  We present an $O(\log d + \log\log_{m/n} n)$-time randomized PRAM algorithm
for computing the connected components of an $n$-vertex, $m$-edge undirected
graph with maximum component diameter $d$. The algorithm runs on an ARBITRARY
CRCW (concurrent-read, concurrent-write with arbitrary write resolution) PRAM
using $O(m)$ processors. The time bound holds with good probability.
  Our algorithm is based on the breakthrough results of Andoni et al. [FOCS'18]
and Behnezhad et al. [FOCS'19]. Their algorithms run on the more powerful MPC
model and rely on sorting and computing prefix sums in $O(1)$ time, tasks that
take $\Omega(\log n / \log\log n)$ time on a CRCW PRAM with $\text{poly}(n)$
processors. Our simpler algorithm uses limited-collision hashing and does not
sort or do prefix sums. It matches the time and space bounds of the algorithm
of Behnezhad et al., who improved the time bound of Andoni et al.
  It is widely believed that the larger private memory per processor and
unbounded local computation of the MPC model admit algorithms faster than that
on a PRAM. Our result suggests that such additional power might not be
necessary, at least for fundamental graph problems like connected components
and spanning forest.
