Communication Complexity of PRAMs

We propose a model. LPRAM. for parallel random access machines uith local memon, that captures bcth the communication and computational requirements in parallel computation. For this model. n? present se\cral interesting resuk including the following: Two n x n matkez can be multiplied in Of n’/p) computation time and O( n“/p’ ‘) communication steps using p processors (for p = Ot n ‘/log’ ’ n) L Furthermore. these bounds are optimal for arithmetic on semiring~ , ,,rng +, x onlgt. It L shown that any algontnm that use) comparisons only and that sorts n words requires fl(n log n/(p log(n/p)I) communication stem for ! s pg n. We also provide an algorithm that sorts n words and uses c)t n log n/p1 computation time and 0( n log n/( p lo& n/p 1) ) communication steps. These bounds also apot) for computing In n-point FIT graph. It is s’lown that computmg any binary tree t with n nodes and hetght h requires R! n/p+ log w + ~4) communication steps, and can always be computed in O(n/p +mint\$ hb) steps. We also present a simple linear-time algorithm that generates a schedule for computing 7 in at most ZD_,.,ttb steps. where D,,(t) represents the minimum communication delay for czzF;tinp 7. It is also shown that various problems that are expressed as DAGs exhibit a communicat;nndelay/,-imputation-time trade-off.