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15.082 and 6.855J

15.082 and 6.855J. Topological Ordering. Preliminary to Topological Sorting. LEMMA. If each node has at least one arc going out, then the first inadmissible arc of a depth first search determines a directed cycle.

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15.082 and 6.855J

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  1. 15.082 and 6.855J Topological Ordering

  2. Preliminary to Topological Sorting • LEMMA. If each node has at least one arc going out, then the first inadmissible arc of a depth first search determines a directed cycle. • COROLLARY 1. If G has no directed cycle, then there is a node in G with no arcs going. And there is at least one node in G with no arcs coming in. • COROLLARY 2. If G has no directed cycle, then one can relabel the nodes so that for each arc (i,j), i < j. 3 1 4 6 7

  3. 1 2 3 4 5 6 7 8 Node Indegree 2 2 3 2 1 1 0 2 7 0 next LIST Initialization 6 1 “Next” will be the label of nodes in the topological order. Determine the indegree of each node LIST is the set of nodes with indegree of 0. 5 8 3 2 7 4

  4. LIST Select a node from LIST 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 5 8 3 2 7 7 4 1 0 1 next 1 2 3 4 5 6 7 8 Node Indegree 2 2 3 1 2 0 1 1 0 2 7 5

  5. LIST Select a node from LIST 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 8 3 2 7 7 4 1 2 0 1 next 1 2 3 4 5 6 7 8 Node Indegree 2 2 1 3 0 1 2 0 1 1 0 0 2 4 5 7 6

  6. LIST Select a node from LIST 3 6 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 8 3 2 7 7 4 1 3 1 2 0 next 1 2 3 4 5 6 7 8 Node Indegree 1 2 2 1 0 3 2 1 0 0 1 0 1 0 2 4 7 5 6 2

  7. LIST Select a node from LIST 3 6 6 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 4 1 3 2 0 next 1 2 3 4 5 6 7 8 Node Indegree 0 1 2 1 0 2 3 0 1 2 1 0 1 0 0 2 4 5 7 2 6 1

  8. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 1 2 5 0 4 3 next 1 2 3 4 5 6 7 8 Node Indegree 2 1 0 2 0 1 2 3 2 1 0 0 1 0 1 0 2 1 4 7 5 1 2 6

  9. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 4 1 6 1 0 2 4 5 3 6 next 1 2 3 4 5 6 7 8 Node Indegree 1 0 2 0 1 2 3 2 1 2 0 1 0 1 1 0 0 1 0 2 6 4 1 2 5 7 8

  10. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 2 5 5 2 2 8 8 3 4 7 7 4 4 1 6 1 4 5 6 2 3 7 0 next 1 2 3 4 5 6 7 8 Node Indegree 2 1 0 0 1 2 0 3 2 1 0 1 2 0 1 0 1 0 2 1 0 8 3

  11. LIST Select a node from LIST 5 3 6 6 1 1 next := next +1order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 8 2 5 5 2 2 8 8 3 3 4 7 7 4 4 1 6 4 1 6 7 8 5 3 2 0 next List is empty. The algorithm terminates with a topological order of the nodes 1 2 3 4 5 6 7 8 Node Indegree 2 0 1 0 1 2 0 2 3 1 2 0 1 0 1 0 1 0 2 0 1 3

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