The linear-array problem in communication complexity resolved

Tiwari(1987) considered the following seen-ario: k+ 1 processors Po,. . ..P~. connected by k links to form a linear array, are to compute a function .f(z, y), x ~ X, y c Y, on a finite domain X x Y, where x is only known to PO, y is only known to Pk; the intermediate processors Pl,. .. . Pk_l do not have any information. The processors compute f(x, y) by exchanging binary messages across the links, according to some protocol @. Let 11~(~) denote the minimal complexity of such a protocol 0, i. e., the total number of bits sent across all links for the worst case input, and let ~(~) = D1 (f) denote the (standard) 2-party communication complexity off. Tiwari proved that Dk(~) ~ k ~(D(f) – O(l)) for almost all functions $ and conjectured this inequality to be true for all $. His conjecture was falsified by Kushilevitz, Linial, and Ostrovsky (1996): they exhibited a function f for which Dk (f) is essentially bounded above by ~kD(f). The best general lower bound known is D~(f) Z k.(~–logk–3). We prove a weakened version of Tiwari's conjecture: Dk(f) ~-y. k. D(f), for arbitrary $ and k, where y > 0.146 is a constant. (The lower bound on ~ can even be improved to 0.275.) Corresponding results for the nondeterministic and randomized versions of the two-party model and the array are also obtained. Applying the general framework provided by Tiwari, we may derive lower bounds for the communicant ion complexity of computing functions in general asynchronous networks on the basis of their two-party communication complexity. Finally, the main result entails that strong lower bounds on the time complexity of a function f : {O, 1}* + {O, 1} on deterministic one-tape Tur-ing machines can be obtained directly by considering the deterministic two-party communication complexity of its restrictions f 1{.,1}2., for n z 1.