The communication complexity of enumeration, elimination, and selection

Abstract

Let f:{0, 1}/sup n//spl times/{0, 1}/sup n//spl rarr/{0, 1}. Assume Alice has x/sub 1/, ..., x/sub k//spl isin/{0, 1}/sup n/, Bob has y/sub 1/, ..., y/sub k//spl isin/{0, 1}/sup n/, and they want to compute f(x/sub 1/, y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/) communicating as few bits as possible. The Direct Sum Conjecture of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing f(x/sub 1/, y/sub 1/), then f(x/sub 2/, y/sub 2/), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits. Since a variant of it implies NC/sup 1//spl ne/NC/sup 2/. We consider three related problems. Enumeration: Alice and Bob output e/spl les/2/sup k/-1 elements of {0, 1}/sup k/: one of which is f(x/sub 1/, y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/). Elimination: Alice and Bob output an element of {0, 1}/sup k/ that is not f(x/sub 1/ y/sub 1/)/spl middot//spl middot//spl middot/f(x/sub k/, y/sub k/) Selection: (k=2) Alice and Bob output i/spl sim/{1,2} such that if f(x/sub 1/, y/sub 1/)=1 V f(x/sub 2/, Y/sub 2/)=1 then f(x/sub i/, y/sub i/)=1. We establish lower bounds on ELIM(f/sup k/) for particular f and connect the complexity of ELIM(f/sup k/), ENUM(k, f/sup k/), and SELECT(f/sup 2/) to the direct sum conjecture and other conjectures.