  We present a deterministic $(1+o(1))$-approximation
$(n^{1/2+o(1)}+D^{1+o(1)})$-time algorithm for solving the single-source
shortest paths problem on distributed weighted networks (the CONGEST model);
here $n$ is the number of nodes in the network and $D$ is its (hop) diameter.
This is the first non-trivial deterministic algorithm for this problem. It also
improves (i) the running time of the randomized $(1+o(1))$-approximation
$\tilde O(n^{1/2}D^{1/4}+D)$-time algorithm of Nanongkai [STOC 2014] by a
factor of as large as $n^{1/8}$, and (ii) the $O(\epsilon^{-1}\log
\epsilon^{-1})$-approximation factor of Lenzen and Patt-Shamir's $\tilde
O(n^{1/2+\epsilon}+D)$-time algorithm [STOC 2013] within the same running time.
Our running time matches the known time lower bound of $\Omega(n^{1/2}/\log n +
D)$ [Elkin STOC 2004] up to subpolynomial factors, thus essentially settling
the status of this problem which was raised at least a decade ago [Elkin SIGACT
News 2004]. It also implies a $(2+o(1))$-approximation
$(n^{1/2+o(1)}+D^{1+o(1)})$-time algorithm for approximating a network's
weighted diameter which almost matches the lower bound by Holzer and Pinsker
[OPODIS 2015]. In achieving this result, we develop two techniques which might
be of independent interest and useful in other settings: (i) a deterministic
process that replaces the "hitting set argument" commonly used for shortest
paths computation in various settings, and (ii) a simple, deterministic,
construction of an $(n^{o(1)}, o(1))$-hop set of size $n^{1+o(1)}$. We combine
these techniques with many distributed algorithmic techniques, some of which
from problems that are not directly related to shortest paths, e.g., ruling
sets [Goldberg et al. STOC 1987], source detection [Lenzen and Peleg PODC
2013], and partial distance estimation [Lenzen and Patt-Shamir PODC 2015].
