preconditioned conjugate gradient methods for three dimensional linear elasticity finite element modelling of three dimensional elasticity problems give rise to large sparse matrices various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques incomplete factorizations based on levels of fill drop tolerance and a two level hierarchical basis are developed various techniques for ensuring that the incomplete factors have positive pivots are presented computational tests are carried out for problems generated using unstructured tetrahedral meshes quadratic basis functions are used the performance of the iterative methods is compared to a standard direct sparse matrix solver problems with up to 70 000 degrees of freedom and small 1 element aspect ratio are considered

high performance pcg solvers for fem structural analysis the preconditioned conjugate gradient algorithm is a well known and powerful method used to solve large sparse symmetric positive definite linear systems such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods it is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods

conditioning analysis of positive definite matrices by approximate factorizations abstract the conditioning analysis of positive definite matrices by approximate lu factorizations is usually reduced to that of stieltjes matrices or even to more specific classes of matrices by means of perturbation arguments like spectral equivalence we show in the present work that a wider class which we call almost stieltjes matrices can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady state diffusion problems

algebraic elimination of slide surface constraints in implicit structural analysis slide surface and contact boundary conditions can be implemented via lagrange multipliers in the algebraic equations in implicit structural analysis this indefinite set of equations is difficult to solve by iterative methods and is often too large to be solved by direct methods when there are m constraints and there exists a set of m variables where each variable is only involved in a single constraint we advocate a direct elimination technique which leaves a sparse positive definite system to solve by iterative methods we prove that the amount of fill in created by this process is independent of the size of the slide surfaces in addition the eigenvalues of the reduced matrix do not differ significantly from the eigenvalues of the unconstrained matrix this method can be extended to the case where constrained surfaces intersect and leads to a graph theoretic approach for determining which variables can be eliminated efficiently for constraints with more general structure published in 2003 by john wiley sons ltd

parallel mic 0 preconditioning of 3d elliptic problems discretized by rannacher turek finite elements novel parallel algorithms for the solution of large fem linear systems arising from second order elliptic partial differential equations in 3d are presented the problem is discretized by rotated trilinear nonconforming rannacher turek finite elements the resulting symmetric positive definite system of equations ax f is solved by the preconditioned conjugate gradient algorithm the preconditioners employed are obtained by the modified incomplete cholesky factorization mic 0 of two kinds of auxiliary matrices b that both are constructed as locally optimal approximations of a in the class of m matrices uniform estimates for the condition number k b 1a are derived two parallel algorithms based on the different block structures of the related matrices b are studied the numerical tests confirm theory in that the algorithm scales as o n 7 6 in the matrix order n

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

communication complexity of simultaneous messages in the multiparty communication game cfl game of chandra furst and lipton proceedings of the 15th annual acm symposium on theory of computing boston ma 1983 pp 94 99 k players collaboratively evaluate a function f x0 xk 1 in which player i knows all inputs except xi the players have unlimited computational power the objective is to minimize communication in this paper we study the simultaneous messages sm model of multiparty communication complexity the sm model is a restricted version of the cfl game in which the players are not allowed to communicate with each other instead each of the k players simultaneously sends a message to a referee who sees none of the inputs the referee then announces the function value we prove lower and upper bounds on the sm complexity of several classes of explicit functions our lower bounds extend to randomized sm complexity via an entropy argument a lemma establishing a tradeoff between average hamming distance and range size for transformations of the boolean cube might be of independent interest our lower bounds on sm complexity imply an exponential gap between the sm model and the cfl model for up to log n 1 epsilon players for any epsilon 0 this separation is obtained by comparing the respective complexities of the generalized addressing function gafg k where g is a group of order n we also combine our lower bounds on sm complexity with the ideas of hastad and goldmann comput complexity 1 1991 pp 113 129 to derive superpolynomial lower bounds for certain depth 2 circuits computing a function related to the gaf function we prove some counterintuitive upper bounds on sm complexity we show that sf gaf mathbb z 2 t 3 has sm complexity o n 0 92 when the number of players is at least c log n for some constant c 0 our sm protocol for sf gaf mathbb z 2 t k has polylog n complexity we also examine a class of functions defined by certain depth 2 circuits this class includes the generalized inner product function and majority of majorities when the number of players is at least 2 log n we obtain polylog n upper bounds for this class of functions

hierarchy theorems for k obdds and k ibdds almost the same types of restricted branching programs or binary decision diagrams bdds are considered in complexity theory and in applications like hardware verification these models are read once branching programs free bdds and certain types of oblivious branching programs ordered and indexed bdds with k layers the complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed bdds of polynomial size

a 3 local characterization of u6 2 and fi22 abstract the unitary group u6 2 often referred to as fi21 and the sporadic simple group fi22 discovered by fischer b fischer finite groups generated by 3 transpositions i invent math 13 1971 232 246 6 are characterized by specifying partial information about the structure of the normalizer of a non trivial 3 central cyclic subgroup

a 3 local characterization of co2 abstract conway s second largest simple group co2 is characterized by the centralizer of an element of order 3 and certain fusion data

incorporation of multipoint constraints into the balancing domain decomposition method and its parallel implementation the domain decomposition method ddm is a major solution algorithm that is used to parallelize the finite element method in the case of implicit structural analysis using the ddm the substructuring based iterative linear solver is a powerful tool when an effective preconditioner such as the balancing domain decomposition bdd method is used in the present study a method by which to incorporate a set of linear multipoint constraints mpc into the bdd method is proposed in this method when an mpc is enforced on the internal degrees of freedom dofs in some subdomains the dofs are converted into interface dofs that is all of the dofs constrained by mpcs become interface dofs then the interface problem with the set of mpcs for the interface dofs is solved by the conjugate projected gradient method in order to combine the above procedure with the preconditioner used in the bdd method the effect of the mpcs for the interface dofs is also imposed on the coarse grid problem of the bdd method using the penalty method a parallel implementation of the present method is also described some illustrative examples are solved and good convergence and parallel performance of the present method are demonstrated copyright 2006 john wiley sons ltd

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

an algorithm for profile and wavefront reduction of sparse matrices summary an algorithm for reducing the profile and wavefront of a sparse matrix is described the scheme is applicable to any sparse matrix which has a symmetric pattern of zeros and may be used to generate efficient labellings for finite element grids in particular it is suitable for generating efficient labellings for profile and frontal solution schemes empirical evidence obtained from analysis of the 30 test problems collected by everstine suggests that the new algorithm is superior to existing methods for profile and wavefront reduction it is fast requires only a small amount of memory and is simple to program

polynomial acceleration of iterative schemes associated with subproper splittings a subproper splitting of a matrix a is a decomposition a b c such that the kernel of a includes that of b while the range of b includes that of a our purpose in the present work is to extend the convergence analysis of polynomial acceleration to the case of iterative schemes associated with subproper splittings in the case of hermitian matrices and consistent systems briefly stated our conclusions show that the regular theory extends to the subproper case provided that convergence to the solution of ax b is understood as convergence to a solution of ax b while b 1 a is understood as b a 0 where b is the moore penrose inverse of b

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

efficient iterative solution of constrained finite element analyses this contribution describes how iterative solvers can meet specific requirements of industrial fe analyses focusing on the case frequently met where the unknowns are subject to linear equality constraints standard iterative methods designed to deal with that kind of problem suffer from a significant overhead with respect to the cpu times involved in the solution of unconstrained problems whereas the performance of direct solvers traditionally used is not affected here we propose a subspace projection method that allows the use of any iterative scheme able to solve the unconstrained problem with the same preconditioner we also highlight that there is no loss of efficiency due to the presence of linear constraints

rounds in communication complexity revisited the k round two party communication complexity was studied in the deterministic model by p h papadimitriou and m sipser proc of the 14th stoc 1982 pp 330 337 and p duris z galil and g schnitger proc of the 16th stoc 1984 pp 81 91 and in the probabilistic model by a c yao proc of the 24th focs 1983 pp 420 428 and b halstenberg and r reischuk proc of the 20th stoc 1988 pp 162 172 this paper presents new lower bounds that give 1 randomization is more powerful than determinism in k round protocols and 2 an explicit function which exhibits an exponential gap between its k and k 1 round randomized complexity this paper also studies the three party communication model and exhibits an exponential gap in 3 round protocols that differ in the starting player finally this paper shows new connections of these questions to circuit complexity that motivate further work in this direction

three results on interactive communication x and y are random variables person p sub x knows x person p sub y knows y and both know the underlying probability distribution of the random pair x y using a predetermined protocol they exchange messages over a binary error free channel in order for p sub y to learn x p sub x may or may not learn y c sub m is the number of information bits that must be transmitted by both persons in the worst case if only m messages are allowed c sub infinity is the corresponding number of bits when there is no restriction on the number of messages exchanged we consider three aspects of this problem c sub 4 it is known that one message communication may require exponentially more bits than the minimum possible for some random pairs c sub 1 2 sup c infinity 1 yet just two messages suffice to reduce communication to almost the minimum for all random pairs c sub 2 or 2 in c sub infinity or c asymptotically this is the largest possible discrepancy amortized complexity the amortized complexity of x y is the limit as k grows of the number of bits required in the worst case for l independent repetitions of x y normalized by k we show that the four message amortized complexity of all random pairs is exactly log mu hence when a random pair is repeated many times no bits can be saved if p sub x knows y in advance

element based preconditioners for elasto plastic problems in geotechnical engineering iterative solvers are widely regarded as the most efficient way to solve the very large linear systems arising from finite element models their memory requirements are small compared to those for direct solvers consequently there is a major interest in iterative methods and particularly the preconditioning necessary to achieve rapid convergence in this paper we present new element based preconditioners specifically designed for linear elasticity and elasto plastic problems the study presented here is restricted to simple associated plasticity but should find wide application in other plasticity models used in geotechnics

numerical experiences of solving elasticity systems by pcg methods abstract in solving systems of linear equations derived from fem structural analysis by the conjugate gradient method an efficient way for preparing effective incomplete factorization preconditioner was published by saint georges 1 in 2 3 we prepared several variants of the cited method due to saint georges 1 and some of the relevant pcg solvers were found more efficient than the original ones 1 here we prepare some additional variants for producing good enough preconditioners by applying our spiral ordering 2 and execute some incomplete factorizations mic ric with the level of fill in dropping strategy which was found the best in 3 in the comparison of the considered solvers we analyze our computer results when systems from elastic frame structures with 3d beam elements were solved

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

solving systems of elastic bar structures by preconditioned conjugate gradient method abstract for solving systems of linear equations derived from structural analysis by conjugate gradient method a new efficient incomplete factorization preconditioning was published by saint georges et al 1 here we present an algorithm for finding a starting point for the ordering applied in 1 based on which a variant of the spiral ordering due to duff et al 2 for an undirected connected graph is formed we test the solvers in 1 and some of its variants when different orderings are applied and for each ordering some incomplete factorization preconditioners are prepared in the comparison of the considered solvers a remarkable reduction in the number of iterations was found by the presented variant of the spiral ordering with ic 0 preconditioner when systems from elastic bar structures with 3d beam elements were solved 1999 elsevier science ltd all rights reserved

a full 3d finite element analysis using adaptive refinement and pcg solver with back interpolation in this paper adaptive refinement finite element analyses were carried out for full 3d problems in order to achieve an optimal computation cost and to eliminate the effects of singular points the adaptive refinement procedure was applied in conjunction with a back interpolation for the construction of a good initial guess for the solution of the linear equations system the combined use of the adaptive refinement procedure the back interpolation scheme and the preconditioned conjugate gradient pcg solver lead to a significant reduction in the operations for the solution of the simultaneous linear equations in the adaptive refinement analysis in many cases the number of iterations needed by the pcg solver to reach a converged solution is independent of the number of equations in the global system the numerical results obtained indicated that for a series of carefully designed adaptive meshes the computational cost required to solve the linear system could be made only proportional to the number of degrees of freedom in the mesh to our knowledge this is the first time that the global stiffness equations in 3d stress analysis have been solved with this efficiency the same set of numerical results also showed that in general the total computational cost of the adaptive refinement procedure can usually be minimized by gradually reducing the target relative error of the solution during successive refinements rather than employing a constant target relative error throughout the whole adaptive analysis

lower bounds for one way probabilistic communication complexity and their application to space complexity abstract we prove three different types of complexity lower bounds for the one way unbounded error and bounded error error probabilistic communication protocols for boolean functions the lower bounds are proved in terms of the deterministic communication complexity of functions and in terms of the notion probabilistic communication characteristic that we define we present boolean functions with the different probabilistic communication characteristics which demonstrates that each of these lower bounds can be more precise than the others depending on the probabilistic communication characteristics of a function our lower bounds are good enough for proving that proper hierarchy for one way probabilistic communication complexity classes depends on a measure of bounded error as the application of lower bounds for probabilistic communication complexity we prove two different types of complexity lower bounds for the one way bounded error error probabilistic space complexity our lower bounds are good enough for proving proper hierarchies for different one way probabilistic space communication complexity classes inside space n namely for bounded error probabilistic computation and for errors of probabilistic computation

algebraic multigrid for moderate order finite elements we investigate the use of algebraic multigrid amg methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with lagrangian finite elements of order at most 4 the resulting system matrices do not have the m matrix property that is required by standard analyses of classical amg and aggregation based amg methods a unified approach is presented that allows us to extend these analyses it uses an intermediate m matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix this constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations provided that jumps are located at elements boundaries for two dimensional problems it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound this analysis validates the application of the amg methods to the

local characteristic p completions of weak bn pairs this paper establishes recognition theorems for the finite lie type groups defined in odd characteristic p the hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak bn pair of rank 2 the groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic p a condition satisfied by the finite lie type groups defined in characteristic p the results presented will find application in one of the ongoing revisions of the finite simple group classification

a 3 local identification of the alternating group of degree 8 the mclaughlin simple group and their automorphism groups we give 3 local identifications of the sporadic simple group mcl and its automorphism group as well as of alt 8 and sym 8

boolean circuits tensor ranks and communication complexity we investigate two methods for proving lower bounds on the size of small depth circuits namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices our methods are combinatorial but we think that our main contribution concerns the algebraic concepts used in this area tensor ranks and rigidity our main results are following i an o n bit protocol for a communication game for computing shifts which also gives an upper bound of o n 2 on the contact rank of the tensor of multiplication of polynomials this disproves some earlier conjectures a related probabilistic construction gives an o n upper bound for computing all permutations and an o n log log n upper bound on the communication complexity of pointer jumping with permutations ii a lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic depth circuits this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model iii an upper bound on boolean circuits of depth 2 for computing shifts and more generally all permutations this shows that such circuits are more efficient than the model based on sending bits along vertex disjoint paths

average case interactive communication x and y are random variables person p sub x knows x person p sub y knows y and both know the joint probability distribution of the pair x y using a predetermined protocol they communicate over a binary error free channel in order for p sub y to learn x p sub x may or may not learn y it is determined how many information bits must be transmitted by both persons on the average the results show that when the arithmetic average number of bits is considered there is no asymptotic advantage to p sub x knowing y in advance and four messages are asymptotically optimum by contrast for the worst case number of bits communication can be significantly reduced if p sub x knows y in advance and it is not known whether a constant number of messages is asymptotically optimum

hat guessing games hat problems have become a popular topic in recreational mathematics in a typical hat problem each of n players tries to guess the color of the hat he or she is wearing by looking at the colors of the hats worn by some of the other players in this paper we consider several variants of the problem united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case we also summarize what is currently known about the worst case analysis of deterministic hat guessing problems with a finite number of players

a property of quantum relative entropy with an application to privacy in quantum communication we prove the following information theoretic property about quantum states substate theorem let and be quantum states in the same hilbert space with relative entropy s c tr log log c then for all 0 there is a state such that the trace distance tr tr 2 and 2o c 2 it states that if the relative entropy of and is small then there is a state close to i e with small trace distance tr that when scaled down by a factor 2o c sits inside or becomes a substate of this result has several applications in quantum communication complexity and cryptography using the substate theorem we derive a privacy trade off for the set membership problem in the two party quantum communication model here alice is given a subset a n bob an input i i n and they need to determine if i i a privacy trade off for set membership in any two party quantum communication protocol for the set membership problem if bob reveals only k bits of information about his input then alice must reveal at least n 2o k bits of information about her input we also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

incomplete factorization based preconditionings for solving the helmholtz equation preconditioning techniques based on incomplete factorization of matrices are investigated to solve highly indefinite complex symmetric linear systems a novel preconditioning is introduced the real part of the matrix is made positive definite or less indefinite by adding properly defined perturbations to the diagonal entries while the imaginary part is unaltered the resulting preconditioning matrix which is obtained by applying standard methods to the perturbed complex matrix turns out to perform significantly better than classical incomplete factorization schemes for realistic values of the gmres restart parameter spectacular reduction of iteration counts is observed a theoretical spectral analysis is provided in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a stieltjes matrix extracted from the indefinite matrix results of numerical experiments are reported which display the efficiency of the new preconditioning copyright 2001 john wiley sons ltd

application of diagonally perturbed incomplete factorization preconditioned conjugate gradient algorithms for edge finite element analysis of helmholtz equations the diagonally perturbed incomplete factorization preconditioning scheme is applied to the conjugate gradient cg method for solving a large system of linear equations resulting from the use of edge based finite element method fem this scheme contains more global information about the coefficient matrix when compared with banded matrix schemes the efficient implementation of this preconditioned cg pcg algorithm is described in detail for complex coefficient matrix equation on several electromagnetic problems the pcg approach converges in cpu time which is 8 6 19 5 times shorter with respect to the cg approach by comparison with other preconditioned techniques the results demonstrate that incomplete factorization preconditioning strategy is especially effective for cg iterative method when edge fem is applied to solve large scale time harmonic electromagnetic field problems

graph distances in the data stream model we explore problems related to computing graph distances in the data stream model the goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory we are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the data stream model in particular we are interested in the trade offs between model parameters such as per data item processing time total space and the number of passes that may be taken over the stream these trade offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature our results include the following 1 spanner construction there exists a single pass tilde o tn 1 1 t space tilde o t 2n 1 t time per edge algorithm that constructs a 2t 1 spanner for t omega log n log log n the algorithm satisfies the semistreaming space restriction of o n operatorname polylog n and has per edge processing time o operatorname polylog n this resolves an open question from j feigenbaum et al theoret comput sci 348 2005 pp 207 216 2 breadth first search bfs trees for any even constant k we show that any algorithm that computes the first k layers of a bfs tree from a prescribed node with probability at least 2 3 requires either greater than k 2 passes or tilde omega n 1 1 k space since constructing bfs trees is an important subroutine in many traditional graph algorithms this demonstrates the need for new algorithmic techniques when processing graphs in the data stream model 3 graph distance lower bounds any t approximation of the distance between two nodes requires omega n 1 1 t space we also prove lower bounds for determining the length of the shortest cycle and other graph properties 4 techniques for decreasing per edge processing we discuss two general techniques for speeding up the per edge computation time of streaming algorithms while increasing the space by only a small factor

study of interlaminar stresses in composite laminates subjected to torsional loading a computational procedure for accurate determination of interlaminar stresses in thick and thin composite laminates subjected to torsional loading is presented a multilevel recursively defined preconditioner in conjunction with the preconditioned conjugate gradient pcg algorithm is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element method the computational procedure is used to model stress fields in thick and thin laminates in torsion using three dimensional and quasi three dimensional models results indicate that the preconditioner developed here can be used to produce an efficient iterative solver for the determination of two and three dimensional stress fields in composite structures

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

reversal complexity revisited we study a generalized version of reversal bounded turing machines where apart from several tapes on which the number of head reversals is bounded by r n there are several further tapes on which head reversals remain unrestricted but size is bounded by s n where n denotes the input length recently m grohe c koch n schweikardt tight lower bounds for query processing on streaming and external memory data theoretical computer science 380 1 2 2007 199 217 m grohe n schweikardt lower bounds for sorting with few random accesses to external memory in proc pods 05 acm press 2005 pp 238 249 such machines were introduced as a formalization of a computation model that restricts random access to external memory and internal memory space here each of the tapes with a restriction on the head reversals corresponds to an external memory device and the tapes of restricted size model internal memory we use st r n s n o 1 to denote the class of all problems that can be solved by deterministic turing machines that comply to the above resource bounds similarly nst and rst respectively are used for the corresponding nondeterministic and randomized classes while previous papers focused on lower bounds for particular problems including sorting the set equality problem and several query evaluation problems the present paper addresses the relations between the r n st classes and classical complexity classes and investigates the structural complexity of the r n st classes our main results are 1 a trade off between internal memory space and external memory head reversals 2 correspondences between the r n st classes and classical time bounded space bounded reversal bounded and circuit complexity classes and 3 hierarchies of r st classes in terms of increasing numbers of head reversals on external memory tapes