  This paper presents significantly improved deterministic algorithms for some
of the key problems in the area of distributed graph algorithms, including
network decomposition, hitting sets, and spanners. As the main ingredient in
these results, we develop novel randomized distributed algorithms that we can
analyze using only pairwise independence, and we can thus derandomize
efficiently. As our most prominent end-result, we obtain a deterministic
construction for $O(\log n)$-color $O(\log n \cdot \log\log\log n)$-strong
diameter network decomposition in $\tilde{O}(\log^3 n)$ rounds. This is the
first construction that achieves almost $\log n$ in both parameters, and it
improves on a recent line of exciting progress on deterministic distributed
network decompositions [Rozho\v{n}, Ghaffari STOC'20; Ghaffari, Grunau,
Rozho\v{n} SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozho\v{n},
Grunau FOCS'22].
