sequential and indexed two dimensional combinatorial template matching allowing rotations we present new and faster algorithms to search for a two dimensional pattern in a two dimensional text allowing any rotation of the pattern this has applications such as image databases and computational biology we consider the cases of exact and approximate matching under several matching models using a combinatorial approach that generalizes string matching techniques we focus on sequential algorithms where only the pattern can be preprocessed as well as on indexed algorithms where the text is preprocessed and an index built on it on sequential searching we derive average case lower bounds and then obtain optimal average case algorithms for all the matching models at the same time these algorithms are worst case optimal on indexed searching we obtain search time polylogarithmic on the text size as well as sublinear time in general for approximate searching

translation representations for automorphic solutions of the wave equation in non euclidean spaces i this paper deals with the spectral theory of the laplace beltrami operator acting on automorphic functions in n dimensional hyperbolic space hn we study discrete subgroups which have a fundamental polyhedron f with a finite number of sides and infinite volume concerning these we have shown previously that the spectrum of contains at most a finite number of point eigenvalues in 1 2 n 1 2 0 and none less than 1 2 n 1 2 here we prove that the spectrum of is absolutely continuous and of infinite multiplicity in 1 2 n 1 2 our approach uses the non euclidean wave equation introduced by faddeev and pavlov energy ef is defined as ut ut u lu where the bracket is the l2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric energy is conserved under the group of operator u t relating initial data to data at time t we construct two isometric representations of the space of automorphic data by l2 r n which transmute the action of u t into translation these representations are given explicitly in terms of integrals of the data over horospheres in part ii we shall show the completeness of these representations utt lu 0 l 1 2 n 1 2

finding maximal 2 dimensional palindromes abstract this paper extends the problem of palindrome searching into a higher dimension addressing two definitions of 2d palindromes the first definition implies a square while the second definition also known as a centrosymmetric factor can be any rectangular shape we present a linear time algorithm for locating all maximal square 2d palindromes in a given 2d text for the second definition of palindromes rect2dp we present two different algorithms given a text of size n n the first algorithm has time o n 3 which is linear in the worst case output size the second algorithm has time o n 2 log n o c c log n where occ is the number of maximal rect2dp in the output this provides a tradeoff in terms of output size if the output size is small the second algorithm is preferable while the first would be more efficient if the output size is n 3

improved dynamic dictionary matching in the dynamic dictionary matching problem a dictionary d contains a set of patterns that can change over time by insertion and deletion of individual patterns the user also presents text strings and asks for all occurrences of any patterns in the text the two main contributions of this paper are 1 a faster algorithm for dynamic string dictionary matching with bounded alphabets and 2 a dynamic dictionary matching algorithm for two dimensional texts and patterns the first contribution is based on an algorithm that solves the general problem of maintaining a sequence of well balanced parentheses under the operations insert delete and find nearest enclosing parenthesis pair the main new idea behind the second contribution is a novel method to efficiently manipulate failure links for two dimensional patterns

efficient retrieval of approximate palindromes in a run length encoded string in this paper we study the palindrome retrieval problem with the input string compressed into run length encoded form given a run length encoded string rle t we show how to preprocess rle t to support subsequent queries of the longest palindrome centered at any specified position and having any specified number of mismatches between its arms we present two algorithms for the problem both taking time and space polynomial in the compressed string size let n denote the number of runs of rle t and let k denote the number of mismatches the first algorithm devised for small k identifies the desired palindrome in o logn min k n time with o nlogn preprocessing time while the second algorithm achieves o log 2n query time independent of k after o n 2logn time preprocessing

riemann roch and abel jacobi theory on a finite graph abstract it is well known that a finite graph can be viewed in many respects as a discrete analogue of a riemann surface in this paper we pursue this analogy further in the context of linear equivalence of divisors in particular we formulate and prove a graph theoretic analogue of the classical riemann roch theorem we also prove several results analogous to classical facts about riemann surfaces concerning the abel jacobi map from a graph to its jacobian as an application of our results we characterize the existence or non existence of a winning strategy for a certain chip firing game played on the vertices of a graph

efficient 2 dimensional approximate matching of half rectangular figures efficient algorithms exist for the approximate two dimensional matching problem for rectangles this is the problem of finding all occurrences of an m m pattern in an n n text with no more than k mismatch insertion and deletion errors in computer vision it is important to generalize this problem to non rectangular figures we make progress towards this goal by defining half rectangular figures of height m and area a the approximate two dimensional matching problem for half rectangular patterns can be solved using a dynamic programming approach in time o an2 we show an o kn2formula formula k2n2 algorithm which combines convolutions with dynamic programming note that our algorithm is superior to previous known solutions for k m13 at the heart of the algorithm are the smaller matching problem and the k aligned ones with location problem these are interesting problems in their own right efficient algorithms to solve both these problems are presented

parallel detection of all palindromes in a string this paper presents two efficient concurrent read concurrent write parallel algorithms that find all palindromes in a given string 1 1 an o log n time n processor algorithm over general alphabets in the case of constant size alphabets the algorithm requires only nlog n processors and thus achieves an optimalspeedup 2 2 an o log log n time n log nloglog n processor algorithm over general alphabets this is the fastest possible time with the number of processors used these new results improve on the known parallel palindrome detection algorithms by using smaller auxiliary space and either by making fewer operations or by achieving a faster running time

two dimensional pattern matching with rotations the problem of pattern matching with rotation is that of finding all occurrences of a two dimensional pattern in a text in all possible rotations we prove an upper and lower bound on the number of such different possible rotated patterns subsequently given an m m array pattern and an n n array text over some finite alphabet we present a new method yielding an o n2m3 time algorithm for this problem

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

space efficient preprocessing schemes for range minimum queries on static arrays given a static array of n totally ordered objects the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on line queries of the form what is the position of a minimum element in the subarray ranging from i to j we focus on two settings where 1 the input array is available at query time and 2 the input array is available only at construction time in setting 1 we show new data structures a of size frac 2n c n theta bigl frac n lg lg n c n lg n bigr bits and query time o c n for any positive integer function c n in o bigl n varepsilon bigr for an arbitrary constant 0 varepsilon 1 or b with o nh k o n bits and o 1 query time where h k denotes the empirical entropy of k th order of the input array in setting 2 we give a data structure of size 2n o n bits and query time o 1 all data structures can be constructed in linear time and almost in place

on the construction of classes of suffix trees for square matrices we provide a uniform framework for the study of index data structures for a two dimensional matrixtext1 n 1 n whose entries are drawn from an ordered alphabet an index fortextcan be informally seen as the two dimensional analog of the suffix tree for a string it allows on line searches and statistics to be performed ontextby representing compactly the n3 square submatrices oftextin optimalo n2 space we identify 4n 1families of indices fortext each containing ni 1 2i 1 isomorphic data structures we also develop techniques leading to a single algorithm that efficiently builds any index in any family ino n2logn time ando n2 space such an algorithm improves in various respects the algorithms for the construction of the pat tree and the lsuffix tree the framework and the algorithm easily generalize tod 2 dimensions moreover as part of our algorithm we provide new algorithmic tools that yield a space efficient implementation of the naming scheme of r karpet al in proceedings fourth symposium on theory of computing pp 125 136 for strings and matrices

a generalization of the suffix tree to square matrices with applications we describe a new data structure the lsuffix tree which generalizes mccreight s suffix tree for a string j assoc comput mach 23 1976 pp 262 272 to a square matrix all matrices have entries from a totally ordered alphabet sigma based on the lsuffix tree we give efficient algorithms for the static versions of the following dual problems that arise in low level image processing and visual databases two dimensional pattern retrieval we have a library of texts s text 1 cdots text r where text i is an n i times n i matrix 1 leq i leq r we may preprocess the library then given an m times m m leq n i 1 leq i leq r pattern matrix pat we want to find all occurrences of pat in text for all text in s let t s sigma i 1 r n i 2 be the size of the library the preprocessing step builds the lsuffix tree for the matrices in s and then transforms it into an index a trie defined over sigma it takes o t s log sigma log t s time and o t s space the index can be queried directly in o m 2 log sigma totocc time where totocc is the total number of occurrences of pat in text for all text in s two dimensional dictionary matching we have a dictionary of patterns dc pat 1 cdots pat s where pat i is of dimension m i times m i 1 leq i leq s we may preprocess the dictionary then given an n times n text matrix text we want to search for all occurrences of patterns in the dictionary in the text let t dc sigma i 1 s m 2 i be the size of the dictionary and let overline t dc be the sum of the m i s the preprocessing consists of building the lsuffix tree for the matrices in dc it takes o t dc log sigma overline t dc log overline t dc time and o t dc space the search step takes o n 2 log sigma log overline t dc totocc time where totocc is the total number of occurrences of patterns in the text both problems have a dynamic version in which the library and the dictionary respectively can be updated by insertion or deletion of square matrices in them in a companion paper we will provide algorithms for the dynamic version

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

on updating suffix tree labels abstract we investigate the problem of maintaining the arc labels in the suffix tree data structure gusfield et al 1992 amir et al 1994 when it undergoes string insertions and deletions in current literature this problem is solved either by a simple accounting strategy to obtain amortized bounds fiala and green 1989 larson 1996 or by a periodical suffix tree reconstruction to obtain worst case bounds according to the global rebuilding technique in overmars 1983 unfortunately the former approach is simple and space efficient at the cost of attaining amortized bounds for the single update the latter is space consuming in practice because it needs to keep two extra suffix tree copies in this paper we obtain a simple realtime algorithm that achieves worst case bounds and only requires small additional space i e a bi directional pointer per suffix tree label we analyze it by introducing a combinatorial coloring problem on the suffix tree arcs

the laplace operator on hyperbolic three manifolds with cusps of non maximal rank where f is a geometrically finite discrete group of orientation preserving isometries which may contain parabolic elements of non maximal rank we prove the existence of a continuum eigenfunction expansion and show that in the spectral parameter the eigenfun ctions have meromorphic continuations to 112 1 these eigenfunctions are eisen stein series for f the continuous spectrum of laplace operators on hyperbolic manifolds was first analysed by number theorists as a means of studying eisenstein series maass and roelke i m r considered the case of finite volume manifolds in two dimen sions increasingly more general hyperbolic manifolds were studied by faddeev if patterson pal lax and phillips lpi lp4 mandouvalos m1 m3 and others recently agmon l a mazzeo and melrose mm and perry p proved the meromorphic continuation of the eisenstein series for geometrically finite groups with fundamental domains of infinite volume provided the groups have no parabolic elements or all the parabolic elements have maximal rank in this case the boundary at infinity which roughly speaking labels the directions at infinity in which scattering can occur is a smooth compact manifold or in the presence of parabolic elements of maximal rank the union of a smooth compact manifold with a finite number of points in fact the work of mazzeo and melrose and recent work of agmon applies to more general manifolds which have a similar structure near infinity however none of the proofs of meromorphic continuation of eigenfun ctions apply when the group has parabolic elements of non maximal rank we will study the laplace operator on a three dimensional hyperbolic manifold when parabolic elements of non maximal rank are present in this case the research supported by the national science and engineering research council of canada research supported in part by n s f grant dms 8911242 research supported in part by n s f grant dms 8802668

a combinatorial geometrical approach to two dimensional robust pattern matching with scaling and rotation the problem of two dimensional pattern matching invariant under a given class of admissible transformations f is to find matches of transformed versions f p of a pattern p in a given text t for all f in f in this paper pattern matching invariant under compositions of real valued scaling and rotation are investigated we give a new discretization technique for this class of transformations and prove sharp lower and upper bounds on the number of different possibilities to transform a pattern in this way subsequently we present the first efficient pattern matching algorithm invariant under compositions of scaling and rotation the algorithm works in time o m 2n 6 for patterns of size m 2 and texts of size n 2 we conclude with an experimental section to support the practical use of our results

scattering theory for the wave equation on a hyperbolic manifold this talk is a survey of work done jointly mainly with peter lax 3 4 5 but also with bettina wilkott and alex woo 10 over the past ten years it deals with the spectrum of the perturbed and unperturbed laplace beltrami operator acting on automorphic functions on an n dimensional hyperbolic space ihn the associated discrete subgroup of motions is assumed to have the finite geometric property but is otherwise unrestricted this means that the fundamental domain when derived by the polygonal method has a finite number of sides its volume may be finite or infinite and it can have cusps of arbitrary rank with the obvious identifications the fundamental domain can be treated as a manifold m

on the sorting complexity of suffix tree construction the suffix tree of a string is the fundamental data structure of combinatorial pattern matching we present a recursive technique for building suffix trees that yields optimal algorithms in different computational models sorting is an inherent bottleneck in building suffix trees and our algorithms match the sorting lower bound specifically we present the following results 1 weiner 1973 who introduced the data structure gave an optimal 0 n time algorithm for building the suffix tree of an n character string drawn from a constant size alphabet in the comparison model there is a trivial o n log n time lower bound based on sorting and weiner s algorithm matches this bound for integer alphabets the fastest known algorithm is the o n log n time comparison based algorithm but no super linear lower bound is known closing this gap is the main open question in stringology we settle this open problem by giving a linear time reduction to sorting for building suffix trees since sorting is a lower bound for building suffix trees this algorithm is time optimal in every alphabet mode in particular for an alphabet consisting of integers in a polynomial range we get the first known linear time algorithm 2 all previously known algorithms for building suffix trees exhibit a marked absence of locality of reference and thus they tend to elicit many page faults i os when indexing very long strings they are therefore unsuitable for building suffix trees in secondary storage devices where i os dominate the overall computational cost we give a linear i o reduction to sorting for suffix tree construction since sorting is a trivial i o lower bound for building suffix trees our algorithm is i o optimal

invariant pattern recognition a review in this document we review and compare some of the classical and modern techniques for solving the problem of invariant pattern recognition such techniques include integral transforms construction of algebraic moments and the use of structured neural networks in all cases we assume that the nature of the invariance group is known a priori many of the methods described apply to specific geometrical transformation groups however some of the techniques are highly general and applicable to large classes of invariance groups we also review some results regarding the existence and structure of invariants under certain kinds of groups

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

order preserving indexing kubica et al 33 and kim et al 29 introduced order preserving pattern matching for a given text the goal is to find its factors having the same shape as a given pattern known results include a linear time algorithm for this problem in case of polynomially bounded alphabet and a generalization to multiple patterns we propose an index that enables order preserving pattern matching queries in time proportional to pattern length the index can be constructed in o n log log n expected time or in o n log 2 log n log log log n worst case time it is an incomplete order preserving suffix tree which may miss a single edge label at each branching node for most applications such incomplete suffix trees provide the same functional power as the complete ones we show a number of their applications including computation of longest common factors longest previously occurring factors and squares in a string in the order preserving setting we also give an o n log n time algorithm constructing complete order preserving suffix trees

translation representations for automorphic solutions of the wave equation in non euclidean spaces the case of finite volume soit un sous groupe discret d automorphismes de h n a polyedre fondamental de volume fini de nombre de cotes fini et a n cusps soit l operateur de laplace beltrami agissant sur des fonctions automorphes par rapport a on demontre que a un spectre absolument continu de multiplicite uniforme n sur n 1 2 2 plus un spectre standard discret

linear series on metrized complexes of algebraic curves a metrized complex of algebraic curves over an algebraically closed field kappa is roughly speaking a finite metric graph gamma together with a collection of marked complete nonsingular algebraic curves c v over kappa one for each vertex v of gamma the marked points on c v are in bijection with the edges of gamma incident to v we define linear equivalence of divisors and establish a riemann roch theorem for metrized complexes of curves which combines the classical riemann roch theorem over kappa with its graph theoretic and tropical analogues from amini and caporaso adv math 240 1 23 2013 baker and norine adv math 215 2 766 788 2007 gathmann and kerber math z 259 1 217 230 2008 and mikhalkin and zharkov tropical curves their jacobians and theta functions contemporary mathematics 203 231 2007 providing a common generalization of all of these results for a complete nonsingular curve x defined over a non archimedean field mathbb k together with a strongly semistable model mathfrak x for x over the valuation ring r of mathbb k we define a corresponding metrized complex mathfrak c mathfrak x of curves over the residue field kappa of mathbb k and a canonical specialization map tau mathfrak c mathfrak x from divisors on x to divisors on mathfrak c mathfrak x which preserves degrees and linear equivalence we then establish generalizations of the specialization lemma from baker algebra number theory 2 6 613 653 2008 and its weighted graph analogue from amini and caporaso adv math 240 1 23 2013 showing that the rank of a divisor cannot go down under specialization from x to mathfrak c mathfrak x as an application we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to eisenbud and harris invent math 85 337 371 1986 using this link we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove among other things that any degeneration of a mathfrak g r d in a regular family of semistable curves is a limit mathfrak g r d on the special fiber

the spectrum of kleinian manifolds abstract we obtain the plancherel theorem for l 2 g where g is a classical simple lie group of real rank one and g is convex cocompact discrete subgroup and deduce its consequences for the spectrum of locally invariant differential operators on bundles over kleinian manifolds as the main tool we develop a geometric version of scattering theory which in particular contains the meromorphic continuation of the eisenstein series for this situation the central role played by invariant distribution sections supported on the limit set is emphasized

order preserving matching we introduce a new string matching problem called order preserving matching on numeric strings where a pattern matches a text if the text contains a substring of values whose relative orders coincide with those of the pattern order preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves solving order preserving matching is closely related to the representation of order relations of a numeric string we define the prefix representation and the nearest neighbor representation of the pattern both of which lead to efficient algorithms for order preserving matching we present efficient algorithms for single and multiple pattern cases for the single pattern case we give an o nlogm time algorithm and optimize it further to obtain o n mlogm time for the multiple pattern case we give an o nlogm time algorithm

spectral analysis of second order elliptic operators on noncompact manifolds on demontre l estimation de mourre et les bornes qui y sont reliees pour une classe d operateurs elliptiques qui inclut des laplaciens pour des quotients cofinis et coinfinis d espaces hyperboliques

determinants of laplacians and isopolar metrics on surfaces of infinite area we construct a determinant of the laplacian for infinite area surfaces which are hyperbolic near infinity and without cusps in the case of a convex co compact hyperbolic metric the determinant can be related to the selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the laplacian in the hyperbolic near infinity case the determinant is analyzed through the zeta regularized relative determinant for a conformal metric perturbation we establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics these results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances

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

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

faster two dimensional pattern matching with rotations the most efficient currently known algorithms for two dimensional pattern matching with rotations have a worst case time complexity of o n 2m 3 where the size of the text is nxn and the size of the pattern is mxm in this paper we present a new algorithm for the problem whose running time is o n 2m 2

resolvent of the laplacian on geometrically finite hyperbolic manifolds for geometrically finite hyperbolic manifolds n 1 we prove the meromorphic extension of the resolvent of laplacian poincare series einsenstein series and scattering operator to the whole complex plane we also deduce the asymptotics of lattice points of in large balls of n 1 in terms of the hausdorff dimension of the limit set of

an alphabet independent approach to two dimensional pattern matching there are many solutions to the string matching problem that are strictly linear in the input size and independent of alphabet size furthermore the model of computation for these algorithms is very weak they allow only simple arithmetic and comparisons of equality between characters of the input in contrast algorithms for two dimensional matching have needed stronger models of computation most notably assuming a totally ordered alphabet the fastest algorithms for two dimensional matching have therefore had a logarithmic dependence on the alphabet size in the worst case this gives an algorithm that runs in o n 2 log m with o m 2 log m preprocessing the authors show an algorithm for two dimensional matching with an o n 2 text scanning phase furthermore the text scan requires no special assumptions about the alphabet i e it runs on the same model as the standard linear time string matching algorithm the pattern preprocessing requires an ordered alphabet and runs with the same alphabet dependency as the previously known algorithms

fast algorithms for finding nearest common ancestors we consider the following problem given a collection of rooted trees answer on line queries of the form what is the nearest common ancester of vertices x and y we show that any pointer machine that solves this problem requires omega log log n time per query in the worst case where n is the total number of vertices in the trees on the other hand we present an algorithm for a random access machine with uniform cost measure and a bound of omega log n on the number of bits per word that requires o 1 time per query and o n preprocessing time assuming that the collection of trees is static for a version of the problem in which the trees can change between queries we obtain an almost linear time and linear space algorithm

generalized pattern matching and periodicity under substring consistent equivalence relations let be a substring consistent equivalence relation scer on strings such that for any two strings x y x y implies that 1 x y and 2 x i j y i j for all 0 i j x examples of scer are parameterized pattern matching and order preserving pattern matching we present a generalized and efficient algorithm for pattern matching with scer also we show analogues of fine and wilf s periodicity lemma hold for scer

rotation and lighting invariant template matching we address the problem of searching for a two dimensional pattern in a two dimensional text or image such that the pattern can be found even if it appears rotated and it is brighter or darker than its occurrence furthermore we consider approximate matching under several tolerance models we obtain algorithms that are almost optimal both in the worst and the average cases simultaneously the complexities we obtain are very close to the best current results for the case where only rotations but not lighting invariance are supported these are the first results for this problem under a combinatorial approach

a mourre estimate and related bounds for hyperbolic manifolds with cusps of non maximal rank we consider the laplace operator on quotients of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries with parabolic subgroups of non maximal rank using methods developed by the first two authors we prove a mourre estimate and commutator estimates on the laplacian which imply absolute continuity of the spectrum and quantitative resolvent estimates these estimates will be used elsewhere to study the scattering matrix and eisenstein series and their meromorphic continuations and should be useful in studying trace formulas for these discrete groups

the laplace operator on a hyperbolic manifold i spectral and scattering theory using techniques of stationary scattering theory for the schrodinger equation we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the laplace beltrami operator on manifolds mn arising as the quotient of hyperbolic n dimensional space by a geometrically finite discrete group of hyperbolic isometries we consider manifolds mn of infinite volume in subsequent papers we will use the techniques developed here to analytically continue eisenstein series for a large class of discrete groups including some groups with parabolic elements

generalizations of suffix arrays to multi dimensional matrices we propose multi dimensional index data structures that generalize suffix arrays to square matrices and cubic matrices giancarlo proposed a two dimensional index data structure the lsuffix tree that generalizes suffix trees to square matrices however the construction algorithm for lsuffix trees maintains complicated data structures and uses a large amount of space we present simple construction algorithms for multi dimensional suffix arrays by applying a new partitioning technique to lexicographic sorting our contributions are the first efficient algorithms for constructing two dimensional and three dimensional suffix arrays directly

scattering theory for twisted automorphic functions the purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane defined by a cofinite but not cocompact discrete group r with an irreducible unitary representation p and satisfying u yz p qy u z the lax phillips approach is used with the wave equation playing a central role incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations r and r for the solution operator the scattering operator which maps r f into r f is unitary and commutes with translation the spectral representation of the scattering operator is a multiplicative operator which can be expressed in terms of the constant term of the eisenstein series when the dimension of p is one the elements of the scattering operator cannot vanish however when dim p 1 this is no longer the case

multiple matching of rectangular patterns we describe the first worst case efficient algorithm for simultaneously matching multiple rectangular patterns of varying sizes and aspect ratios in a rectangular text efficient means significantly more efficient asymptotically than applying known algorithms that handle one height or width or aspect ratio at a time for each height our algorithm features an interesting use of multidimensional range searching as well as new adaptations of several known techniques for two dimensional string matching we also extend our algorithm to a dynamic setting where the set of patterns can change over time

simple and flexible detection of contiguous repeats using a suffix tree we study the problem of detecting all occurrences of primitive tandem repeats and tandem arrays in a string we first give a simple time and space optimal algorithm to find all tandem repeats and then modify it to become a time and space optimal algorithm for finding only the primitive tandem repeats both of these algorithms are then extended to handle tandem arrays the contribution of this paper is both pedagogical and practical giving simple algorithms and implementations based on a suffix tree using only standard tree traversal techniques

finding patterns and periods in cartesian tree matching abstract we introduce a new metric of match called cartesian tree matching which means that two strings match if they have the same cartesian trees based on cartesian tree matching we define single pattern matching multiple pattern matching and indexing problems we propose a linear time algorithm for single pattern matching and randomized linear time algorithms for multiple pattern matching and indexing we also define three types of periods called full period initial period and general period we propose o n time o n log log n time and o n log n time algorithms for finding all the full periods initial periods and general periods of a string of length n respectively our efficient algorithms use a representation of the cartesian tree called parent distance representation

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

usefulness of the karp miller rosenberg algorithm in parallel computations on strings and arrays abstract the karp miller rosenberg 1972 algorithm was one of the first efficient almost linear sequential algorithms for finding repeated patterns and for string matching in the area of efficient sequential computations on strings it was soon superseded by more efficient and more sophisticated algorithms we show that the karp miller rosenberg algorithm kmr must be considered as a basic technique in parallel computations for many problems variations of kmr give the known most efficient parallel algorithms the representation of the set of basic factors subarrays of a string array produced by the algorithm is an extremely useful data structure in parallel algorithms on strings and arrays this gives also a general unifying framework for a large variety of problems we show that the following problems for strings and arrays can be solved by almost optimal parallel algorithms pattern matching longest repeated factor subarray longest common factor subarray maximal symmetric factor subarray also the following problems for strings can be solved within the same complexity bounds finding squares testing even palstars and compositions of k palindromes for k 2 3 4 computing lyndon factorization and building minimal pattern matching automata in the model without concurrent writes the parallel time is o log n 2 with n processors and in the model with concurrent writes the time for most of the problems is o log n with n processors for two problems related to the one dimensional case longest repeated factor and longest common factor there were designed parallel algorithms using suffix trees apostolico et al 1988 however our data structure is simpler and furthermore for the two dimensional case suffix trees do not work the complexity of our algorithms does not depend on the size of the alphabet except for the computation of pattern matching automata

alphabet independent two dimensional witness computation we study two dimensional periodicity introduced by amir and benson we characterize periods of a two dimensional array namely the vectors such that two copies of the array one shifted by the vector over the other overlap without a mismatch using this characterization we design an alphabet independent linear time algorithm for two dimensional witness computation i e an o m 2 time algorithm that finds periods of an m times m array as well as witnesses against nonperiods of the array among the vectors whose length is less than m 4 the constant in the o notation does not depend on the alphabet size combined with the alphabet independent text processing algorithm of amir benson and farach siam j comput 23 1994 pp 313 323 this leads to the first alphabet independent linear time algorithm for two dimensional pattern matching

two and higher dimensional pattern matching in optimal expected time algorithms with optimal expected running time are presented for searching the occurrences of a two dimensional m x m pattern p in a two dimensional n x n text t over an alphabet of size c the algorithms are based on placing in the text a static grid of test points determined only by n m and c not dynamically by earlier test results using test strings read from the test points the algorithms eliminate as many potential occurrences of p as possible the remaining potential occurrences are separately checked for actual occurrences a suitable choice of the test point set leads to algorithms with expected running time o n2logc m2 m2 using the uniform bernoulli model of randomness this is shown to be optimal by a generalization of a one dimensional lower bound result by yao experimental results show that the algorithms are efficient in practice too the method is also generalized for the k mismatches problem the resulting algorithm has expected running time o kn2logc m2 m2 provided that k leq m lfloor m lceil log c m 2 rceil rfloor 1 2 all algorithms need preprocessing of p which takes time and space o m2 the text processing can be done on line using a rather small window the algorithms easily generalize to d dimensional matching for any d

density and location of resonances for convex co compact hyperbolic surfaces let x vargamma backslash mathbb h 2 be a convex co compact hyperbolic surface and let be the hausdorff dimension of the limit set let x be the hyperbolic laplacian we show that the density of resonances of the laplacian x in rectangles bigl sigma leq mathrm re s leq delta big vert mathrm im s big vert leq t bigr is less than o t1 in the limit t where frac delta 2 this improves the previous fractal weyl upper bound of zworski invent math 136 2 353 409 1999 and goes in the direction of a conjecture stated in jakobson and naud geom funct anal 22 2 352 368 2012

a linear time solution to the single function coarsest partition problem abstract the problem of finding the coarsest partition of a set s with respect to another partition of s one or more functions on s has several applications one of which is the state minimization of finite state automata in 1971 hopcroft presented an algorithm to solve the many function coarsest partition problem for sets of n elements in o n log n time and o n space in 1974 aho hopcroft and ullman presented an o n log n algorithm that solves the special case of this problem for only one function both these algorithms use a negative strategy that repeatedly refines the original partition until a solution is found we present a new algorithm to solve the single function coarsest partition problem in o n time and space using a different constructive approach our algorithm can be applied to the automated manufacturing of woven fabric

fast parallel and serial multidimensional approximate array matching abstract consider the multidimensional array matching problem where differences between characters of the pattern and characters of the text are permitted a difference may be due to a mismatch between a text and pattern character superfluous text character or superfluous pattern character given a d dimensional array of size nd text and a d dimensional array of size md pattern we present the following algorithms for a given k find all occurrences of the pattern in the text with at most k differences our serial algorithm runs in time o nd dk k2 and the parallel algorithm runs in time o d dlog n k k2 using nd processors if superfluous characters are not allowed and the only permitted errors are mismatches we solve the problem serially in time o nddk and in parallel in time o d dlogn k using nd processors we present an alternate algorithm for the mismatches problem which runs serially in time o 2dndlog2 m and in parallel in time o d log n using nd processors this algorithm is more efficient for large k we also give an efficient solution to the close match problem here a mismatch weight function f 0 1 is assigned the weight function gives weight to the mismatches some mismatches being worse than others we present a serial algorithm for finding all appearances of the pattern in the text with a bounded total error in time o 2dnd log2 m our parallel algorithm is again of time complexity o d log n using nd processors

average complexity of exact and approximate multiple string matching we show that the average number of characters examined to search for r random patterns of length m in a text of length n over a uniformly distributed alphabet of size cannot be less than n log rm m when we permit up to k insertions deletions and or substitutions of characters in the occurrences of the patterns the lower bound becomes n k log rm m this generalizes previous single pattern lower bounds of yao for exact matching and of chang and marr for approximate matching and proves the optimality of several existing multipattern search algorithms