approximation we propose a randomized algorithm to compute the capacity of a finite state channel with a markovian input when the mutual information rate of the channel is concave with respect to the chosen parameterization the proposed algorithm proves to be convergent to the capacity of the channel almost surely with the derived convergence rate we also discuss the convergence behavior of the algorithm without the concavity assumption

a simple paradigm for graph recognition application to cographs and distance hereditary graphs an easy way for graph recognition algorithms is to use a two step process first compute a characteristic feature as if the graph belongs to that class second check whether the computed characteristic feature as if the graph belongs to that class second check whether the computed separating them may yield new and much more easily understood algorithms in this paper we apply that paradigm to the cograph and distance hereditary graph recognition problems

entropy of hidden markov processes via cycle expansion hidden markov processes hmp is one of the basic tools of the modern probabilistic modeling the characterization of their entropy remains however an open problem here the entropy of hmp is calculated via the cycle expansion of the zeta function a method adopted from the theory of dynamical systems for a class of hmp this method produces exact results both for the entropy and the moment generating function the latter allows to estimate via the chernoff bound the probabilities of large deviations for the hmp more generally the method offers a representation of the moment generating function and of the entropy via convergent series

tradeoffs between the excess code length exponent and the excess distortion exponent in lossy source coding lossy compression of a discrete memoryless source dms with respect to a single letter distortion measure is considered we study the best attainable tradeoff between the exponential rates of the probabilities that the codeword length and that the cumulative distortion exceed respective thresholds for two main cases the first scenario examined is that where the source is corrupted by a discrete memoryless channel dmc prior to reaching the coder in the second part of this work we examine the universal setting where the noise free source is an unknown member p sub spl theta of a given family p sub spl theta spl theta spl isin spl theta here inspired by an approach which was proven fruitful previously in the context of composite hypothesis testing we allow the constraint on the excess code length exponent to be spl theta dependent corollaries are derived for some special cases of interest including marton s 1974 classical source coding exponent and its generalization to the case where the constraint on the rate of the code is relaxed from an almost sure constraint to a constraint on the excess code length exponent

an o n 2 log n algorithm for the hamiltonian cycle problem on circular arc graphs a circular arc family f is a collection of arcs on a circle a circular arc graph is the intersection graph of an arc family a hamiltonian cycle hc in a graph is a cycle that passes through every vertex exactly once this paper presents an o n 2 log n algorithm to determine whether a given circular arc graph contains an hc this algorithm is based on two subroutines for interval graphs i a linear time greedy algorithm for the node disjoint path cover problem and ii a linear time hc algorithm if the given graph does not contain an hc this paper can produce a proof either through the deletion of an appropriate cutset or through the failure to obtain a specific type of hc

on analytic properties of entropy rate entropy rate is a real valued functional on the space of discrete random sources for which it exists however it lacks existence proofs and or closed formulas even for classes of random sources which have intuitive parameterizations a good way to overcome this problem is to examine its analytic properties relative to some reasonable topology a canonical choice of a topology is that of the norm of total variation as it immediately arises with the idea of a discrete random source as a probability measure on sequence space it is shown that both upper and lower entropy rate hence entropy rate itself if it exists are lipschitzian relative to this topology which by well known facts is close to differentiability an application of this theorem leads to a simple and elementary proof of the existence of entropy rate of random sources with finite evolution dimension this class of sources encompasses arbitrary hidden markov sources and quantum random walks

on competitive prediction and its relation to rate distortion theory consider the normalized cumulative loss of a predictor f on the sequence x sup n x sub 1 x sub n denoted l sub f x sup n for a set of predictors g let l g x sup n min sub f spl isin g l sub f x sup n denote the loss of the best predictor in the class on x sup n given the stochastic process x x sub 1 x sub 2 we look at el g x sup n termed the competitive predictability of g on x sup n our interest is in the optimal predictor set of size m i e the predictor set achieving min sub g spl les m el g x sup n when m is subexponential in n simple arguments show that min sub g spl les m el g x sup n coincides for large n with the bayesian envelope min sub f el sub f x sup n we investigate the behavior for large n of min sub g spl les e sup nr el g x sup n which we term the competitive predictability of x at rate r we show that whenever x has an autoregressive representation via a predictor with an associated independent and identically distributed i i d innovation process its competitive predictability is given by the distortion rate function of that innovation process indeed it will be argued that by viewing g as a rate distortion codebook and the predictors in it as codewords allowed to base the reconstruction of each symbol on the past unquantized symbols the result can be considered as the source coding analog of shannon s classical result that feedback does not increase the capacity of a memoryless channel for a general process x we show that the competitive predictability is lower bounded by the shannon lower bound slb on the distortion rate function of x and upper bounded by the distortion rate function of any not necessarily memoryless innovation process through which the process x has an autoregressive representation thus the competitive predictability is also precisely characterized whenever x can be autoregressively represented via an innovation process for which the slb is tight the error exponent i e the exponential behavior of min sub g spl les exp nr pr l g x sup n d is also characterized for processes that can be autoregressively represented with an i i d innovation process

on generalized iterated function systems defined on l sum of a metric space abstract miculescu and mihail in 2008 introduced a concept of a generalized iterated function system gifs in short a particular extension of classical ifs instead of families of selfmaps of a metric space x they considered families of mappings defined on finite cartesian product x m it turned out that a great part of the classical hutchinson barnsley theory has natural counterpart in this gifss case recently secelean extended these considerations to mappings defined on the space x of all bounded sequences of elements of x and obtained versions of the hutchinson barnsley theorem for appropriate families of such functions in the paper we study some further aspects of secelean s setting in particular we introduce and investigate a bit more restrictive framework and we show that some problems of the theory have more natural solutions within such a case finally we present an example which shows that this extended theory of gifss gives us fractal sets that cannot be obtained by any ifss or even by any gifss

a unified approach for encoding clean and noisy sources by means of waveform and autoregressive model vector quantization data compression by vector quantization is considered for sources which have been degraded by noise it is shown that by appropriately modifying the given distortion measure the problem becomes a standard quantization problem for the noisy source and the modified distortion measure for the special case of sources corrupted by statistically independent additive noise the authors provide sufficient conditions on the original distortion measure and probability distributions of the source and the noise for convergence of the generalized lloyd algorithm in designing the quantizers the results are specialized to waveform and autoregressive model vector quantization using the weighted quadratic and the itakura saito distortion measures respectively

capacity and coding for the gilbert elliott channels the gilbert elliott channel a varying binary symmetric channel with crossover probabilities determined by a binary state markov process is treated in general such a channel has a memory that depends on the transition probabilities between the states a method of calculating the capacity of this channel is introduced and applied to several examples and the question of coding is addressed in the conventional usage of varying channels a code suitable for memoryless channels is used in conjunction with an interleaver with the decoder considering the deinterleaved symbol stream as the output of a derived memoryless channel the transmission rate is limited by the capacity of this memoryless channel which is often considerably less than the capacity of the original channel a decision feedback decoding algorithm that completely recovers this capacity loss is introduced it is shown that the performance of a system incorporating such an algorithm is determined by an equivalent genie aided channel the capacity of which equals that of the original channel the calculated random coding exponent of the genie aided channel indicates a considerable increase in the cutoff rate over that of the conventionally derived memoryless channel

decomposition of binary random fields and zeros of partition functions on montre qu il existe une relation etroite entre le probleme de decomposition pour des champs eleatoires binaires etudies dans un article recent de hajek et berger 1987 et le probleme de mecanique statistique de determination de regions independantes du zero pour des fonctions de partition