  We consider the problem of computing all pairs shortest paths (APSP) and
shortest paths for k sources in a weighted graph in the distributed CONGEST
model. For graphs with non-negative integer edge weights (including zero
weights) we build on a recent pipelined algorithm to obtain
$\tilde{O}(\lambda^{1/4}\cdot n^{5/4})$ in graphs with non-negative integer
edge-weight at most $\lambda$, and $\tilde{O}(n \cdot \bigtriangleup^{1/3})$
rounds for shortest path distances at most $\bigtriangleup$. Additionally, we
simplify some of the procedures in the earlier APSP algorithms for non-negative
edge weights in [HNS17,ARKP18]. We also present results for computing h-hop
shortest paths and shortest paths from $k$ given sources.
  In other results, we present a randomized exact APSP algorithm for graphs
with arbitrary edge weights that runs in $\tilde{O}(n^{4/3})$ rounds w.h.p. in
n, which improves the previous best $\tilde{O}(n^{3/2})$ bound, which is
deterministic. We also present an $\tilde{O}(n/\epsilon^2)$-round deterministic
$(1+\epsilon)$ approximation algorithm for graphs with non-negative $poly(n)$
integer weights (including zero edge-weights), improving results in
[Nanongkai14,LP15] that hold only for positive integer weights.
