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ENGM 732 Formalization of Network Flows

ENGM 732 Formalization of Network Flows. Network Flow Models. Terminology. 1. Loop. 4. 5. Graph. 1. 3. 2. Terminology. 4. 5. Graph. 1. 3. 2. 4. 5. Multigraph. 1. 3. 2. Terminology. 4. 5. Graph. 1. 3. 2. 4. 5. Directed Network. 1. 3. 2. Terminology.

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ENGM 732 Formalization of Network Flows

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  1. ENGM 732Formalization of Network Flows Network Flow Models

  2. Terminology 1 Loop 4 5 Graph 1 3 2

  3. Terminology 4 5 Graph 1 3 2 4 5 Multigraph 1 3 2

  4. Terminology 4 5 Graph 1 3 2 4 5 Directed Network 1 3 2

  5. Terminology • Walk – alternating sequence of nodes and lines beginning and ending with nodes • Closed walk – same node begins and ends walk • Open walk – ends at node other than beginning node • Trail – walk with no line repeated • Path - a walk with no node repeated

  6. Terminology • Walk – alternating sequence of nodes and lines beginning and ending with nodes • Open walk – ends at node other than beginning node 4 5

  7. Terminology • Closed walk – same node begins and ends walk • Trail – walk with no line repeated 4 1 2

  8. Terminology • Path - a walk with no node repeated 4 1 2

  9. Terminology - Trees [0] [ bi ] (ck, hk) (1,-1) 2 6 (3,5) (2,-1) 1 4 [3] [-5] (2,1) (1,2) 1 4 5 3 7 (5,3) (2,1) 5 2 3 [0] 5 Root = 5 NT = {5} MT = 

  10. Terminology - Trees [0] [ bi ] (ck, hk) (1,-1) 2 6 (3,5) (2,-1) 1 4 [3] [-5] (2,1) (1,2) 1 4 5 3 7 (5,3) (2,1) 5 2 3 [0] 2 Root = 5 NT = {1,2,3,4,5} MT = {1,2,5,7} 1 1 4 5 7 2 5 3 Spanning Tree

  11. Terminology - Trees [0] [ bi ] (ck, hk) (1,-1) 2 6 (3,5) (2,-1) 1 4 [3] [-5] (2,1) (1,2) 1 4 5 3 7 (5,3) (2,1) 5 2 3 [0] 2 4 1 1 5 7 5 3 Root = 5 NT = {1,2, 5} MT = {1,7} Root = 3 NT = {3, 4} MT = {5} Spanning Forest

  12. Expanded Network 2 6 1 4 Directed Network 1 4 5 3 7 5 2 o(-k) = t(k) t(-k) = o(k) 3 8 -6 2 -4 6 1 4 -7 -1 Expanded Network 1 4 5 3 -3 7 -5 -2 5 2 -8 3 8

  13. Expanded Network (2,2,-1) 2 6 (fk , ck , hk) 1 4 1 4 5 3 7 Network with flow 5 2 3 8 -6 2 -4 1 4 -7 -1 Marginal Network Consists of a subset that are admissible arcs 1 4 5 3 -3 7 -5 -2 5 2 -8 3 8

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