how long are the arms in dla the author shows that the maximal length of the arms of dla can grow at most at a rate of n2 3 in dimension 2 and at a rate of n2 d in dimension d 2 here n denotes the volume or mass of the aggregate

entropy and data compression schemes some new ways of defining the entropy of a process by observing a single typical output sequence as well as a new kind of shannon mcmillan breiman theorem are presented this provides a new and conceptually very simple ways of estimating the entropy of an ergodic stationary source as well as new insight into the workings of such well known data compression schemes as the lempel ziv algorithm

profiles of tries tries from retrieval are one of the most popular data structures on words they are pertinent to the internal structure of stored words and several splitting procedures used in diverse contexts the profile of a trie is a parameter that represents the number of nodes either internal or external with the same distance from the root it is a function of the number of strings stored in a trie and the distance from the root several if not all trie parameters such as height size depth shortest path and fill up level can be uniformly analyzed through the external and internal profiles although profiles represent one of the most fundamental parameters of tries they have hardly been studied in the past the analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior we present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source we first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root it appears that profiles of tries exhibit several fascinating phenomena when moving from the root to the leaves of a trie the growth of the expected profiles varies near the root the external profiles tend to zero at an exponential rate and then the rate gradually rises to being logarithmic the external profiles then abruptly tend to infinity first logarithmically and then polynomially they then tend polynomially to zero again furthermore the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially while symmetric tries are nonoscillating in contrast to most shape parameters of random tries studied previously such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form also the widest levels in symmetric tries contain a linear number of nodes differing from the order n sqrt log n for asymmetric tries n being the size of the trees finally it is observed that profiles satisfy central limit theorems when the variance goes unbounded while near the height they are distributed according to poisson laws as a consequence of these results we find typical behaviors of the height shortest path fill up level and depth these results are derived here by methods of analytic algorithmics such as generating functions mellin transform poissonization and de poissonization the saddle point method singularity analysis and uniform asymptotic analysis

digital search trees revisited several algorithms have been proposed which build search trees using digital properties of the search keys a general approach to the study of the average case performance of such algorithms is discussed with particular attention to the analysis of the digital search tree structures of coflman and eve specifically the method leads to the solution of a problem left open by knuth finding the average number of nodes in digital search trees with both sons null the paper may be of interest as a survey and tutorial treatment of the analysis of the three primary digital tree search methods digital search trees radix search tries and patricia tries

multi particle diffusion limited aggregation we consider a stochastic aggregation model on mathbb z d start with particles distributed according to the product bernoulli measure with parameter mu in addition start with an aggregate at the origin non aggregated particles move as continuous time simple random walks obeying the exclusion rule whereas aggregated particles do not move the aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it this evolution is called multi particle diffusion limited aggregation our main result states that if on d 1 the initial density of particles is large enough then with positive probability the aggregate has linearly growing arms that is there exists a constant c 0 so that at time t the aggregate contains a point of distance at least ct from the origin for all t the key conceptual element of our analysis is the introduction and study of a new growth process consider a first passage percolation process called type 1 starting from the origin whenever type 1 is about to occupy a new vertex with positive probability instead of doing it it gives rise to another first passage percolation process called type 2 which starts to spread from that vertex each vertex gets occupied only by the process that arrives to it first this process may have three phases extinction type 1 gets eventually surrounded by type 2 coexistence infinite clusters of both types emerge and strong survival type 1 produces an infinite cluster which entraps all type 2 clusters understanding the various phases of this process is of mathematical interest on its own right we establish the existence of a strong survival phase and use this to show our main result

geometry based symbolic approximation for fast sequence matching on manifolds in this paper we consider the problem of fast and efficient indexing techniques for sequences evolving in non euclidean spaces this problem has several applications in the areas of human activity analysis where there is a need to perform fast search and recognition in very high dimensional spaces the problem is made more challenging when representations such as landmarks contours and human skeletons etc are naturally studied in a non euclidean setting where even simple operations are much more computationally intensive than their euclidean counterparts we propose a geometry and data adaptive symbolic framework that is shown to enable the deployment of fast and accurate algorithms for activity recognition dynamic texture recognition motif discovery toward this end we present generalizations of key concepts of piece wise aggregation and symbolic approximation for the case of non euclidean manifolds we show that one can replace expensive geodesic computations with much faster symbolic computations with little loss of accuracy in activity recognition and discovery applications the framework is general enough to work across both euclidean and non euclidean spaces depending on appropriate feature representations without compromising on the ultra low bandwidth high speed and high accuracy the proposed methods are ideally suited for real time systems and low complexity scenarios

asymptotical growth of a class of random trees we study three rules for the development of a sequence of finite subtrees t n of an infinite m ary tree t independent realizations omega n of a stationary ergodic process omega on m letters are used to trace out paths in t in the first rule t n is formed by adding a node to t n 1 at the first location where the path defined by omega n leaves t n 1 the second and third rules are similar but more complicated for each rule the height l n of the added node is shown to grow in probability as ln n divided by h the entropy per symbol of the generic process a typical retrieval time has the same behavior on the other hand lim inf nl n ln n sigma 1 lim sup n l n ln n sigma 2 a s where the constants sigma 1 sigma 2 are in general different depend on the rule in use and sigma 1 1 h sigma 2 it is proven along the way that the height of t n grows as sigma 2 ln n with probability one

the birational type of the moduli space of even spin curves abstract we determine the kodaira dimension of the moduli space s g of even spin curves for all g precisely we show that s g is of general type for g 8 and has negative kodaira dimension for g 8

asymptotic behavior of the lempel ziv parsing scheme and digital search trees the lempel ziv parsing scheme finds a wide range of applications most notably in data compression and algorithms on words it partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase the parameter of interest is the number mn of phrases that one can construct from a sequence of length n in this paper for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of mn which turns out to be normal this proves a long standing open problem in fact to obtain this result we solved another open problem namely that of establishing the limiting distribution of the internal path length in a digital search tree the latter is a consequence of an asymptotic solution of a multiplicative differential functional equation often arising in the analysis of algorithms on words interestingly enough our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability and analytical techniques such as mellin transform differential functional equations de poissonization and so forth in concluding remarks we indicate a possibility of extending our results to markovian models

on entropy of pyramid structures an information theoretic analysis of multiresolution pyramid structures is presented the analysis is carried out using the concept of spectral entropy which for gaussian sources is linearly related to the differential entropy the spectral entropy is used to define the spectral roughness measure that in turn is an indicator of the amount of memory in a source the more the memory in a source the greater is the value of its spectral roughness measure the spectral roughness measure also plays an important role in lower bounding the rate distortion function the main result is that the spectral roughness measure of a pyramid structure is less than that of the original fullband process this means that substantial reduction in bits can be obtained by merely representing a source as a pyramid simulations using one dimensional and two dimensional sources verifying these claims are presented

a formal theory for optimal and information theoretic syntactic pattern recognition abstract in this paper we present a foundational basis for optimal and information theoretic syntactic pattern recognition we do this by developing a rigorous model m for channels which permit arbitrarily distributed substitution deletion and insertion syntactic errors more explicitly if a is any finite alphabet and a the set of words over a we specify a stochastically consistent scheme by which a string u a can be transformed into any y a by means of arbitrarily distributed substitution deletion and insertion operations the scheme is shown to be functionally complete and stochastically consistent apart from the synthesis aspects we also deal with the analysis of such a model and derive a technique by which pr y u the probability of receiving y given that u was transmitted can be computed in cubic time using dynamic programming one of the salient features of this scheme is that it demonstrates how dynamic programming can be applied to evaluate quantities involving complex combinatorial expressions and which also maintain rigid probability consistency constraints experimental results which involve dictionaries with strings of lengths between 7 and 14 with an overall average noise of 39 75 demonstrate the superiority of our system over existing methods apart from its straightforward applications in string generation and recognition we believe that the model also has extensive potential applications in speech uni dimensional signal processing and computational molecular biology

on minimal decompositions of low rank symmetric tensors abstract in this paper we study the set of decompositions of symmetric tensors of low rank of any dimension and of arbitrary order and the waring loci of points which appear in a decomposition of the tensor we describe a new and complete stratification of the set of symmetric tensors of rank less than 5 by studying hilbert function and regularity of ideals of points contained in the apolar ideal for each stratum we describe a procedure to compute a minimal apolar set of points exploiting the algebraic properties of the waring loci

a note on the height of binary search trees let h n be the height of a binary search tree with n nodes constructed by standard insertions from a random permutation of 1 n it is shown that h n log n c 4 31107 in probability as n where c is the unique solution of c log 2 e c 1 c 2 also for all p 0 lim n e h p n log p n c p finally it is proved that s n log n c 0 3733 in probability where c is defined by c log 2 e c 1 c 1 and s n is the saturation level of the same tree that is the number of full levels in the tree

on the rationality of the moduli space of luroth quartics we prove that the moduli space mathfrak m l of luroth quartics in mathbb p 2 i e the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of mathrm pgl 3 mathbb c is rational as is the related moduli space of bateman seven tuples of points in mathbb p 2

one dimensional long range diffusion limited aggregation i we examine diffusion limited aggregation generated by a random walk on zz with long jumps we derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has under various regularity conditions on the tail of the step distribution we prove that the diameter grows as n o 1 n o 1 with an explicitly given the growth rate of the aggregate is shown to have three phase transitions when the walk steps have finite third moment finite variance and conjecturally finite half moment

spin curves and scorza quartics in the paper takagi and zucconi in on blow ups of the quintic del pezzo 3 fold and varieties of power sums of quartic hypersurfaces pp 1 40 preprint submitted 2008 we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces in this paper we show that these quartics coincide with the scorza quartics of general pairs of trigonal curves and ineffective theta characteristics among other applications we give an affirmative answer to the conjecture of dolgachev and kanev on the existence of the scorza quartics for any general pairs of curves and ineffective theta characteristics we also give descriptions of the moduli spaces of trigonal even spin curves

average profile of the generalized digital search tree and the generalized lempel ziv algorithm the goal of this research is threefold i to analyze generalized digital search trees ii to derive the average profile i e phrase length of a generalization of the well known parsing algorithm due to lempel and ziv and iii to provide analytic tools to analyze asymptotically certain partial differential functional equations often arising in the analysis of digital trees in the generalized lempel ziv parsing scheme one partitions a sequence of symbols from a finite alphabet into phrases such that the new phrase is the shortest substring seen in the past by at most b 1 phrases b 1 corresponds to the original lempel ziv scheme such a scheme can be analyzed through a generalized digital search tree in which every node is capable of storing up to b strings in this paper we investigate the depth of a randomly selected node in such a tree and the length of a randomly selected phrase in the generalized lempel ziv scheme these findings and some recent results allow us to compute the average redundancy of the generalized lempel ziv code and compare it to the ordinary lempel ziv code leading to an optimal value of b analytic techniques of precise analyses of algorithms are used to establish most of these conclusions

optimal off line detection of repetitions in a string an algorithm is presented to detect within optimal time o n log n and space o n off line on a ram all of the distinct repetitions in a given textstring on a finite alphabet the proposed strategy is self contained as it depends more heavily on algorithmic design considerations than on the combinatorial properties of the output it is based on a new data structure the leaf tree which is particularly suited to exploit simple properties of the suffix tree associated with the string to be analyzed

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

renewal theory in the analysis of tries and strings we give a survey of a number of simple applications of renewal theory to problems on random strings and tries insertion depth size insertion mode and imbalance of tries variations for b tries and patricia tries khodak and tunstall codes

rationality of the moduli spaces of plane curves of sufficiently large degree we prove that the moduli space of plane curves of degree d is rational for all sufficiently large d

laws of large numbers and tail inequalities for random tries and patricia trees we consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space if hn is the height of this tree we show that hn e hn tends to one in probability additional tail inequalities are given for the height depth size internal path length and profile of these trees and ordinary tries that apply without any conditions on the string distributions they need not even be identically distributed

renyi entropies and large deviations for the first match function we define the first match function t n mathcal c n rightarrow 1 ldots n where mathcal c is a finite alphabet for two copies of x 1 n in mathcal c n this function gives the minimum number of steps one has to slide one copy of x 1 n to get a match with the other one for ergodic positive entropy processes saussol and coauthors proved the almost sure convergence of t n n we compute the large deviation properties of this function we prove that this limit is related to the renyi entropy function which is also proved to exist our results hold under a condition easy to check which defines a large class of processes we provide some examples

asymptotic properties of data compression and suffix trees recently wyner and ziv see ibid vol 35 p 1250 8 1989 have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is 1 h log n in probability where h is the entropy of the alphabet this finding was used to obtain several insights into certain universal data compression schemes most notably the lempel ziv data compression algorithm wyner and ziv have also conjectured that their result can be extended to a stronger almost sure convergence in this paper we settle this conjecture in the negative in the so called right domain asymptotic that is during a dynamic phase of expanding the data base we prove under an additional assumption involving mixing conditions that the length of a typical repeated subword oscillates almost surely a s between 1 h sub 1 log n and 1 h sub 2 log n where d

second phase changes in random m ary search trees and generalized quicksort convergence rates we study the convergence rate to normal limit law for the space requirement of random m ary search trees while it is known that the random variable is asymptotically normally distributed for 3 m 26 and that the limit law does not exist for m 26 we show that the convergence rate is o n 1 2 for 3 m 19 and is o n 3 3 2 where 4 3 3 2 is a parameter depending on m for 20 m 26 our approach is based on a renemen t to the method of moments and applicable to other recursive random variables we briey mention the applications to quicksort proper and the generalized quicksort of hennequin where more phase changes are given these results provide natural concrete examples for which the berry esseen bounds are not necessarily proportional to the reciprocal of the standard deviation local limit theorems are also derived

the moduli space of genus four even spin curves is rational abstract using the mori theory for threefolds we prove that the moduli space of pairs of smooth curves of genus four and theta characteristics without global sections is a rational variety

the functional law of the iterated logarithm for stationary strongly mixing sequences let x i i z be a strictly stationary and strongly mixing sequence of real valued mean zero random variables let a n n 0 be the sequence of strong mixing coefficients we define the strong mixing function by t t and we denote by q the quantile function of x 0 assume that 0 1 1 t q 2 t dt where f 1 denotes the inverse of the monotonic function f the main result of this paper is that the functional law of the iterated logarithm lil holds whenever x i i z satisfies moreover it follows from doukhan massart and rio that for any positive there exists a stationary sequence x i i z with strong mixing coefficients n of the order of n such that the bounded lil does not hold if condition is violated the proof of the functional lil is mainly based on new maximal exponential inequalities for strongly mixing processes which are of independent interest

average profile and limiting distribution for a phrase size in the lempel ziv parsing algorithm consider the parsing algorithm developed by lempel and ziv 1978 that partitions a sequence of length n into variable phrases blocks such that a new block is the shortest substring not seen in the past as a phrase in practice the following parameters are of interest number of phrases the size of a phrase the number of phrases of given size and so forth in this paper we focus on the size of a randomly selected phrase and the average number of phrases of a given size the so called average profile of phrase sizes these parameters can be efficiently analyzed through a digital search tree representation for a memoryless source with unequal probabilities of symbols generation the so called asymmetric bernoulli model we prove that the size of a typical phrase is asymptotically normally distributed with mean and variance explicitly computed in terms of digital search trees we prove the normal limiting distribution of the typical depth i e the length of a path from the root to a randomly selected node the latter finding is proved by a technique that belongs to the toolkit of the analytical analysis of algorithms and it seems to be novel in the context of data compression

a generalized suffix tree and its un expected asymptotic behaviors suffix trees find several applications in computer science and telecommunications most notably in algorithms on strings data compressions and codes despite this very little is known about their typical behaviors in a probabilistic framework a family of suffix trees further called b suffix trees built from the first n suffixes of a random word is considered in this family a noncompact suffix tree i e such that every edge is labeled by a single symbol is represented by b 1 and a compact suffix tree i e without unary nodes is asymptotically equivalent to b to infty as n to infty several parameters of b suffix trees are studied namely the depth of a given suffix the depth of insertion the height and the shortest feasible path some new results concerning typical i e almost sure behaviors of these parameters are established these findings are used to obtain several insights into certain algorithms on words molecular biology and universal data compression schemes

theory of powders starting from the observations that powders have a large number of particles and reproducible properties we show how statistical mechanics applies to powders the volume of the powder plays the role of the energy in conventional statistical mechanics with the hypothesis that all states of a specified volume are equally probable we introduce a variable x the compactivity which is analogous to the temperature in thermodynamics some simple models are considered which demonstrate how the problems involved can be tackled using the concept of compactivity there is an increasing interest in applying the methods of statistical mechanics and of transport theory to systems which are neither atomistic nor in equilibrium but which still fulfil a remaining tenet of statistical physics which is that systems can be completely defined by a very small number of parameters and can be constructed in a reproducible way powders fall into this category if a powder consists for example of uniform cubes of salt and is poured into a container falling at low density uniformly from a great height one expects a salt powder of a certain density repeating the preparation reproduces the same density a treatment such as shaking the powder by a definite routine produces a new density and the identical routine applied to another sample of the initial powder will result in the same final density clearly a maxwell demon could arrange the little cubes of nacl to make a material of different properties to that of our experiment but if such demonics are ignored and we restrict ourselves to extensive operations such as stirring shaking compressing all actions which do not act on grains individually then well defined states of the powder result in this paper we will set up a framework for describing the state of the powder basing our development on anologies with statistical mechanics some attempts have been made to apply information theory ideas directly to powders

the geometry of the moduli space of odd spin curves the spin moduli spacesg is the parameter space of theta characteristics spin structures on stable curves of genus g it has two connected components s g ands g depending on the parity of the spin structure we establish a complete birational classication by kodaira dimension of the odd components g of the spin moduli space we show thats g is uniruled for g 12 and even unirational for g 8 in this range introducing the concept of cluster for the mukai variety whose one dimensional linear sections are general canonical curves of genus g we construct new birational models ofs g these we then use to explicitly describe the birational structure of s g for instance s 8 is birational to a locally trivial p 7 bundle over the moduli space of elliptic curves with seven pairs of marked points for g 12 we prove thats g is a variety of general type in genus 12 this requires the construction of a counterexample to the slope conjecture on eective divisors on the moduli space of stable curves of genus 12

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

a sandwich proof of the shannon mcmillan breiman theorem let x t be a stationary ergodic process with distribution p admitting densities p x 0 ldots x n 1 relative to a reference measure m that is finite order markov with stationary transition kernel let i m p denote the relative entropy rate then n 1 log p x 0 ldots x n 1 rightarrow i m p mathrm a s p we present an elementary proof of the shannon mcmillan breiman theorem and the preceding generalization obviating the need to verify integrability conditions and also covering the case i m p infty a sandwich argument reduces the proof to direct applications of the ergodic theorem

behavior near the extinction time in self similar fragmentations ii finite dislocation measures we study a markovian model for the random fragmentation of an object at each time the state consists of a collection of blocks each block waits an exponential amount of time with parameter given by its size to some power alpha independently of the other blocks every block then splits randomly into sub blocks whose relative sizes are distributed according to the so called dislocation measure we focus here on the case where alpha 0 in this case small blocks split intensively and so the whole state is reduced to dust in a finite time almost surely we call this the extinction time in this paper we investigate how the fragmentation process behaves as it approaches its extinction time in particular we prove a scaling limit for the block sizes which as a direct consequence gives us an expression for an invariant measure for the fragmentation process in an earlier paper we considered the same problem for another family of fragmentation processes the so called stable fragmentations the results here are similar but we emphasize that the methods used to prove them are different our approach in the present paper is based on markov renewal theory and involves a somewhat unusual spine decomposition for the fragmentation which may be of independent interest

asymptotic behavior of the height in a digital search tree and the longest phrase of the lempel ziv scheme we study the height of a digital search tree dst built from n random strings generated by an unbiased memoryless source i e all symbols are equally likely we shall argue that the height of such a tree is equivalent to the length of the longest phrase in the lempel ziv parsing scheme that partitions a random sequence into n phrases we also analyze the longest phrase in the lempel ziv scheme in which a string of fixed length m is parsed into a random number of phrases in the course of our analysis we shall identify four natural regions of the height distribution and characterize them asymptotically for large n in particular for the region where most of the probability mass is concentrated the asymptotic distribution of the height exhibits an exponential of a gaussian distribution with an oscillating term around the most probable value k 1 lfloor log 2 n sqrt 2 log 2 n log 2 sqrt 2 log 2 n frac 1 log 2 frac 1 2 rfloor 1 more precisely we shall prove that the asymptotic distribution of a dst is concentrated on either the one point k1 or the two points k1 1 and k1 which actually proves slightly modified kesten s conjecture quoted in probab theory related fields 79 1988 pp 509 542 finally we compare our findings for dst with the asymptotic distributions of the height for other digital trees such as tries and patricia tries we derive these results by a combination of analytic methods such as generating functions laplace transform the saddle point method and ideas of applied mathematics such as linearization asymptotic matching and the wkb method our analysis makes certain assumptions about the forms of some of the asymptotic expansions as well as their asymptotic matching we also present detailed numerical verification of our results

on edwards theory of powders abstract we briefly review a statistical mechanics approach for the description of granular materials following the original ideas developed by edwards we show that edwards assumptions works very well for a class of lattice models and we discuss some recent analytic results obtained in such a framework on their phase diagram jamming transition and mixing segregation properties

universal limit laws for depths in random trees random binary search trees b ary search trees median of 2k 1 trees quadtrees simplex trees tries and digital search trees are special cases of random split trees for these trees we offer a universal law of large numbers and a limit law for the depth of the last inserted point as well as a law of large numbers for the height

upper bounds for the growth rate of dla abstract we show that the maximal length of the arms in dla can grow at most at a rate n 2 d 1 where n denotes the volume or mass of the aggregate and d is the dimension a logarithmic correction factor is needed if d 3 we also give a lower bound for the number of sites in a sphere which are eventually occupied by the aggregate

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

the height and size of random hash trees and random pebbled hash trees the random hash tree and the n tree were introduced by ehrlich in 1981 in the random hash tree n data points are hashed to values x1 xn independently and identically distributed random variables taking values that are uniformly distributed on 0 1 place the xi s in n equal sized buckets as in hashing with chaining for each bucket with at least two points repeat the same process keeping the branch factor always equal to the number of bucketed points if hn is the height of tree obtained in this manner we show that hn log2 n to 1 in probability we also show that the expected number of nodes in the random hash tree and random pebbled hash tree is asymptotic to 2 3020238 n and 1 4183342 n respectively

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

a diffusion limit for a class of randomly growing binary trees binary trees are grown by adding one node at a time an available node at height i being added with probability proportional to c i c 1 we establish both a strong law of large numbers and a central limit theorem for the vector x t x i t where x i t is the proportion of nodes of height i that are available at time t we show in fact that there is a deterministic process x i t such that sum x i t x i t text converges to 0 a text s text and such that if c2 tfrac 1 2 z i n t 2 n 2 x n 1 tc n x n 1 tc n and z n t z i n t then z n t converges weakly to a gaussian diffusion z t the results are applied to establish asymptotic normality in the unbiased coin tossing case for an entropy estimation procedure due to j ziv and to obtain results on the growth of the maximum height of the tree

on the shortest distance between orbits and the longest common substring problem abstract in this paper we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions the decay of the shortest distance depends on the correlation dimension for irrational rotations we prove a different behaviour depending on the irrational exponent of the angle of the rotation for random processes this problem corresponds to the longest common substring problem we extend the result of 6 on sequence matching to mixing processes with exponential decay

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

an algorithm for source coding subject to a fidelity criterion based on string matching a practical suboptimal universal block source coding scheme subject to a fidelity criterion is proposed the algorithm is an extension of the lempel ziv algorithm and is based on string matching with distortion it is shown that given average distortion d 0 the algorithm achieves a rate of exceeding r d 2 for a large class of sources and distortion measures tighter bounds on the rate are derived for discrete memoryless sources and for memoryless gaussian sources

an improved algorithm for approximate string matching given a text string a pattern string and an integer k a new algorithm for finding all occurrences of the pattern string in the text string with at most k differences is presented both its theoretical and practical variants improve upon the known algorithms

diffusion limited aggregation on the hyperbolic plane we consider an analogous version of the diffusion limited aggregation model defined on the hyperbolic plane we prove that almost surely the aggregate viewed at time infinity will have a positive density

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

loci of curves with subcanonical points in low genus inside the moduli space of curves of genus three with one marked point we consider the locus of hyperelliptic curves with a marked weierstrass point and the locus of non hyperelliptic curves with a marked hyperflex these loci have codimension two we compute the classes of their closures in the moduli space of stable curves of genus three with one marked point similarly we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point these loci naturally arise in the study of minimal dimensional strata of abelian differentials

a note on the height of suffix trees consider a random word in which the individual symbols are drawn from a finite or infinite alphabet with symbol probabilities p i and let h n be the height of the suffix tree constructed from the first n suffixes of this word it is shown that h n is asymptotically close to 2 log n log 1 sum i p i 2 in many respects the difference is o log log n in probability and the ratio tends to one almost surely and in the mean

optimal data compression algorithm abstract a practical source coding scheme based on approximate string matching is proposed it is an approximate fixed length string matching data compression combined with a block coder based on the empirical distribution a lemma on approximate string matching which is an extension of the kac lemma is proved it is shown based on the lemma that the deterministic algorithm converts the stationary and ergodic source u into an output process v and under the assumption that v is a stationary process after the scheme has run for an infinite time the optimal compression ratio r d is achieved this reduces the problem of the universal lossy coder to the proof of stationarity of the output process v in the proposed algorithm the main advantages of the proposed method are the asymptotic sequential behavior of the encoder and the simplicity of the decoder

patricia tries again revisited the patricia trie is a simple modification of a regular trie by eliminating unary branching nodes the patricia achieves better performance than regular tries however the question is how much on the average is the patricia better this paper offers a thorough answer to this question by considering some statistics of the number of nodes examined in a successful search and an unsuccessful search in the patricia tries it is shown that for the patricia containing n records the average of the successful search length s n asymptotically becomes 1 h 1 ln n o 1 and the variance of s n is either var s n c ln n 0 1 for an asymmetric patricia or var s n 0 1 for a symmetric patricia where h 1 is the entropy of the alphabet over which the patricia is built and c is an explicit constant higher moments of s n are also assessed the number of nodes examined in an unsuccessful search u n is studied only for binary symmetric patricia tries we prove that the m th moment of the unsuccessful search length eu m n satisfies lim n eu m n log m 2 n 1 and the variance of u n is var u n 0 87907 these results suggest that patricia tries are very well balanced trees in the sense that a random shape of patriciatries resembles the shape of complete trees that are ultimately balanced trees

diffusion limited aggregation on a tree we study the following growth model on a regular d ary tree points at distance n adjacent to the existing subtree are added with probabilities proportional to n where 1 is a positive real parameter the heights of these clusters are shown to increase linearly with their total size this complements known results that show the height increases only logarithmically when 1 results are obtained using stochastic monotonicity and regeneration results which may be of independent interest our motivation comes from two other ways in which the model may be viewed as a problem in first passage percolation and as a version of diffusion limited aggregation dla adjusted so that fingering occurs

on growing random binary trees abstract a sequence tn n 1 of nested binary trees generated by an infinite sequence of i i d random variables is studied two absolute constants 1 2 are shown to exist 0 37 lim h n ln n 1 lim h n ln n 2 with probability one here hn and hn are respectively the lengths of the shortest and the longest branches of the tree tn

a statistical mechanics approach to the inherent states of granular media abstract we consider a statistical mechanics approach to granular systems by following the original ideas developed by edwards we use the concept of inherent states defined as the stable configurations in the potential energy landscape introduced in the context of glasses under simplifying assumptions the equilibrium inherent states can be characterized by a configurational temperature 1 we link to edwards compactivity and address the problem of its experimental measure we also discuss the possibility to describe the time dependent distribution probability in the inherent states with an appropriate master equation

analysis of digital tries with markovian dependency a complete characterization of a digital tree also called a trie is presented from the depth viewpoint in a markovian framework that is under the assumption that symbols in a key are markov dependent the main findings show that asymptotically as the number of keys n tends to infinity the average depth becomes ed sub n approximately 1 h sub 1 log n c and the variance is var d sub n approximately alpha log n c where h sub 1 is the entropy of the markovian dependent alphabet alpha is a parameter of the probabilistic model and c and c are constants the symmetric independent model has alpha 0 hence in this case var d sub n o 1 limiting distribution is also derived for the depth d sub n and in particular it is shown that d sub n tends to the normal distribution in all cases except the symmetric independent model these results extend all previous analyses since most of them have been limited to independent models

some results on tries with adaptive branching we study a modification of digital trees or tries with adaptive multi digit branching such tries can dynamically adjust degrees of their nodes by choosing the number of digits to be processed per lookup while we do not specify any particular method for selecting the degrees of nodes we assume that such selection can be accomplished by examining the number of strings remaining in each sub tree and or estimating parameters of the input distribution we call this class of digital trees adaptive multi digit tries or amd tries and provide a preliminary analysis of their expected behavior in a memoryless model we establish the following results 1 there exist amd tries attaining a constant expected time of a successful search 2 there exist amd tries consuming a linear in the number of strings inserted amount of space 3 both constant search time and linear space usage can be attained if the memoryless source is symmetric we accompany our analysis with a brief survey of several known types of adaptive trie structures and show how our analysis extends and or complements previous results

efficient pattern matching for rna secondary structures we propose efficient methods to address key pattern matching problems in rna secondary structures using the notion of structural strings a structural string s string is composed of constant symbols and parameter symbols from the alphabets and respectively an individual symbol in the alphabet may be considered a complement of another unique symbol in the notion of matching constants parameters and complements is referred to as the structural matching s match problem which is helpful in matching rna and previously was solved by the structural suffix tree sst other approaches to rna matching that do not openly consider the s match include the use of affix data structures in this paper we provide new data structures and algorithms to address the s match problem specifically we introduce the structural suffix array and structural longest common prefix array and then identify how to s match with these data structures our new s matching solution is then used as the framework to answer various combinatorial queries encountered in matching rna secondary structures

birational aspects of the geometry of varieties of sums of powers abstract varieties of sums of powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms the study of these varieties dates back to sylvester and hilbert but only few of them for special degrees and number of variables are concretely identified in this paper we aim to understand a general birational behavior of vsp to do this we birationally embed these varieties into grassmannians and prove the rational connectedness of many vsp in arbitrary degrees and number of variables

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression theorems concerning the entropy of a stationary ergodic information source are derived and used to obtain insight into the workings of certain data compression coding schemes in particular the lempel siv data compression algorithm

fluctuations of the front in a one dimensional model of x y 2x we consider a model of the reaction x y to 2x on the integer lattice in which y particles do not move while x particles move as independent continuous time simple symmetric random walks y particles are transformed instantaneously to x particles upon contact we start with a fixed number a ge 1 of y particles at each site to the right of the origin and define a class of configurations of the x particles to the left of the origin having a finite l 1 norm with a specified exponential weight starting from any configuration of x particles to the left of the origin within such a class we prove a central limit theorem for the position of the rightmost visited site of the x particles

domination between trees and application to an explosion problem we define a notion of stochastic domination between trees where one tree dominates another if when the vertices of each are labeled with independent identically distributed random variables one tree is always more likely to contain a path with a specified property sufficient conditions for this kind of domination are 1 more symmetry and 2 earlier branching we apply these conditions to the problem of determining how fast a tree must grow before first passage percolation on the tree exhibits an explosion that is to say infinitely many vertices are reached in finite time for a tree in which each vertex at distance n 1 from the root has f n offspring f nondecreasing an explosion occurs with exponentially distributed passage times if and only if f n 1

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

koszul property of projections of the veronese cubic surface abstract let v p 9 be the veronese cubic surface we classify the projections of v to p 8 whose coordinate rings are koszul in particular we obtain a purely theoretical proof of the koszulness of the pinched veronese a result obtained originally by caviglia using filtrations deformations and computer assisted computations to this purpose we extend to certain complete intersections results of conca herzog trung and valla concerning homological properties of diagonal algebras