Eulerian Cycles in directed graphs

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Presentation transcript:

Eulerian Cycles in directed graphs Network Optimization Eulerian Cycles in directed graphs Get ahold of a network, and use the same network to illustrate the shortest path problem for communication newtorks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.) James Orlin 2010

The initial network 2 3 1 Every node has the same number of arcs coming in as going out. 4 5 6 9 7 8 10 11 12 15 13 14

Determine a tree directed into node 1 2 3 1 Determine a bfs (or dfs) tree directed into node 1. 4 5 6 9 7 8 In the tree, each node has one arc coming out except node 1. 10 11 12 15 13 14

Order the arcs coming out of each node 2 3 1 The tree arc out of node j should be the last arc of A(j) 4 5 6 9 7 8 10 11 12 15 13 14

Create the walk Start at node 1 and create a walk. 2 3 1 Start at node 1 and create a walk. 4 5 6 9 7 8 Visit nodes of A(j) in order. 10 11 12 15 13 14

Continue the walk Continue the walk. Visit nodes of A(j) in order. 2 3 1 Continue the walk. 4 5 6 9 7 8 Visit nodes of A(j) in order. 10 11 12 15 13 14

Continue the walk Continue the walk. Visit nodes of A(j) in order. 2 3 1 Continue the walk. 4 5 6 9 7 8 Visit nodes of A(j) in order. 10 11 12 15 13 14

Continue the walk Continue the walk. Visit nodes of A(j) in order. 2 3 1 Continue the walk. 4 5 6 9 7 8 Visit nodes of A(j) in order. 10 11 12 15 13 14

15.082J / 6.855J / ESD.78J Network Optimization MITOpenCourseWare http://ocw.mit.edu 15.082J / 6.855J / ESD.78J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.