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Lectures on Network Flows

Lectures on Network Flows. COMP 523: Advanced Algorithmic Techniques Lecturer: Dariusz Kowalski. Overview. Previous lectures: Dynamic programming Weighted interval scheduling Sequence alignment These lectures: Network flows Applications: largest matching in bipartite graphs. Network.

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Lectures on Network Flows

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  1. Lectures on Network Flows COMP 523: Advanced Algorithmic Techniques Lecturer: Dariusz Kowalski Lectures on Network Flows

  2. Overview Previous lectures: • Dynamic programming • Weighted interval scheduling • Sequence alignment These lectures: • Network flows • Applications: largest matching in bipartite graphs Lectures on Network Flows

  3. Network A directed graph G = (V,E) such that • each directed edge e has its nonnegative capacity denoted by ce • there is a node s (source) with no incoming edges • there is a node t (target) with no outgoing edges u 20 10 u,v - internal nodes 30 t s 10 20 v Lectures on Network Flows

  4. Flow s-t flow in G = (V,E) is a function f from E to R+ • capacity condition: for each e, 0  f(e) ce • conservation condition: for each internal node v, ∑e in vf(e) =∑e out vf(e) • there is a node t (target) with no outgoing edges Property: ∑e in tf(e) =∑e out sf(e) u u Network: 20 10 Flow: 20 10 30 t 10 t s s 10 20 10 20 Lectures on Network Flows v v

  5. Useful definitions Given s-t flow f in G = (V,E) and any subset of nodes S • f in(S) = ∑e in Sf(e) • f out(S) = ∑e out Sf(e) Property: f in(t) = f out(s) Example: f in(u,v) = f out(u,v) = 30 u u Network: 20 10 Flow: 20 10 30 t s 10 t s 10 20 10 20 Lectures on Network Flows v v

  6. Problem(s) • What is the maximum value of f in(t) (flow) for a given graph G = (V,E) ? • How to compute it efficiently? Assumption: capacities are positive integers. Example: f in(t) = f out(s) = 30 u u Network: Flow: 20 10 20 10 30 t s 10 t s 10 20 10 20 Lectures on Network Flows v v

  7. Residual graph Assume that we are given a flow f ingraph G. Residual graph Gf • The same nodes, internal and s,t • For each edge e in E with ce > f(e) we put weight ce - f(e) (residual capacity) • For each edge e = (u,v) in E we put weight f(e) to the backward edge (v,u) (residual capacity) u u Residual Graph: Network: u 20 0 Flow: 20 10 20 10 20 t s 30 t 20 10 t s s 0 20 20 10 10 20 v Lectures on Network Flows v v

  8. Augmenting path & augmentation Assume that we are given a flow f ingraph G, andthe corresponding residual graph Gf • Find a new flow in residual graph - through a path with no repeating nodes, and value equal to the minimum capacity on the path (augmenting path) • Update residual graph along the path New residual graph: u New flow: u Network: 10 20 u 20 10 10 10 20 10 20 t s 30 t 10 10 20 t s 20 s 10 v 10 20 20 Lectures on Network Flows v v

  9. Ford-Fulkerson Algorithm • Initialize f(e) = 0 for all e • While there is s-t path P in residual graph • Augment f through path P and get new f and new residual graph Augmentf through path P: • Find minimum capacity on the path • Go through the path and modify weights u New residual graph: Network: u u New flow: 20 10 10 10 20 20 30 t 10 20 10 10 20 t s t s s 10 10 10 20 20 20 v v v Lectures on Network Flows

  10. Analysis Correctness: maximum flow - proof later termination - each time the flow is an integer and advances by at least 1 (assumption about integer capacities) Time: O(mC) • at most C iterations, where C is the value of the maximum flow, m is the number of edges • each iteration takes O(m+n) steps - use DFS to find path P u New residual graph: u u Network: New flow: 20 10 10 10 20 20 30 t 10 20 10 10 20 t s t s s 10 10 10 20 20 20 v v v Lectures on Network Flows

  11. Reminder: Depth-First Search (DFS) Given a directed graph G = (V,E) of diameter D and the root node Goal: find a directed rooted spanning tree such that each edge in graph G corresponds to the ancestor relation in the tree Recursive idea of the algorithm: Repeat until no neighbor of the root is free • Select a free out-neighbor v of the root and add the edge from root to v to partial DFS • Recursively find a DFS’ in graph G restricted to free nodes and node v as the root’ and add it to partial DFS root’ root root’’ Lectures on Network Flows

  12. Implementing DFS Structures: • Adjacency list • List (stack) S • Array Discovered[1…n] Algorithm: • Set S= {root} • Consider the top element v in S • For each out-neighbor w of node v, if w is not Discovered then put w into the stack S and start the next iteration for w as the top element • Otherwise remove v from the stack, add edge (z,v) to partial DFS, where z is the current top element, and start the next iteration for z as the top element Remark: after considering the out-neighbor of node v we remove this neighbor from adjacency list to avoid considering it many times! root’ root root’’ Lectures on Network Flows

  13. Flows vs. Cuts (A,B) - cut in graph G: • A,B is a partition of nodes, s in A, t in B c(A,B)= ∑e out Ac(e) = ∑e in Bc(e) is the capacity of this cut Property: Minimum cut is equal to the maximum flow Example: c(A,B)= 50 u u A B 20 10 20 10 30 t 30 t s s 10 20 10 20 v v Lectures on Network Flows

  14. Max Flow vs. Min Cut For any setAcontaining s we proceed in 3 steps: • value(f)= ∑v in A ∑e out vf(e) - ∑v in A ∑e in vf(e) • value(f)= f out(A) - f in(A) • c(A,B)  f out(A) - f in(A) = value(f) Conclusion: Min-cut  value(f) Example:c(A,B)= 50 andf out(A) = 30 u u A B 20 10 20 10/10 30 t 30/10 t s s 10 20 10/10 20 v v Lectures on Network Flows

  15. FF-algorithm gives Max-flow Suppose FF-algorithm stopped with flow f : • Directed DFS tree rooted in s does not contain t • It means that cut-capacity between nodes in DFS and the remaining ones is 0 in residual graph • It follows that each edge in this cut has been reversed by augmenting flow, which means that c(DFS,DFS’) = value(f) • Using Min-cut  value(f) we get thatfis maximum u u 20 10 20 10 Residual graph: 30 t 10 20 t s s DFS 10 20 10 20 Lectures on Network Flows v Min-cut v

  16. Conclusions Network flow algorithms: • Ford-Fulkerson algorithm in time O(mC) • Correspondence between max-flows and min-cuts Lectures on Network Flows

  17. Exercises READING: • Chapter 7, Sections 7.1, 7.2, and 7.3 EXERCISES: • Modify FF-algorithm to work in time O(m2 log C) • Find an augmentation scheme to guarantee time O(mn) in FF-algorithm independently from integer C Lectures on Network Flows

  18. Overview Previous lecture: • Network flows • Ford-Fulkerson algorithm • Max-flows versus Min-cuts This lecture: • Rational and real capacities in network • Applications: largest matching in bipartite graphs • Application to disjoint paths problem Lectures on Network Flows

  19. Network Given a directed graph G = (V,E) such that • each directed edge e has its nonnegative capacity denoted by ce • there is a node s (source) with no incoming edges • there is a node t (target) with no outgoing edges u 20 10 u,v - internal nodes 30 t s 10 20 v Lectures on Network Flows

  20. Flow s-t flow in G = (V,E) is a function f from E to R+ • capacity condition: for each e, 0  f(e) ce • conservation condition: for each internal node v, ∑e in vf(e) =∑e out vf(e) • there is a node t (target) with no outgoing edges Property: ∑e in tf(e) =∑e out sf(e) u u Network: 20 10 Flow: 20 10 30 t 10 t s s 10 20 10 20 Lectures on Network Flows v v

  21. Problem What is a maximum value f in(t) = f out(s) (flow) for a given graph G = (V,E) ? How to compute it efficiently? Assumption: capacities are positive integers. Example: f in(t) = f out(s) = 30 u u Network: Flow: 20 10 20 10 30 t s 10 t s 10 20 10 20 Lectures on Network Flows v v

  22. Ford-Fulkerson Algorithm • Initialize f(e) = 0 for all e • While there is s-t path P in residual graph • Augment f through path P and get new f and new residual graph Augmentf through path P: • Find minimum capacity on the path • Go through the path and modify weights u New residual graph: Network: u u New flow: 20 10 10 10 20 20 30 t 10 20 10 10 20 t s t s s 10 10 10 20 20 20 v v v Lectures on Network Flows

  23. Non-integer capacities • Rational capacities - rescale them to integers by multiplying by the common multiply • Real capacities - difficult to handle: • Min-cut = Max-flow • FF-algorithm may work very slowly u New residual graph: u u New flow: Graph: 10 10 10 20 20 20 t 30 t 10 20 10 10 20 s t s s 10 10 10 20 20 20 v v v Lectures on Network Flows

  24. Flows vs. Cuts (A,B) - cut in graph G: • A,B is a partition of nodes, s in A, t in B c(A,B)= ∑e out Ac(e) = ∑e in Bc(e) is a capacity of the cut Property: Minimum cut is equal to the maximum flow Example: c(A,B)= 50 u u A B 20 10 20 10 30 t 30 t s s 10 20 10 20 v v Lectures on Network Flows

  25. Applications - largest matching Input: bipartite graph G=(V,W,E) Goal: largest set of non-incident edges (with different ends) Solution using flow algorithms: • Lets direct edges from V to W, introduce s connected to all nodes in V, t connected from all nodes in W • Capacities are 1 for all edges • Max-flow is the largest matching t s Lectures on Network Flows

  26. Applications - disjoint paths Input: network graph G=(V,E) Goal: largest set of edge-disjoint paths from s to t Solution: Using flow algorithms, where each edge has capacity 1 t s Lectures on Network Flows

  27. Disjoint paths cont. Suppose that k is the largest number of edge-disjoint paths from s to t. • It is also a flow: • capacity condition is clear since we push flow with value 1 through each path, and • conservation is satisfied since if a path comes into a node it also goes out Conclusion: the largest number of edge-disjoint paths from s to t is not bigger than Max-flow t s Lectures on Network Flows

  28. Disjoint paths cont. Suppose that x is the value of Max-flow produced by FF-algorithm. How to construct x edge-disjoint paths from s to t ? By induction on the number of edges in the flow. For 0 edges trivial (nothing to do, no even a path) Assume j edges in the flow. Take one of them (s,v) and continue going using edges in the flow: • We go to t - done, since we have path, remove it from the graph and continue by induction • We make a cycle - reduce the flow along the cycle to zero, and the obtained is a flow having the same value but smaller number of edges - next continue by induction t s Lectures on Network Flows

  29. Complexity Time of FF-algorithm: O(mn) ( since C = O(n) ) Time of extracting paths: each edge is considered once while extracting one path, hence total time O(mn) Total time: O(mn) Additional memory: O(m+n) for keeping paths t s Lectures on Network Flows

  30. Conclusions Network flow algorithms: • Rational capacities are easy to deal with • Real capacities are difficult to compute - although we can alternatively check Min-cut • Application to the largest matching problem in time O(mn) (n = C since capacities are 1) • Application to the edge-disjoint paths problem in time O(mn) (n = C since capacities are 1) Lectures on Network Flows

  31. Textbook and Exercises READING: • Chapter 7, Sections 7.5, 7.6 and 7.7 EXERCISES: • Find a network with real capacities for which FF-algorithm works very slowly • Prove formally that FF-algorithm gives the largest matching in the last application • Design the algorithm for finding the largest number of edge-disjoint paths from s to t in undirected network • Design the algorithm for finding the largest number of node-disjoint paths from s to t in both directed and undirected networks Lectures on Network Flows

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