Juraj Hromkovič

Dissemination of Information in Interconnection Networks (Broadcasting & Gossiping)

Considerable attention in recent theoretical computer science is devoted to parallel computing. Here, we would like to present a special part of this large topic, namely, the part devoted to an abstract study of the dissemination of information in interconnection networks. The importance of this research area lies in the fact that the ability of a network to effectively disseminate information is an important qualitative measure for the suitabilty of the network for parallel computing. This follows simply from the observation that the communication among processes working in parallel is one of the main parts of the whole parallel computation. So, the effectivity of information exchange among processors essentially influences the effectivity of the whole computation process.

A comparison of two lower-bound methods for communication complexity

The methods “Rank” and “Fooling Set” for proving lower bounds on the deterministic communication complexity of Boolean functions are compared. The main results are as follows. (i) The Rank method provides the lower bound n on communication complexity for almost all Boolean functions of 2n variables, whereas the Fooling Set method provides only the lower bound d(n) ≤ log2n + log2 10. A specific sequence of Boolean functions {f2n} n=1 ∞ of 2n variables, is constructed, such that the Rank method provides exponentially higher lower bounds for f2n than the Fooling Set method. (ii) A specific sequence of Boolean functions {f2n} n=1 ∞ is constructed such that the Fooling Set method provides a lower bound of n for h2n, whereas the Rank method provides only (log2 3)/2 · n ≈ 0.79 · n as a lower bound. (iii) It is proved that lower bounds obtained by the Fooling Set method are better by at most a factor of two compared with lower bounds obtained by the Rank method.

Translating Regular Expressions into Small epsilon-Free Nondeterministic Finite Automata

It is proved that every regular expression of size n can be converted into an equivalent nondeterministic finite automaton (NFA) of size O(n(log n)2) in polynomial time. The best previous conversions result in NFAs of worst case size Θ(n2). Moreover, the nonexistence of any linear conversion is proved: we give a language L n described by a regular expression of size O(n) such that every NFA accepting L n is of size Ω(n log n).

Nondeterministic communication with a limited number of advice bits

We present a new technique for differentiating deterministic from nondeterministic communication complexity. As a consequence we give almost tight lower bounds for the nondeter- ministic communication complexity with a restricted number ofadvice bits. In particular, f or any

The bisection problem for graphs of degree 4 (configuring transputer systems)

It is well known that for each k≥3 there exists such a constant c k and such an infinite sequence {Gn} ∞ n=8 of k-degree graphs (each G n has exactly n vertices) that the bisection width of G n is at least c k ·n. It this paper some upper bounds on cks are found. Let σk(n) be the maximum of bisection widths of all k-bounded graphs of n vertices. We prove that

Optimal algorithms for dissemination of information in generalized communication modes

\sigma _k \left( n \right) \leqslant \frac{{\left( {k - 2} \right)}}{4} \cdot n + o\left( n \right)

Two Lower Bounds on Distributive Generation of Languages

for all k=2r, r≥2. This result is improved for k=4 by constructing two algorithms A and B, where for a given 4-degree-bounded graph G n of n vertices (i) A constructs a bisection of G n involving at most n/2+4 edges for even n≤76 (i.e., σ4(n)≤n/2+4 for even n≤76) (ii) B constructs a bisection of G n involving at most n/2+2 edges for n≥256 (i.e. σ4(n)≤n/2+2 for n≥256).

On Embedding Interconnection Networks into Rings of Processors

The study of synchronized alternating machines has enabled to characterize several natural complexity classes. It is known that synchronized alternating space SASPACE(S(n))= ?c>0NSPACE(ncS(n)) for any (space-constructible) function S(n) Hromkovic?et al. (1991)]. In particular, context-sensitive languages are characterized by two-way synchronized alternating finite automata. Furthermore, PSPACE is characterized by synchronized alternating multihead finite automata and NLOG by synchronized alternating two-way finite automata with parallelism bounded by a constant. In the present paper we prove analogous characterizations for deterministic space classes using a restricted form of synchronization - globally deterministic synchronization. This enables to study the well-known open problems concerning nondeterminism versus determinism as problems about synchronization. We also show that globally deterministic synchronization is strictly more powerful than deterministic synchronization.