p uniform circuit complexity much complexity theoretic work on parallelism has focused on the class nc which is defined in terms of logspace uniform circuits yet p uniform circuit complexity is in some ways a more natural setting for studying feasible parallelism in this paper p uniform nc punc is characterized in terms of space bounded auxpdas and alternating turing machines with bounded access to the input the notions of general purpose and special purpose computation are considered and a general purpose parallel computer for punc is presented it is also shown that nc punc if all tally languages in p are in nc this implies that the nc punc question and the nc p question are both instances of the aspace s n aspace time s n s n o 1 question as a corollary it follows that nc punc implies pspace dtime 2 no 1

on the degrees of bases of free modulesover a polynomial ring let k be an infinite field a the polynomial ring k x 1 x n and f in a n times m a matrix such that rm im f subset a n is a free in particular quillen suslin theorem implies that rm ker f is also free let d be the maximum of the degrees of the entries of f and s the rank of f we show that there exists a basis v 1 ldots v m of a m such that v 1 ldots v m s is a basis of rm ker f f v m s 1 ldots f v m is a basis of rm im f and the degrees of their coordinates are of order m s sd o n 4

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

node and element resequencing using the laplacian of a finite element graph part i general concepts and algorithm a finite element graph feg is defined here as a nodal graph g a dual graph g or a communication graph g associated with a generic finite element mesh the laplacian matrix l g l g or l g used for the study of spectral properties of an feg is constructed from usual vertex and edge connectivities of a graph an automatic algorithm based on spectral properties of an feg g g or g is proposed to reorder the nodes and or elements of the associated finite element mesh the new algorithm is called spectral peg resequencing sfr this algorithm uses global information in the graph it does not depend on a pseudoperipheral vertex in the resequencing process and it does not use any kind of level structure of the graph moreover the sfr algorithm is of special advantage in computing environments with vector and parallel processing capabilities nodes or elements in the mesh can be reordered depending on the use of an adequate graph representation associated with the mesh if g is used then the nodes in the mesh are properly reordered for achieving profile and wavefront reduction of the finite element stiffness matrix if either g or g is used then the elements in the mesh are suitably reordered for a finite element frontai solver a unified approach involving fegs and finite element concepts is presented conclusions are inferred and possible extensions of this research are pointed out in part ii of this work 1 the computational implementation of the sfr algorithm is described and several numerical examples are presented the examples emphasize important theoretical numerical and practical aspects of the new resequencing method

programs over aperiodic monoids abstract we consider a model of computation where the execution of a program on an input corresponds to calculating a product in a finite monoid this model has recently been formalized to give an algebraic point of view on certain types of boolean circuits we focus our discussion on computations over aperiodic monoids in particular we exhibit examples of such monoids over which it is not possible to count modulo p

two applications of inductive counting for complementation problems following the recent independent proofs of immerman siam j comput 17 1988 pp 935 938 and szelepcsenyi bull european assoc theoret comput sci 33 1987 pp 96 100 that nondeterministic space bounded complexity classes are closed under complementation two further applications of the inductive counting technique are developed first an errorless probabilistic algorithm for the undirected graph s t connectivity problem that runs in o log n space and polynomial expected time is given then it is shown that the class logcfl is closed under complementation the latter is a special case of a general result that shows closure under complementation of classes defined by semi unbounded fan in circuits or equivalently nondeterministic auxiliary pushdown automata or tree size bounded alternating turing machines as one consequence it is shown that small numbers of role switches in two person pebbling can be eliminated

a generalization of spira s theorem and circuits with small segregators or separators spira showed that any boolean formula of size s can be simulated in depth o log s we generalize spira s theorem and show that any boolean circuit of size s with segregators or separators of size f s can be simulated in depth o f s log s this improves and generalizes a simulation of polynomial size boolean circuits of constant treewidth k in depth o k 2 log n by jansen and sarma our results imply that the class of languages computed by non uniform families of polynomial size circuits with constant size segregators equals non uniform n c 1 as a corollary we show that the boolean circuit value problem for circuits with constant size segregators is in deterministic space log 2 n our results also imply that the planar circuit value problem which is known to be p complete is in space n log n and that the layered circuit value and synchronous circuit value problems which are both p complete are in space n

competitive self stabilizing k clustering in this paper we give a silent self stabilizing algorithm for constructing a k clustering of any asynchronous connected network with unique ids our algorithm stabilizes in o n rounds using o log k log n space per process where n is the number of processes in the general case our algorithm constructs o nk k clusters if the network is a unit disk graph udg then our algorithm is 7 2552k o 1 competitive that is the number of k clusters constructed by the algorithm is at most 7 2552k o 1 times the minimum possible number of k clusters in any k clustering of the same network more generally if the network is an quasi unit disk graph qudg with approximation ratio then our algorithm is 7 2552 2k o competitive in case of tree networks our algorithm computes a k clustering with the minimum number of clusters our solution is based on the self stabilizing construction of a data structure called an mis tree a spanning tree of the network whose processes at even levels form a maximal independent set of the network the mis tree construction we use called lfmis is the time bottleneck of our k clustering algorithm as it takes n rounds in the worst case while the rest of the algorithm takes o d rounds where d is the diameter of the network we would like to improve that time to be o d but we show that our distributed mis tree construction is a p complete problem

using maximal independent sets to solve problems in parallel by using an o log n 2 time erew pram algorithm for a maximal independent set problem mis we show the following two results 1 given a graph the maximal vertex induced subgraph satisfying a hereditary graph property can be found in time 0 t n log n 2 using a polynomial number of processors where is the maximum of diameters of minimal graphs violating and t n is the time needed to decide whether a graph with n vertices satisfies 2 given a family c c1 c m of subsets of a finite set s 1 n with s u i 1 m ci a minimal set cover for s can be computed on an erew pram in time o log n m 2 using a polynomial number of processors where max ci i 1 m and max dj j 1 n

on zero testing and interpolation of k sparse multivariate polynomials over finite fields abstract given a black box which will produce the value of a k sparse multivariate polynomial for any given specific argument one may ask for optimal strategies 1 to distinguish such a polynomial from the zero polynomial 2 to distinguish any two such polynomials from one other and 3 to uniformly reconstruct the polynomial from such an information source while such strategies are known already for polynomials over fields of characteristic zero the equally important but considerably more complicated case of a finite field k of small characteristic is studied in the present paper the result is that the time complexity of such strategies depends critically on the degree m of the extension field of k from which the arguments are to be chosen e g if m equals the number n of variables then 1 can be solved by k 1 and 2 as well as 3 by 2 k 1 queries while in case m 1 essentially 2 log n log k queries are needed

a unified set of resequencing algorithms a general algorithm for resequencing equations and finite element meshes is presented this algorithm incorporates in a unified form the most successful resequencing schemes namely the sloan gibbs king gk and gibbs poole stockmeyer gps algorithms as well it provides a context within which the development of new algorithms may take place with a minimum of change it is easily programmed and has been shown to speed up the execution of the original gk algorithm

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

a hybrid algorithm for reducing matrix bandwidth a hybrid algorithm for reducing the bandwidth of symmetric matrices is described in terms of a finite element grid the new algorithm produces significantly lower bandwidths than either the commonly used gibbs poole stockmeyer gps or cuthill mckee cm algorithms with run times comparable to the original cm algorithm the new hybrid algorithm uses the gps algorithm as a preprocessor to provide a good initial node numbering for the author s node shuffling algorithm which accounts for variable degrees of freedom per node dof node the hybrid algorithm was tested on the 30 benchmark problems that were compiled by everstine and on 10 supplemental problems with variable dof node bandwidths and cpu times are presented for all problems

parallel algorithms in graph theory planarity testing we present efficient o log 2 n parallel algorithms for two classical graph problems planarity testing and finding triconnected components the algorithms use only a polynomial number of processors previous algorithms used omega n operations regardless of the number of available processors

parallel evaluation of arithmetic circuits abstract in this paper a generic algorithm designed for the parallel evaluation of arithmetic circuits is given this algorithm can be used in the domain of vlsi design in order to get tight upper bounds on the computing time of a circuit it can also be used in automatic parallelization of numerical programs as a guide for the detection of some predefinite schemes such as dot products or reductions more generally the theoretical algorithm presented in section 2 evaluates very quickly arithmetic straight line programs and its evaluation time serves as a good upper bound this algorithm generalizes miller ramachandran and kaltofen s algorithm 1988 in the sense it deals with a great variety of algebraic structures semi rings rings or lattices our contribution resides on the one hand in a new bound for the evaluation of circuits over lattices which improves previous results miller and teng 1987 and on the other hand in the unified formulation for the evaluation algorithm this algorithm runs in o min log n log d log n ha log n log n parallel time d being the algebraic degree in an extended sense of the circuit and ha the maximal number of alternances of and on a path of the circuit if the and operations define a lattice with m n processors where m n is the number of processors necessary for the multiplication of two n n matrices in the structure in o log n parallel time after presenting this algorithm its efficiency is shown on particular cases taking as input a simple and sequential algorithm it can be used as a compiler to produce a sorting circuit as fast as cole s circuit with logarithmic depth or an adder equivalent to brent and kung s adder in terms of size and depth these academic examples confirm the relevance of the algorithm presented here in the area of conception of fast vlsi arithmetic operators

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

amplifying lower bounds by means of self reducibility we observe that many important computational problems in nc1 share a simple self reducibility property we then show that for any problem a having this self reducibility property a has polynomial size tc0 circuits if and only if it has tc0 circuits of size n1 for every 0 counting the number of wires in a circuit as the size of the circuit as an example of what this observation yields consider the boolean formula evaluation problem bfe which is complete for nc1 and has the self reducibility property it follows from a lower bound of impagliazzo paturi and saks that bfe requires depth d tc0 circuits of size n1 d if one were able to improve this lower bound to show that there is some constant 0 independent of the depth d such that every tc0 circuit family recognizing bfe has size at least n1 then it would follow that tc0 nc1 we show that proving lower bounds of the form n1 is not ruled out by the natural proof framework of razborov and rudich and hence there is currently no known barrier for separating classes such as acc0 tc0 and nc1 via existing natural approaches to proving circuit lower bounds we also show that problems with small uniform constant depth circuits have algorithms that simultaneously have small space and time bounds we then make use of known time space tradeoff lower bounds to show that sat requires uniform depth d tc0 and ac0 6 circuits of size n1 c for some constant c depending on d

the complexity of computing the number of strings of given length in context free languages abstract computing the number of strings of given length contained in a language is related to classical problems of combinatorics formal languages and computational complexity here we study the complexity of this problem in the case of context free languages it is shown that for unambiguous context free languages such a computation is easy and can be carried out by efficient parallel algorithms on the contrary for some context free languages of ambiguity degree two the problem becomes intractable these results are related to other classical subjects concerning counting problems exponential time recognizable languages and sparse sets

parallei graph algorithms that are efficient on average the following three problems concerning random graphs can be solved in logn o 1 expected time using linearly many processors 1 finding the lexicographically first maximal independent set 2 coloring the vertices using a number of colors that is almost surely within twice the chromatic number and 3 finding a hamiltonian circuit

simulating threshold circuits by majority circuits we prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial size depth 2 majority circuit in general we show that a polynomial size depth d threshold circuit can be simulated uniformly by a polynomial size majority circuit of depth d 1 goldmann hastad and razborov showed in comput complexity 2 1992 pp 277 300 that a nonuniform simulation exists our construction answers two open questions posed by them we give an explicit construction whereas they use a randomized existence argument and we show that such a simulation is possible even if the depth d grows with the number of variables n their simulation gives polynomial size circuits only when d is constant

bandwidth reduction by simulated annealing the method of simulated annealing sa is investigated in the concrete problem of bandwidth reductio we develop a very efficient way to compute the transitions and this allows long annealing sessions monte carlo runs in reasonable time enabling meaningful experimenting it is demonstrated empirically that the neighbourhood relation in the space of states of the system greatly influences the convergence of the annealing a pure sa algorithm is developed and used to obtain new lower bounds for the bandwidth in the 30 everstine problems used as a benchmark for testing bandwidth reduction algorithms an hybrid algorithm is also developed to be used in practice it consists of an annealing session without previous melting run in cascade after the cuthill mckee algorithm the running times are of the same order as those of the deterministic algorithms now in use and improvements of typically 20 per cent are obtained on the 30 everstine problems

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

independent sets versus perfect matchings some structural relationships between matchings and independent sets are presented one consequence is that any nc algorithm for generating a uniform random perfect matching of a bipartite graph must differ from the only known polynomial time random matching algorithm due to broder 1986 and jerrum and sinclair 1988 unless nc p efficient parallel algorithms for emulating random walk on general graphs and some implicitly represented large scale graphs are also presented

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

near minimal matrix profiles and wavefronts for testing nodal resequencing algorithms the speeds of algorithms that are specifically designed to solve sparse matrix equations depend on the ordering of the unknowns because it is difficult to know what a good ordering is many resequencing algorithms have been developed to reorder the equations in a manner that minimizes the execution time of the solver being used there is no theoretical way of evaluating resequencing algorithms but four widely used algorithms cuthill mckee gibbs poole stockmeyer levy gibbs king have been compared with one another on the basis of their performance on a set of benchmark test problems this paper reports what we believe to be are minimal or near minimal matrix profiles and wavefronts for the benchmark problems comparisons of the minimal results with those produced by the widely used resequencing algorithms show that they produce profiles typically a few tens of per cent greater than minimal but 50 per cent to 100 per cent greater on two problem types the algorithm that produced the near minimal results used a simulated annealing technique and is far too slow for general use

parallel complexity of the regular code problems abstract the regular code problem rcp seeks to decide whether a right linear grammar g generates a code i e whether or not l g is free over l g here l g is the language generated by g the regular free monoid problem rfmp seeks to decide whether a right linear grammar g generates a free monoid both problems can be reduced to the linear context free grammar emptiness problem which in turn can be reduced to matrix inversion in the case of rcp the reductions give rise to an nc algorithm in the rfmp case the reduction yields only an exptime algorithm

on deciding trace equivalences for processes abstract in this paper we investigate the complexity of deciding trace maximal trace and trace equivalences for finite state processes and recursively defined processes specified by normed context free grammars cfgs in greibach normal form gnf the main results are as follows 1 1 trace maximal trace and trace equivalences for processes specified by normed gnf cfgs are all undecidable for this class of processes the regularity problem with respect to trace maximal trace or trace equivalence is also undecidable moreover all these undecidability results hold even for locally unary processes for processes specified by unary gnf cfgs the maximal trace equivalence is 2 p complete while the trace equivalence is nl complete and the trace equivalence is decidable in polynomial time by a dynamic programming algorithm 2 2 trace maximal trace and trace equivalences for finite state processes are pspace complete this holds even for locally unary finite state processes for unary finite state processes the maximal trace equivalence is co np complete while trace and trace equivalences are nl complete

complexity theory of parallel time and hardware abstract the parallel resources time and hardware and the complexity classes defined by them are studied using the aggregate model the equivalence of complexity classes defined by sequential space and uniform aggregate hardware is established aggregate time is related to bounded fanin circuit depth and similarly aggregate hardware is related to circuit width interelationships between aggregate time and hardware follow as corollaries aggregate time is related to the sequential resource reversal simultaneous relationships from aggregate hardware and time to sequential space and reversal are shown and conversely and these are used as evidence for an extended parallel computation thesis these simultaneous relationships provide new characterizations for the simultaneous parallel complexity class nc and for the complementary class sc the evaluation of monotone planar circuits is shown to be in nc in fact in logcfl

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

the iterated mod problem abstract the iterated mod problem is this given a b1 b2 bn all integers or all polynomials in q x is a mod b1 mod b2 mod bn 0 when the inputs are integers we prove the problem p complete with respect to log space reductions whereas in the polynomial case we prove the problem is in nc the significance of these results lies primarily in the similarity between the iterated mod problem and the euclidean algorithm we also show that the superincreasing knapsack problem is p complete using a very similar proof

ranking and formal power series abstract we prove that the ranking problem for unambiguous context free languages is nc 1 reducible to the value problem for algebraic formal power series in noncommuting variables in the particular case of regular languages we show that the problem of ranking is nc 1 reducible to the problem of counting the number of strings of given length in suitable regular languages as a consequence ranking problems for regular languages are nc 1 reducible to integer division and hence computable by log space uniform boolean circuits of polynomial size and depth o log n log log n or by p uniform boolean circuits of polynomial size and depth o log n

a p complete graph partition problem abstract the different than majority labelling dtml problem has a simple polynomial time sequential algorithm the labelling given by this algorithm is called the lexicographical first dtml in this paper we show that the lf dtml problem is p complete furthermore we show that even when restricted to planar graphs the lf dtml problem remains p complete

computing a context free grammar generating series the parallel complexity of computing context free grammar generating series is investigated it is known that this problem is in div but in terms of n rather than n where n is the index of the desired coefficient and is the grammar size a new method is presented which is in div in terms of 22o n evidence is provided that any direct application of elimination theory to this problem leads to a space and time resource factor that is nearly exponential in grammar size

automatic element reordering for finite element analysis with frontal solution schemes this paper describes an element reordering algorithm which is suitable for use with a frontal solution package the procedure is shown to generate efficient element numberings for a wide variety of test examples in an effort to obtain an optimum elimination order the algorithm first renumbers the nodes and then uses this result to resequence the elements this intermediate step is necessary because of the nature of the frontal solution procedure which assembles variables on an element by element basis but eliminates them node by node to renumber the nodes a modified version of the king1 algorithm is used in order to minimize the number of nodal numbering schemes that need to be considered the starting nodes are selected automatically by using some concepts from graph theory once the optimum numbering sequence has been ascertained the elements are then reordered in an ascending sequence of their lowest numbered nodes this ensures that the new elimination order is preserved as closely as possible for meshes that are composed of a single type of high order element it is only necessary to consider the vertex nodes in the renumbering process this follows from the fact that mesh numberings which are optimal for low order elements are also optimal for high order elements significant economies in the reordering strategy may thus be achieved a computer implementation of the algorithm written in fortran iv is given

a note on the complexity of deciding bisimilarity of normed unary processes we give nc1 reduction of bisimulation equivalence for general normed context free processes to the same problem for normed unary context free processes

on deciding readiness and failure equivalences for processes in this paper we study the complexity of deciding readiness and failure equivalences for finite state processes and recursively defined processes specified by normed context free grammars cfgs in greibach normal form gnf the results are as follows 1 readiness and failure equivalences for processes specified by normed gnf cfgs are both undecidable for this class of processes the regularity problem with respect to failure or readiness equivalence is also undecidable moreover all these undecidability results hold even for locally unary processes in the unary case these problems become decidable in fact they are p2 complete we also show that with respect to bisimulation equivalence the regularity for processes specified by normed gnf cfgs is nl complete 2 readiness and failure equivalences for finite state processes are pspace complete this holds even for locally unary finite state processes these two equivalences are co np complete for unary finite state processes further for acyclic finite state processes readiness and failure equivalences are co np complete and they are nl complete in the unary case 3 for finite tree processes we show that finite trace readiness and failure equivalences are all l complete further the results remain true for the unary case our results provide a complete characterization of the computational complexity of deciding readiness and failure equivalences for several important classes of processes

membership testing in commutative tranformation semigroups abstract given a finite set x of states a finite set of commuting transformations of x generators and another transformation f of x we analyze the complexity of deciding whether f can be obtained by composition of the generators looking first at the action of a commutative semigroup of transformations of a finite set we obtain an algorithm for membership testing valid for arbitrary commutative semigroups we then show that the complexity of the problem varies with the threshold of the semigroup polynomial time nc 3 in parallel with threshold zero or one and np complete otherwise

bounded size dictionary compression sc k completeness and nc algorithms we study the parallel complexity of a bounded size dictionary version lru deletion heuristic of the lz2 compression algorithm the unbounded version was shown to be p complete when the size of the dictionary is o logk n the problem of computing the lz2 compression is shown to be hard for the class of problems solvable simultaneously in polynomial time and o logk n space that is sck we also introduce a variation of this heuristic that turns out to be an sck complete problem the original heuristic belongs to sck 1 in virtue of these results we argue that there are no practical parallel algorithms for lz2 compression with lru deletion heuristic or any other heuristic deleting dictionary elements in a continuous way for simpler heuristics swap restart freeze practical parallel algorithms are given

a comparison of boundary graph grammars and context free hypergraph grammars context free hypergraph grammars and boundary graph grammars of bounded nonterminal degree have the same power both for generating sets of graphs and for generating sets of hypergraphs arbitrary boundary graph grammars have more graph generating power than context free hypergraph grammars but they have the same hypergraph generating power to obtain these results several normal forms for boundary graph grammars are given it is also shown that the class of boundary graph languages is closed under the operation of edge contraction where the label of the edge indicates whether or not the edge should be contracted

the complexity of planarity testing we clarify the computational complexity of planarity testing by showing that planarity testing is hard for l and lies in sl this nearly settles the question since it is widely conjectured that l sl the upper bound of sl matches the lower bound of l in the context of non uniform circuit complexity since l poly is equal to sl poly similarly we show that a planar embedding when one exists can be found in flsl previously these problems were known to reside in the complexity class ac1 via the o log n time crcw pram algorithm of ramachandran and reif although planarity checking for degree three graphs had been shown to be in sl chicago j theoret comput sci 1995 j acm 31 2 1984 401

highly parallel computations modulo a number having only small prime factors abstract highly parallel algorithms computing the inverse discrete roots or a large power modulo a number that has only small prime factors are presented the elaborated uniform families of boolean circuits simultaneously achieve depth o log n and size o n0 1 for p uniformity and depth o log n log log n and size o n0 1 for log space uniformity

parallel algorithms for matrix normal forms abstract here we offer a new randomized parallel algorithm that determines the smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field the algorithm has two important advantages over our previous one the multipliers relating the smith form to the input matrix are computed and the algorithm is probabilistic of las vegas type i e always finds the correct answer the smith form algorithm is also a good sequential algorithm our algorithm reduces the problem of smith form computation to two hermite form computations thus the smith form problem has complexity asymptotically that of the hermite form problem we also construct fast parallel algorithms for jordan normal form and testing similarity of matrices both the similarity and non similarity problems are in the complexity class rnc for the usual coefficient fields i e they can be probabilistically decided in polylogarithmic time using polynomially many processors

simple and effective bandwidth reduction algorithm an automatic algorithm is presented for reducing the bandwidth of symmetric interconnectivity matrices the technique takes an arbitrary input scheme and reorders the nodes so as to reduce the bandwidth a large number of examples is presented to illustrate the reliability and cost effectiveness of the method

identifying the minimal transversals of a hypergraph and related problems the paper considers two decision problems on hypergraphs hypergraph saturation and recognition of the transversal hypergraph and discusses their significance for several search problems in applied computer science hypergraph saturation i e given a hypergraph cal h decide if every subset of vertices is contained in or contains some edge of cal h is shown to be co np complete a certain subproblem of hypergraph saturation the saturation of simple hypergraphs i e sperner families is shown to be under polynomial transformation equivalent to transversal hypergraph recognition i e given two hypergraphs cal h 1 cal h 2 decide if the sets in cal h 2 are all the minimal transversals of cal h 1 the complexity of the search problem related to the recognition of the transversal hypergraph the computation of the transversal hypergraph is an open problem this task needs time exponential in the input size it is unknown whether an output polynomial algorithm exists for several important subcases for instance if an upper or lower bound is imposed on the edge size or for acyclic hypergraphs output polynomial algorithms are presented computing or recognizing the minimal transversals of a hypergraph is a frequent problem in practice which is pointed out by identifying important applications in database theory boolean switching theory logic and artificial intelligence ai particularly in model based diagnosis

an optimal parallel algorithm for formula evaluation a new approach to buss s textbf nc 1 algorithm proc 19th acm symposium on theory of computing association for computing machinery new york 1987 pp 123 131 for evaluation of boolean formulas is presented this problem is shown to be complete for textbf nc 1 over textbf ac 0 reductions this approach is then used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits

a tradeoff theorem for space and reversal in this note the following result is proved let m be an off line turing machine n the input length s the work space and r the total number of reversals of the work tape heads then rs o n implies rs o 1

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

the complexity of acyclic conjunctive queries this paper deals with the evaluation of acyclic booleanconjunctive queries in relational databases by well known resultsof yannakakis 1981 this problem is solvable in polynomial time its precise complexity however has not been pinpointed so far weshow that the problem of evaluating acyclic boolean conjunctivequeries is complete for logcfl the class of decision problems thatare logspace reducible to a context free language since logcfl iscontained in ac1 and nc2 the evaluation problem of acyclic booleanconjunctive queries is highly parallelizable we present a paralleldatabase algorithm solving this problem with alogarithmic number ofparallel join operations the algorithm is generalized to computingthe output of relevant classes of non boolean queries we also showthat the acyclic versions of the following well known database andai problems are all logcfl complete the query output tuple problemfor conjunctive queries conjunctive query containment clausesubsumption and constraint satisfaction the logcfl completenessresult is extended to the class of queries of bounded tree widthand to other relevant query classes which are more general than theacyclic queries

word problems and membership problems on compressed words we consider a compressed form of the word problem for finitely presented monoids where the input consists of two compressed representations of words over the generators of a monoid m and we ask whether these two words represent the same monoid element of m words are compressed using straight line programs i e context free grammars that generate exactly one word for several classes of finitely presented monoids we obtain completeness results for complexity classes in the range from p to expspace as a by product of our results on compressed word problems we obtain a fixed deterministic context free language with a pspace complete compressed membership problem the existence of such a language was open so far finally we will investigate the complexity of the compressed membership problem for various circuit complexity classes

an alternative pseudoperipheral node finder for resequencing schemes many resequencing algorithms for reducing the bandwidth profile and wavefront of sparse symmetric matrices have been published in finite element applications the sparsity of a matrix is related to the nodal ordering of the finite element mesh some of the most successful algorithms which are based on graph theory require a pair of starting pseudoperipheral nodes these nodes located at nearly maximal distance apart are determined using heuristic schemes this paper presents an alternative pseadoperipheral node finder which is based on the algorithm developed by gibbs poole and stockmeyer this modified scheme is suitable for nodal reordering of finite meshes and provides more consistency in the effective selection of the starting nodes in problems where the selection becomes arbitrary due to the number of candidates for these starting nodes this case arises in particular for square meshes the modified scheme was implemented in gibbs poole stockmeyer gibbs king and sloan algorithms test problems of these modified algorithms include 1 everstine s 30 benchmark problems 2 sets of square rectangular and annular cylindrical finite element meshes with quadrilateral and triangular elements and 3 additional examples originating from mesh refinement schemes the results demonstrate that the modifications to the original algorithms contribute to the improvement of the reliability of all the resequencing algorithms tested herein for the nodal reordering of finite element meshes

the maximum flow problem is log space complete for p the space complexity of the maximum flow problem is investigated it is shown that the problem is log space complete for deterministic polynomial time thus the maximum flow problem probably has no algorithm which needs only o logk n storage space for any constant k another consequence is that there is probably no fast parallel algorithm for the maximum flow problem

multilist layering complexity and applications abstract a natural combinatorial generalization of the convex layer problem termed multilist layering is introduced it is observed to be p complete in the general case when the number of lists or layer size are bounded by s n multilist layering is shown to be logspace hard for the class of problems solvable simultaneously in polynomial time and space s n on the other hand simultaneous polynomial time and o s n log n space solutions in the above cases are provided thus a natural almost complete problem for steve s classes sc 1 sc 2 4 is in particular obtained also nc algorithms for multilist layering when the number of lists or the layer size is bounded by a constant are given as a result the first nc solutions sc solutions respectively for the convex layer problem where the number of orientations or the layer size are bonded by a constant polylog bounded respectively are derived

tight complexity bounds for term matching problems abstract in this paper we study the sequential and parallel complexity of various important term matching problems these problems occur frequently in applications such as term rewriting functional programming and logic programming first we obtain non trivial lower bounds on the parallel complexity of the uninterpreted term matching problem we also establish the tightness of these bounds for some representations and several models we then characterize completely the sequential complexity of associative and commutative matching when the number of occurrences of variables is varied specifically we show that even if each variable is restricted to at most two occurrences in the terms both associative matching and commutative matching are np complete interestingly for the important restriction of boolean terms we show that commutative matching is np complete whereas associative matching has a linear time sequential algorithm for linear terms we significantly improve the existing upper bound for associative commutative matching and present a new algorithm for associative matching designing direct parallel algorithms for associative commutative matching of linear terms appears to be a difficult task despite this we have been able to resolve questions about the parallel complexity of these problems using complexity theoretic techniques finally an interesting consequence of the research reported here is the demonstration of a tighter relationship between associative commutative matching for linear terms and bipartite matching on both sequential and parallel models

parallel matrix and graph algorithms matrix multiplication algorithms for cube connected and perfect shuffle computers are presented it is shown that in both these models two n times n matrices can be multiplied in o n m log m time when n 2 m 1 leqq m leqq n processing elements pes are available when only m 2 1 leqq m leqq n pes are available two n times n matrices can be multiplied in o n 2 m m n m 2 61 time it is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms

parallel construction of perfect matchings and hamiltonian cycles on dense graphs the intensive study of fast parallel and distributed algorithms for various routing and communications problems on graphs with good expanding properties ref pu1 ref pu2 ref ss is being carried out recently the parallel solutions for expanders that already exist ref pu 2 required an extensive randomization and an application of a randomized subroutine for the maximum matching problem in this paper we attack the problem of fast parallel algorithms for constructing perfect matchings and hamiltonian cycles on dense graph networks undirected graphs of minimal degree v over 2 somewhat surprisingly we design fast deterministic parallel algorithms for constructing both the perfect matchings and the hamiltonian cycles on the dense graphs the algorithm for constructing perfect matchings on dense graphs works in o log 2 n parallel time and o n 8 processors or o log 4 n parallel time and o n 4 processors on a crew pram the algorithm for constructing hamiltonian cycles on dense graphs works in o log 5 n parallel time and o n 4 processors on a crew pram our method of the parallel solution involves a development of new dense graph combinatorics suitable for fast parallelization

multiplication division and shift instructions in parallel random access machines abstract we prove that polynomial time on a parallel random access machine pram with unit cost multiplication and division or on a pram with unit cost shifts is equivalent to polynomial space on a turing machine pspace this extends the result that polynomial time on a basic pram is equivalent to pspace to hold when the pram is allowed multiplication or division or unrestricted shifts it also extends to the pram the results that polynomial time on a random access machine ram with multiplication is equivalent to pspace and that polynomial time on a ram with shifts that is a vector machine is equivalent to pspace we obtain simulations of uniform circuits by rams with enhanced instruction sets and use the enhanced rams to simulate prams with enhanced instruction sets

logspace and logtime leaf languages the computation tree of a nondeterministic machinemwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order we may characterize problems by considering for a particular leaf language y the set of allxfor which the leaf string ofmis contained iny in this way in the context of polynomial time computation leaf languages were shown to capture many complexity classes in this paper we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation we show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages capturing also many classes withinp in contrast logtime leaf languages basically behave like logtime reducibilities both cases are more subtle to handle than the polynomial time case we also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language we indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

the isomorphism problem for k trees is complete for logspace we show that for k constant k tree isomorphism can be decided in logarithmic space by giving an o klogn space canonical labeling algorithm the algorithm computes a unique tree decomposition uses colors to fully encode the structure of the original graph in the decomposition tree and invokes lindell s tree canonization algorithm as a consequence the isomorphism the automorphism as well as the canonization problem for k trees are all complete for deterministic logspace completeness for logspace holds even for simple structural properties of k trees we also show that a variant of our canonical labeling algorithm runs in time o k 1 n where n is the number of vertices yielding the fastest known fpt algorithm for k tree isomorphism

designing programs that check their work a program correctness checker is an algorithm for checking the output of a computation that is given a program and an instance on which the program is run the checker certifies whether the output of the program on that instance is correct this paper defines the concept of a program checker it designs program checkers for a few specific and carefully chosen problems in the class fp of functions computable in polynomial time problems in fp for which checkers are presented in this paper include sorting matrix rank and gcd it also applies methods of modern cryptography especially the idea of a probabilistic interactive proof to the design of program checkers for group theoretic computations two structural theorems are proven here one is a characterization of problems that can be checked the other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable all problems in the class are checkable