Stable topological sort

Question

Let say I have a graph where the nodes is stored in a sorted list. I now want to topological sort this graph while keeping the original order where the topological order is

Answer 1

Here's an easy iterative approach to topological sorting: continually remove a node with in-degree 0, along with its edges.

To achieve a stable version, just modify to: continually remove the smallest-index node with in-degree 0, along with its edges.

In pseudo-python:

# N is the number of nodes, labeled 0..N-1
# edges[i] is a list of nodes j, corresponding to edges (i, j)

inDegree = [0] * N
for i in range(N):
   for j in edges[i]:
      inDegree[j] += 1

# Now we maintain a "frontier" of in-degree 0 nodes.
# We take the smallest one until the frontier is exhausted.
# Note: You could use a priority queue / heap instead of a list,
#       giving O(NlogN) runtime. This naive implementation is
#       O(N^2) worst-case (when the order is very ambiguous).

frontier = []
for i in range(N):
    if inDegree[i] == 0:
        frontier.append(i)

order = []
while frontier:
    i = min(frontier)
    frontier.remove(i)
    for j in edges[i]:
       inDegree[j] -= 1
       if inDegree[j] == 0:
           frontier.append(j)

 # Done - order is now a list of the nodes in topological order,
 # with ties broken by original order in the list.