  In the decremental $(1+\epsilon)$-approximate Single-Source Shortest Path
(SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$,
undergoing edge deletions, and a distinguished source $s \in V$, and we are
asked to process edge deletions efficiently and answer queries for distance
estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage,
such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+
\epsilon)\mathbf{dist}_G(s,v)$. In the decremental $(1+\epsilon)$-approximate
All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for
distance estimates $\widetilde{\mathbf{dist}}_G(u,v)$ for every $u,v \in V$. In
this article, we consider the problems for undirected, unweighted graphs.
  We present a new \emph{deterministic} algorithm for the decremental
$(1+\epsilon)$-approximate SSSP problem that takes total update time $O(mn^{0.5
+ o(1)})$. Our algorithm improves on the currently best algorithm for dense
graphs by Chechik and Bernstein [STOC 2016] with total update time
$\tilde{O}(n^2)$ and the best existing algorithm for sparse graphs with running
time $\tilde{O}(n^{1.25}\sqrt{m})$ [SODA 2017] whenever $m = O(n^{1.5 -
o(1)})$.
  In order to obtain this new algorithm, we develop several new techniques
including improved decremental cover data structures for graphs, a more
efficient notion of the heavy/light decomposition framework introduced by
Chechik and Bernstein and the first clustering technique to maintain a dynamic
\emph{sparse} emulator in the deterministic setting.
  As a by-product, we also obtain a new simple deterministic algorithm for the
decremental $(1+\epsilon)$-approximate APSP problem with near-optimal total
running time $\tilde{O}(mn /\epsilon)$ matching the time complexity of the
sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai
[FOCS 2013].
