Graphs: find a sink in less than O(|V|) - or show it can't be done

Question

I have a graph with n nodes as an adjacency matrix.

Is it possible to detect a sink in less than O(n) time?

If yes, how? If no

Answer 1

This page answers your exact question. The linear time algorithm is

   def find-possible-sink(vertices):
     if there's only one vertex, return it
     good-vertices := empty-set
     pair vertices into at most n/2 pairs
     add any left-over vertex to good-vertices
     for each pair (v,w):
       if v -> w:
         add w to good-vertices
       else:
         add v to good-vertices
     return find-possible-sink(good-vertices)

   def find-sink(vertices):
     v := find-possible-sink(vertices)
     if v is actually a sink, return it
     return "there is no spoon^H^H^H^Hink"