A Communication-Time Tradeoff

We show a nontrivial tradeoff between the communication c and time t required to compute a collection of values whose dependencies form a grid, i.e., value $(i,j)$ depends on the values $(i - 1,j)$ and $(i,j - 1)$. No matter how we share the responsibility for computing the nodes of the $n \times n$ grid among processors, the law $(c + n)t = \Omega (n^3 )$ must hold. Further, there must be a single path through the grid along which there are t communication steps, where $(d + 1)t = \Omega (n^2 )$. Depending on the machine organization, either law may be the more significant.