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

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

global bifurcation of steady gravity water waves with critical layers we construct families of two dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed in particular establishing the existence of waves of large amplitude a riemann hilbert problem approach is used to recast the governing equations as a one dimensional elliptic pseudodifferential equation with a scalar constraint the structural properties of this formulation which arises as the euler lagrange equation of an energy functional enable us to develop a theory of analytic global bifurcation

computing logcfl certificates the complexity class logcfl consists of all languages or decision problems which are logspace reducible to a context free language since logcfl is included in ac1 the problems in logcfl are highly parallelizable by results of ruzzo jcss 21 1980 218 the complexity class logcfl can be characterized as the class of languages accepted by alternating turing machines atms which use logarithmic space and have polynomially sized accepting computation trees we show that for each such atm m recognizing a language a in logcfl it is possible to construct an llogcfl transducer tm such that tm on input w e a outputs an accepting tree for m on w it follows that computing single logcfl certificates is feasible in functional ac1 and is thus highly parallelizable wanke j algorithms 16 1994 470 has recently shown that for any fixed k deciding whether the treewidth of a graph is at most k is in the complexity class logcfl as an application of our general result we show that the task of computing a tree decomposition for a graph of constant treewidth is in functional logcfl and thus in ac1 we also show that the following tasks are all highly parallelizable computing a solution to an acyclic constraint satisfaction problem computing an m coloring for a graph of bounded treewidth computing the chromatic number and minimal colorings for graphs of bounded tree width

periodic traveling gravity water waves with discontinuous vorticity we use elliptic theory to prove the existence of steady two dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity this is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context

nonparametric estimation for a class of levy processes we consider estimation for a class of levy processes modelled as a sum of a drift a symmetric stable process and a compound poisson process we propose a nonparametric approach to estimating unknown parameters of our model including the drift the scale and index parameters in the stable law the mean of the poisson process and the underlying jump size distribution we show that regression and nonparametric deconvolution methods based on the empirical characteristic function can be used for inference interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution

steady periodic water waves with constant vorticity regularity and local bifurcation this paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed using conformal mappings the free boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable the new formulation leads to a regularity result and by use of bifurcation theory to the existence of waves of small amplitude even in the presence of stagnation points in the flow

on stratified steady periodic water waves with linear density distribution and stagnation points abstract we use bifurcation theory to construct small periodic gravity stratified water waves with density which depends linearly upon the pseudostream function as a special feature the density may also decrease with depth and the waves we obtain may posses two different critical layers with cat s eye vortices within the vortex the density of the fluid has an extremum at the stagnation point

rank and select revisited and extended the deep connection between the burrows wheeler transform bwt and the so called rank and select data structures for symbol sequences is the basis of most successful approaches to compressed text indexing rank of a symbol at a given position equals the number of times the symbol appears in the corresponding prefix of the sequence select is the inverse retrieving the positions of the symbol occurrences it has been shown that improvements to rank select algorithms in combination with the bwt turn into improved compressed text indexes this paper is devoted to alternative implementations and extensions of rank and select data structures first we show that one can use gap encoding techniques to obtain constant time rank and select queries in essentially the same space as what is achieved by the best current direct solution and sometimes less second we extend symbol rank and select to substring rank and select giving several space time trade offs for the problem an application of these queries is in position restricted substring searching where one can specify the range in the text where the search is restricted to and only occurrences residing in that range are to be reported in addition arbitrary occurrences are reported in text position order several byproducts of our results display connections with searchable partial sums chazelle s two dimensional data structures and grossi et al s wavelet trees

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

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

optimal bounds for decision problems on the crcw pram optimal o log n log log n lower bounds on the time for crcw prams with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven a strict time hierarchy of explicit boolean functions of n bits on such machines that holds up to o log n log log n time is also exhibited that is for every time bound t within this range a function is exhibited that can be easily computed using polynomial resources in time t but requires more than polynomial resources to be computed in time t 1 finally it is shown that almost all boolean functions of n bits require log n log log n o 1 time when the number of processors is at most polynomial in n the bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

an algebraic theory of graph reduction we show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules the method is constructive in the sense that we describe an algorithm which will produce from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth k a set of rewrite rules that reduces any member of the class to one of finitely many graphs in a number of steps bounded by the size of the graph this reduction system corresponds to an algorithm that runs in time linear in the size of the graph

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

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

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

optimal parallel algorithms on planar graphs abstract few existing parallel graph algorithms achieve optimality when applied to very sparse graphs such as planar graphs we describe optimal pram algorithms for the connected components spanning tree biconnected components and strong orientation problems that work on classes of undirected graphs including planar graphs and graphs of bounded genus the running times achieved for n vertex input graphs are o log n on the crcw pram and o log n log n on the erew pram we also give non optimal randomized erew pram algorithms using o log n time and n processors and non uniform deterministic erew pram algorithms using o log n time and o n 2 processors

an efficient parallel algorithm for the single function coarsest partition problem we describe an efficient parallel algorithm to solve the single function coarsest partition problem the algorithm runs in o log n time using o n log log n operations on the arbitrary crcw pram the previous best known algorithms run in o log2 n time using o n log2n operations on the crew pram and o log n time using o n log n operations on the arbitrary crcw pram our solution is based on efficient algorithms for solving several subproblems that are of independent interest in particular we present efficient parallel algorithms to find a minimal starting point of a circular string with respect to lexicographic ordering and to sort lexicographically a list of strings of different lengths

reduction algorithms for graphs of small treewidth this paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth s arnborg b courcelle a proskurowski and d seese j assoc comput mach 40 1134 1164 1993 have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth using a finite set of reduction rules these algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first we show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth including all problems definable in monadic second order logic we also show that a variant of these reduction algorithms can be used to solve constructive optimization problems in o n time for example optimization and construction variants of independent set and hamiltonian completion number can be solved in this way on graphs of small treewidth additionally we show that the results of h l bodlaender and t hagerup siam j comput 27 1725 1746 1998 can be applied to our reduction algorithms which results in parallel reduction algorithms that use o n operations and o log n log n time on an erew pram or o log n time on a crcw pram

recognizing hyperelliptic graphs in polynomial time abstract based on analogies between algebraic curves and graphs baker and norine introduced divisorial gonality a graph parameter for multigraphs related to treewidth multigraph algorithms and number theory various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs called stable gonality and stable divisorial gonality we consider so called hyperelliptic graphs multigraphs of gonality 2 in any meaning of graph gonality and provide a safe and complete set of reduction rules for such multigraphs this results in an algorithm to recognize hyperelliptic graphs in time o m n log n where n is the number of vertices and m the number of edges of the multigraph a corollary is that we can decide with the same runtime whether a two edge connected graph g admits an involution such that the quotient g is a tree

existence of capillary gravity water waves with piecewise constant vorticity abstract in this paper we construct small amplitude periodic capillary gravity water waves with a piecewise constant vorticity distribution they describe water waves traveling on superposed linearly sheared currents that have different vorticities this is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump the solutions of the diffraction problem found by using local bifurcation theory are the desired solutions of the hydrodynamical problem

efficient parallel algorithms for testing k connectivity and finding disjoint s t paths in graphs an efficient parallel algorithm for testing whether a graph g is k vertex connected is presented the algorithm runs in o k 2 log n time and uses n k 2 kc n m processors on a crow pram where n and m are the number of vertices and edges of g and c n m is the number of processors required to compute the connected components of g in logarithmic time for fixed k the algorithm runs in logarithmic time and uses nc n m processors to develop our algorithm an efficient parallel algorithm is designed for the following disjoint s t paths problem given a graph g and two specified vertices s and t find k vertex disjoint paths between s and t if they exist if no such paths exist find a set of at most k 1 vertices whose removal disconnects s and t the parallel algorithm for this problem runs in o k 2 log n time and uses kc n m processors the way to modify the algorithm to find k edge disjoint paths if they exist is shown this yields an efficient parallel algorithm for testing w

fast parallel space allocation estimation and integer sorting abstract the following problems are shown to be solvable in o log n time with optimal speedup with high probability on a randomized crcw pram using o n space space allocation given nonnegative integers allocate nonoverlapping blocks of consecutive memory cells of sizes from a base segment of consecutive memory cells estimation given integers in the range 1 compute good estimates of the number of occurrences of each value in the range 1 semisorting given integers in the range 1 store the integers 1 in an array of cells such that for all 1 all elements of 1 and occur together separated only by empty cells integer chain sorting given integers in the range 1 construct a linked list containing the integers 1 such that for all 1 if precedes in the list then moreover given slightly superlinear processor and space bounds these problems or variations of them can be solved in constant time with high probability as a corollary of the integer chain sorting result it follows that n integers in the range 1 n can be sorted in o log n log log n time with optimal speedup with high probability

upper bounds for sorting integers on random access machines two models of random access machines suitable for sorting integers are presented our main results show that i a ram with addition subtraction multiplication and integer division can sort n integers in the range 0 2 cn in o n log c n steps ii a ram with addition subtraction and left and right shifts can sort any n integers in linear time iii a ram with addition subtraction and left and right shifts can sort n integers in the range 0 n c in o n log c n steps where all intermediate results are bounded in value by the largest input

the complexity of parallel prefix problems on small domains abstract we establish non trivial lower bounds for several prefix problems in the crcw pram model the chaining problem is given a binary input for each 1 in the input to find the index of the nearest 1 to its left our main result is that for an input of n bits solving the chaining problem using o n processors requires inverse ackerman time this matches the previously known upper bound we also give a reduction to show that the same lower bound applies to a parenthesis matching problem again matching the previously known upper bound we also give reductions to show that similar lower bounds hold for the prefix maxima and the range maxima problem

sorting strings and constructing digital search trees in parallel we describe two simple optimal work parallel algorithms for sorting a list l x1 x2 xm of m strings over an arbitrary alphabet where i 1m xi n and two elements of can be compared in unit time using a single processor the first algorithm is a deterministic algorithm that runs in o log2m log log m time and the second is a randomized algorithm that runs in o logm time both algorithms use o m log m n operations compared to the best known parallel algorithms for sorting strings our algorithms offer the following improvements 1 1 the total number of operations used by our algorithms is optimal while all previous parallel algorithms use a nonoptimal number of operations 2 2 we make no assumption about the alphabet while the previous algorithms assume that the alphabet is restricted to 1 2 no 1 3 3 the computation model assumed by our algorithms is the common crcw pram unlike the known algorithms that assume the arbitrary crcw pram 4 4 our algorithms use o m log m n space while the previous parallel algorithms use o n1 e space where e is a positive constant we also present optimal work parallel algorithms to construct a digital search tree for a given set of strings and to search for a string in a sorted list of strings we use our parallel sorting algorithms to solve the problem of determining a minimal starting point of a circular string with respect to lexicographic ordering our solution improves upon the previous best known result to solve this problem

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

optimal parallel 5 colouring of planar graphs we show that a 5 colouring of the vertices of an n vertex planar graph may be computed in o log n log n time by an exclusive read exclusive write parallel ram with o n log n log n processors our algorithm while faster than all previously known methods is at the same time the first parallel 5 colouring algorithm to exhibit an optimal speedup optimality is achieved through a method based on the accelerating cascades technique and of independent interest it should be emphasized that although input to the algorithm is a planar graph we do not require a planar embedding to be given as part of the input other results concern the colouring of graphs of bounded genus and the construction of search structures for triangular planar subdivisions

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

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

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

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

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

delaunay triangulations in o sort n time and more we present several results about delaunay triangulations dts and convex hulls in transdichotomous and hereditary settings i the dt of a planar point set can be computed in expected time o sort n on a word ram where sort n is the time to sort n numbers we assume that the word ram supports the shuffle operation in constant time ii if we know the ordering of a planar point set in x and in y direction its dt can be found by a randomized algebraic computation tree of expected linear depth iii given a universe u of points in the plane we construct a data structure d for delaunay queries for any p s u d can find the dt of p in expected time o vpv log log vuv iv given a universe u of points in 3 space in general convex position there is a data structure d for convex hull queries for any p s u d can find the convex hull of p in expected time o vpv log log vuv 2 v given a convex polytope in 3 space with n vertices which are colored with g 2 colors we can split it into the convex hulls of the individual color classes in expected time o n log log n 2 the results i iii generalize to higher dimensions where the expected running time now also depends on the complexity of the resulting dt we need a wide range of techniques most prominently we describe a reduction from dts to nearest neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling

towards optimal parallel bucket sorting we present a simple deterministic parallel algorithm that runs on a crcw pram and sorts n integers of size polynomial in n in time o log n using o n log log nlog n processors it is closer to optimality than any previously known deterministic algorithm that solves the stated restricted sorting problem in polylog time

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

dispersion relations for fixed mean depth flows with two discontinuities in vorticity abstract we derive here the dispersion relation for small amplitude steady two dimensional periodic water waves that propagate over a flat bed with a specified and fixed mean depth the water flow has a discontinuous vorticity distribution reflecting an isolated rotational layer of fluid we then use the dispersion relation to obtain necessary and sufficient conditions for local bifurcation to occur moreover a stability analysis of the laminar flows is presented

local asymptotic mixed normality for semimartingale experiments we give conditions for local asymptotic mixed normality of experiments when the observed process is a semimartingale and the observation time increases to infinity as a consequence we obtain asymptotic efficiency of various estimators several special models for counting process s diffusion processes and diffusions with jumps are studied

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

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

simple fast parallel hashing by oblivious execution a hash table is a representation of a set in a linear size data structure that supports constant time membership queries we show how to construct a hash table for any given set of n keys in o lg lg n parallel time with high probability using n processors on a weak version of a concurrent read concurrent write parallel random access machine crcw pram our algorithm uses a novel approach of hashing by oblivious execution based on probabilistic analysis the algorithm is simple and has the following structure partition the input set into buckets by a random polynomial of constant degree for t 1 to o lg lg n do allocate mt memory blocks each of size kt let each bucket select a block at random and try to injectively map its keys into the block using a random linear function buckets that fail carry on to the next iteration the crux of the algorithm is a careful a priori selection of the parameters mt and kt the algorithm uses only o lg lg n random words and can be implemented in a work efficient manner

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

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

equivalence between priority queues and sorting we present a general deterministic linear space reduction from priority queues to sorting implying that if we can sort up to n keys in s n time per key then there is a priority queue supporting delete and insert in o s n time and find min in constant time conversely a priority queue can trivially be used for sorting first insert all keys to be sorted then extract them in sorted order by repeatedly deleting the minimum asymptotically this settles the complexity of priority queues in terms of that of sorting previously at soda 96 such a result was presented by the author for the special case of monotone priority queues where the minimum is not allowed to decrease besides nailing down the complexity of priority queues to that of sorting and vice versa our result yields several improved bounds for linear space integer priority queues with find min in constant time deterministically we get o log log n update time using a sorting algorithm of han from stoc 02 this improves the o log log n log log log n update time of han from soda 01 randomized we get o slog log n expected update time using a randomized sorting algorithm of han and thorup from focs 02 this improves the o log log n expected update time of thorup from soda 96 deterministically in ac0 without multiplication for any e 0 we get o log log n 1pe update time using an ac0 sorting algorithm of han and thorup from focs 02 this improves the o log log n 2 update time of thorup from soda 98 randomized in ac0 we get o log log n expected update time using a randomized ac0 sorting algorithm of thorup from soda 97 this improves the o log log n 1pe expected update time of thorup also from soda 97 the above bounds assume that each integer is stored in a single word and that word operations take unit time as in the word ram

undirected single source shortest paths with positive integer weights in linear time the single source shortest paths problem sssp is one of the classic problems in algorithmic graph theory given a positively weighted graph g with a source vertex s find the shortest path from s to all other vertices in the graph since 1959 all theoretical developments in sssp for general directed and undirected graphs have been based on dijkstra s algorithm visiting the vertices in order of increasing distance from s thus any implementation of dijkstra s algorithm sorts the vertices according to their distances from s however we do not know how to sort in linear time here a deterministic linear time and linear space algorithm is presented for the undirected single source shortest paths problem with positive integer weights the algorithm avoids the sorting bottleneck by building a hierarchical bucketing structure identifying vertex pairs that may be visited in any order

dynamic ordered sets with exponential search trees we introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully dynamic linear space data structures this leads to an optimal bound of o slog n log log n for searching and updating a dynamic set x of n integer keys in linear space searching x for an integer y means finding the maximum key in x which is smaller than or equal to y this problem is equivalent to the standard text book problem of maintaining an ordered set the best previous deterministic linear space bound was o log n log log n due to fredman and willard from stoc 1990 no better deterministic search bound was known using polynomial space we also get the following worst case linear space trade offs between the number n the word length w and the maximal key u our results are generalized to finger searching and string searching providing optimal results for both in terms of n