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Given a directed conected graph which representation is its adjacency matrix $A$, design an algorithm to detect a sink in $\mathcal{O}(V)$ time, being $V$ the number of vertices.

As definitions can vary, in this context, a sink is defined as a vertex with $0$ exit degree and $V-1$ enter degree.

Obviously, the problem is reduced to find a $j\in\{1,\dots,V\}$ such as $a_{ji}=0$ for all $i$ and $a_{ij}=1$ for all $i\not=j$, thats what I tried. However, this solution is in $\mathcal{O}(V^2)$.

Any idea?

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Assumption :There is at most one row which contains all zero's.

You want to find a row of the adjacency matrix $A$ whose all entries are zero's. Simple observation if say the desired row is $k$th than in the $k$th column of adjacency matrix $A$ will contain all ones except the [k,k] index (it is easy to verify ). Let $n$ is the number of rows and columns in the matrix $A$.

Algorithm :

  1. Start from the top right corner of the matrix $A$.
  2. If $a_{i,j} = 0$ and $i \neq j$ then it's not your required column, skip this column mean move to $ j-1 $th column (see you have skipped one column in this step, so the number of columns for your search has been reduced).
  3. if $a_{i,j} = 1$ and $i \neq j$ then it is your required column, but it is not required row (think why ? ) so change the row move to $ i+1 $th keep the column same.

Running time. : In each step you either reducing the number of rows or number of columns, so maximum number of steps is going to be at max $2n$ (number of rows + number of columns ).

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  • $\begingroup$ It seems to me that you are you assuming the graph is complete in the sense that every pair of vertices are conected by a single edge. Am I wrong? $\endgroup$ – Álvaro G. Tenorio Aug 27 '17 at 10:00
  • $\begingroup$ No I am not assuming that given graph is complete. I am only assuming there is at most one sink node $\endgroup$ – user35837 Aug 27 '17 at 10:10
  • $\begingroup$ When you discard a row (step 3 ok) and then step 2 fails, when going again to step 2 you only check $a_{ij}=1$ for the non discarded rows. It does not affect the correctness of the algorithm? $\endgroup$ – Álvaro G. Tenorio Aug 27 '17 at 10:13
  • $\begingroup$ It does not affect the correctness of the algorithm , because I am carefully ( based on condition ) skipping the rows and columns $\endgroup$ – user35837 Aug 27 '17 at 10:21
  • $\begingroup$ Ok, i got it, tricky but elegant, thanks! $\endgroup$ – Álvaro G. Tenorio Aug 27 '17 at 10:36
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I believe this page provides an answer. The following pseudocode is from that page:

   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"
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    $\begingroup$ Please get rid of the source code and replace it with ideas, pseudo code and arguments of correctness. See here and here for related meta discussions. $\endgroup$ – Raphael♦ Aug 27 '17 at 7:55
  • $\begingroup$ I have some unclear points. What's the meaning of "pair vertices into at most n/2 pairs"? Take an arbitrary pairing of "good-vertices"? What happens if there is no edge betwen v and w in a pair (v,w)? $\endgroup$ – Álvaro G. Tenorio Aug 27 '17 at 9:49

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