Before revision:  First, we obtain an O(  | E| + k^2| V| \log{|V|)-time algorithm, which in particular is faster than all existing MPC algorithms when k = o(\sqrt{|V|}/\log{|V|}) and |E| = \omega(k|V|) but |E| = o(|V|^2).
After revision:  First, we obtain an O( | V| \log{|V|)-time algorithm, which in particular is faster than all existing MPC algorithms when k = o(\sqrt{|V|}/\log{|V|}) and |E| = \omega(k|V|) but |E| = o(|V|^2).