Network Optimization Topological Ordering Preliminary to Topological Sorting

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Network Optimization Topological Ordering

Network Optimization Topological Ordering

Preliminary to Topological Sorting LEMMA. If each node has at least one arc going

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. 1 4 6 3 7 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. 2

Initialization Determine the indegree di of each node i. 5 LIST = {i :

Initialization Determine the indegree di of each node i. 5 LIST = {i : di = 0} 7 Node Indegree 6 1 2 8 3 4 1 2 3 4 5 6 7 8 2 2 3 2 1 1 0 2 3 7 LIST

Initialization “Next” will be the label of nodes in the topological order. 5 7

Initialization “Next” will be the label of nodes in the topological order. 5 7 6 1 2 8 3 4 1 next Node Indegree 1 2 3 4 5 6 7 8 2 2 3 2 1 1 0 2 4 7 LIST

Select a node from LIST Select Node 7. Order(7) : = 1 Delete node

Select a node from LIST Select Node 7. Order(7) : = 1 Delete node 7. 5 77 6 1 2 8 3 4 1 0 1 next Node Indegree 1 2 3 4 5 6 7 8 2 2 3 2 1 1 0 2 5 7 LIST

Updates update “next” update indegrees 5 update LIST 77 6 1 2 8 3

Updates update “next” update indegrees 5 update LIST 77 6 1 2 8 3 4 1 0 1 2 next Node Indegree 1 2 3 4 5 6 8 2 2 3 1 2 1 0 1 2 6 7 5 LIST

Select node 5 Select Node 5. Order(5) : = 2 Delete node 5. 2

Select node 5 Select Node 5. Order(5) : = 2 Delete node 5. 2 77 5 6 1 2 8 3 4 1 0 1 2 next Node Indegree 1 2 3 4 5 6 8 2 2 3 1 2 0 1 2 7 7 5 LIST

Updates update “next” update indegrees 2 update LIST 77 5 6 1 2 8

Updates update “next” update indegrees 2 update LIST 77 5 6 1 2 8 3 4 1 0 1 2 3 next Node Indegree 1 2 3 4 6 8 2 2 1 3 0 1 2 1 0 2 8 4 7 5 6 LIST

Select Node 6 (or 4) Select Node 6. Order(6) : = 3 Delete node

Select Node 6 (or 4) Select Node 6. Order(6) : = 3 Delete node 6. 2 77 5 3 6 1 2 8 3 4 1 0 1 2 3 next Node Indegree 1 2 3 4 6 8 2 2 1 3 0 1 2 0 2 9 4 7 5 6 LIST

Updates update “next” update indegrees 2 update LIST 77 5 3 6 1 2

Updates update “next” update indegrees 2 update LIST 77 5 3 6 1 2 8 3 4 1 0 1 2 3 4 next Node Indegree 1 2 3 4 8 2 1 0 3 0 1 2 2 10 4 7 5 2 LIST

Select Node 2 (or 4) Select Node 2. 3 6 Order(2) : = 4

Select Node 2 (or 4) Select Node 2. 3 6 Order(2) : = 4 Delete node 2. 2 77 5 1 2 4 8 3 4 1 0 1 2 3 4 next Node Indegree 1 2 3 4 8 2 1 0 3 0 1 2 2 11 4 7 5 6 2 LIST

Updates update “next” update indegrees 3 6 2 update LIST 77 5 1 2

Updates update “next” update indegrees 3 6 2 update LIST 77 5 1 2 4 8 3 4 1 0 1 2 3 4 5 next Node Indegree 1 3 4 8 2 1 0 3 0 1 2 2 12 4 7 5 61 LIST

Select node 4 (or 1) Select Node 4. 3 6 Order(4) : = 5

Select node 4 (or 1) Select Node 4. 3 6 Order(4) : = 5 Delete node 4. 2 77 5 1 2 4 8 3 4 1 5 0 1 2 3 4 5 next Node Indegree 1 3 4 8 2 1 0 3 0 2 13 4 7 5 61 LIST

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 1

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 1 2 4 8 3 4 5 0 1 2 3 4 5 6 next Node Indegree 1 3 8 2 1 0 3 2 2 1 14 5 7 61 LIST

Select Node 1. 3 6 Order(1) : = 6 Delete node 1. 2 77

Select Node 1. 3 6 Order(1) : = 6 Delete node 1. 2 77 1 5 6 1 2 4 8 3 4 5 0 1 2 3 4 5 6 next Node Indegree 1 3 8 0 3 2 2 1 15 5 7 61 LIST

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 6

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 6 1 2 4 8 3 4 5 0 1 2 3 4 5 6 7 next Node Indegree 3 8 3 2 1 0 16 5 7 8 LIST

Select Node 8. 3 6 Order(8) : = 7 Delete node 8. 2 77

Select Node 8. 3 6 Order(8) : = 7 Delete node 8. 2 77 1 5 6 1 7 8 2 4 3 4 5 0 1 2 3 4 5 6 7 next Node Indegree 3 8 3 2 1 0 17 5 7 8 LIST

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 6

Updates update “next” update indegrees 3 6 2 update LIST 77 1 5 6 1 7 8 2 4 3 4 5 0 1 2 3 4 5 6 7 8 next Node Indegree 3 3 2 1 0 5 7 3 18 LIST

Select node 3 Select Node 3. 3 6 Order(3) : = 8 Delete node

Select node 3 Select Node 3. 3 6 Order(3) : = 8 Delete node 3. 2 77 1 5 6 1 7 8 2 4 8 3 4 5 0 1 2 3 4 5 6 7 8 next Node Indegree 3 0 List is empty. The algorithm terminates with a topological order of the nodes 19 5 7 3 LIST

MITOpen. Course. Ware http: //ocw. mit. edu 15. 082 J / 6. 855 J

MITOpen. Course. Ware http: //ocw. mit. edu 15. 082 J / 6. 855 J / ESD. 78 J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http: //ocw. mit. edu/terms.

Computing a Topological Ordering 1 A topological orderingComputing a Topological Ordering 1 A topological ordering
Event Ordering Event Ordering Time and Ordering TheEvent Ordering Event Ordering Time and Ordering The
Event Ordering Event Ordering Time and Ordering TheEvent Ordering Event Ordering Time and Ordering The
Topological Ordering public MaybeListNode topological Order ListNode orderedTopological Ordering public MaybeListNode topological Order ListNode ordered
Topological Sort Introduction Definition of Topological Sort TopologicalTopological Sort Introduction Definition of Topological Sort Topological
Topological Sort Introduction Definition of Topological Sort TopologicalTopological Sort Introduction Definition of Topological Sort Topological
Topological Sort Introduction Definition of Topological Sort TopologicalTopological Sort Introduction Definition of Topological Sort Topological
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Topological Sort Introduction Definition of Topological Sort TopologicalTopological Sort Introduction Definition of Topological Sort Topological
Topological Sort Topological Sort Sorting technique over DAGsTopological Sort Topological Sort Sorting technique over DAGs
Graph Algorithms Topological Sort The topological sorting problemGraph Algorithms Topological Sort The topological sorting problem
Chapter 8 Sorting Sorting Ordering or Sequencing SinceChapter 8 Sorting Sorting Ordering or Sequencing Since
Sorting Algorithms Sorting Sorting is a process thatSorting Algorithms Sorting Sorting is a process that
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Linear Sorting Comparison based sorting Any sorting algorithmLinear Sorting Comparison based sorting Any sorting algorithm
SORTING PENGURUTAN SORTING PENGURUTAN Sorting adalah proses mengaturSORTING PENGURUTAN SORTING PENGURUTAN Sorting adalah proses mengatur
Array Sorting Searching Sorting Algorithms A sorting algorithmArray Sorting Searching Sorting Algorithms A sorting algorithm
Sorting Chapter 4 Sorting Sorting is one ofSorting Chapter 4 Sorting Sorting is one of