We present a new class of asymptotically good, linear error-correcting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both randomized and explicit constructions for some of these codes. Experimental results demonstrate the extremely good performance of the randomly chosen codes. 1. Introduction We present a new class of error correcting codes derived from expander graphs. These codes have the advantage that they can be decoded very efficiently. That makes them particularly suitable for devices which must decode cheaply, such as compact disk players and remote satellite receivers. We hope that the connection we draw between expander graphs and error correcting codes will stimulate research in both fields. 1.1. Error correcting codes An error correcting code is a mapping from messages to codewords such that the mappi...

. Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every nondeterministic computational task S(x; y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S 0 such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in polylogarithmic Monte Carlo time; and (iii) a witness satisfying S 0 can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in error-correcting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N O(log log N)...

We present a new class of asymptotically good, linear error-correcting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmic-depth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized and explicit constructions of these codes.

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For more than 35 years, the fastest known method for integer multiplication has been the Schönhage-Strassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log ∗ n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.

A "Pointer Machine" is many things. Authors who consider referring to this term are invited to read the following note first. 1 Introduction In a 1992 paper by Galil and the author we referred to a "pointer machine " model of computation. A subsequent survey of related literature has produced over twenty references to papers having to do with "pointer machines", naturally containing a large number of cross-references. These papers address a range of subjects that range from the model considered in the above paper to some other ones which are barely comparable. The fact that such different notions have been discussed under the heading of "pointer machines" has produced the regrettable effect that cross references are sometimes found to be misleading. Clearly, it is easy for a reader who does not follow a paper carefully to misinterpret its claims when a term that is so ill-defined is used. This note is an attempt to rectify the situation. We start with a survey of the different notions...

: The aspect of purity versus impurity that we address involves the absence versus presence of mutation: the use of primitives (RPLACA and RPLACD in Lisp, set-car! and set-cdr! in Scheme) that change the state of pairs without creating new pairs. It is well known that cyclic list structures can be created by impure programs, but not by pure ones. In this sense, impure Lisp is "more powerful" than pure Lisp. If the inputs and outputs of programs are restricted to be sequences of atomic symbols, however, this difference in computability disappears. We shall show that if the temporal sequence of input and output operations must be maintained (that is, if computations must be "online "), then a difference in complexity remains: for a pure program to do what an impure program does in n steps, O(n log n) steps are sufficient, and in some cases\Omega\Gamma n log n) steps are necessary. * This research was partially supported by an NSERC Operating Grant. 1. Introduction The programming la...

y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.

Modern imperative object-oriented design methods and languages take a rigorous approach to compatibility and reusability mainly from an interface and specification point of view --- if at all. Beside functional specification, however, users select classes from libraries based on performance characteristics, too. This article develops an appropriate fundamental approach towards performance estimation, measurement and metering in OO approaches. We use examples written in the Sather language to demonstrate the concepts of so-called OO-machines, which lend themselves to performance metrics, and a calculus for reasoning about performance. A language binding of these concepts is then sketched in the form of cost annotations that allow programmers to file classes in libraries well-documented with cost related specifications. These annotations can optionally be used for instrumenting code that meters cost and checks whether the taken measurements are consistent with the given specification. In...

We present a new data structure of size O(M) for solving the vEB problem. When this data structure is used in combination with a new memory topology it provides an O(1) worst case time solution. 1