Some contributions of the study of abstract communication complexity to other areas of computer science

Abstract

Communication Complexity · 3 —Size of distinct models of branching programs —Depth of decision trees —Data structure problems. To illustrate the progress covered by the above list we mention two specific contributions. The first superlinear lower bound on the size of planar Boolean circuits computing a specific Boolean function and the first superpolylogarithmic lower bounds on the depth of monotone Boolean circuits have been established. The big success of communication complexity application should not to be surprising because we have information transfer in all computing models (for instance, between two parts of input data , between some parts (processors) of a parallel computing model, between two time moments, etc.). So, you can cut hardware, time, or both in your computing model, and then apply lower bounds on the communication complexity of your computing problem. In this way you have a lower bound on the information transfer that must be realized in the computing model considered in order to compute the given task. The appropriate choice of the cut is crucial for obtaining good lower bounds. One of the perspectives is to extend the applications for proving lower bounds for multilective and/or non-oblivious computing models. This is one of the hardest tasks of special importance in complexity theory. The recent results show that using Ramsey theory and communication complexity over overlapping (not disjoint) partitions of inputs one has good chances to achieve progress in this hard topic too. 3. NONDETERMINISTIC AND RANDOMIZED COMPUTATIONS One of the central principal questions of current theoretical computer science is which computational power have nondeterministic and randomized computations, especially in the comparison with the deterministic one. The fundamental questions about polynomial time computations (like P versus NP, P versus ZPP, P versus R) are long-stated open problems. For communication complexity the research has been successful and the relation between determinism, nondeterminism and randomness has been fixed. This has essentially contributed to the understanding of the nature of randomness and nondeterminism. Some of the main results are the following ones: (1) There are exponential gaps between —determinism and Monte Carlo randomness —nondeterminism and bounded error probabilism. (2) Deterministic communication can be bounded by at most twice the product of nondeterministic communication of the language and its complement. This implies an at most quadratic gap between determinism and Las Vegas randomization. A language having this quadratic gap has been found. (3) There is a linear gap between determinism and Las Vegas randomness for oneway communication complexity. (4) O(log n) random bits are sufficient to reach the full power of randomized communication for Las Vegas and Monte Carlo (error-bounded) protocols. (5) In contrast to 4. there exist high thresholds on the amount of nondeterminism (for some computing problems the deterministic communication complexity is