  Consider the problem of maintaining source sink
reachability($st$-Reachability), single source reachability(SSR) and strongly
connected component(SCC) in an edge decremental directed graph. In particular,
we design a randomized algorithm that maintains with high probability:
  1) $st$-Reachability in $\tilde{O}(mn^{4/5})$ total update time. 2)
$st$-Reachability in a total update time of $\tilde{O}(n^{8/3})$ in a dense
graph. 3) SSR in a total update time of $\tilde{O}(m n^{9/10})$. 4) SCC in a
total update time of $\tilde{O}(m n^{9/10})$. For all the above problems, we
improve upon the previous best algorithm (by Henzinger et. al. (STOC 2014)).
  Our main focus is maintaining $st$-Reachability in an edge decremental
directed graph (other problems can be reduced to $st$-Reachability). The
classical algorithm of Even and Shiloach (JACM 81) solved this problem in
$O(1)$ query time and $O(mn)$ total update time. Recently, Henzinger,
Krinninger and Nanongkai (STOC 2014) designed a randomized algorithm which
achieves an update time of $\tilde{O}(m n^{0.98})$ and broke the long-standing
$O(mn)$ bound of Even and Shiloach. However, they designed four algorithms $A_i
(1\le i \le 4)$ such that for graphs having total number of edges between $m_i$
and $m_{i+1}$ ($m_{i+1} > m_i$), $A_i$ outperforms other three algorithms. That
is, one of the four algorithms may be faster for a particular density range of
edges, but it may be too slow asymptotically for the other ranges. Our main
contribution is that we design a {\it single} algorithm which works for all
types of graphs. Not only is our algorithm faster, it is much simpler than the
algorithm designed by Henzinger et.al. (STOC 2014).
