The Linear-Array Conjecture in Communication Complexity is False

A linear array network consists of k + 1 processors P 0 ; P 1 ; : : : ; P k with links only between P i and P i+1 (0 i ! k). It is required to compute some boolean function f(x; y) in this network, where initially x is stored at P 0 and y is stored at P k . Let D k (f) be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, D k (f) k \Delta D(f ), where D(f) is the standard two-party communication complexity of f . Tiwari proved that for almost all functions D k (f) k(D(f) \Gamma O(1)) and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which D k (f) is essentially at most 3 4 k \Delta D(f ). Our construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are suc..

The Linear-Array Conjecture in Communication Complexity is False

A linear array network consists of k + 1 processors P 0;P 1;:::;Pk with links only between Pi and Pi+1 (0 i<k). It is required to compute some boolean function f(x; y) in this network, where initially x is stored at P 0 and y is stored at Pk. Let Dk(f) be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, Dk(f) k D(f), where D(f) is the standard two-party communication complexity off. Tiwari proved that for almost all functions Dk(f) k(D(f) , O(1)) and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an in nite family of functions for which Dk(f) is essentially at most 3 k D(f). Our construction also leads 4 to progress on another major problem in this area: It is easy to bound the two-party communication complexity ofanyfunction, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice as large as the best lower bound obtainable this way