Pavol Duris

Lower bounds on communication complexity

Abstract We prove the following four results on communication complexity: (1) For every k ≥ 2, the language of encodings of directed graphs of out-degree one that contain a path of length k + 1 from the first vertex to the last vertex can be recognized by exchanging O(k log n)1 bits using a simple k-round protocol and requires the exchange of Ω( n 1 2 (k 4 log 3 n) ) bits by any (k − 1)-round protocol. (2) For every k ≥ 1 and for infinitely many n ≥ 1, there exists a collection of sets Lkn ⊆ {0, 1}2n that can be recognized by exchanging O(k log n) bits using a k-round protocol, and any (k − 1)-round protocol recognizing Lkn requires the exchange of Ω( n k ) bits. (3) Given a set L ⊆ {0, 1}2n, there is a set L ⊆ {0, 1} 8n such that any (k-round) protocol recognizing L can be transformed to a (k-round) fixed-partition protocol recognizing L with the same communication complexity, and vice versa. (4) For every integer function f, 1 ≤ f(n) ≤ n, there are languages recognizable by a one-round deterministic protocol exchanging f(n) bits, but not by any nondeterministic protocol exchanging f(n) − 1 bits. The first two results show in an incomparable way an exponential gap between (k − 1)-round and k-round protocols, settling a conjecture by Papadimitriou and Sipser. The third result shows that as long as we are interested in existence proofs, a fixed partition of the input is not a restriction. The fourth result extends a result by Papadimitriou and Sipser who showed that for every integer function f, 1 ≤ f(n) ≤ n, there is a language accepted by a deterministic protocol exchanging f(n) bits but not by any deterministic protocol exchanging f(n) − 1 bits.

Las Vegas Versus Determinism for One-way Communication Complexity, Finite Automata, and Polynomial-time Computations

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main aim of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. An at most polynomial gap has been established for the combinational complexity of circuits and for the communication complexity of two-party protocols. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, finite automata and polynomialtime relativized Turing machine computation. (i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight. (ii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.

A time-space tradeoff for language recognition

We define a languageL and show that its time and space complexitiesT andS must satisfyT2S ≥cn3 even allowing machines with multiple (non random) access to the input.

Two nonlinear lower bounds

We prove the following lower bounds for <underline>on line</underline> computation. 1) Simulating two tape <underline>nondeterministic</underline> machines by one tape machines requires Ω(n log log n) time. 2) Simulating k tape (deterministic) machines by machines with k pushdown stores requires Ω(n log<supscrpt>1/(k+1)</supscrpt>n) time.

On reversal-bounded counter machines and on pushdown automata with a bound on the size of the pushdown store

The two main results of the paper are: (1) proving a fine hierarchy of reversal-bounded counter machine languages; and (2) showing that a tape is better than a pushdown store for two-way machines, in the case where their size is sublinear.

Two lower bounds in asynchronous distributed computation

We introduce new techniques for deriving lower bounds on the message complexity in asynchronous distributed computation. These techniques combine the choice of specific patterns of communication delays and crossing sequence arguments with consideration of the speed of propagation of messages, together with careful counting of messages in different parts of the network. They enable us to prove the following results, settling two open problems: An Ω(n log* n) lower bound for the number of messages sent by an asynchronous algorithm for computing any nonconstant function on a bidirectional ring of n anonymous processors. An Ω(n log n) lower bound for the average number of messages sent by any maximum finding algorithm on a ring of n processors, in case n is known.

On Multipartition Communication Complexity

We study k-partition communication protocols, an extension of the standard two-party best-partition model to k input partitions. The main results are as follows. 1. A strong explicit hierarchy on the degree of non-obliviousness is established by proving that, using k+1 partitions instead of k may decrease the communication complexity from Θ(n) to Θ(log k). 2. Certain linear codes are hard for k-partition protocols even when k may be exponentially large (in the input size). On the other hand, one can show that all characteristic functions of linear codes are easy for randomized OBDDs. 3. It is proven that there are subfunctions of the triangle-freeness function and the function ⊕ CLIQUEn,3 which are hard for multipartition protocols. As an application, truly exponential lower bounds on the size of nondeterministic read-once branching programs for these functions are obtained, solving an open problem of Razborov [17].

Two tapes are better than one for nondeterministic machines

It is known that k tapes are no better than two tapes for nondeterministic machines. We show here that two tapes are better than one. In fact, we show that two pushdown stores are better than one tape. Also, k tapes are no better than two for nondeterministic reversal-bounded machines; and we show that even two reversal-bounded pushdown stores are better than one reversal-bounded tape. We also show that for one-tape nondeterministic machines, unrestricted operation is better than reversal-bounded operation.

Power of Cooperation and Multihead Finite Systems

We consider systems of finite automata performing together computation on an input string. Each automaton has its own read head that moves independently of the other heads, but the automata cooperate in making state transitions. Computational power of such devices depends on the number of states of automata, the number of automata, and the way they cooperate. We concentrate our attention on the last issue. The first situation that we consider is that each automaton has a full knowledge on the states of all automata (multihead automata). The other extreme is that each automaton (called also a processor) has no knowledge of the states of other automata; merely, there is a central processing unit that may “freeze” any automaton or let it proceed its work (so called multiprocessor automata). The second model seems to be severely restricted, but we show that multihead and multiprocessor automata have similar computational power. Nevertheless, we show a separation result.

A time-space tradeoff for language recongnition

We define a language L and show that its time and space complexities T and S must satisfy T2S ≥ cn3 even allowing machines with multiple (non random) access to the input.