Trade Offs Between Depth And Width in Parallel Computation Rev Ed

Cover Trade Offs Between Depth And Width in Parallel Computation Rev Ed
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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