1 / 28

Maximum Flow

Maximum Flow. Haidong Xue (Haydon ) 10/28/2013. What is a flow network?. A no- selfloop , weak connected directed graph (V, E) Among vertices in V T here is one and only one source vertex There is one and only one sink vertex For each vertex v, there is a path

sagira
Download Presentation

Maximum Flow

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Maximum Flow HaidongXue (Haydon) 10/28/2013

  2. What is a flow network? • A no-selfloop, weak connected directed graph (V, E) • Among vertices in V • There is one and only one source vertex • There is one and only one sink vertex • For each vertex v, there is a path • No antiparallel edges in E • if • A Capacity is defined on each edge: V E

  3. What is a flow network? • A flow on a flow network • i.e. does not exceed (capasity constraint) • (flow conservation) Note: In some definition of flow networks, negative flows are allowed; in these slides, we keep consistent with the textbook

  4. Red: flow Green: capacity What is a flow network? • Are they flow networks with valid flows? 17/12 12/12 12/12 20/20 15/20 15/20 11/16 11/16 11/16 1/4 1/4 6/4 7/7 7/7 4/9 4/9 4/9 Not connected Violate the capacity constraint 4/4 4/4 13/13 8/13 10/13 11/14 11/14 12/12 15/20 11/16 1/4 7/7 4/9 4/4 8/13 11/14 Multiple Sinks Violate the flow conservation rule

  5. Red: flow Green: capacity What is a flow network? • Are they flow networks with valid flows? 0/7 12/12 12/12 12/12 15/20 15/20 15/20 11/16 11/16 11/16 s t 1/4 1/4 1/4 7/7 7/7 7/7 0/10 4/9 4/9 4/9 have a self-loop have an antiparallel edge 4/4 4/4 4/4 8/13 8/13 8/13 11/14 11/14 11/14 1 source; 1 sink; no self-loop; no antiparallel edge; no flow exceed capacity; for all non-source non-sink vertices, total flow is 0. Page 710 Fig 26.1(b)

  6. What if the real problem has antiparallel edges or multiple sinks or sources? v1 v1 10 V’ 4 Why do we disallow antiparallel edges? 10 4 10 Add an intermediate vertex to eliminate antiparallel edges v2 v2 s1 10 3  12 s2 t1  5 15  6 8   t s3 t2 s 20 14  13 7  s4 t3  11 18 2 s5 Add a super-source and a super sink

  7. What is the max-flow problem? 12/12 15/20 11/16 s t 1/4 7/7 4/9 The flow of a flow network, : the net output of the source i.e., - 4/4 8/13 11/14 In the above example, = 11+8=19

  8. What is the max-flow problem? Given a flow network, there could be more than one valid flows: = 17 The maximum-flow problem: Given a flow network, find a valid flow that has the maximum 12/12 10/12 12/12 19/20 15/20 15/20 9/16 12/16 11/16 s s s t t t 1/4 1/4 4 7/7 7/7 7/7 9 4/9 2/9 = 19 4/4 2/4 4/4 11/13 8/13 8/13 11/14 9/14 11/14 = 23 (it is a max-flow)

  9. Find a maximum flow • Input: a flow network • Output: a flow net work with max-flow 12/12 12 19/20 20 16 12/16 s s t t 4 4 7 7/7 9 9 4 4/4 13 11/13 14 11/14

  10. Find a maximum flow- residual networks • What are the possible modifications can be made on an edge that does not break the capacity constraint? 8/12 4/20 4/16 s t 4/4 7 4/9 4/4 4/13 4/14 Increase it at most by 10 = c-f=14-4; Decrease it at most by 4=f. To represent a possible modifications, residual networks are introduced

  11. Find a maximum flow - residual networks A flow network with a valid flow 12/12 15/20 11/16 s t 1/4 7/7 4/9 4/4 8/13 A residual network of where 11/14 5 15 12 5 11 3 s t 5 1 4 7 8 4 11 5 3

  12. Find a maximum flow – augmenting paths 12/12 15/20 11/16 s t 1/4 7/7 4/9 An augmenting path is a simple path from s to t in the residual network 4/4 8/13 11/14 5 15 12 5 15 12 5 11 3 s t 5 5 1 4 7 11 3 s t 5 8 1 4 7 8 4 11 5 4 11 5 3 3

  13. Find a maximum flow – augmenting paths An augmenting path is a simple path from s to t in the residual network 5 15 12 4/ 5 11 3 s t 5 1 7 4/ 4 8 4 11 4/ 5 Guaranteed by Corollary 26.3 , modifying according to this flow on does not break the flow conservation rule 3 The residual capacity of an augmenting path: The flow defined by

  14. Find a maximum flow – augmenting paths 5 15 12 12/12 15/20 4/ 5 11/16 11 3 s t 5 1 7 4/ 4 8 s t 1/4 7/7 4/9 4 11 4/ 5 3 Modifyaccording to 4/4 8/13 11/14 12/12 11/16 19/20 s t 1/4 7/7 0/9 4/4 12/13 11/14

  15. Find a maximum flow • Corollary 26.3 guarantees that: • is a valid flow in • Theorem 26.6 (Max-flow min-cut theorem) further tells that: • If there is no augmenting path in , in is a maximum flow Do you have any idea to develop an algorithm to find a max-flow?

  16. Find a maximum flow – the basic Ford-Fulkerson algorithm FORD-FULKERSON() for each ; whilethere exist pathfrom toin for each in p if+ else Initialize to 0 Repeatedly find a , and apply the modification until there is no augmenting path Is it correct? What is the time complexity?

  17. Find a maximum flow – Analysis of F-F • Is it correct in general? • No! • A counter example: http://en.wikipedia.org/wiki/Ford%E2%80%93Fulkerson_algorithm#Non-terminating_example • Is it correct with integer capacities? • Yes. • To find a, if using DFS or BFS that is O(|V|+|E|)=O(|E|) • At most, need to it times • Time complexity:

  18. Find a maximum flow – Edmonds-Karp algorithm EDMONDS-KARP() for each ; while there exist a BFS pathfrom toin for each in p if+ else

  19. Find a maximum flow – Analysis of E-K • Theorem 26.8 • The total number of flow augmentations performed by the algorithm is O(|V||E|) • Time complexity

  20. Find a maximum flow – Example 12 20 16 s t 4 7 9 12 12 12 4 13 20 20 20 16 16 16 14 s s s t t t 4 4 4 7 7 7 9 9 9 12/ 12 12/ 8 12/ 4 4 4 4 13 13 13 12 12 s t 4 7 14 14 14 9 4 13 14 12 12/ 8 12/ 4 12/ 12 12 s t 4 7 9 4 4 4 4/ 9 4/ 10 4/

  21. Find a maximum flow – Example 12 12/ 12/ 8 12/ 4 12/ 12/ 12 12 s t 4 7 9 4 4 12 12 4 4/ 4/ 9 4/ 20 20 10 4/ 16 16 s s t t 4 4 7 7 9 9 12 19/ 1 4 4 4 12 13 13 19 s t 4 7 7/ 9 14 14 11 11 4 2 11/ 11/ 3 A max-flow cannot be reached by

  22. Maximum bipartite matching • A problem can be converted to the max-flow problem • Bipartite Graph • A undirected graph (V, E) • , • All edges in E go betweenand • Every vertex in V has at least one incident edge

  23. Maximum bipartite matching L L L R R R

  24. Maximum bipartite matching A matching , L L L R R R The maximum bipartite matching problem: find a M with the maximum

  25. Maximum bipartite matching Convert it to a max-flow problem L R L R s t Set each edge with a capacity of 1 By Lemma 26.9,

  26. Summary • Max-flow problem • Flow network • The flow of • The residual network • A augmenting path of • , • Ford-Fulkerson, Edmonds-Karps • Maximum bipartite Matching • Bipartite graph • Matching • Convert a bipartite graph to a flow net work

  27. FYI • Your next assignment has already been posted • Chaoyang’s will be holding extended office hours on Wednesdays from 12:00pm-4:00pm • ACM has an event tomorrow Room 460, University Center Distributed Content Delivery Systems with Real Time Route Optimization Free knowledge, pizza, soda, donut, and a door price

  28. Pop Quiz Write down your official name clearly on the right-top corner in the form: Last Name, First Name. (e.g. Xue, Haidong)

More Related