Showing papers on "Communication complexity published in 1989"

Privacy and communication complexity

Abstract: Each of two parties P/sub 1/ and P/sub 2/ holds an n-bit input, x and y, respectively They wish to compute privately the value of f(x,y) Two questions are considered: (1) Which functions can be privately computed? (2) What is the communication complexity of protocols that privately compute a function f (in the case in which such protocols exist)? A complete combinatorial characterization of privately computable functions is given This characterization is used to derive tight bounds on the rounds complexity of any privately computable function and to design optimal private protocols that compute these functions It is shown that for every 1 >

Communication Complexity: A New Approach to Circuit Depth

Abstract: "Communication Complexity "describes a new intuitive model for studying circuit networks that captures the essence of circuit depth. Although the complexity of boolean functions has been studied for almost 4 decades, the main problems the inability to show a separation of any two classes, or to obtain nontrivial lower bounds remain unsolved. The communication complexity approach provides clues as to where to took for the heart of complexity and also sheds light on how to get around the difficulty of proving lower bounds.Karchmer's approach looks at a computation device as one that separates the words of a language from the non-words. It views computation in a top down fashion, making explicit the idea that flow of information is a crucial term for understanding computation. Within this new setting, "Communication Complexity "gives simpler proofs to old results and demonstrates the usefulness of the approach by presenting a depth lower bound for "st"-connectivity. Karchmer concludes by proposing open problems which point toward proving a general depth lower bound.Mauricio Karchmer received his doctorate from Hebrew University and is currently a Postdoctoral Fellow at the University of Toronto. Communication Complexity received the 1988 ACM Doctoral Dissertation Award.

On the communication complexity of generalized 2-D convolution on array processors

Abstract: Several parallel convolution algorithms for array processors with N/sup 2/ processing elements (PEs) connected by mesh, hypercube, and shuffle-exchange topologies, respectively, are presented. The computation time complexity is the same for array processors with different interconnection networks. The communication time complexity, however, varies from network to network, and is the main focus. It is shown that by using inter-PE communication networks efficiently, each PE requires only a small local memory, many unnecessary data transmissions are eliminated, and the overall time complexity (including computation and communication) of algorithms is reduced to O(M/sup 2/). >

Multiparty communication complexity

Abstract: A given Boolean function has its input distributed among many parties. The aim is to determine which parties to talk to and what information to exchange with each of them in order to evaluate the function while minimizing the total communication. It is shown that it is possible to obtain the Boolean answer deterministically with only a polynomial increase in communication with respect to the information lower bound given by the nondeterministic communication complexity of the function. >

The bit complexity of randomized leader election on a ring

Abstract: The inherent bit complexity of leader election on asynchronous unidirectional rings of processors is examined under various assumptions about global knowledge of the ring. If processors have unique identities with a maximum of $m$ bits, then the expected number of communication bits sufficient to elect a leader with probability 1, on a ring of (unknown) size $n$ is $O(nm)$. If the ring size is known to within a multiple of 2, then the expected number of communication bits sufficient to elect a leader with probability 1 is $O(n \log n)$. These upper bounds are complemented by lower bounds on the communication complexity of a related problem called solitude verification that reduces to leader election in $O(n)$ bits. If processors have unique identities chosen from a sufficiently large universe of size $s$, then the average, over all choices of identities, of the communication complexity of verifying solitude is $\Omega (n \log s)$ bits. When the ring size is known only approximately, then $\Omega (n \log n)$ bits are required for solitude verification. The lower bounds address the complexity of certifying solitude. This is modelled by tbe best case behaviour of non-deterministic solitude verification algorithms.

Analysis and benchmarking of two parallel sorting algorithms: hyperquicksort and quickmerge

Abstract: We analyze the computational and communication complexity of four sorting algorithms as implemented on a hypercube multicomputer: two variants of hyperquicksort and two variants of quickmerge. Based upon this analysis, machine-specific parameters can be used to determine when each algorithm requires less communication time than the others. We present benchmark results of the four algorithms on a 64-processor NCube/7. The benchmarking provides experimental evidence that hyperquicksort divides the values to be sorted more evenly among the processors than quickmerge. Because it does a better job balancing work between processors, hyperquicksort proves to be uniformly superior to quickmerge.

Comparison of two-dimensional FFT methods on the hypercube

Abstract: Complex two-dimensional FFTs up to size 256 x 256 points are implemented on the Intel iPSC/System 286 hypercube with emphasis on comparing the effects of data mapping, data transposition or communication needs, and the use of distributed FFTs. Two new implementations of the 2D-FFT include the Local-Distributed method which performs local FFTs in one direction followed by distributed FFTs in the other direction, and a Vector-Radix implementation that is derived from decimating the DFT in two-dimensions instead of one. In addition, the Transpose-Split method involving local FFTs in both directions with an intervening matrix transposition and the Block 2D-FFT involving distributed FFT butterflies in both directions are implemented and compared with the other two methods. Timing results show that on the Intel iPSC/System 286, there is hardly any difference between the methods, with the only differences arising from the efficiency or inefficiency of communication. Since the Intel cannot overlap communication and computation, this forces the user to buffer data. In some of the methods, this causes processor blocking during communication. Issues of vectorization, communication strategies, data storage and buffering requirements are investigated. A model is given that compares vectorization and communication complexity. While timing results show that the Transpose-Split method is in general slightly faster, our model shows that the Block method and Vector-Radix method have the potential to be faster if the communication difficulties were taken care of. Therefore if communication could be “hidden” within computation, the latter two methods can become useful with the Block method vectorizing the best and the Vector-Radix method having 25% fewer multiplications than row-column 2D-FFT methods. Finally the Local-Distributed method is a good hybrid method requiring no transposing and can be useful in certain circumstances. This paper provides some general guidelines in evaluating parallel distributed 2D-FFT implementations and concludes that while different methods may be best suited for different systems, better implementation techniques as well as faster algorithms still perform better when communication become more efficient.

Reduction techniques for selection in distributed files

Abstract: The problem of selecting the Kth smallest element of a set of N elements distributed among d sites of a communication network is examined. A reduction technique is a distributed algorithm that transforms this problem to an equivalent one where either K or N (or both) are reduced. A collection of distributed reduction techniques is presented; the combined use of the algorithms offers new solutions for the selection problem in shout-echo networks and in a class of point-to-point networks. The communication complexity of these solutions is analyzed and shown to represent an improvement on the multiplicative constant of existing bounds for those networks. >

Results on communication complexity classes

Abstract: The authors consider deterministic, probabilistic, nondeterministic, and alternating complexity classes defined by polylogarithmic communication. They give a simple technique allowing translation of most known separation and containment results for complexity classes of the fixed-partition model to the more difficult optimal partition model, for which few results were previously known. They demonstrate that a certain natural language (block equality) in Sigma /sub 2//sup cc/ is also, unexpectedly, in Pi /sub 2//sup cc/. >

Multi-Level Logic Synthesis Using Communication Complexity

Abstract: We present a new multi-level logic synthesis technique based on minimizing communication complexity. Intuitively, we believe this approach is viable because for many types of circuits lower bounds on the area needed to implement those circuits have been obtained considering only communication complexity. It performs especially well for functions which are hierarchically decomposable (e.g., adders, parity generators, comparators, etc.). Unlike many other multi-level logic synthesis techniques, a lower bound can be computed to determine how well the synthesis was performed. We also present a new multi-level logic synthesis program based on the techniques described for reducing communication complexity.

The communication complexity of several problems in matrix computation

Abstract: The communication complexity of a function f measures the communication resources required for computing f. In the design of VLSI systems, where savings on the chip area and computation time are desired, this complexity dictates an area × time2 lower bound. We investigate the communication complexity of singularity testing, where the problem is to determine whether a given square matrix M is singular. We show that, for n × n matrices of k-bit integers, the communication complexity of Singularity Testing is Θ(k n2). Our results imply tight bounds for a wide variety of other problems in numerical linear algebra. Among those problems are determining the rank and computing the determinant, as well as the computation of several matrix decompositions. Another important corollary concerns the solvability of systems of linear equations. This problem is to decide whether a linear system A x = b has a solution. When A is an n × n matrix of k-bit integers and b a vector of n k-bit integers, its communication complexity is Θ(k n2).

Tradeoffs between communication and space

Abstract: This paper initiates the study of communication complexity when the processors have limited work space. The following tradeoffs between number C of communications steps and space S are proved: For multiplying two n × n matrices in the arithmetic model with two-way communication, CS = T(n2).For convolution of two degree n polynomials in the arithmetic model with two-way communication, CS = T(n2).For multiplying an n × n matrix by an n-vector in the Boolean model with one-way communication, CS = T(n2).In contrast, the discrete Fourier transform and sorting can be accomplished in O(n) communication steps and O(log n) space simultaneously, and the search problems of Karchmer and Wigderson associated with any language in NCk can be solved in O(logkn) communication steps and O(logkn) space simultaneously.

Distributed orthogonal factorization

Abstract: We describe several algorithms for computing the orthogonal factorization on distributed memory multiprocessors. One of the algorithms is based on Givens rotations, two others employ column Householder transformations but with different communication schemes: broadcast and pipelined ring. A fourth algorithm is a hybrid; it uses Househlolder transformations and Givens rotations in separate phases. We present expressions for the arithmetic and communication complexity of each algorithm. The algorithms were implemented on an iPSC-286 and the observed times agree well with our analyses.

On the Communication Complexity of Planarity

Abstract: We prove θ(n log n) bound for the deterministic communication complexity of the graph property planarity.

Communication complexity in distributed algebraic computation

Abstract: Consideration is given to a situation in which two processors, P/sub 1/ and P/sub 2/, are to evaluate a collection of functions f/sub 1/, . . . f/sub s/ of two vector variables x, y under the assumption that processor P/sub 1/ (respectively, P/sub 2/) has access only to the value of the variable x (respectively, y) and the functional form of f/sub 1/,. . ., f/sub s/. Bounds on the communication complexity (the amount of information that has to be exchanged between the processors) are provided. An almost optimal bound is derived for the case of one-way communication when the functions are polynomials. Lower bounds for the case of two-way communication that improve on earlier bounds are also derived. As an application, the case in which x and y are n*n matrices and f(x,y) is a particular entry of the inverse of x+y is considered. Under a certain restriction on the class of allowed communication protocols, an Omega (n/sup 2/) lower bound is obtained. The results are based on certain tools from classical algebraic geometry and field extension theory. >