In fact, most parallel algorithms for finding MST (including those CRCW
PRAM algorithms) are based on a similar approach.(2) These parallel algorithms are "sequential" in the sense that the computation of [B.sub.i] starts only after [B.sub.i-1] is available (see Figure 1(a)).
In Section 4 we show that each step of a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] machine can be implemented in O(logp) amortized time with high probability on a p processor hypercube or CRCW PRAM.
In the CRCW (concurrent read concurrent write) PRAM this assumption is made directly in the model, and in the hypercube we assume the network supports combining within the switches to support these concurrent accesses [Ranade 1989].
We remark here that with the CRCW PRAM model of computation there is no point in looking for lower bounds for an unbounded number of processors, since any computable function can be computed in constant time given enough processors.
A strong version of the model allows concurrent reading and concurrent writing (CRCW) of a word by more than one processor, while other variations are restricted to exclusive writing (CREW) or to exclusive reading or writing (EREW).
The approach here can be extended to get an O(n) work, O(log n) time, Las Vegas type algorithm for CRCW
PRAM [Farach and Muthukrishnan 1996]; previously best-known optimal algorithms required [Omega]([log.sup.2] n) time.
A formal shared memory model allowing arbitrary patterns of simultaneous accesses is the concurrent read concurrent write (CRCW
) PRAM (see ).
Following the nomenclature introduced by Vishkin , the CRCW
(Concunent Read, Concurrent Write) PRAM allows memory access conflicts.
Optimal Bounds for Decision Problems on the CRCW
However, once the off-line embedding is performed for one G, simulation of another G' can be performed on-line by using G to simulate G' For example, by embedding a butterfly or other universal network into H off-line, we can then use H to simulate any CRCW
PRAM algorithm in an on-line fashion with delay O([log.sup.2]n/[Alpha]) and constant size queues.