On the power of multiple reads in a chip

Abstract

A chip algorithm is called r-multilective if it reads its input bits r times. In this paper we relate the r-round communication complexity of a language L and the area requirement by an r-multilective chip algorithm of a related language $\tilde L$. More specifically

$$A(\tilde L) \geqslant C_{2r} (L)/(2r - 1).$$

Improving known lower bounds on the r-round communication complexity of certain languages we obtain several hierarchies of r-multilective languages depending on the parameter r.

For example: If 0 < r < s, then there are constants 0 < c, c′ such that for infinitely many n there is a language L _n ⊕ {0, 1}ⁿ such that there is an s-multilective algorithm recognizing L _n using area at most clog n and any r-multilective algorithm recognizing L _n requires area at least c′n/log n.

Similar results have been known only for s = 2 and r = 1 and have been open in other cases.

Part of this research was done when the first author visited the University of Paderborn, Germany, supported by DFG grant ME 873/1–3.