A characterization of average case communication complexity

It is well known that the average case deterministic communication complexity is bounded below by an entropic quantity, which one would now call deterministic information complexity. In this paper we show a corresponding upper bound. We also improve known lower bounds for the public coin Las Vegas communication complexity by a constant factor.     

Partition arguments in multiparty communication complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x₁,…,x_k∈{0,1}ⁿ communicate to compute the value f(x₁,…,x_k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem.In this paper, we study the power of partition arguments. Our two main results are very different in nature:

(i)

For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Ω(n), while partition arguments can only yield an Ω(logn) lower bound. The same holds for nondeterministiccommunication complexity.

(ii)

For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the “log-rank conjecture” in communication complexity. We also observe that, in the case of computing relations (search problems), very large gaps do exist.

We conclude with two results on the multiparty “fooling set technique”, another method for obtaining communication complexity lower bounds.     

The BNS-Chung criterion for multi-party communication complexity

The "Number on the Forehead" model of multi-party communication complexity was first suggested by Chandra, Furst and Lipton. The best known lower bound, for an explicit function (in this model), is a lower bound of , where n is the size of the input of each player, and k is the number of players (first proved by Babai, Nisan and Szegedy). This lower bound has many applications in complexity theory. Proving a better lower bound, for an explicit function, is a major open problem. Based on the result of BNS, Chung gave a sufficient criterion for a function to have large multi-party communication complexity (up to ). In this paper, we use some of the ideas of BNS and Chung, together with some new ideas, resulting in a new (easier and more modular) proof for the results of BNS and Chung. This gives a simpler way to prove lower bounds for the multi-party communication complexity of a function. Received: December 12, 1997.     

Communication complexity towards lower bounds on circuit depth

Karchmer, Raz, and Wigderson (1995) discuss the circuit depth complexity of n-bit Boolean functions constructed by composing up to d = log n/log log n levels of k = log n-bit Boolean functions. Any such function is in AC¹ . They conjecture that circuit depth is additive under composition, which would imply that any (bounded fan-in) circuit for this problem requires depth. This would separate AC¹ from NC¹. They recommend using the communication game characterization of circuit depth. In order to develop techniques for using communication complexity to prove circuit depth lower bounds, they suggest an intermediate communication complexity problem which they call the Universal Composition Relation. We give an almost optimal lower bound of dk—O(d ²(k log k)^1/2) for this problem. In addition, we present a proof, directly in terms of communication complexity, that there is a function on k bits requiring circuit depth. Although this fact can be easily established using a counting argument, we hope that the ideas in our proof will be incorporated more easily into subsequent arguments which use communication complexity to prove circuit depth bounds. Received: July 30, 1999.     

Average-case analysis of algorithms using Kolmogorov complexity

Analyzing the average-case complexity of algorithms is a very practical but very difficult problem in computer science.In the past few years,we have demonstrated that Kolmogorov complexity is an improtant tool for analyzing the average-case complexity of algorithms.We have developed the incompressibility method.In this paper,sereral simple examples are used to further demonstrate the power and simplicity of such method.We prove bounds on the average-case number of stacks(queues)required for sorting sequential or parallel Queuesort or Stacksort.     

An information statistics approach to data stream and communication complexity

We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving conditional information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.Our paradigm leads to two main results:(1) An improved lower bound for the multi-party set-disjointness problem in the general communication complexity model, and a nearly optimal lower bound in the one-way communication model. As a consequence, we show that for any real k>2, approximating the kth frequency moment in the data stream model requires essentially Ω(n1−2/k) space; this resolves a conjecture of Alon et al. (J. Comput. System Sci. 58(1) (1999) 137).(2) A lower bound for the L_p approximation problem in the general communication model; this solves an open problem of Saks and Sun (in: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), 2002, pp. 360-369). As a consequence, we show that for p>2, approximating the L_p norm to within a factor of n^ε in the data stream model with constant number of passes requires Ω(n1−4ε−2/p) space.     

Fourier analysis for probabilistic communication complexity

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.     

Lower bounds for decision problems in imaginary,norm-Euclidean quadratic integer rings

We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound.     

Kolmogorov Complexity for Possibly Infinite Computations

In this paper we study the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortest input that produce a desired output via a possibly non-halting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if the machine is allowed to overwrite its output, this complexity coincides with the classical Kolmogorov complexity for halting computations relative to the first jump of the halting problem. However, on machines that cannot erase their output –called monotone machines–, we prove that our complexity for non effective computations and the classical Kolmogorov complexity separate as much as we want. We also consider the prefix-free complexity for possibly infinite computations. We study several properties of the graph of these complexity functions and specially their oscillations with respect to the complexities for effective computations.     

A note on monotone complexity and the rank of matrices

We shall give simpler proofs of some lower bounds on monotone computations. We describe a simple condition on combinatorial structures, such that the rank of the matrix associated with these structures gives lower bounds on monotone span program size and monotone formula size. We also prove an upper bound on the rank of the corresponding matrices, and show that such structures can be constructed from self-avoiding families. As a corollary, we obtain an upper bound on the size of self-avoiding families, which solves a problem posed by Babai and Gál [Combinatorica 19 (3) (1999) 301-319].     

Computational complexity of queries based on itemsets

We investigate determining the exact bounds of the frequencies of conjunctions based on frequent sets. Our scenario is an important special case of some general probabilistic logic problems that are known to be intractable. We show that despite the limitations our problems are also intractable, namely, we show that checking whether the maximal consistent frequency of a query is larger than a given threshold is NP-complete and that evaluating the Maximum Entropy estimate of a query is PP-hard. We also prove that checking consistency is NP-complete.     

Predictive complexity and information

The notions of predictive complexity and of corresponding amount of information are considered. Predictive complexity is a generalization of Kolmogorov complexity which bounds the ability of any algorithm to predict elements of a sequence of outcomes. We consider predictive complexity for a wide class of bounded loss functions which are generalizations of square-loss function. Relations between unconditional KG(x) and conditional KG(x|y) predictive complexities are studied. We define an algorithm which has some “expanding property”. It transforms with positive probability sequences of given predictive complexity into sequences of essentially bigger predictive complexity. A concept of amount of predictive information IG(y:x) is studied. We show that this information is noncommutative in a very strong sense and present asymptotic relations between values IG(y:x), IG(x:y), KG(x) and KG(y).     

On the complexity of a two-point boundary value problem in different settings

We study the complexity of a two-point boundary value problem. We concentrate on the linear problem of order k with separated boundary conditions. Right-hand side functions are assumed to be r times differentiable with all derivatives bounded by a constant. We consider three models of computation: deterministic with standard and linear information, randomized and quantum. In each setting, we construct an algorithm for solving the problem, which allows us to establish upper complexity bounds. In the deterministic setting, we show that the use of linear information gives us a speed-up of at least one order of magnitude compared with the standard information. For randomized algorithms, we show that the speed-up over standard deterministic algorithms is by 1/2 in the exponent. For quantum algorithms, we can achieve a speed-up by one order of magnitude. We also provide lower complexity bounds. They match upper bounds in the deterministic setting with the standard information, and almost match upper bounds in the randomized and quantum settings. In the deterministic setting with the linear information, a gap still remains between the upper and lower complexity bounds.     

Operational state complexity of nested word automata

We introduce techniques to prove lower bounds for the number of states needed by finite automata operating on nested words. We study the state complexity of Boolean operations and obtain lower bounds that are tight within an additive constant. The results for union and complementation differ from corresponding bounds for ordinary finite automata. For reversal and concatenation, we establish lower bounds that are of a different order than the worst-case bounds for ordinary finite automata.     

On complexity of single-minded auction

We consider complexity issues for a special type of combinatorial auctions, the single-minded auction, where every agent is interested in only one subset of the commodities.First, we present a matching bound on the communication complexity for the single-minded auction under a general communication model. Next, we prove that it is NP-hard to decide whether Walrasian equilibrium exists in a single-minded auction. Finally, we establish a polynomial size duality theorem for the existence of Walrasian equilibrium for the single-minded auction.     

Some bounds on multiparty communication complexity of pointer jumping

We introduce the model of conservative one-way multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping.? The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node. In a conservative protocol, the ith player can see only the node reached by following the first i–1 edges and the edges on the jth layer for each j > i. This is a restriction of the general model where the ith player sees edges of all layers except for the ith one. In a one-way protocol, each player communicates only once in a prescribed order: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard.?Our main results are the following bounds on k-party conservative one-way communication complexity of pointer jumping with k layers:? (1) A lower bound of order for any .?(2) Matching upper and lower bounds of order for . received March 22, 1996     

Bounds on tradeoffs between randomness and communication complexity

The power of randomness in improving the efficiency (or even possibility) of computations has been demonstrated in numerous contexts. A fundamental question ishow much randomness is required for these improvements, or how does the improvement grow as a function of the amount of randomness allowed. This quantitative question, restricted to the context of communication complexity, is the focus of our paper.We prove general lower bounds on the amount of randomness used in randomized protocols for computing a functionf, the input of which is split between two parties. The bounds depend on the number of bits communicated and the deterministic communication complexity off. Four models for communication complexity are considered: the random input of the parties may be public or private, and the communication may be one-way or two-way. (Unbounded advantage is allowed.)The bounds are shown to be tight; i.e., we demonstrate functions and protocols for these functions which meet the above bounds up to a constant factor. We do this for all the models, for all values of the deterministic communication complexity, and for all possible quantities of bits communicated.     

Property Testing Lower Bounds via Communication Complexity

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of ??(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.