improved parallel construction of wavelet trees and rank select structures abstract existing parallel algorithms for wavelet tree construction have a work complexity of o n log this paper presents parallel algorithms for the problem with improved work complexity our first algorithm is based on parallel integer sorting and has either o n log log n log log n log log n work and polylogarithmic depth or o n log log n work and sub linear depth we also describe another algorithm that has o n log log n work and o log n depth we then show how to use similar ideas to construct variants of wavelet trees arbitrary shaped binary trees and multiary trees as well as wavelet matrices in parallel with lower work complexity than prior algorithms finally we show that the rank and select structures on binary sequences and multiary sequences which are stored on wavelet tree nodes can be constructed in parallel with improved work bounds matching those of the best existing sequential algorithms for constructing rank and select structures

on parallel integer merging the problem of merging two sorted arrays a a1 a2 an1 and b b1 b2 bn2 is considered for input elements that are drawn from a domain of integers 1 s we present an algorithm that runs in o log log log s time using n log log log s crew pram processors optimal speed up and o ns space where n n1 n2 for input elements that are drawn from a domain of integers 1 n we present a second algorithm that runs in o n time where n is the inverse of ackermann s function using n n crew pram processors and linear space this second algorithm is non uniform however it can be made uniform at a price of a certain loss of speed or by using a crcw pram

order preserving pattern matching with k mismatches we consider a generalization of the recently introduced order preserving pattern matching the difference between the standard pattern matching and the order preserving variant is that instead of looking for an exact copy of the pattern we only require that the relative order between the elements is the same in our generalization we additionally allow up to k mismatches between the pattern of length m and the text of length n and the goal is to construct an efficient algorithm for small values of k our solution detects an order preserving occurrence with up to k mismatches in o n loglogm kloglogk time

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

a really simple approximation of smallest grammar in this paper we present a really simple linear time algorithm constructing a context free grammar of size 4 g log 3 2 n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c algorithms with such an approximation guarantee and running time are known however all of them were non trivial and their analyses were involved the here presented algorithm computes the lz77 factorisation and transforms it in phases to a grammar in each phase it maintains an lz77 like factorisation of the word with at most factors as well as additional o letters where was the size of the original lz77 factorisation in one phase in a greedy way by a left to right sweep and a help of the factorisation we choose a set of pairs of consecutive letters to be replaced with new symbols i e nonterminals of the constructed grammar we choose at least 2 3 of the letters in the word and there are o many different pairs among them hence there are o log n phases each of them introduces o nonterminals to a grammar a more precise analysis yields a bound 4 log n as g this yields the desired bound g 4 g log n g

phase space functions and correspondence rules it is shown that positive quantum joint distributions in phase space lead to quantum mechanical operators depending on the state to be considered it is also shown that a simple requirement of canonical invariance leads to the weyl correspondence rule as the only allowed one

improved deterministic parallel integer sorting abstract we consider the problem of deterministic sorting of integers on a parallel ram pram the best previous result t hagerup 1987 inform and comput 75 39 51 states that n integers of size polynomial in n can be sorted in time o log n on a priority crcw pram with o n log log n log n processors we prove that n integers drawn from a set 0 m 1 can be sorted on an arbitrary crcw pram in time o log n log log n log log m with a time processor product of o n log log m in particular if m n log n o 1 the time and number of processors used are o log n log log n and o n log log n 2 log n respectively this improves the previous result in several respects the new algorithm is faster it works on a weaker pram model and it is closer to optimality for input numbers of superpolynomial size if log log m o log n log log n the new algorithm is optimally fast for any polynomial number of processors and if log log m 1 1 log log n and log log m 0 log n it has optimal speedup relative to the fastest known sequential algorithm the space needed is o nme for arbitrary but fixed e 0 the sorting algorithm derives its speed from a fast solution to a special list ranking problem of possible independent interest the monotonic list ranking problem in monotonic list ranking each list element has an associated key and the keys are known to increase monotonically along the list we show that monotonic list ranking problems of size n can be solved optimally in time o log n log log n we also discuss and attempt to solve some of the problems arising in the precise description and implementation of parallel recursive algorithms as part of this effort we introduce a new pram variant the allocated pram

a survey of string orderings and their application to the burrows wheeler transform abstract for over 20 years the data clustering properties and applications of the efficient burrows wheeler transform have been researched lexicographic suffix sorting is induced during the transformation and more recently a new direction has considered alternative ordering strategies for suffix arrays and thus the transforms in this survey we look at these distinctly ordered bijective and linear transforms for arbitrary alphabets we discuss the v bwt derived from v order and the d bwt based on lex extension order the binary case yields a pair of transforms the binary rouen b bwt defined using binary block order lyndon words are relevant to implementing the original transform the new transforms are defined for analogous structures v words indeterminate lyndon words and b words respectively there is plenty of scope for further non lexicographic transforms as indicated in the conclusion

chattering avoidance by second order sliding mode control relying on the possibility of generating a second order sliding motion by using as control the first derivative of the control signal instead of the actual control a new solution to the problem of chattering elimination in variable structure control vsc is presented such a solution inspired by the classical bang bang optimal control strategy is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second order uncertain system with unavailable velocity then the applicability of the algorithm is extended via suitable modifications to the case of nonlinear systems with uncertainties of more general types the proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold

improved fast integer sorting in linear space we present a fast deterministic algorithm for integer sorting in linear space our algorithm sorts n integers in the range 0 1 2 m 1 in linear space in o n log log n log log log n time when log m log2 n 0 we can further achieve o n log log n time this improves the o n log log n 2 time bound given in m thorup 1998 in proc 1998 acm siam symp on discrete algorithms soda 98 pp 550 555 this result is obtained by combining our new technique with that of thorup s signature sorting a andersson t hagerup s nilsson and r raman 1995 in proc 1995 symposium on theory of computing pp 427 436 a andersson s result 1996 in proc 1996 ieee symp on foundations of computer science pp 135 141 r raman s result 1996 lecture notes in computer science vol 1136 pp 121 137 springer verlag berlin new york and our previous result y han and x shen 1999 in proc 1999 tenth annual acm siam symposium on discrete algorithms soda 99 baltimore md january pp 419 428 are also used for the design of our algorithms we provide an approach and techniques which are totally different from previous approaches and techniques for the problem as a consequence our technique can be extended to apply to nonconservative sorting and parallel sorting our nonconservative sorting algorithm sorts n integers in 0 1 m 1 in time o n log log n 2 log k log log log n using word length k log m n where k log n our erew parallel algorithm sorts n integers in 0 1 m 1 in o log n 2 time and o n log log n 2 log log log n operations provided log m log n 2

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

efficient construction of maximal and minimal representations of motifs of a string two substrings of a given text string are called synchronous occurrence equivalent if their sets of occurrence locations are translates of each other linear time algorithms are given for the problems of finding a shortest and a longest substring that is synchronous with a given substring we also introduce approximate variants of the motif discovery problem and give polynomial time algorithms for finding longest and shortest substrings whose suitably translated occurrence location set contains or respectively is contained in a given set of locations the fft technique used here also leads to an o nlogn algorithm for finding the maximum content gapped motif that is synchronous with a given set of locations the previously known algorithm for this problem is only quadratic

faster suffix sorting we propose a fast and memory efficient algorithm for lexicographically sorting the suffixes of a string a problem that has important applications in data compression as well as string matching our algorithm eliminates much of the overhead of previous specialized approaches while maintaining their robustness for degenerate inputs for input size n our algorithm operates in only two integer arrays of size n and has worst case time complexity o nlogn we demonstrate experimentally that our algorithm has stable performance compared with other approaches

efficient computation of longest single arm gapped palindromes in a string abstract in this paper we introduce new types of approximate palindromes called single arm gapped palindromes shortly sagps a sagp contains a gap in either its left or right arm which is in the form of either w g u c u r w r or w u c u r g w r where w and u are non empty strings w r and u r are respectively the reversed strings of w and u g is a string called a gap and c is either a single character or the empty string here we call wu and u r w r the arm of the sagp and u v the length of the arm we classify sagps into two groups those which have u c u r as a maximal palindrome type 1 and the others type 2 we propose several algorithms to compute type 1 sagps with longest arms occurring in a given string based on suffix arrays then we propose a linear time algorithm to compute all type 1 sagps with longest arms based on suffix trees also we show how to compute type 2 sagps with longest arms in linear time we also perform some preliminary experiments to show practical performances of the proposed methods

parallel algorithms with optimal speedup for bounded treewidth we describe the first parallel algorithm with optimal speedup for constructing minimum width tree decompositions of graphs of bounded treewidth on n vertex input graphs the algorithm works in o log n 2 time using o n operations on the erew pram we also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth including all decision problems expressible in monadic second order logic on n vertex input graphs the algorithms use o n operations together with o log n log n time on the erew pram or o log n time on the crcw pram

on multi input chattering free second order sliding mode control a solution to the problem of eliminating the chattering effect which is always associated with practical implementations of variable structure control is presented with reference to a class of uncertain multi input nonlinear systems the solution procedure relies on the application of an original control approach capable of enforcing a second order sliding mode i e a sliding regime on a surface s x t 0 in the system state space with s spl dot x t identically equal to zero a regime enforced by a control signal depending on s x t but directly acting only on s spl uml x t such an approach in its original formulation only applies to single input nonlinear systems with particular types of uncertainties in the present paper its validity is extended to multi input nonlinear systems characterized by uncertainties of more general nature covering a wide class of real processes

indexing weighted sequences neat and efficient in a weighted sequence for every position of the sequence and every letter of the alphabet a probability of occurrence of this letter at this position is specified weighted sequences are commonly used to represent imprecise or uncertain data for example in molecular biology where they are known under the name of position weight matrices given a probability threshold 1 z we say that a string p of length m occurs in a weighted sequence x at position i if the product of probabilities of the letters of p at positions i i m 1 in x is at least 1 z in this article we consider an indexing variant of the problem in which we are to pre process a weighted sequence to answer multiple pattern matching queries we present an o nz time construction of an o nz sized index for a weighted sequence of length n that answers pattern matching queries in the optimal o m occ time where occ is the number of occurrences reported the cornerstone of our data structure is a novel construction of a family of z strings that carries the information about all the strings that occur in the weighted sequence with a sufficient probability we thus improve the most efficient previously known index by amir et al theor comput sci 2008 with size and construction time o nz2 log z preserving optimal query time on the way we develop a new more straightforward index for the so called property matching problem we provide an open source implementation of our data structure and present experimental results using both synthetic and real data our construction allows us also to obtain a significant improvement over the complexities of the approximate variant of the weighted index presented by biswas et al at edbt 2016 and an improvement of the space complexity of their general index we also present applications of our index

output tracking control of uncertain nonlinear second order systems abstract the solution of a tracking problem for a secondorder nonlinear system with uncertain dynamics and incomplete state measurement is obtained by means of a procedure directly inspired by the solution of the classical minimum time optimal control problem two different types of uncertainty are considered in the paper in the first case a constant bound on the uncertain dynamics is assumed to be known in the second case the bound is a function of both the measurable and the unmeasurable state variable of the system in both cases the possibility of applying the proposed control algorithms is proved to be determined by a proper choice of the control signal features the resulting system is characterized by a suitable feedback switching logic and the convergence of the system trajectory to the desired one or to a vicinity of this latter is proved also in the uncertain case

analysis of chattering in systems with second order sliding modes a systematic approach to the chattering analysis in systems with second order sliding modes is developed the neglected actuator dynamics is considered to be the main cause of chattering in real systems the magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second order sliding mode control generalized suboptimal 2 smc g so algorithms is evaluated sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding poincare maps for linear systems driven by 2 smc g so algorithms analysis tools based on the frequency domain methods are developed the first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions the second technique represents a modified tsypkin s method and provides for a relatively simple theoretically exact approach to evaluate the periodic motion parameters examples of analysis and simulation results are given throughout this paper

output feedback sliding mode control in the presence of unknown disturbances this paper examines the use of a sliding mode controller and an equivalent output injection sliding mode observer to control a nonlinear plant in the presence of an unknown disturbance with only the measured output available for feedback the observer reconstructs the full state in finite time and provides an estimate of the unknown disturbance the controller ensures asymptotic reference trajectory tracking or alternatively asymptotic convergence to the origin for the plant asymptotic stability of the closed loop system is proved under a set of nonrestrictive assumptions on the plant and the disturbance these results are verified numerically through simulation

born jordan quantization and the uncertainty principle the weyl correspondence and the related wigner formalism lie at the core of traditional quantum mechanics we discuss here an alternative quantization scheme the idea of which goes back to born and jordan and which has recently been revived in another context namely time frequency analysis we show in particular that the uncertainty principle does not enjoy full symplectic covariance properties in the born and jordan scheme as opposed to what happens in the weyl quantization

higher order sliding mode controllers with optimal reaching higher order sliding mode hosm control design is considered for systems with a known permanent relative degree in this paper we introduce the robust fuller s problem that is a robust generalization of the fuller s problem a standard optimal control problem for a chain of integrators with bounded control by solving the robust fuller s problem it is possible to obtain feedback laws that are hosm algorithms of generic order and in addition provide optimal finite time reaching of the sliding manifold a common difficulty in the use of existing hosm algorithms is the tuning of design parameters our methodology proves useful for the tuning of hosm controller parameters in order to assure desired performances and prevent instabilities the convergence and stability properties of the proposed family of controllers are theoretically analyzed simulation evidence demonstrates their effectiveness

binary block order rouen transform novel twin binary burrows wheeler type transforms are introduced the transforms are defined for lyndon like b words which apply binary block order we call this approach the b bwt rouen transform these bijective rouen transforms and inverses are computed in linear time preliminary experimental results indicate potential value of binary transforms we introduce bijective burrows wheeler type transforms for binary strings 1 the original method by burrows and wheeler 4 is based on lexicographic order for general alphabets and the transform is defined to be the last column of the ordered bwt matrix this new approach applies binary block order b order which yields not one but twin transforms one based on lyndon words the other on a repetition of lyndon words these binary b bwt transforms are constructed here for b words analogous structures to lyndon words a key computation in the transforms is the application of a linear time suffix sorting technique such as 18 21 22 27 to sort the cyclic rotations of a binary input string into their b order moreover like the original lexicographic transform we show that computing the b bwt inverses is also achieved in linear time by using straightforward combinatorial arguments

technical communique finite time output stabilization with second order sliding modes in this note a class of discontinuous feedback laws that switch over branches of parabolas in the auxiliary state plane is analyzed conditions are provided under which controllers belonging to this class are second order sliding mode algorithms they ensure uniform global finite time output stability for uncertain systems of relative degree two

the proper quantum analogue of the birkhoff gustavson method of normal forms a deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation this completes the programme of robnik 1984 to develop a purely algebraic method for the quantization of non integrable hamiltonian systems about a point of stable equilibrium

second order sliding mode control for nonlinear affine systems with quantized uncertainty abstract this paper deals with the design of a second order sliding mode sosm control algorithm able to enhance the closed loop performance depending on the current working conditions the novelty of the proposed approach is the design of a nonsmooth switching line based on the quantization of the uncertainties affecting the system the quantized uncertainty levels allow one to define nested box sets in the auxiliary state space i e the space of the sliding variable and its first time derivative and select suitable control amplitudes for each set in order to guarantee the convergence of the sliding variable to the sliding manifold in a finite time the proposed algorithm is theoretically analyzed proving the existence of an upperbound of the reaching time to the origin through the considered quantization levels

improved parallel integer sorting without concurrent writing we show thatnintegers in the range 1 ncan be sorted stably on an erew pram usingo t time ando n lognloglogn logn 2 t operations for arbitrary givent lognloglogn and on a crew pram usingo t time ando n logn logn 2t logn operations for arbitrary givent logn in addition we are able to sortnarbitrary integers on a randomized crew pram within the same resource bounds with high probability in each case our algorithm is a factor of almost logn closer to optimality than all previous algorithms for the stated problem in the stated model and our third result matches the operation count of the best previous sequential algorithm we also show thatnintegers in the range 1 mcan be sorted ino logn 2 time witho n operations on an erew pram using a nonstandard word length ofo lognloglognlogm bits thereby greatly improving the upper bound on the word length necessary to sort integers with a linear time processor product even sequentially our algorithms were inspired by and in one case directly use the fusion trees of fredman and willard

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

a bijective variant of the burrows wheeler transform using v order in this paper we introduce the v transform v bwt a variant of the classic burrows wheeler transform the original bwt uses lexicographic order whereas we apply a distinct total ordering of strings called v order v order string comparison and lyndon like factorization of a string x x 1 n x x 1 n into v words have recently been shown to be linear in their use of time and space daykin et al 2011 18 here we apply these subcomputations along with n n suffix sorting ko and aluru 2003 26 to implement linear v sorting of all the rotations of a string when it is known that the input string x is a v word we compute the v transform in n n time and space and also outline an efficient algorithm for inverting the v transform and recovering x we further outline a bijective algorithm in the case that x is arbitrary we propose future research into other variants of transforms using lex extension orderings daykin et al 2013 19 motivation for this work arises in possible applications to data compression

fast bwt in small space by blockwise suffix sorting abstract we present a new space and time efficient algorithm for computing the burrow wheeler transform bwt for any choice of a parameter v 3 n 2 3 the computation of bwt for a text of length n takes o n log n v n worst case time and o n log n v n average case time using o n log n v bits of space in addition to the text and the bwt for example if v log 2 n the time is o n log 2 n in the worst case and o n log n on an average with the additional space requirement of o n bits the algorithm is alphabet independent it uses only character comparisons and the complexities do not depend on the alphabet size unless v does a practical implementation is 2 3 times slower than one of the fastest and most space efficient previous algorithms while needing only one third of the main memory the algorithm is based on suffix arrays but unlike any other algorithm it can construct the suffix array a small block at a time without storing the rest of the suffix array anywhere

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

fully functional suffix trees and optimal text searching in bwt runs bounded space indexing highly repetitive texts such as genomic databases software repositories and versioned text collections has become an important problem since the turn of the millennium a relevant compressibility measure for repetitive texts is r the number of runs in their burrows wheeler transforms bwts one of the earliest indexes for repetitive collections the run length fm index used o r space and was able to efficiently count the number of occurrences of a pattern of length m in a text of length n in o m log log n time with current techniques however it was unable to locate the positions of those occurrences efficiently within a space bounded in terms of r in this article we close this long standing problem showing how to extend the run length fm index so that it can locate the occ occurrences efficiently in o occ log log n time within o r space by raising the space to o r log log n our index counts the occurrences in optimal time o m and locates them in optimal time as well o m occ by further raising the space by an o w log factor where is the alphabet size and w e log n is the ram machine size in bits we support count and locate in o m log w and o m log w occ time which is optimal in the packed setting and had not been obtained before in compressed space we also describe a structure using o r log n r space that replaces the text and extracts any text substring of length e in the almost optimal time o log n r e log w within that space we similarly provide access to arbitrary suffix array inverse suffix array and longest common prefix array cells in time o log n r and extend these capabilities to full suffix tree functionality typically in o log n r time per operation our experiments show that our o r space index outperforms the space competitive alternatives by 1 2 orders of magnitude in time competitive implementations of the original fm index are outperformed by 1 2 orders of magnitude in space and or 2 3 in time

on demand string sorting over unbounded alphabets on demand string sorting is the problem of preprocessing a set of strings to allow subsequent queries for finding the k lexicographically smallest strings and afterward the next k etc this on demand variant strongly resembles the search engine queries which give you the best k ranked pages recurringly we present a data structure that supports this in o n preprocessing time where n is the number of strings and answer queries in o logn time there is also a cost of o n time amortized over all operations where n is the total length of the strings our data structure is a heap of strings which supports heapify and delete mins as it turns out implementing a full heap with all operations is not that simple for the sake of completeness we propose a heap with full operations based on balanced indexing trees that supports the heap operations in optimal times

rank select on dynamic compressed sequences and applications operations rank and select over a sequence of symbols have many applications to the design of succinct and compressed data structures managing text collections structured text binary relations trees graphs and so on we are interested in the case where the collections can be updated via insertions and deletions of symbols two current solutions stand out as the best in the tradeoff of space versus time when considering all the operations one solution by makinen and navarro achieves compressed space i e nh 0 o email protected bits and o email protected worst case time for all the operations where n is the sequence length s is the alphabet size and h 0 is the zero order entropy of the sequence the other solution by lee and park achieves o logn email protected amortized time and uncompressed space i e nlog email protected o n o email protected bits in this paper we show that the best of both worlds can be achieved we combine the solutions to obtain nh 0 o email protected bits of space and o logn email protected worst case time for all the operations apart from the best current solution to the problem we obtain several byproducts of independent interest applicable to partial sums text indexes suffix arrays the burrows wheeler transform and others

approximation of grammar based compression via recompression in this paper we present a simple linear time algorithm constructing a context free grammar of size o g log n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c otherwise additional cost of o n log is needed algorithms with such an approximation guarantee and running time are known the novelty of this paper is a particular simplicity of the algorithm as well as the analysis of the algorithm which uses a general technique of recompression recently introduced by the author furthermore contrary to the previous results this work does not use the lz representation of the input string in the construction nor in the analysis

linear work suffix array construction suffix trees and suffix arrays are widely used and largely interchangeable index structures on strings and sequences practitioners prefer suffix arrays due to their simplicity and space efficiency while theoreticians use suffix trees due to linear time construction algorithms and more explicit structure we narrow this gap between theory and practice with a simple linear time construction algorithm for suffix arrays the simplicity is demonstrated with a c implementation of 50 effective lines of code the algorithm is called dc3 which stems from the central underlying concept of difference cover this view leads to a generalized algorithm dc that allows a space efficient implementation and moreover supports the choice of a space time tradeoff for any v 1 n it runs in o vn time using o n v space in addition to the input string and the suffix array we also present variants of the algorithm for several parallel and hierarchical memory models of computation the algorithms for bsp and erew pram models are asymptotically faster than all previous suffix tree or array construction 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

alphabet independent linear time construction of compressed suffix arrays using o n log n bit working space the suffix array is a fundamental index data structure in string algorithms and bioinformatics and the compressed suffix array csa and the fm index are its compressed versions many algorithms for constructing these index data structures have been developed recently hon et al w k hon k sadakane w k sung breaking a time and space barrier in constructing full text indices in proceedings of the 44th annual ieee symposium on foundations of computer science 2003 pp 251 260 proposed a construction algorithm using o nloglog time and o nlog bit working space which is the fastest algorithm using o nlog bit working space in this paper we give an efficient algorithm to construct the index data structures our algorithm constructs the suffix array the csa the fm index and burrows wheeler transform using alphabet independent o n time and bit working space where log32 our algorithm takes less time and more space than hon et al s algorithm our algorithm uses least working space among alphabet independent linear time algorithms

linear time computation of prefix table for weighted strings applications the prefix table of a string is one of the most fundamental data structures of algorithms on strings it determines the longest factor at each position of the string that matches a prefix of the string it can be computed in time linear with respect to the size of the string and hence it can be used efficiently for locating patterns or for regularity searching in strings a weighted string is a string in which a set of letters may occur at each position with respective occurrence probabilities weighted strings also known as position weight matrices or uncertain strings naturally arise in many biological contexts for example they provide a method to realise approximation among occurrences of the same dna segment in this article given a weighted string x of length n and a constant cumulative weight threshold 1 z defined as the minimal probability of occurrence of factors in x we present an o n time algorithm for computing the prefix table of x furthermore we outline a number of applications of this result for solving various problems on non standard strings and present some preliminary experimental results

parallel computation of the burrows wheeler transform in compact space abstract the burrows wheeler transform bwt has become since its introduction a key tool for representing large text collections in compressed space while supporting indexed searching on a text of length n over an alphabet of size it requires o n lg bits of space instead of the o n lg n bits required by classical indexes a challenge for its adoption is to build it within o n lg bits as well there are some sequential algorithms building it within this space but no such a parallel algorithm in this paper we present a pram crew algorithm to build the bwt using o n lg n work o lg 3 n lg depth and o n lg bits

optimal and sublogarithmic time randomized parallel sorting algorithms this paper assumes a parallel ram random access machine model which allows both concurrent reads and concurrent writes of a global memory the main result is an optimal randomized parallel algorithm for integer sort i e for sorting n integers in the range 1 n this algorithm costs only logarithmic time and is the first known that is optimal the product of its time and processor bounds is upper bounded by a linear function of the input size also given is a deterministic sublogarithmic time algorithm for prefix sum in addition this paper presents a sublogarithmic time algorithm for obtaining a random permutation of n elements in parallel and finally sublogarithmic time algorithms for general sort and integer sort are presented our sub logarithmic general sort algorithm is also optimal

the pseudopalindromic completion of regular languages pseudopalindromes are words that are fixed points for some antimorphic involution in this paper we discuss a newer word operation that of pseudopalindromic completion in which symbols are added to either side of the word such that the new obtained words are pseudopalindromes this notion represents a particular type of hairpin completion where the length of the hairpin is at most one we give precise descriptions of regular languages that are closed under this operation and show that the regularity of the closure under the operation is decidable

chattering analysis a formal mathematical definition of chattering is proposed chattering phenomena are classified into three types in particular the first type is harmless and cannot be avoided chattering properties of various control approaches are considered the dangerous second and third types of chattering phenomena are proved to be removable by proper use of high order sliding modes hosm fast stable actuators and sensors only generate the first type of chattering in hosm systems and practically never affect the sliding motion computer simulation confirms the theoretical results

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

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

a quick tour on suffix arrays and compressed suffix arrays suffix arrays are a key data structure for solving a run of problems on texts and sequences from data compression and information retrieval to biological sequence analysis and pattern discovery in their simplest version they can just be seen as a permutation of the elements in 1 2 n encoding the sorted sequence of suffixes from a given text of length n under the lexicographic order yet they are on a par with ubiquitous and sophisticated suffix trees over the years many interesting combinatorial properties have been devised for this special class of permutations for instance they can implicitly encode extra information and they are a well characterized subset of the n permutations this paper gives a short tutorial on suffix arrays and their compressed version to explore and review some of their algorithmic features discussing the space issues related to their usage in text indexing combinatorial pattern matching and data compression

compressed property suffix trees property matching is a biologically motivated problem where the task is to find those occurrences of an online pattern p in a string text t of size n such that the matched text part satisfies some conceptual property the property of a string is a set p of possibly overlapping intervals s 1 f 1 s 2 f 2 corresponding to the part of text and an occurrence of a pattern p t i i p 1 is a valid output only if t i i p 1 is completely contained in at least one interval s j f j email protected the indexing version of this problem was introduced by a amir 2008 where the text is preprocessed in o email protected nloglogn time and an o nlogn bits index named property suffix tree pst is maintained pst can perform property matching in o p email protected occ p time where occ p is the number of occurrences of p in t satisfying the property t kopelowitz 2010 considered the dynamic version of this problem where intervals can be added or deleted however all these indexes take space linear to the size of text o nlogn bits which can be much more than the size of the text email protected bits in this paper we propose the first index for property matching occupying space close to the entropy compressed space requirement of the text our compressed index takes csa n email protected o 1 bits space and performs query answering in o t p email protected 1 occ p t s a time where csa is the size of compressed suffix array of t t p be the time for searching a pattern of length p in csa t s a is the time for computing the suffix array value and e 0 is a constant we also introduce a dynamic index which takes csa o n p logn bits space and performs query answering in o t p 1 occ p logn t s a logn loglogn time and can update insert delete an interval s f in o f s logn t s a time

the parametrisation of quantisation rules equivalent to operator orderings and the effect of different rules on the physical spectrum the author recalls the problem of choosing an ordering for the terms in a hamiltonian of the form h p q of finite dimension in p q he demonstrates that most orderings which provide one with a hermitian operator are in fact redundant and he gives a general form for ordering rules in terms of arbitrary parameters he concludes that differences in the choice of an ordering contribute differences to the physics which are only of order 2 and higher he provides an example of a hamiltonian for which the spectrum is explicitly dependent on the choice of quantisation rule operator ordering he compares the method of characterising operator ordering rules with that of cohen 1966 1970 finally he generalises cohen s method to the case of all linear quantisation rules and shows how his rule is a special case

optimal parallel algorithms for forest and term matching abstract forest matching is a fundamental step in solving various problems defined on terms such as term matching we describe the first optimal speedup parallel algorithm for solving the forest matching problem our algorithm runs in time o log n using n log n processors on a crcw pram given a forest of n nodes as input we use this algorithm to design the first optimal speedup parallel algorithm for solving the term matching problem we also extend these algorithms to run on the weaker crew pram with optimal speedup as well this will involve a simple randomization scheme for simulating concurrent writes through a use of hashing

searching for gapped palindromes we study the problem of finding in a given word all maximal gapped palindromes verifying two types of constraints that we call long armed and length constrained palindromes for each of the two classes we propose an algorithm that runs in time o n s for a constant size alphabet where s is the number of output palindromes both algorithms can be extended to compute biological gapped palindromes within the same time bound

an o n semipredictive universal encoder via the bwt we provide an o n algorithm for a nonsequential semipredictive encoder whose pointwise redundancy with respect to any unbounded depth tree source is o 1 bits per state above rissanen s lower bound this is achieved by using the burrows wheeler transform bwt an invertible permutation transform that has been suggested for lossless data compression first we use the bwt only as an efficient computational tool for pruning context trees and encode the input sequence rather than the bwt output second we estimate the minimum description length mdl source by incorporating suffix tree methods to construct the unbounded depth context tree that corresponds to the input sequence in o n time third we point out that a variety of previous source coding methods required superlinear complexity for determining which tree source state generated each of the symbols of the input we show how backtracking from the bwt output to the input sequence enables to solve this problem in o n worst case complexity

multi input control affine systems static feedback equivalent to a triangular form and their flatness abstractin this paper we give a complete geometric characterisation of systems locally static feedback equivalent to a triangular form compatible with the chained form for m 1 respectively with the m chained form for m 2 where the number of controls is m 1 they are x flat systems we provide a system of first order pde s to be solved in order to find all x flat outputs for m 1 respectively all minimal x flat outputs for m 2 we illustrate our results by examples in particular by an application to a mechanical system the coin rolling without slipping on a moving table