  We present the first non-trivial fully dynamic algorithm maintaining exact
single-source distances in unweighted graphs. This resolves an open problem
stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019].
Previous fully dynamic single-source distances data structures were all
approximate, but so far, non-trivial dynamic algorithms for the exact setting
could only be ruled out for polynomially weighted graphs (Abboud and
Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main
case for which neither a subquadratic dynamic algorithm nor a quadratic lower
bound was known.
  Our dynamic algorithm works on directed graphs, is deterministic, and can
report a single-source shortest paths tree in subquadratic time as well. Thus
we also obtain the first deterministic fully dynamic data structure for
reachability (transitive closure) with subquadratic update and query time. This
answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019].
  Finally, using the same framework we obtain the first fully dynamic data
structure maintaining all-pairs $(1+\epsilon)$-approximate distances within
non-trivial sub-$n^\omega$ worst-case update time while supporting optimal-time
approximate shortest path reporting at the same time. This data structure is
also deterministic and therefore implies the first known non-trivial
deterministic worst-case bound for recomputing the transitive closure of a
digraph.
