Communication Complexity

The first section starts with the basic definitions following mainly the notations of the book written by E. Kushilevitz and N. Nisan [8]. This whole thesis started from this book and some parts follow the proofs from there but I always tried to simplify and improve them, I even discovered and corrected a false claim. But most of the results are mine, supervised by Zoltán Király. At the beginning of each part I will indicate if a result is not mine. At the end of the first section I examine tree-balancing. At the very end I introduce a function that I will use only in the later sections. In the second section I summarize the well-known lower bound methods and prove the exact complexity of certain functions. In the first part of the third section I introduce the random complexity and prove the basic lemmas about it. In the second part I prove a better lower bound for the complexity of all random functions. In the third part I introduce and compare several upper bounds for the complexity of the identity function. In the fourth section I examine the well-known Direct-sum conjecture. I introduce a different model of computation then prove that it is the same as the original one up to a constant factor. This new model is used to bound the Amortized Time Complexity of a function by the number of the leaves of its protocol-tree. After this I examine the Direct-sum problem in case of Partial Information and in the Random case. In the last section I introduce the well-known hierarchy classes, the reducibility and the completeness of series of functions. Then I define the class PSPACE and Oracles in the communication complexity model and prove some basic claims about them. There are plenty questions left open in this paper, I hope I can manage to make some progress in some of them during the forthcoming years. I would like to thank Zoltán Király, my supervisor, for his guidance, help and inspiration. This thesis would be nowhere without him.