In the past thirty years, Communication Complexity has emerged as a foundational tool to proving lower bounds in many areas of computer science. Its power comes from its generality, but this generality comes at a price---no superlinear communication lower bound is possible, since a player may communicate his entire input. However, what if the players are limited in their ability to recall parts of their interaction? We introduce memory models for 2-party communication complexity. Our general model is as follows: two computationally unrestricted players, Alice and Bob, each have s(n) bits of memory. When a player receives a bit of communication, he "compresses" his state. This compression may be an arbitrary function of his current memory contents, his input, and the bit of communication just received; the only restriction is that the compression must return at most s(n) bits. We obtain memory hierarchy theorems (also comparing this general model with its restricted variants), and show super-linear lower bounds for some explicit (non-boolean) functions. Our main conceptual and technical contribution concerns the following variant. The communication is one-way, from Alice to Bob, where Bob controls two types of memory: (i) a large, oblivious memory, where updates are only a function of the received bit and the current memory content, and (ii) a smaller, non-oblivious/general memory, where updates can be a function of the input given to Bob. We exhibit natural protocols where this semi-obliviousness shows up. For this model we also introduce new techniques through which certain limitations of space-bounded computation are revealed. One of the main motivations of this work is in understanding the difference in the use of space when computing the following functions: Equality (EQ), Inner Product (IP), and connectivity in a directed graph (REACH). When viewed as communication problems, EQ can be decided using 0 non-oblivious bits (and log2 n oblivious bits), IP requires exactly 1 non-oblivious bit, whereas for REACH we obtain the same lower bound as for IP and conjecture that the actual bound is Omega(log2 n). In fact, proving that 1 non-oblivious bit is required becomes technically sophisticated, and the question even for 2 non-oblivious bits for any explicit boolean function remains open.
Proceedings of the 4th Conference on Innovations in Theoretical Computer Science , Itcs '13, 2013, p. 159-172
Conference on Innovations in Theoretical Computer Science, 2013