  We present a simple polylogarithmic-time deterministic distributed algorithm
for network decomposition. This improves on a celebrated $2^{O(\sqrt{\log
n})}$-time algorithm of Panconesi and Srinivasan [STOC'92] and settles a
central and long-standing question in distributed graph algorithms. It also
leads to the first polylogarithmic-time deterministic distributed algorithms
for numerous other problems, hence resolving several well-known and decades-old
open problems, including Linial's question about the deterministic complexity
of maximal independent set [FOCS'87; SICOMP'92]---which had been called the
most outstanding problem in the area.
  The main implication is a more general distributed derandomization theorem:
Put together with the results of Ghaffari, Kuhn, and Maus [STOC'17] and
Ghaffari, Harris, and Kuhn [FOCS'18], our network decomposition implies that
$$\mathsf{P}\textit{-}\mathsf{RLOCAL} = \mathsf{P}\textit{-}\mathsf{LOCAL}.$$
That is, for any problem whose solution can be checked deterministically in
polylogarithmic-time, any polylogarithmic-time randomized algorithm can be
derandomized to a polylogarithmic-time deterministic algorithm. Informally, for
the standard first-order interpretation of efficiency as polylogarithmic-time,
distributed algorithms do not need randomness for efficiency.
  By known connections, our result leads also to substantially faster
randomized distributed algorithms for a number of well-studied problems
including $(\Delta+1)$-coloring, maximal independent set, and Lov\'{a}sz Local
Lemma, as well as massively parallel algorithms for $(\Delta+1)$-coloring.
