The University of Chicago Communication Complexity and Information Complexity a Dissertation Submitted to the Faculty of the Division of the Physical Sciences in Candidacy for the Degree of Doctor of Philosophy Department of Computer Science by Denis Pankratov

Shannon introduced information theory in 1948. In Shannon’s model, the central question is to minimize the number of bits required to transmit a message from one party to another over a (possibly noisy) communication channel. Yao introduced communication complexity in 1979. In Yao’s model, the central question is to minimize the amount of communication required to compute a function of an input distributed between two parties that communicate over a perfect channel. In spite of the fact that communication complexity and information theory try to quantify communication in various contexts, communication complexity developed without the influence of information theory. This changed recently when the notion of information complexity was introduced. The current definition of internal information complexity is due to Bar-Yossef et al. (2004), but similar definitions in restricted communication settings can be traced back to Chakrabarti et al. (2001) and Ablayev (1996). Information complexity enabled the use of information-theoretic tools in communication complexity theory. Prior to the results presented in this thesis, information complexity was mainly used for proving lower bounds and direct-sum theorems in the setting of communication complexity. We present three results that demonstrate new connections between information complexity and communication complexity. In the first contribution we thoroughly study the information complexity of the smallest nontrivial two-party function: the AND function. While computing the communication complexity of AND is trivial, computing its exact information complexity presents a major technical challenge. In overcoming this challenge, we reveal that information complexity gives rise to rich geometrical structures. Our analysis of information complexity relies on new analytic techniques and new characterizations of communication protocols. We also uncover a connection of information complexity to the theory of elliptic partial differential equations. Once we compute the exact information complexity of AND, we can compute exact communication complexity of