Question

I am trying to find a fast algorithm with modest space requirements to solve the following problem.

For each vertex of a DAG find the sum of its in-degree and out-degree in the DAG’s transitive closure.

Given this DAG:

I expect the following result:

Vertex #   Reacability Count  Reachable Vertices in closure
   7             5            (11, 8, 2, 9, 10)
   5             4            (11, 2, 9, 10)
   3             3            (8, 9, 10)
  11             5            (7, 5, 2, 9, 10)
   8             3            (7, 3, 9)
   2             3            (7, 5, 11)
   9             5            (7, 5, 11, 8, 3)
  10             4            (7, 5, 11, 3)

It seems to me that this should be possible without actually constructing the transitive closure. I haven’t been able to find anything on the net that exactly describes this problem. I’ve got some ideas about how to do this, but I wanted to see what the SO crowd could come up with.

Answer 1

I have constructed a viable solution to this question. I base my solution on a modification of the topological sorting algorithm. The algorithm below calculates only the in-degree in the transitive closure. The out-degree can be computed in the same fashion with edges reversed and the two counts for each vertex summed to determine the final “reachability count”.

for each vertex V
   inCount[V] = inDegree(V)   // inDegree() is O(1)
   if inCount[V] == 0
      pending.addTail(V)

while pending not empty
   process(pending.removeHead())

function process(V)
   for each edge (V, V2)
      predecessors[V2].add(predecessors[V])   // probably O(|predecessors[V]|)
      predecessors[V2].add(V)
      inCount[V2] -= 1
      if inCount[V2] == 0
          pending.add(V2)
   count[V] = sizeof(predecessors[V])         // store final answer for V
   predecessors[V] = EMPTY                    // save some memory

Assuming that the set operations are O(1), this algorithm runs in O(|V| + |E|). It is more likely, however, that the set union operation predecessors[V2].add(predecessors[V]) makes it somewhat worse. The additional steps required by the set unions depends on the shape of the DAG. I believe the worst case is O(|V|^2 + |E|). In my tests this algorithm has shown better performance than any other I have tried so far.

Furthermore, by disposing of predecessor sets for fully processed vertices, this algorithm will typically use less memory than most alternatives. It is true, however, that the worst case memory consumption of the above algorithm matches that of constructing the transitive closure, but that will not be true for most DAGs.