efficient checkers for number theoretic computations blum has recently proposed a new approach to the problem of program correctness the program checker here efficient program checkers are presented for two important number theory problems integer gcd and modular exponentiation the former result answers an open problem posed by blum the latter result paves the way for more robust cryptographic programs as modular exponentiation is the basis for several cryptosystems taken together these results lay the foundation for more reliable number theoretic programs

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

3 query locally decodable codes of subexponential length locally decodable codes ldcs allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword even if a constant fraction of the codeword is damaged in a recent work j acm 55 2008 article 1 yekhanin constructs a 3 query ldc with subexponential length however this construction requires a conjecture that there are infinitely many mersenne primes in this paper we give the first unconditional constant query ldc construction with subexponential codeword length in addition our construction reduces codeword length

some improvements to the flux type a posteriori error estimators abstract this paper presents explicit equilibrated fluxes for the local pure neumann problems used by the a posteriori error estimator in ainsworth and oden m ainsworth j t oden a procedure for a posteriori error estimation for h p finite element methods comput methods appl mech engrg 101 1992 73 96 we modify the target function for minimization to obtain new equilibrated fluxes in a weighted form we investigate the performance of the new equilibrated fluxes in two dimensional spectral hp element approximations of an elliptic equation using conforming triangle elements the new fluxes appear more robust and lead to substantial computational savings compared to the original numerical procedure in ainsworth and oden ainsworth and oden et al 1992

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

circuit lower bounds for merlin arthur classes we show that for each k 0 mathsf ma 1 mathsf ma with 1 bit of advice does not have circuits of size n k this implies the first superlinear circuit lower bounds for the promise versions of the classes mathsf ma mathsf am and mathsf zpp parallel mathsf np we extend our main result in several ways for each k we give an explicit language in mathsf ma cap mathsf coma 1 which does not have circuits of size n k we also adapt our lower bound to the average case setting i e we show that mathsf ma 1 cannot be solved on more than 1 2 1 n k fraction of inputs of length n by circuits of size n k furthermore we prove that mathsf ma does not have arithmetic circuits of size n k for any k as a corollary to our main result we obtain that derandomization of mathsf ma o 1 implies the existence of pseudorandom generators computable using o 1 bits of advice

undirected connectivity in log space we present a deterministic log space algorithm that solves st connectivity in undirected graphs the previous bound on the space complexity of undirected st connectivity was log4 3 obtained by armoni ta shma wigderson and zhou jacm 2000 as undirected st connectivity is complete for the class of problems solvable by symmetric nondeterministic log space computations the class sl this algorithm implies that sl e l where l is the class of problems solvable by deterministic log space computations independent of our work and using different techniques trifonov stoc 2005 has presented an o log n log log n space deterministic algorithm for undirected st connectivity our algorithm also implies a way to construct in log space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph specifically we give log space constructible universal traversal sequences for graphs with restricted labeling and log space constructible universal exploration sequences for general graphs

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

random self reducibility of complete sets this paper generalizes the previous formal definitions of random self reducibility it is shown that even under a very general definition sets that are complete for any level of the polynomial hierarchy are not nonadaptively random self reducible unless the hierarchy collapses in particular np complete sets are not nonadaptively random self reducible unless the hierarchy collapses at the third level by contrast we show that sets complete for the classes pp and text mod m text p are random self reducible

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

on the security of pseudorandomized information theoretically secure schemes in this paper we discuss a naive method of randomness reduction for cryptographic schemes which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator prg we propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases even against an adversary with computationally unbounded attack algorithm hence the prg based randomness reduction can be effective even for information theoretically secure cryptographic schemes especially when the amount of information received by the adversary is small in comparison to a preceding result of dubrov and ishai stoc 2006 our result removes the requirement of generalized notion of nb prgs and is effective for more general kinds of protocols we give some numerical examples to show the effectiveness of our result in practical situations and we also propose a further idea for improving the effect of the prg based randomness reduction

decoding frequency permutation arrays under chebyshev distance a frequency permutation array fpa of length n m and distance d is a set of permutations on a multiset over m symbols where each symbol appears exactly times and the distance between any two elements in the array is at least d fpa generalizes the notion of permutation array in this paper under the chebyshev distance we first prove lower and upper bounds on the size of fpa then we give several constructions of fpas and some of them come with efficient encoding and decoding capabilities moreover we show one of our designs is locally decodable i e we can decode a message bit by reading at most 1 symbols which has an interesting application to private information retrieval

eigenvalues and expansion of regular graphs the spectral method is the best currently known technique to prove lower bounds on expansion ramanujan graphs which have asymptotically optimal second eigenvalue are the best known explicit expanders the spectral method yielded a lower bound of k 4 on the expansion of linear sized subsets of k regular ramanujan graphs we improve the lower bound on the expansion of ramanujan graphs to approximately k 2 moreover we construct a family of k regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k 2 this shows that k 2 is the best bound one can obtain using the second eigenvalue method we also show an upper bound of roughly 1 k 1 on the average degree of linear sized induced subgraphs of ramanujan graphs this compares positively with the classical bound 2 k 1 as a byproduct we obtain improved results on random walks on expanders and construct selection networks respectively extrovert graphs of smaller size respectively degree than was previously known

locally testable and locally correctable codes approaching the gilbert varshamov bound one of the most important open problems in the theory of error correcting codes is to determine the tradeoff between the rate r and minimum distance delta of a binary code the best known tradeoff is the gilbert varshamov bound and says that for every delta in 0 1 2 there are codes with minimum distance delta and rate r r mathsf gv delta 0 for a certain simple function r mathsf gv cdot in this paper we show that the gilbert varshamov bound can be achieved by codes which support local error detection and error correction algorithms specifically we show the following results 1 local testing for all delta in 0 1 2 and all r there exist codes with length n rate r and minimum distance delta that are locally testable with mathsf quasipolylog n query complexity 2 local correction for all epsilon 0 for all delta sufficiently large and all r there exist codes with length n rate r and minimum distance delta that are locally correctable from delta 2 o 1 fraction errors with o n epsilon query complexity furthermore these codes have an efficient randomized construction and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity our results on locally correctable codes also immediately give locally decodable codes with the same parameters our local testing result is obtained by combining thommesen s random concatenation technique and the best known locally testable codes by kopparty et al our local correction result which is significantly more involved also uses random concatenation along with a number of further ideas the guruswami sudan indyk list decoding strategy for concatenated codes alon edmonds luby distance amplification and the local list decodability local list recoverability and local testability of reed muller codes curiously our final local correction algorithms go via local list decoding and local testing algorithms this seems to be the first time local testability is used in the construction of a locally correctable code

on the complexity of checking semantic equivalences between pushdown processes and finite state processes simulation preorder equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory their standard definitions are often called strong simulation bisimulation while weak simulation bisimulation abstracts from internal t actions we study the computational complexity of checking these strong and weak semantic preorders equivalences between pushdown processes and finite state processes we present a complete picture of the computational complexity of these problems and also study fixed parameter tractability in two important input parameters x the size of the finite control of the pushdown process and y the size of the finite state process all simulation problems are generally exptime complete and only become polynomial if both parameters x and y are fixed weak bisimulation equivalence is pspace complete but becomes polynomial if and only if parameter x is fixed strong bisimulation equivalence is pspace complete but becomes polynomial if either parameter x or y is fixed

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

upper bounds on matching families in bbz_ pq n matching families are one of the major ingredients in the construction of locally decodable codes ldcs and the best known constructions of ldcs with a constant number of queries are based on matching families the determination of the largest size of any matching family in zmn where zm is the ring of integers modulo m is an interesting problem in this paper we show an upper bound of o pq 0 625n 0 125 for the size of any matching family in zpqn where p and q are two distinct primes our bound is valid when n is a constant p and p q 1 our result improves an upper bound of dvir and coworkers

linear time encodable decodable codes with near optimal rate we present an explicit construction of linear time encodable and decodable codes of rate r which can correct a fraction 1 r spl epsiv 2 of errors over an alphabet of constant size depending only on spl epsiv for every 0 0 the error correction performance of these codes is optimal as seen by the singleton bound these are near mds codes such near mds linear time codes were known for the decoding from erasures our construction generalizes this to handle errors as well concatenating these codes with good constant sized binary codes gives a construction of linear time binary codes which meet the zyablov bound and also the more general blokh zyablov bound by resorting to multilevel concatenation our work also yields linear time encodable decodable codes which match forney s error exponent for concatenated codes for communication over the binary symmetric channel the encoding decoding complexity was quadratic in forney s result and forney s bound has remained the best constructive error exponent for almost 40 years now in summary our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding but in addition have linear time encoding and decoding algorithms

a nonadaptive nc checker for permutation group intersection we design a nonadaptive nc checker for permutation group intersection sharpening a result of blum and kannan j acm 43 1995 269 291 additionally we also get nonadaptive nc checkers for some related group theoretic problems

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

2 server pir with subpolynomial communication a 2 server private information retrieval pir scheme allows a user to retrieve the ith bit of an n bit database replicated among two noncommunicating servers while not revealing any information about i to either server in this work we construct a 2 server pir scheme with total communication cost no log log n log n this improves over current 2 server protocols which all require n1 3 communication our construction circumvents the n1 3 barrier of razborov and yekhanin 2007 which holds for the restricted model of bilinear group based schemes covering all previous 2 server schemes the improvement comes from reducing the number of servers in existing protocols based on matching vector codes from 3 or 4 servers to 2 this is achieved by viewing these protocols in an algebraic way using polynomial interpolation and extending them using partial derivatives

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

the complexity of counting quantifiers on equality languages an equality language is a relational structure with infinite domain whose relations are first order definable in equality we classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers in doing this we find ourselves in various worlds in which dichotomies or trichotomies subsist

explicit codes achieving list decoding capacity error correction with optimal redundancy in this paper we present error correcting codes that achieve the information theoretically best possible tradeoff between the rate and error correction radius specifically for every 0 r 1 and epsiv 0 we present an explicit construction of error correcting codes of rate that can be list decoded in polynomial time up to a fraction 1 r epsiv of worst case errors at least theoretically this meets one of the central challenges in algorithmic coding theory our codes are simple to describe they are folded reed solomon codes which are in fact exactly reed solomon rs codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols given the ubiquity of rs codes this is an appealing feature of our result and in fact our methods directly yield better decoding algorithms for rs codes when errors occur in phased bursts the alphabet size of these folded rs codes is polynomial in the block length we are able to reduce this to a constant depending on epsiv using existing ideas concerning ldquolist recoveryrdquo and expander based codes concatenating the folded rs codes with suitable inner codes we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard gmd decoding

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

uniform random sampling not recommended for large graph size estimation abstract the norm of data size estimation is to use uniform random samples whenever possible there have been tremendous efforts in obtaining uniform random samples using methods such as metropolis hasting random walk or importance sampling 2 this paper shows that on the contrary to the common practice uniform random sampling should be avoided when pps probability proportional to size sampling is available for large data to develop intuition of the sampling process we discuss the sampling and estimation problem in the context of graph the size is the number of nodes in the graph uniform random sampling corresponds to uniform random node rn sampling and pps sampling is approximated by random edge re sampling in this setting we show that for large graphs re sampling outperforms rn sampling with a ratio proportional to the normalized graph degree variance this result is particularly important in the era of big data when data are typically large and scale free 3 resulting in large degree variance we derive the result by giving the variances of rn and re estimators each step of the derivation is supported and demonstrated by simulation studies assuming power law distributions then we use 18 real world networks to verify the result furthermore we show that the performance of random walk rw sampling is data dependent and can be significantly worse than rn and re more specifically rw can estimate online social networks but not web graphs due to the difference of the graph conductance

bidimensional optical solitons in a quadratic medium the modulation evolution of a short localized optical pulse in a crystal belonging to one of the classesn 42m n 43m 3 m 6 mm and with a non vanishing second order nonlinearity is considered in 2c 1 dimensions the partial differential system accounting for it can be reduced to the completely integrable davey stewartson system if some conditions are satisfied the first integrability condition represents a balance between the third order kerr effect and the cascaded second order nonlinearities while the second condition is an equilibrium between the dispersion and the kinetic factor of the electro optic optical rectification wave interaction for anomalous dispersion the obtained davey stewartson system is of the type i that admits localized soliton solutions lump solution algebraically decaying in all directions exist in any case satisfying the above conditions

checking approximate computations of polynomials and functional equations a majority of the results on self testing and correcting deal with programs which purport to compute the correct results precisely we relax this notion of correctness and show how to check programs that compute only a numerical approximation to the correct answer the types of programs that we deal with are those computing polynomials and functions defined by certain types of functional equations we present results showing how to perform approximate checking self testing and self correcting of polynomials settling in the affirmative a question raised by p gemmell et al proceedings of the 23rd acm symposium on theory of computing 1991 pp 32 42 r rubinfeld and m sudan proceedings of the third annual acm siam symposium on discrete algorithms orlando fl 1992 pp 23 43 r rubinfeld and m sudan siam j comput 25 1996 pp 252 271 we obtain this by first building approximate self testers for linear and multilinear functions we then show how to perform approximate checking self testing and self correcting for those functions that satisfy addition theorems settling a question raised by r rubinfeld siam j comput 28 1999 pp 1972 1997 in both cases we show that the properties used to test programs for these functions are both robust in the approximate sense and stable finally we explore the use of reductions between functional equations in the context of approximate self testing our results have implications for the stability theory of functional equations

tripartite to bipartite entanglement transformation by stochastic local operations and classical communication and the structure of matrix spaces we study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication slocc it is known that the tripartite to bipartite slocc convertibility is characterized by the maximal schmidt rank of the given tripartite state i e the largest schmidt rank over those bipartite states lying in the support of the reduced density operator in this paper we further study this problem and exhibit novel results in both multi copy and asymptotic settings utilizing powerful results from the structure of matrix spaces in the multi copy regime we observe that the maximal schmidt rank is strictly super multiplicative i e the maximal schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal schmidt ranks we then provide a full characterization of those tripartite states whose maximal schmidt rank is strictly super multiplicative when taking tensor product with itself notice that such tripartite states admit strict advantages in tripartite to bipartite slocc transformation when multiple copies are provided in the asymptotic setting we focus on determining the tripartite to bipartite slocc entanglement transformation rate computing this rate turns out to be equivalent to computing the asymptotic maximal schmidt rank of the tripartite state defined as the regularization of its maximal schmidt rank despite the difficulty caused by the super multiplicative property we provide explicit formulas for evaluating the asymptotic maximal schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces including the study of matrix semi invariants these formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under slocc in the asymptotic setting applying the recent progress on the non commutative rank problem we can verify this condition in deterministic polynomial time

on helping by robust oracle machines abstract the concept of helping by robust oracle turing machines introduced recently by schoning is extended to the notion of one sided helping and its relations to the structural properties of np sets are investigated various results on who can help whom have been obtained on sets in bpp in r in up and sparse sets sets which do not help any set are studied and two types of sets defined by structural properties are demonstrated to help no sets except those in p sest which help themselves are also studied and are shown to be related to self reducible sets several previously known structural results on np sets are reproved from the point of view of helping

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

locally decodable codes from nice subsets of finite fields and prime factors of mersenne numbers a k query locally decodable code ldc encodes an n bit message x as an n bit codeword c x such that one can probabilistically recover any bit x i of the message by querying only k bits of the codeword c x even after some constant fraction of codeword bits has been corrupted the major goal of ldc related research is to establish the optimal trade off between length and query complexity of such codes recently vast improvements in upper bounds for the length of ldcs were achieved via constructions that rely on existence of certain special nice subsets of finite fields in this work we extend the constructions of ldcs from nice subsets we argue that further progress on upper bounds for ldcs via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of mersenne numbers specifically we show that every mersenne number m 2 t 1 that has a prime factor p m gamma yields a family of k gamma query ldcs of length exp n 1 t conversely if for some fixed k and all epsilon 0 one can use the nice subsets technique to obtain a family of k query ldcs of length exp n epsilon then infinitely many mersenne numbers have prime factors larger than currently known

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

a shifted method for solving symmetric eigenvalue problems abstract in this paper we present a shifted method using householder reflections to calculate the largest eigenvalue of a full symmetric matrix differing from the ql or qr algorithm the shifted method transforms each column independently at each stage of the iterative process it is therefore very suitable for a parallel processing machine this algorithm is the generalization of a scheme due to c d lebudde for the positive definite problems 1 2 however our algorithm not only speeds up the convergence but also is suitable for any symmetric matrix the algorithm can be used to determine the first few largest eigenvalues we give the convergence analysis and convergence rate some numerical results which show the efficiency of the algorithm are given

on monotone formula composition of perfect zero knowledge languages we investigate structural properties of interactive perfect zero knowledge pzk proofs specifically we look into the closure properties of pzk languages under monotone boolean formula composition this gives rise to new protocol techniques we show that interactive pzk for random self reducible rsr and for co rsr languages is closed under monotone boolean formula composition namely we present pzk proofs for monotone boolean formulae whose atoms are statements about membership in a pzk language which is rsr or whose complement is rsr we also discuss extensions recent applications and generalizations of the techniques

computing pure bayesian nash equilibria in games with finite actions and continuous types we extend the well known fictitious play fp algorithm to compute pure strategy bayesian nash equilibria in private value games of incomplete information with finite actions and continuous types g facts we prove that if the frequency distribution of actions fictitious play beliefs converges then there exists a pure strategy equilibrium strategy that is consistent with it we furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type this algorithm can also be used to compute pure e nash equilibria when distributions are not fully converged we then apply our algorithm to find equilibria in an important and previously unsolved game simultaneous sealed bid second price auctions where various types of items e g substitutes or complements are sold finally we provide an analytical characterisation of equilibria in games with linear utilities specifically we show how equilibria can be found by solving a system of polynomial equations for a special case of simultaneous auctions we also solve the equations confirming the results obtained numerically

towards 3 query locally decodable codes of subexponential length a q query locally decodable code ldc encodes an n bit message x as an n bit codeword c x such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword c x even after some constant fraction of codeword bits has been corrupted we give new constructions of three query ldcs of vastly shorter length than that of previous constructions specifically given any mersenne prime p e 2t 1 we design three query ldcs of length n e exp o n1 t for every n based on the largest known mersenne prime this translates to a length of less than exp o n10 7 compared to exp o n1 2 in the previous constructions it has often been conjectured that there are infinitely many mersenne primes under this conjecture our constructions yield three query locally decodable codes of length n e exp no 1 log log n for infinitely many n we also obtain analogous improvements for private information retrieval pir schemes we give 3 server pir schemes with communication complexity of o n10 7 to access an n bit database compared to the previous best scheme with complexity o n1 5 25 assuming again that there are infinitely many mersenne primes we get 3 server pir schemes of communication complexity no 1 log logn for infinitely many n previous families of ldcs and pir schemes were based on the properties of low degree multivariate polynomials over finite fields our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set

subcubic equivalences between path matrix and triangle problems we say an algorithm on n n matrices with integer entries in l m mr or n node graphs with edge weights from l m m is truly subcubic if it runs in o n3 poly log m time for some 0 we define a notion of subcubic reducibility and show that many important problems on graphs and matrices solvable in o n3 time are equivalent under subcubic reductions namely the following weighted problems either all have truly subcubic algorithms or none of them do the all pairs shortest paths problem on weighted digraphs apsp detecting if a weighted graph has a triangle of negative total edge weight listing up to n2 99 negative triangles in an edge weighted graph finding a minimum weight cycle in a graph of non negative edge weights the replacement paths problem on weighted digraphs finding the second shortest simple path between two nodes in a weighted digraph checking whether a given matrix defines a metric verifying the correctness of a matrix product over the min semiring finding a maximum subarray in a given matrix therefore if apsp cannot be solved in n3 e time for any e 0 then many other problems also need essentially cubic time in fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization verifying a matrix product over the same structure and corresponding triangle detection problems over the structure these equivalences simplify prior work on subcubic algorithms for all pairs path problems since it now suffices to give appropriate subcubic triangle detection algorithms other consequences of our work are new combinatorial approaches to boolean matrix multiplication over the or and semiring abbreviated as bmm we show that practical advances in triangle detection would imply practical bmm algorithms among other results building on our techniques we give two improved bmm algorithms a derandomization of the combinatorial bmm algorithm of bansal and williams focs 09 and an improved quantum algorithm for bmm

error correcting data structures we study data structures in the presence of adversarial noise we want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object even if the data structure has been corrupted by a constant fraction of errors we measure the efficiency of a data structure in terms of its length the number of bits in its representation and query answering time measured by the number of bit probes to the possibly corrupted representation the main issue is the trade off between these two this new model is the common generalization of static data structures and locally decodable error correcting codes ldcs we prove a number of upper and lower bounds on various natural error correcting data structure problems in particular we show that the optimal length of t probe error correcting data structures for the membership problem where we want to store subsets of size s from a universe of size n such that membership queries can be answered effic

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

low end uniform hardness versus randomness tradeoffs for am impagliazzo and wigderson proceedings of the 39th annual ieee symposium on foundations of computer science ieee computer society washington dc 1998 pp 734 743 proved a hardness versus randomness tradeoff for bpp in the uniform setting which was subsequently extended to give optimal tradeoffs for the full range of possible hardness assumptions in slightly weaker settings gutfreund shaltiel and ta shma comput complexity 12 2003 pp 85 130 proved a uniform hardness versus randomness tradeoff for am but that result worked only on the high end of possible hardness assumptions in this work we give uniform hardness versus randomness tradeoffs for am that are near optimal for the full range of possible hardness assumptions following gutfreund shaltiel and ta shma we do this by constructing a hitting set generator hsg for am with resilient reconstruction our construction is a recursive variant of the miltersen vinodchandran hsg comput complexity 14 2005 pp 256 279 the only known hsg construction with this required property the main new idea is to have the reconstruction procedure operate implicitly and locally on superpolynomially large objects using tools from pcps low degree testing self correction together with a novel use of extractors that are built from reed muller codes for a sort of locally computable error reduction as a consequence we obtain gap theorems for am and am cap coam that state roughly that either am or am cap coam protocols running in time t n can simulate all of exp arthur merlin games are powerful or else all of am or am cap coam can be simulated in nondeterministic time s n arthur merlin games can be derandomized for a near optimal relationship between t n and s n as in gutfreund shatiel and ta shma the case of am cap coam yields a particularly clean theorem that is of special interest due to the wide array of cryptographic and other problems that lie in this class

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

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

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

high rate locally correctable and locally testable codes with sub polynomial query complexity locally correctable codes lccs and locally testable codes ltcs are error correcting codes that admit local algorithms for correction and detection of errors those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword the fundamental question about lccs and ltcs is to determine the optimal tradeoff among their rate distance and query complexity in this work we construct the first lccs and ltcs with constant rate constant relative distance and sub polynomial query complexity specifically we show that there exist lccs and ltcs with block length n constant rate which can even be taken arbitrarily close to 1 and constant relative distance whose query complexity is exp o slog n for lccs and log n o log log n for ltcs in addition to having small query complexity our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by lccs or ltcs specifically over large but constant size alphabet our codes approach the singleton bound that is they have almost the best possible relationship between their rate and distance over the binary alphabet our codes meet the zyablov bound such tradeoffs between the rate and the relative distance were previously not known for any o n query complexity our results on lccs also immediately give locally decodable codes with the same parameters

computing pure nash and strong equilibria in bottleneck congestion games bottleneck congestion games properly model the properties of many real world network routing applications they are known to possess strong equilibria a strengthening of nash equilibrium to resilience against coalitional deviations in this paper we study the computational complexity of pure nash and strong equilibria in these games we provide a generic centralized algorithm to compute strong equilibria which has polynomial running time for many interesting classes of games such as e g matroid or single commodity bottleneck congestion games in addition we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics using unilateral improvement dynamics in matroid games pure nash equilibria can be reached efficiently in contrast computing even a single coalitional improvement move in matroid and single commodity games is strongly np hard in addition we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics they continue to hold even for convergence to approximate equilibria

a note on decision versus search for graph automorphism we show that for any graphg knontrivial automorphisms ofg if as many exist can be computed in time g o logk with nonadaptive queries to ga the decision problem for graph automorphism as a consequence we show that some problems related to ga are actually polynomial time truth table equivalent to ga one of these results provides an answer to an open question of lubiw siam j comput 10 1981 11 21

high rate codes with sublinear time decoding locally decodable codes are error correcting codes that admit efficient decoding algorithms any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword the tradeoff between the rate of a code and the locality efficiency of its decoding algorithms has been well studied and it has widely been suspected that nontrivial locality must come at the price of low rate a particular setting of potential interest in practice is codes of constant rate for such codes decoding algorithms with locality o k were known only for codes of rate 1 where k is the length of the message furthermore for codes of rate 1 2 no nontrivial locality had been achieved in this article we construct a new family of locally decodable codes that have very efficient local decoding algorithms and at the same time have rate approaching 1 we show that for every 0 and 0 for infinitely many k there exists a code c which encodes messages of length k with rate 1 and is locally decodable from a constant fraction of errors using o k queries and time these codes which we call multiplicity codes are based on evaluating multivariate polynomials and their derivatives multiplicity codes extend traditional multivariate polynomial codes they inherit the local decodability of these codes and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance

propagation of optical localized pulses in crystals a dimensional model and its reduction to the nls equation we derive the equations that describe the evolution of the modulation of a short localized optical pulse in a bulk medium presenting a nonzero second order nonlinear susceptibility far from the the phase matching we give the equations in the general case and for the symmetry classes of crystals and 3m a wave interaction between the wave and some dc field due to optical rectification and the electro optic effect strongly modifies the obtained system with regard to the three dimensional nls equation an exhaustive list of the cases where the derived model can be reduced to the one dimensional nls equation is given this reduction is possible for only a few special choices of the polarization and of the modulation direction in a way corresponding to the symmetry of the crystal the reduction of the same system to two coupled one dimensional nls equations is also investigated with regard to the integrable systems of this type an additional nonlinear term arises in all but a few particular cases

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

robust algorithms a different approach to oracles abstract a new notion of an oracle machine being helped by an oracle set is introduced it is required that the oracle machine is robust i e it always computes the same set independent of the oracle the main result states that the class of sets that can be computed by deterministic polynomial time algorithms being helped by some oracle set is exactly np co np some connections to probabilistic classes are also investigated

an interactive method for the eigenvalue problem for matrices abstract the algorithm described in this article uses householder reflections to obtain the largest eigenvalue of a full positive definite matrix the algorithm can be used to obtain all the eigenvalues of a symmetric matrix and may be suitable for a parallel processing machine we also consider the differential equations analogue of this method and prove the convergence of that method together with the convergence rates finally some numerical examples are given this algorithm is very suitable to determine the eigenvalues in an interactive environment unlike the ql algorithm the full matrix is operated upon at each stage of the iterative process hence one could use the apl programming language to write a very brief code to implement this program

computation of equilibriain noncooperative games this paper presents algorithms for finding equilibria of mixed strategy in multistage noncooperative games of incomplete information like probabilistic blindfold chess where at every opportunity a player can perform different moves with some probability these algorithms accept input games in extensive form our main result is an algorithm for computing aeguential equilibrium which is the most widely accepted notion of equilibrium for mixed strategies of noncooperative probabilistic games in mainstream economic game theory previously there were no known algorithms for computing sequential equilibria strategies except for the special case of single stage games the computational aspects of passage from a recursive presentation of a game to its extensive form are also discussed for nontrivial inputs the concatenation of this procedure with the equilibrium computation is time intensive but has low spatial requirements given a recursively represented game with a position space bound s n and a log space computable next move relation we can compute an example mixed strategy satisfying the sequential equilibria condition all in space bound o s n 2 furthermore in space o s n 3 we can compute the connected components of mixed strategies satisfying sequential equilibria

green s theorem and isolation in planar graphs we show a simple application of email protected s theorem from multivariable calculus to the isolation problem in planar graphs in particular we give a log space construction of a skew symmetric polynomially bounded edge weight function for directed planar graphs such that the weight of any simple cycle in the graph is non zero with respect to this weight function as a direct consequence of the above weight function we are able to isolate a directed path between two fixed vertices in a directed planar graph we also show that given a bipartite planar graph we can obtain an edge weight function using the above function in log space which isolates a perfect matching in the given graph earlier this was known to be true only for grid graphs which is a proper subclass of planar graphs we also look at the problem of obtaining a straight line embedding of a planar graph in log space although we do not quite achieve this goal we give a piecewise straight line embedding of the given planar graph in log space

local correctability of expander codes in this work we present the first local decoding algorithm for expander codes this yields a new family of constant rate codes that can recover from a constant fraction of errors in the codeword symbols and where any symbol of the codeword can be recovered with high probability by reading n e symbols from the corrupted codeword where n is the block length of the code expander codes introduced by sipser and spielman are formed from an expander graph g v e of degree d and an inner code of block length d over an alphabet each edge of the expander graph is associated with a symbol in a string in e will be a codeword if for each vertex in v the symbols on the adjacent edges form a codeword in the inner code we show that if the inner code has a smooth reconstruction algorithm in the noiseless setting then the corresponding expander code has an efficient local correction algorithm in the noisy setting instantiating our construction with inner codes based on finite geometries we obtain novel locally decodable codes with rate approaching one this provides an alternative to the multiplicity codes of kopparty saraf and yekhanin stoc 11 and the lifted codes of guo kopparty and sudan itcs 13

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

kernels for below upper bound parameterizations of the hitting set and directed dominating set problems in the hitting set problem we are given a collection f of subsets of a ground set v and an integer p and asked whether v has a p element subset that intersects each set in f we consider two parameterizations of hitting set below tight upper bounds p m k and p n k in both cases k is the parameter we prove that the first parameterization is fixed parameter tractable but has no polynomial kernel unless conp np poly the second parameterization is w 1 complete but the introduction of an additional parameter the degeneracy of the hypergraph h v f makes the problem not only fixed parameter tractable but also one with a linear kernel here the degeneracy of h v f is the minimum integer d such that for each x v the hypergraph with vertex set v x and edge set containing all edges of f without vertices in x has a vertex of degree at most d in nonblocker directed nonblocker we are given an undirected graph a directed graph g on n vertices and an integer k and asked whether g has a set x of n k vertices such that for each vertex y x there is an edge arc from a vertex in x to y nonblocker can be viewed as a special case of directed nonblocker replace an undirected graph by a symmetric digraph dehne et al proc sofsem 2006 proved that nonblocker has a linear order kernel we obtain a linear order kernel for directed nonblocker

two query pcp with subconstant error we show that the np complete language 3sat has a pcp verifier that makes two queries to a proof of almost linear size and achieves subconstant probability of error eeo 1 the verifier performs only projection tests meaning that the answer to the first query determines at most one accepting answer to the second query the number of bits representing a symbol in the proof depends only on the error e previously by the parallel repetition theorem there were pcp theorems with two query projection tests but only arbitrarily small constant error and polynomial size there were also pcp theorems with subconstant error and almost linear size but a constant number of queries that is larger than 2 as a corollary we obtain a host of new results in particular our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem a partial list includes the following 1 3sat cannot be efficiently approximated to within a factor of 7 8po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 7 8pe for any constant e 0 under polynomial reductions hastad 2 3lin cannot be efficiently approximated to within a factor of 1 2po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 1 2pe for any constant e 0 under polynomial reductions hastad 3 a pcp theorem with amortized query complexity 1 p o 1 and amortized free bit complexity o 1 previously the best known amortized query complexity and free bit complexity were 1pe and e respectively for any constant e 0 samorodnitsky and trevisan one of the new ideas that we use is a new technique for doing the composition step in the classical proof of the pcp theorem without increasing the number of queries to the proof we formalize this as a composition of new objects that we call locally decode reject codes ldrc the notion of ldrc was implicit in several previous works and we make it explicit in this work we believe that the formulation of ldrcs and their construction are of independent interest