We identify and study a natural and frequently occurring subclass of Concurrent-Read, Exclusive-Write Parallel Random Access Machines (CREW-PRAMs). Called Concurrent-Read, Owner-Write, or CROW-PRAMs, these are machines in which each global memory location is assigned a unique "owner" processor, which is the only processor allowed to write into it. Considering the difficulties that would be involved in physically realizing a full CREW-PRAM model, it is interesting to observe that in fact, most known CREW-PRAM algorithms satisfy the CROW restriction or can be easily modified to do so. This paper makes three main contributions. First, we formally define the CROW-PRAM model and demonstrate its stability

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics. Keywords: Cluster algorithms, computational complexity, diffusion limited aggregation, Ising model, Metropolis algorithm, P-completeness 1

We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems. In particular, we prove that deciding if a given integer is square-free and testing co-primality of two integers by unbounded fan-in circuits of bounded depth requires superpolynomial size. 1 Introduction In recent years spectral techniques based on the Abstract Harmonic Analysis on the hypercube have been shown to represent a very useful tool for obtaining lower complexity bounds. Various links between Fourier coefficients of Boolean functions and their complexity characteristics have been studied in a number of works, see [1, 2, 3, 4, 6, 8, 13, 19, 20, 22, 23]. In particular, these spectral techniques have been successfully applied to the parity function and to threshold functions. Institut fur Informatik, Technische Universitat Munchen, D-80290 Munchen, Germany. bernasco@informatik.tu-mue...

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This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale as O(log 2 N ), where N is the system size, and the number of processors required scale as a polynomial in N . The algorithms are based on fast parallel procedures for finding minimum weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist. Keywords: Ballistic deposition, Computationa...

Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain a linear lower bound on the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. \Lambda Supported by DFG grant Me 1077/14-1.

the date of receipt and acceptance should be inserted later Abstract We study the power of reliable anonymous distributed systems, where processes do not fail, do not have identifiers, and run identical programmes. We are interested specifically in the relative powers of systems with different communication mechanisms: anonymous broadcast, read-write registers, or read-write registers plus additional shared-memory objects. We show that a system with anonymous broadcast can simulate a system of shared-memory objects if and only if the objects satisfy a property we call idemdicence; this result holds regardless of whether either system is synchronous or asynchronous. Conversely, the key to simulating anonymous broadcast in anonymous shared memory is the ability to count: broadcast can be simulated by an asynchronous shared-memory system that uses only counters, but readwrite registers by themselves are not enough. We further examine the relative power of different types and sizes of bounded counters and conclude with a non-robustness result.

Abstract. We say a polynomial P over ZM strongly M-represents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P one-sidedly M-represents F if F(x) = 0 ⇐ ⇒ P(x) ≡ 0 (mod M) for all x ∈ {0, 1} n. Lower bounds are obtained on the degree and the number of monomials of polynomials over Z M, which strongly or one-sidedly M-represent the Boolean function deciding if a given nbit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function. 1

Abstract We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymtotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximative polynomial representations of this function. *Supported by DFG grant Me 1077/14-1.#

A parallel algorithm for diffusion-limited aggregation (DLA) is described and analyzed from the perspective of computational complexity. The dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random-access machine (PRAM) model of parallel computation according to T ∼ Lz, where L is the cluster size, T is the running time, and the algorithm uses a number of processors polynomial in L. It is argued that z = D −D2/2, where D is the fractal dimension and D2 is the second generalized dimension. Simulations of DLA are carried out to measure D2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm. It is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of DLA. 1