Trade Offs Between Depth And Width in Parallel Computation Rev Ed

U Vishkin

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• . (xu. YOi).
i?«-z?*-iuKi, . Yi, ). ••. (i^. YOi- It is easy to see that for every 0-^t^T (1) \I)'\^\D*-^\+t, li?°|=0 and hence \D* \£t{t -i-i)/ 2.
(2) £^ =I(Z?* ), Z?' -'ci?^ £^ C£^ -^ (3) f « ?'0.
In particular we have: Lemmaai: £•''^0 and |Z3^|^r(r+l)/2 Remark. The definition above generates a set E'^ of "easy to analyze" inputs. Regardless of the function being computed. Therefore we believe that this tech- nique can be used to prove lower bounds for the computation of other functions in ...this model Lemraa 2. 2: Let M be an CRCIV PRAM(l) computing a function /, and let £'^ be defined as above for M. Then for every x. Y zE''', f ix)=f (y).
A rigorous proof of this lemma is given in the appendix. The idea is to show inductively on t, that any processor which writes at time t on some input in £*, will have exactly the same computation through time t on every input in £'^.
Proof at theorem. 2. 1: Recall that M computes a A: -sensitive everywhere function / in time T.

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