Showing papers on "Communication complexity published in 1995"

Private information retrieval

Abstract: We describe schemes that enable a user to access k replicated copies of a database (k/spl ges/2) and privately retrieve information stored in the database. This means that each individual database gets no information on the identity of the item retrieved by the user. For a single database, achieving this type of privacy requires communicating the whole database, or n bits (where n is the number of bits in the database). Our schemes use the replication to gain substantial saving. In particular, we have: A two database scheme with communication complexity of O(n/sup 1/3/). A scheme for a constant number, k, of databases with communication complexity O(n/sup 1/k/). A scheme for 1/3 log/sub 2/ n databases with polylogarithmic (in n) communication complexity.

On data structures and asymmetric communication complexity

Abstract: In this paper we consider two-party communication complexity, the ``asymmetric case'', when the input sizes of the two players differ significantly. Most of previous work on communication complexity only considers the total number of bits sent, but we study trade-offs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs. ] 1998 Academic Press

Discrete Neural Computation: A Theoretical Foundation

Abstract: 1. Introduction. 2. Linear Threshold Element. 3. Computing Symmetric Functions. 4. Depth Efficient Arithmetic Circuits. 5. Depth-Size Tradeoffs. 6. Computing with Small Weights. 7. Rational Approximation and Optimal Size Circuits. 8. Geometric Framework and Spectral Analysis. 9. Limitations of AND-OR Circuits. 10. Lower Bounds via Communication Complexity. 11. Hopfield Network. Glossary. Bibliography. Index.

Amortized Communication Complexity

Abstract: In this work we study the direct-sum problem with respect to communication complexity: Consider a relation $f$ defined over $\{0,1\}^{n} \times \{0,1\}^{n}$. Can the communication complexity of simultaneously computing $f$ on $\cal l$ instances $(x_1,y_1),\ldots,(x_{\cal l},y_{\cal l})$ be smaller than the communication complexity of computing $f$ on the $\cal l$ instances, separately? Let the amortized communication complexity of $f$ be the communication complexity of simultaneously computing $f$ on $\cal l$ instances, divided by $\cal l$. We study the properties of the amortized communication complexity. We show that the amortized communication complexity of a relation can be smaller than its communication complexity. More precisely, we present a partial function whose (deterministic) communication complexity is $\Theta(\log n)$ and its amortized (deterministic) communication complexity is $O(1)$. Similarly, for randomized protocols, we present a function whose randomized communication complexity is $\Theta(\log n)$ and its amortized randomized communication complexity is $O(1)$. We also give a general lower bound on the amortized communication complexity of any function $f$ in terms of its communication complexity $C(f)$: for every function $f$ the amortized communication complexity of $f$ is $\Omega \left (\sqrt{C(f)} - \log n \right)$.

On rank vs. communication complexity

Abstract: This paper concerns the open problem of Lovasz and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.

Super-logarithmic depth lower bounds via the direct sum in communication complexity

Abstract: Is it easier to solve two communication problems together than separately? This question is related to the complexity of the com- position of boolean functions. Based on this relationship, an approach to separating NC 1 from P is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotone NC 1 from monotone P.

Fourier analysis for probabilistic communication complexity

Abstract: We use Fourier analysis to get general lower bounds for the probabilistic communication complexity of large classes of functions. We give some examples showing how to use our method in some known cases and for some new functions. Our main tool is an inequality by Kahn, Kalai, and Linial, derived from two lemmas of Beckner.

Fractional Covers and Communication Complexity

Abstract: It is possible to view communication complexity as the minimum solution of an integer programming problem. This integer programming problem is relaxed to a linear programming problem and from it information regarding the original communication complexity question is deduced. A particularly appealing avenue this opens is the possibility of proving lower bounds on the communication complexity (which is a minimization problem) by exhibiting upper bounds on the maximization problem defined by the dual of the linear program. This approach works very neatly in the case of nondeterministic communication complexity. In this case a special case of Lovasz's fractional cover measure is obtained. Through it the amortized nondeterministic communication complexity is completely characterized. The power of the approach is also illustrated by proving lower and upper bounds on the nondeterministic communication complexity of various functions. In the case of deterministic complexity the situation is more complicated. Two attempts are discussed and some results using each of them are obtained. The main result regarding the first attempt is negative: one cannot use this method for proving superpolynomial lower bounds for formula size. The main result regarding the second attempt is a "direct-sum" theorem for two-round communication complexity.

Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity

Abstract: The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider weaker forms of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. We then give two applications of such weaker forms. First, we show that our lower bound on a variant of rigidity implies lower bounds on size-depth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a recent result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain weaker forms of rigidity to conclude several separation results in communication complexity theory. Our results complement and strengthen a result of Razborov.

On the "log rank"-conjecture in communication complexity

Abstract: We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove: Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.

Many-to-many personalized communication with bounded traffic

Abstract: This paper presents solutions for the problem of many-to-many personalized communication, with bounded incoming and outgoing traffic, on a distributed memory parallel machine. We present a two-stage algorithm that decomposes the many-to-many communication with possibly high variance in message size into two communications with low message size variance. The algorithm is deterministic and takes time 2t/spl mu/(+lower order terms) when t/spl ges/0(p/sup 2/+p/spl tau///spl mu/) Here t is the maximum outgoing or incoming traffic at any processor, /spl tau/ is the startup overhead and /spl mu/ is the inverse of the data transfer rate. Optimality is achieved when the traffic is large, a condition that is usually satisfied in practice on coarse-grained architectures. The algorithm was implemented on the Connection Machine CM-5. The implementation used the low latency communication primitives (active messages) available on the CM-5, but the algorithm as such is architecture-independent. An alternate single-stage algorithm using distributed random scheduling for the CM-5 was implemented and the performance of the two algorithms were compared. >

Improved Efficient Arguments (Preliminary Version)

Abstract: We consider complexity of perfect zero-knowledge arguments [4]. Let T denote the time needed to (deterministically) check a proof and let L denote an appropriate security parameter. We introduce new techniques for implementing very efficient zero-knowledge arguments. The resulting argument has the following features: ? The arguer can, if provided with the proof that can be deterministically checked in O(T) time, run in time O(TLO(1)). The best previous bound was O(T1+?LO(1)). ? The protocol can be simulated in time O(LO(1)). The best previous bound was O(T1+?LO(1)). ? A communication complexity of O(LlgL), where L is the security parameter against the prover. The best previous known bound was O(LlgT).This can be based on fairly general algebraic assumptions, such as the hardness of discrete logarithms.Aside from the quantitative improvements, our results become qualitatively different when considering arguers that can run for some super-polynomial but bounded amount of time. In this scenario, we give the first arguments zero-knowledge arguments and the first "constructive" arguments in which the complexity of arguing a proof is tightly bounded by the complexity of verifying the proof.We obtain our results by a hybrid construction that combines the best features of different PCPs. This allows us to obtain better bounds than the previous technique, which only used a single PCP. In our proof of soundness we exploit the error correction capabilities as well as the soundness of the known PCPs.

New systolic array implementation of the 2-D discrete cosine transform and its inverse

Abstract: A new systolic array without matrix transposition hardware is proposed to compute the two-dimensional discrete cosine transform (2-D DCT) based on the row-column decomposition. This architecture uses N/sup 2/ multipliers to evaluate N/spl times/N-point DCTs at a rate of one complete transform per N clock cycles, where N is even. It possesses the features of regularity and modularity, and is thus well suited to VLSI implementation. As compared to existing pipelined regular architectures for the 2-D DCT, the proposed one has better throughput performance, smaller area-time complexity, and lower communication complexity. The new idea for the 2-D DCT is also extended to derive a similar systolic array for the 2-D inverse discrete cosine transform (IDCT). Simulation results demonstrate that the proposed 2-D DCT and IDCT architectures have good fixed-point error performance for both real image and random data. As a consequence, they are useful for applications where very high throughput rates are required. >

A fault-tolerant algorithm for replicated data management

Abstract: We examine the tradeoff between message overhead and data availability that arises in the design of fault-tolerant algorithms for replicated data management in distributed systems. We propose a property called asymptotically high resiliency which is useful for evaluating the fault-tolerance of replica control algorithms and distributed mutual exclusion algorithms. We present a new algorithm for replica control that can be tailored (through a design parameter) to achieve the desired balance between low message overhead and high data availability. Further, we show that for a message overhead of O(/spl radic/(Nlog N)), our algorithm can achieve asymptotically high resiliency.

New models and algorithms for future networks

Abstract: In future networks, transmission and switching capacity will dominate processing capacity. The authors investigate the way in which distributed algorithms should be changed in order to operate efficiently in this new environment. They introduce a class of new models for distributed algorithms which make explicit the difference between switching and processing. Based on these new models they define new message and time complexity measures which, they believe, capture the costs in many high-speed networks more accurately then traditional measures. In order to explore the consequences of the new models, they examine three problems in distributed computation. For the problem of maintaining network topology they devise a broadcast algorithm which takes O(n) messages and O(log n) time for a single broadcast in the new measure. For the problem of leader election they present a simple algorithm that uses O(n) messages and O(n) time. The third problem, distributed computation of a "globally sensitive" function, demonstrates some important features and tradeoffs in the new models and emphasizes and differences with the traditional network model. The results of the present paper influenced later research, as well as the design of IBM Networking Broadband Services (NBBS). >

On randomized one-round communication complexity

Abstract: We present several results regarding randomized one-round communication complexity. Our results include a connection to the VC-dimension, a study of the problem of computing the inner product of two real valued vectors, and a relation between “simultaneous” protocols and one-round protocols.

An approximation algorithm for scheduling dependent tasks on m processors with small communication delays

Abstract: This paper defines and studies an approximation algorithm for scheduling tasks with small communication delays on parallel processors. In a first step, a schedule for the relaxed problem instance with an unlimited number of processors is generated. Then this solution is used to solve the resource conflicts during the scheduling phase on m processors, with a rather unusual feature: a feasible task may be tactically delayed, even inducing idleness on a processor in order to wait for a more important task. The relative worst case performance of this algorithm is analysed with respect to the ratio between communication times and processing times and to the performance of the relaxed solution. It improves significantly the best known performance ratio of an algorithm for this problem (7/3 against 3).

Toward high communication performance through compiled communications on a circuit switched interconnection network

Abstract: This paper discusses a new principle of interconnection network for massively parallel architectures in the field of numerical computation. The principle is motivated by an analysis of the application features and the need to design new kind of communication networks combining very high bandwidth, very low latency, performance independence to communication pattern or network load and a performance improvement proportional to the hardware performance improvement. Our approach is to associate compiled communications and a circuit switched interconnection network. This paper presents the motivations for this principle, the hardware and software issues and the design of a first prototype. The expected performance are a sustained aggregate bandwidth of more than 500 GBytes/s and an overall latency less than 270 ns, for a large implementation (4K inputs) with the current available technology. >

On the average communication complexity of asynchronous distributed algorithms

Abstract: We study the communication complexity of asynchronous distributed algorithms. Such algorithms can generate excessively many messages in the worst case. Nevertheless, we show that, under certain probabilistic assumptions, the expected number of messages generated per time unit is bounded by a polynomial function of the number of processors under a very general model of distributed computation. Furthermore, for constant-degree processor graphs, the expected number of generated messages is only O(nT), where n is the number of processors and T is the running time. We conclude that (under our model) any asynchronous algorithm with good time complexity will also have good communication complexity, on the average.

Aligning parallel arrays to reduce communication

Abstract: Axis and stride alignment is an important optimization in compiling data-parallel programs for distributed-memory machines. We previously developed an optimal algorithm for aligning array expressions. Here, we examine alignment for more general program graphs. We show that optimal alignment is NP-complete in this setting, so we study heuristic methods. This paper makes two contributions. First, we show how local graph transformations can reduce the size of the problem significantly without changing the best solution. This allows more complex and effective heuristics to be used. Second, we give a heuristic that can explore the space of possible solutions in a number of ways. We show that some of these strategies can give better solutions than a simple greedy approach proposed earlier. Our algorithms have been implemented; we present experimental results showing their effect on the performance of some example programs running on the CM-5. >

On the Power of Circuits with Gates of Low L 1 Norms

Abstract: We examine the power of Boolean functions with low L1 norms in several settings. In a large part of the recent literature, the degree of a polynomial which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function. However, some functions with low communicational complexity (AND, OR, PARITY, ID) have high degree, but small L1 norms. So, in conjunction with communication complexity, instead of the degree, the L1 norm can be an important measure of hardness. We conjecture that the randomized communication complexity of any Boolean function is bounded by the polylogarithm of its L1 norm. We can prove only a weaker statement: we present a two-party, randomized, common-coin communication protocol for computing functions with O(L21δ) bits of communication, with error-probability of exp(−c°) (even with large degree or exponential number of terms). Then we present several applications of this theorem for circuit lower bounds (both for bounded- and unbounded depth), and a decision-tree lower bound.

Subquadratic zero-knowledge

Abstract: We improve on the communication complexity of zero-knowledge proof systems. Let C be a Boolean circuit of size n. Previous zero-knowledge proof systems for the satisfiability of C require the use of Ω(kn) bit commitments in order to achieve a probability of undetected cheating below 2 -k . In the case k = n, the communication complexity of these protocols is therefore Ω(n 2 ) bit commitments. In this paper, we present a zero-knowledge proof system for achieving the same goal with only O(n 1+en + k√n 1+en ) bit commitments, where e n goes to zero as n goes to infinity. In the case k = n, this is O(n√n 1+en ). Moreover, only O(k) commitments need ever be opened, which is interesting if it is substantially less expensive to commit to a bit than to open a commitment.

Communication Complexity of Sum-Type Functions Invariant under Translation

Abstract: The communication complexity of a function f denotes the number of bits that two processors have to exchange in order to compute f(x, y), when each processor knows one of the variables x and y, respectively. In this paper the deterministic communication complexity of sum-type functions, such as the Hamming distance and the Lee distance, is examined. Here f: X × X ? G, where X is a finite set and G is an Abelian group, and the sum-type function fn:Xn × Xn ? G is defined by fn((x1, ..., xn), (y1, ..., yn)) = ?ni=1f(xi, yi) Since the functions examined are also translation-invariant, their function matrices are simultaneously diagonalizable and the corresponding eigenvalues can be calculated. This allows to apply a rank lower bound for the communication complexity. The best results are obtained for G = Z/2Z. For prime numbers |X| in this case the communication complexity of all non-trivial sum-type functions is determined exactly. Exact results are also obtained for the parity of the Hamming distance and the parity of the Lee distance. For the Hamming distance and the Lee distance exact results are only obtained for special parameters n and |X|.

On the message and time complexity of protocols for reliable broadcasts/multicasts in networks with omission failures

Abstract: This paper presents a fundamental study on the message and time complexity of reliable broadcast/multicast protocols in point-to-point networks subject to omission failures. We assume a weakly synchronous model in which there is a known upper bound on the delay in delivering a message from one process to another. A faulty process may omit sending or receiving messages; this characterizes a common faulty behavior in networking applications. We focus our study on the number of messages and the maximum amount of time required of any fault-tolerant protocol in failure-free executions. We present protocols that, in an n-process network subject to at most t faulty processes, guarantee the delivery of a message from any process to all other nonfaulty processes. In particular, when no failure occurs in the network, our protocols require (n+t-1) messages and at most (n-1+upper bound [log/sub 2/(t+1)])/spl middot//spl delta/ units of time, where /spl delta/ is the maximum time required to deliver a message from a process to another. Moreover, we show that our protocols are optimal with respect to message and time complexity. The new insights provided by the lower bound proofs yield a graph-theoretic characterization of all message and time optimal reliable broadcast/multicast protocols in failure-free executions. We further discuss the implications of our results on the support of multicast services in high-speed switching networks such as ATM. >

Digit pipelined arithmetic on fine-grain array processors

Abstract: In this paper, we present a novel scheme for performing fixed-point arithmetic efficiently on fine-grain, massively parallel, programmable architectures including both custom and FPGA-based systems. We achieve anO(n) speedup, wheren is the operand precision, over the bit-serial methods of existing fine-grain systems such as the DAP, the MPP and the CM2, within the constraints of regular, near neighbor communication and only a small amount of on-chip memory. This is possible by means of digit pipelined algorithms which avoid broadcast and which operate in a fully systolic manner by pipelining at the digit level. A base 4, signed-digit, fully redundant number system and on-line techniques are used to limit carry propagation and minimize communication costs. p ]Although our algorithms are digit-serial, we are able to match the performance of the bit-parallel methods, while retaining low communication complexity. Reconfigurable hardware systems built using field programmable gate arrays (FPGA's) can share in the speed benefits of these algorithms. By using the organization of logic blocks suggested in this paper, problems of placement and routing that exist in such systems can be avoided. Since the algorithms are amenable to pipelining, very high throughput can be obtained.

A study of the communication cost of the FFT on torus multicomputers

Abstract: The computation of a one-dimensional FFT on a c-dimensional torus multicomputer is analyzed. Different approaches are proposed which differ in the way they use the interconnection network. The first approach is based on the multidimensional index mapping technique for the FFT computation. The second approach starts from a hypercube algorithm and then embeds the hypercube onto the torus. The third approach reduces the communication cost of the hypercube algorithm by pipelining the communication operations. A novel methodology to pipeline the communication operations on a torus is proposed. Analytical models are presented to compare the different approaches. This comparison study shows that the best approach depends on the number of dimensions of the torus and the communication start-up and transfer times. The analytical models allow us to select the most efficient approach for the available machine. >

Communication complexity of matrix computation over finite fields

Abstract: We investigate the communication complexity of singularity testing in a finite field, where the problem is to determine whether a given square matrixM is singular. We show that, forn×n matrices whose entries are elements of a finite field of sizep, the communication complexity of this problem is Θ(n2 logp). Our results imply tight bounds for several other problems likedetermining the rank andcomputing the determinant.

Automatically increasing fault tolerance in distributed systems

Abstract: Developing fault-tolerant distributed protocols is a difficult task. The difficulty of this task increases with the severity of the failures to be tolerated. One way to deal with this difficulty is to develop protocols tolerant of benign failures and then transform these protocols into ones that are tolerant of more severe failures. This transformation mechanism is called a translation. This dissertation considers a variety of processor failures and synchrony models. The failures studied range from simple stopping failures to arbitrary faulty behavior. The synchrony models range from systems in which processors are fully synchronized (synchronous systems) to systems in which processors are not synchronized at all (asynchronous systems). For all synchrony models, the dissertation gives general definitions of translations and of measures to evaluate their performance. The two measures considered are communication complexity and fault-tolerance. Communication complexity is the communication overhead incurred when using a translation. Fault-tolerance is the maximum proportion of processors that can be faulty without affecting the correctness of the translations. For synchronous systems, this dissertation presents a complete study of the relationship between fault-tolerance and round complexity of translations. It develops new translations that are optimal and proves that some previously developed translations are optimal. For asynchronous systems, it proves that some previously developed translations are optimal. For systems that are only partially synchronous this dissertation discusses some of the issues involved in designing efficient translations.