Chapter 7. Network Flow. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
|
|
- Angelina Bennett
- 6 years ago
- Views:
Transcription
1 Chapter 7 Network Flow Slide by Kevin Wayne. Copyright 5 Pearon-Addion Weley. All right reerved.
2 Soviet Rail Network, 55 Reference: On the hitory of the tranportation and maximum flow problem. Alexander Schrijver in Math Programming, :,.
3 Maximum Flow and Minimum Cut Max flow and min cut. Two very rich algorithmic problem. Cornertone problem in combinatorial optimization. Beautiful mathematical duality. Nontrivial application / reduction. Data mining. Open-pit mining. Project election. Airline cheduling. Bipartite matching. Baeball elimination. Image egmentation. Network reliability. Ditributed computing. Egalitarian table matching. Security of tatitical data. Network intruion detection. Multi-camera cene recontruction. Many many more... Network connectivity.
4 Minimum Cut Problem Flow network. Abtraction for ubtance flowing through the edge. G = (V, E) = directed graph, no parallel edge. Two ditinguihed node: = ource, t = target. c(e) = capacity of edge e ource 5 t target capacity 5 5 7
5 Cut Def. An -t cut i a partition (A, B) of V with A and t B. Def. The capacity of a cut (A, B) i: cap( A, B) = c(e) e out of A t A Capacity = = 5
6 Cut Def. An -t cut i a partition (A, B) of V with A and t B. Def. The capacity of a cut (A, B) i: cap( A, B) = c(e) e out of A t A Capacity = =
7 Minimum Cut Problem Min -t cut problem. Find an -t cut of minimum capacity t A Capacity = + + = 7
8 Flow Def. An -t flow i a function that atifie: For each e E: f (e) c(e) (capacity) For each v V {, t}: f (e) = f (e) (conervation) e in to v e out of v Def. The value of a flow f i: v( f ) = f (e) e out of t capacity 5 5 flow 7 Value =
9 Flow Def. An -t flow i a function that atifie: For each e E: f (e) c(e) (capacity) For each v V {, t}: f (e) = f (e) (conervation) e in to v e out of v Def. The value of a flow f i: v( f ) = f (e) e out of t capacity 5 5 flow 7 Value =
10 Maximum Flow Problem Max flow problem. Find -t flow of maximum value t capacity 5 5 flow 7 Value =
11 Flow and Cut Flow value lemma. Let f be any flow, and let (A, B) be any -t cut. Then, the net flow ent acro the cut i equal to the amount leaving. e out of A f (e) e in to A f (e) = v( f ) t A Value =
12 Flow and Cut Flow value lemma. Let f be any flow, and let (A, B) be any -t cut. Then, the net flow ent acro the cut i equal to the amount leaving. e out of A f (e) e in to A f (e) = v( f ) t A Value = =
13 Flow and Cut Flow value lemma. Let f be any flow, and let (A, B) be any -t cut. Then, the net flow ent acro the cut i equal to the amount leaving. e out of A f (e) e in to A f (e) = v( f ) t A Value = =
14 Flow and Cut Flow value lemma. Let f be any flow, and let (A, B) be any -t cut. Then f (e) f (e) = v( f ). e out of A e in to A Pf. v( f ) = f (e) e out of by flow conervation, all term except v = are = v ( f (e) f (e) ) A e out of v e in to v = f (e) e out of A f (e). e in to A
15 Flow and Cut Weak duality. Let f be any flow, and let (A, B) be any -t cut. Then the value of the flow i at mot the capacity of the cut. Cut capacity = Flow value t A Capacity = 5
16 Flow and Cut Weak duality. Let f be any flow. Then, for any -t cut (A, B) we have v(f) cap(a, B). Pf. v( f ) = f (e) e out of A e out of A f (e) c(e) e out of A = cap(a, B) e in to A f (e) A 7 B t
17 Certificate of Optimality Corollary. Let f be any flow, and let (A, B) be any cut. If v(f) = cap(a, B), then f i a max flow and (A, B) i a min cut. Value of flow = Cut capacity = Flow value t A
18 Toward a Max Flow Algorithm Greedy algorithm. Start with f(e) = for all edge e E. Find an -t path P where each edge ha f(e) < c(e). Augment flow along path P. Repeat until you get tuck. t Flow value =
19 Toward a Max Flow Algorithm Greedy algorithm. Start with f(e) = for all edge e E. Find an -t path P where each edge ha f(e) < c(e). Augment flow along path P. Repeat until you get tuck. X X t X Flow value =
20 Toward a Max Flow Algorithm Greedy algorithm. Start with f(e) = for all edge e E. Find an -t path P where each edge ha f(e) < c(e). Augment flow along path P. Repeat until you get tuck. locally optimality global optimality t t greedy = opt =
21 Reidual Graph Original edge: e = (u, v) E. capacity Flow f(e), capacity c(e). u 7 v flow Reidual edge. "Undo" flow ent. e = (u, v) and e R = (v, u). reidual capacity Reidual capacity: u v c f (e) = { c(e)-f(e) if e E reidual capacity f(e R ) if e R E Reidual graph: G f = (V, E f ). Reidual edge with poitive reidual capacity. E f = {e : f(e) < c(e)} {e R : f(e) > }.
22 Ford-Fulkeron Algorithm G: capacity 5 t
23 Ford-Fulkeron Algorithm G: flow capacity 5 t Flow value =
24 Ford-Fulkeron Algorithm G: X X flow capacity X 5 t Flow value = G f : reidual capacity 5 t
25 Ford-Fulkeron Algorithm G: X X X X 5 t Flow value = G f : 5 t 5
26 Ford-Fulkeron Algorithm G: X X X X 5 t Flow value = G f : 7 5 t
27 Ford-Fulkeron Algorithm G: X X X X 5 t Flow value = G f : 5 t 7
28 Ford-Fulkeron Algorithm X G: X 7 X X X 5 t Flow value = G f : 5 t
29 Ford-Fulkeron Algorithm G: 7 5 t Flow value = G f : 7 5 t
30 Ford-Fulkeron Algorithm G: 7 5 t Cut capacity = Flow value = G f : 7 5 t
31 Augmenting Path Algorithm Augment(f, c, P) { b bottleneck(p,c,f) foreach e P { } if (e E) f(e) f(e) + b ele f(e R ) f(e R ) - b } return f min reidual capacity of edge on P forward edge revere edge Ford-Fulkeron(G,, t, c) { foreach e E f(e) G f reidual graph } while (there exit augmenting path P in G f ) { f Augment(f, c, P) update G f (along path P) } return f
32 Max-Flow Min-Cut Theorem Augmenting path theorem. Flow f i a max flow iff there are no augmenting path. Max-flow min-cut theorem. [Ford-Fulkeron 5] The value of the max flow i equal to the value of the min cut. Proof trategy. We prove both imultaneouly by howing the following are equivalent: (i) There exit a cut (A, B) uch that v(f) = cap(a, B). (ii) Flow f i a max flow. (iii) There i no augmenting path relative to f. (i) (ii) Thi wa the corollary to weak duality lemma. (ii) (iii) We how contrapoitive: Let f be a flow. If there exit an augmenting path, then we can improve f by ending flow along path.
33 Proof of Max-Flow Min-Cut Theorem (iii) (i) Let f be a flow with no augmenting path. Let A be et of vertice reachable from in reidual graph. By definition of A, A. By definition of G f, t A. v( f ) = f (e) e out of A = c(e) e out of A = cap(a, B) e in to A f (e) (otherwie connect outide A) A B t original network
34 Running Time Aumption. All capacitie are integer between and C. Invariant. Every flow value f(e) and every reidual capacitie c f (e) remain an integer throughout the algorithm. Theorem. The algorithm terminate in at mot v(f*) nc iteration. Pf. Each augmentation increae value by at leat. Corollary. If C =, Ford-Fulkeron run in O(mn) time. Integrality theorem. If all capacitie are integer, then there exit a max flow f for which every flow value f(e) i an integer. Pf. Since algorithm terminate, theorem follow from invariant.
35 7. Chooing Good Augmenting Path
36 Ford-Fulkeron: Exponential Number of Augmentation Q. I generic Ford-Fulkeron algorithm polynomial in input ize? m, n, and log C A. No. If C=larget capacity, then algorithm can take C iteration. X C C X X C C t X X X t C C X C iteration C C X X
37 Chooing Good Augmenting Path Ue care when electing augmenting path. Some choice lead to exponential algorithm. Clever choice lead to polynomial algorithm. If capacitie are irrational, algorithm not guaranteed to terminate! Goal: chooe augmenting path o that: Can find augmenting path efficiently. Few iteration. Chooe augmenting path with: [Edmond-Karp 7, Dinitz 7] Max bottleneck capacity. Sufficiently large bottleneck capacity. Fewet number of edge. 7
38 Capacity Scaling Intuition. Chooing path with highet bottleneck capacity increae flow by max poible amount. Don't worry about finding exact highet bottleneck path. Maintain caling parameter Δ. Let G f (Δ) be the ubgraph of the reidual graph coniting of only edge with capacity at leat Δ. t t 7 7 G f G f ()
39 Capacity Scaling Scaling-Max-Flow(G,, t, c, C) { foreach e E f(e) Δ larget power of C G f reidual graph } while (Δ ) { G f (Δ) Δ-reidual graph while (there exit augmenting path P in G f (Δ)) { f augment(f, c, P) update G f (Δ) (along P) } Δ Δ / } return f
40 Ford-Fulkeron Algorithm G: flow capacity 5 t Flow value =
41 Ford-Fulkeron Algorithm with Capacity Scaling G: flow capacity 5 t Flow value = Δ = G f : reidual capacity 5 t
42 Ford-Fulkeron Algorithm with Capacity Scaling G: X X flow capacity X 5 t Flow value = G f (): reidual capacity 5 t
43 Ford-Fulkeron Algorithm with Capacity Scaling G: 5 t Flow value = Δ = G f : 5 t
44 Ford-Fulkeron Algorithm with Capacity Scaling G: 5 t Flow value = Δ = G f (): 5 t
45 Ford-Fulkeron Algorithm with Capacity Scaling G: X X X X 5 t Flow value = G f (): 5 t 5
46 Ford-Fulkeron Algorithm with Capacity Scaling G: 5 t Flow value = Δ = G f : 5 t
47 Ford-Fulkeron Algorithm with Capacity Scaling G: 5 t Flow value = Δ = G f (): 5 t 7
48 Ford-Fulkeron Algorithm with Capacity Scaling X G: X 5 X X X 5 t Flow value = G f (): 5 t
49 Ford-Fulkeron Algorithm with Capacity Scaling G: 5 5 t Flow value = 7 Δ = G f : 5 5 t
50 Ford-Fulkeron Algorithm with Capacity Scaling G: X X5 7 X 5 t Flow value = 7 G f (): 5 5 t 5
51 Ford-Fulkeron Algorithm with Capacity Scaling G: 7 5 t Flow value = Δ = G f : 7 5 t 5
52 Ford-Fulkeron Algorithm with Capacity Scaling G: 7 5 t Flow value = Δ = G f (): 7 5 t 5
53 Ford-Fulkeron Algorithm with Capacity Scaling G: 7 5 t Flow value = G f (): 7 5 t 5
54 Ford-Fulkeron Algorithm with Capacity Scaling G: 7 5 t Cut capacity = Flow value = G f (): 7 5 t A = {vertice reachable from in G f ()} 5
55 Capacity Scaling: Correctne Aumption. All edge capacitie are integer between and C. Integrality invariant. All flow and reidual capacity value are integral. Correctne. If the algorithm terminate, then f i a max flow. Pf. By integrality invariant, when Δ = G f (Δ) = G f. Upon termination of Δ = phae, there are no augmenting path. 55
56 Capacity Scaling: Running Time Lemma. Let f be the flow at the end of a Δ-caling phae. Then the value of the maximum flow i at mot v(f) + m Δ. Pf. (almot identical to proof of max-flow min-cut theorem) We how that at the end of a Δ-phae, there exit a cut (A, B) uch that cap(a, B) v(f) + m Δ. Chooe A to be the et of node reachable from in G f (Δ). By definition of A, A. By definition of G f (Δ), t A. v( f ) = f (e) e out of A (c(e) e out of A = c(e) e in to A Δ) f (e) e in to A e out of A e out of A e in to A cap(a, B) - m Δ Δ Δ Δ A B t original network 5
57 Capacity Scaling: Running Time Lemma. The outer while loop repeat + log C time. Pf. Initially C/ < Δ C < Δ. Δ decreae by a factor of each iteration. Lemma. Let f be the flow at the end of a Δ-caling phae. Then the value of the maximum flow i at mot v(f) + m Δ. proof on previou lide Lemma. There are at mot m augmentation per caling phae. Pf. Initially, each of the m edge can carry at mot C<Δ flow. In general, conider the ituation at the beginning of a Δ-phae. Each augmentation in a Δ-phae will increae v(f) by at leat Δ. Let f be the flow at the end of the previou caling (Δ-)phae. Lemma v(f*) v(f) + m (Δ). Theorem. The caling max-flow algorithm find a max flow in O(m log C) augmentation. It can be implemented to run in O(m log C) time. 57
58 Chapter 7 Network Flow Slide by Kevin Wayne. Copyright 5 Pearon-Addion Weley. All right reerved. 5
Algorithm Design and Analysis
Algorithm Deign and Analyi LECTURES 1-1 Network Flow Flow, cut Ford-Fulkeron Min-cut/max-flow theorem Adam Smith // A. Smith; baed on lide by E. Demaine, C. Leieron, S. Rakhodnikova, K. Wayne Detecting
More informationChapter 7. Network Flow. CS 350: Winter 2018
Chapter 7 Network Flow CS 3: Winter 1 1 Soviet Rail Network, Reference: On the hitory of the tranportation and maximum flow problem. Alexander Schrijver in Math Programming, 1: 3,. Maximum Flow and Minimum
More informationToday. Flow review. Augmenting paths. Ford-Fulkerson Algorithm. Intro to cuts (reason: prove correctness)
Today Flow review Augmenting path Ford-Fulkeron Algorithm Intro to cut (reaon: prove correctne) Flow Network = ource, t = ink. c(e) = capacity of edge e Capacity condition: f(e) c(e) Conervation condition:
More informationRunning Time. Assumption. All capacities are integers between 1 and C.
Running Time Assumption. All capacities are integers between and. Invariant. Every flow value f(e) and every residual capacities c f (e) remains an integer throughout the algorithm. Theorem. The algorithm
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LETURE 28 Network Flow hoosing good augmenting paths apacity scaling algorithm A. Smith /4/2008 A. Smith; based on slides by K. Wayne and S. Raskhodnikova Pre-break: Ford-Fulkerson
More informationSoviet Rail Network, 1955
Sovie Rail Nework, 1 Reference: On he hiory of he ranporaion and maximum flow problem. Alexander Schrijver in Mah Programming, 1: 3,. Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic
More information! Abstraction for material flowing through the edges. ! G = (V, E) = directed graph, no parallel edges.
Sovie Rail Nework, haper Nework Flow Slide by Kevin Wayne. opyrigh Pearon-ddion Weley. ll righ reerved. Reference: On he hiory of he ranporaion and maximum flow problem. lexander Schrijver in Mah Programming,
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LETURE 2 Network Flow Finish bipartite matching apacity-scaling algorithm Adam Smith 0//0 A. Smith; based on slides by E. Demaine,. Leiserson, S. Raskhodnikova, K. Wayne Marriage
More informationSoviet Rail Network, 1955
Ch7. Network Flow Soviet Rail Network, 955 Reference: On the history of the transportation and maximum flow problems. Alexander Schrijver in Math Programming, 9: 3, 2002. 2 Maximum Flow and Minimum Cut
More informationAgenda. Soviet Rail Network, We ve done Greedy Method Divide and Conquer Dynamic Programming
Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Now Flow Networks, Max-flow Min-cut and Applications c Hung Q. Ngo (SUNY at Buffalo) CSE 531 Algorithm Analysis and Design 1 / 52
More informationAlgorithms and Theory of Computation. Lecture 11: Network Flow
Algorithms and Theory of Computation Lecture 11: Network Flow Xiaohui Bei MAS 714 September 18, 2018 Nanyang Technological University MAS 714 September 18, 2018 1 / 26 Flow Network A flow network is a
More informationCS4800: Algorithms & Data Jonathan Ullman
CS800: Algorithm & Data Jonathan Ullman Lecture 17: Network Flow Chooing Good Augmenting Path Mar 0, 018 Recap Directed graph! = #, % Two pecial node: ource & and ink = ' Edge capacitie ( ) 9 5 15 15 ource
More information6/3/2009. CS 244 Algorithm Design Instructor: t Artur Czumaj. Lecture 8 Network flows. Maximum Flow and Minimum Cut. Minimum Cut Problem.
Maximum Flow and Minimum Cu CS lgorihm Deign Inrucor: rur Czumaj Lecure Nework Max and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy.
More informationMaximum Flow and Minimum Cut
// Sovie Rail Nework, Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy. Nework Flow Flow nework.
More informationMax Flow, Min Cut COS 521. Kevin Wayne Fall Soviet Rail Network, Cuts. Minimum Cut Problem. Flow network.
Sovie Rail Nework, Max Flow, Min u OS Kevin Wayne Fall Reference: On he hiory of he ranporaion and maximum flow problem. lexander Schrijver in Mah Programming, :,. Minimum u Problem u Flow nework.! Digraph
More informationChapter Landscape of an Optimization Problem. Local Search. Coping With NP-Hardness. Gradient Descent: Vertex Cover
Coping With NP-Hardne Chapter 12 Local Search Q Suppoe I need to olve an NP-hard problem What hould I do? A Theory ay you're unlikely to find poly-time algorithm Mut acrifice one of three deired feature
More informationSoviet Rail Network, 1955
7.1 Nework Flow Sovie Rail Nework, 19 Reerence: On he hiory o he ranporaion and maximum low problem. lexander Schrijver in Mah Programming, 91: 3, 00. (See Exernal Link ) Maximum Flow and Minimum Cu Max
More informationAlgorithm Design and Analysis
Algorihm Deign and Analyi LECTURES 17 Nework Flow Dualiy of Max Flow and Min Cu Algorihm: Ford-Fulkeron Capaciy Scaling Sofya Rakhodnikova S. Rakhodnikova; baed on lide by E. Demaine, C. Leieron, A. Smih,
More informationdirected weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time
Network Flow 1 The Maximum-Flow Problem directed weighted graphs as flow networks the Ford-Fulkerson algorithm termination and running time 2 Maximum Flows and Minimum Cuts flows and cuts max flow equals
More informationAlgorithms: Lecture 12. Chalmers University of Technology
Algorithms: Lecture 1 Chalmers University of Technology Today s Topics Shortest Paths Network Flow Algorithms Shortest Path in a Graph Shortest Path Problem Shortest path network. Directed graph G = (V,
More informationChapter 7. Network Flow. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 7 Network Flow Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 7.5 Bipartite Matching Matching Matching. Input: undirected graph G = (V, E). M E is a matching
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 7 Network Flow Application: Bipartite Matching Adam Smith 0//008 A. Smith; based on slides by S. Raskhodnikova and K. Wayne Recently: Ford-Fulkerson Find max s-t flow
More informationGreedy. I Divide and Conquer. I Dynamic Programming. I Network Flows. Network Flow. I Previous topics: design techniques
Algorihm Deign Technique CS : Nework Flow Dan Sheldon April, reedy Divide and Conquer Dynamic Programming Nework Flow Comparion Nework Flow Previou opic: deign echnique reedy Divide and Conquer Dynamic
More informationUndirected Graphs. V = { 1, 2, 3, 4, 5, 6, 7, 8 } E = { 1-2, 1-3, 2-3, 2-4, 2-5, 3-5, 3-7, 3-8, 4-5, 5-6 } n = 8 m = 11
Undirected Graphs Undirected graph. G = (V, E) V = nodes. E = edges between pairs of nodes. Captures pairwise relationship between objects. Graph size parameters: n = V, m = E. V = {, 2, 3,,,, 7, 8 } E
More informationThe Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationMax Flow: Algorithms and Applications
Max Flow: Algorithms and Applications Outline for today Quick review of the Max Flow problem Applications of Max Flow The Ford-Fulkerson (FF) algorithm for Max Flow Analysis of FF, and the Max Flow Min
More informationAlgorithms. Algorithms 6.4 MAXIMUM FLOW
Algorithms ROBERT SEDGEWICK KEVIN WAYNE 6.4 MAXIMUM FLOW Algorithms F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE http://algs4.cs.princeton.edu introduction Ford Fulkerson algorithm maxflow mincut
More informationThe Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationLecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004
18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem
More informationCSC 373: Algorithm Design and Analysis Lecture 12
CSC 373: Algorithm Design and Analysis Lecture 12 Allan Borodin February 4, 2013 1 / 16 Lecture 12: Announcements and Outline Announcements Term test 1 in tutorials. Need to use only two rooms due to sickness
More information10 Max-Flow Min-Cut Flows and Capacitated Graphs 10 MAX-FLOW MIN-CUT
10 Max-Flow Min-Cut 10.1 Flows and Capacitated Graphs Previously, we considered weighted graphs. In this setting, it was natural to thinking about minimizing the weight of a given path. In fact, we considered
More informationThe max flow problem. Ford-Fulkerson method. A cut. Lemma Corollary Max Flow Min Cut Theorem. Max Flow Min Cut Theorem
The max flow problem Ford-Fulkerson method 7 11 Ford-Fulkerson(G) f = 0 while( simple path p from s to t in G f ) 10-2 2 1 f := f + f p output f 4 9 1 2 A cut Lemma 26. + Corollary 26.6 Let f be a flow
More informationMatching. Slides designed by Kevin Wayne.
Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node appear in a mo edge in M. Max maching: find a max cardinaliy maching. Slide deigned by Kevin Wayne. Biparie Maching Biparie
More informationMaximum Flow. Contents. Max Flow Network. Maximum Flow and Minimum Cut
Conen Maximum Flow Conen. Maximum low problem. Minimum cu problem. Max-low min-cu heorem. Augmening pah algorihm. Capaciy-caling. Shore augmening pah. Princeon Univeriy COS Theory o Algorihm Spring Kevin
More informationToday: Max Flow Proofs
Today: Max Flow Proof COSC 58, Algorihm March 4, 04 Many of hee lide are adaped from everal online ource Reading Aignmen Today cla: Chaper 6 Reading aignmen for nex cla: Chaper 7 (Amorized analyi) In-Cla
More informationAlgorithm Design and Analysis
Algorihm Deign and Analyi LECTURE 0 Nework Flow Applicaion Biparie maching Edge-dijoin pah Adam Smih 0//0 A. Smih; baed on lide by E. Demaine, C. Leieron, S. Rakhodnikova, K. Wayne La ime: Ford-Fulkeron
More information7. NETWORK FLOW II. bipartite matching disjoint paths extensions to max flow 3D Memory Reliability survey design airline scheduling image segmentation
7. NETWORK FLOW II bipartite matching disjoint paths extensions to max flow 3D Memory Reliability survey design airline scheduling image segmentation Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison
More informationAlgorithms. Algorithms 6.4 MAXIMUM FLOW
Algorithms ROBERT SEDGEWICK KEVIN WAYNE 6.4 MAXIMUM FLOW Algorithms F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE http://algs4.cs.princeton.edu introduction Ford-Fulkerson algorithm maxflow-mincut
More informationBalanced Network Flows
revied, June, 1992 Thi paper appeared in Bulletin of the Intitute of Combinatoric and it Application 7 (1993), 17-32. Balanced Network Flow William Kocay* and Dougla tone Department of Computer cience
More informationProblem Set 8 Solutions
Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem
More informationAlgorithms 6.4 MAXIMUM FLOW. overview Ford-Fulkerson algorithm analysis Java implementation applications
6.4 MAXIMUM FLOW Algorithms F O U R T H E D I T I O N overview Ford-Fulkerson algorithm analysis Java implementation applications R O B E R T S E D G E W I C K K E V I N W A Y N E Algorithms, 4 th Edition
More information4. Connectivity Connectivity Connectivity. Whitney's s connectivity theorem: (G) (G) (G) for special
4. Connectivity 4.. Connectivity Vertex-cut and vertex-connectivity Edge-cut and edge-connectivty Whitney' connectivity theorem: Further theorem for the relation of and graph 4.. The Menger Theorem and
More informationAlgorithms. Algorithms 6.4 MIN CUT / MAX FLOW
Algorithms ROBERT SEDGEWICK KEVIN WAYNE 6.4 MIN CUT / MAX FLOW Algorithms F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE http://algs4.cs.princeton.edu introduction Ford Fulkerson algorithm maxflow
More information7.5 Bipartite Matching
7. Bipartite Matching Matching Matching. Input: undirected graph G = (V, E). M E is a matching if each node appears in at most edge in M. Max matching: find a max cardinality matching. Bipartite Matching
More informationBogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
More informationFlow Network. The following figure shows an example of a flow network:
Maximum Flow 1 Flow Network The following figure shows an example of a flow network: 16 V 1 12 V 3 20 s 10 4 9 7 t 13 4 V 2 V 4 14 A flow network G = (V,E) is a directed graph. Each edge (u, v) E has a
More informationTwo Applications of Maximum Flow
Two Applications of Maximum Flow The Bipartite Matching Problem a bipartite graph as a flow network maximum flow and maximum matching alternating paths perfect matchings 2 Circulation with Demands flows
More informationLINEAR ALGEBRA METHOD IN COMBINATORICS. Theorem 1.1 (Oddtown theorem). In a town of n citizens, no more than n clubs can be formed under the rules
LINEAR ALGEBRA METHOD IN COMBINATORICS 1 Warming-up example Theorem 11 (Oddtown theorem) In a town of n citizen, no more tha club can be formed under the rule each club have an odd number of member each
More informationChapter 7. Network Flow. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 7 Network Flow Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 7.5 Bipartite Matching Matching Matching. Input: undirected graph G = (V, E). M E is a matching
More informationBipartite Matchings and Stable Marriage
Bipartite Matchings and Stable Marriage Meghana Nasre Department of Computer Science and Engineering Indian Institute of Technology, Madras Faculty Development Program SSN College of Engineering, Chennai
More informationProblem set 1. (c) Is the Ford-Fulkerson algorithm guaranteed to produce an acyclic maximum flow?
CS261, Winter 2017. Instructor: Ashish Goel. Problem set 1 Electronic submission to Gradescope due 11:59pm Thursday 2/2. Form a group of 2-3 students that is, submit one homework with all of your names.
More informationFlows and Cuts. 1 Concepts. CS 787: Advanced Algorithms. Instructor: Dieter van Melkebeek
CS 787: Advanced Algorithms Flows and Cuts Instructor: Dieter van Melkebeek This lecture covers the construction of optimal flows and cuts in networks, their relationship, and some applications. It paves
More informationChapter 6. Dynamic Programming. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 6 Dynamic Programming Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 6.8 Shortest Paths Shortest Paths Shortest path problem. Given a directed graph G = (V,
More informationMyriad of applications
Shortet Path Myriad of application Finding hortet ditance between location (Google map, etc.) Internet router protocol: OSPF (Open Shortet Path Firt) i ued to find the hortet path to interchange package
More informationCMPSCI 611: Advanced Algorithms
CMPSCI 611: Advanced Algorithms Lecture 12: Network Flow Part II Andrew McGregor Last Compiled: December 14, 2017 1/26 Definitions Input: Directed Graph G = (V, E) Capacities C(u, v) > 0 for (u, v) E and
More informationCS 170: Midterm Exam II University of California at Berkeley Department of Electrical Engineering and Computer Sciences Computer Science Division
1 1 April 000 Demmel / Shewchuk CS 170: Midterm Exam II Univerity of California at Berkeley Department of Electrical Engineering and Computer Science Computer Science Diviion hi i a cloed book, cloed calculator,
More informationList Coloring Graphs
Lit Coloring Graph February 6, 004 LIST COLORINGS AND CHOICE NUMBER Thomaen Long Grotzch girth 5 verion Thomaen Long Let G be a connected planar graph of girth at leat 5. Let A be a et of vertice in G
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should
More informationSOME RESULTS ON INFINITE POWER TOWERS
NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite
More informationMaximum flow problem CE 377K. February 26, 2015
Maximum flow problem CE 377K February 6, 05 REVIEW HW due in week Review Label setting vs. label correcting Bellman-Ford algorithm Review MAXIMUM FLOW PROBLEM Maximum Flow Problem What is the greatest
More informationTheoretical Computer Science. Optimal algorithms for online scheduling with bounded rearrangement at the end
Theoretical Computer Science 4 (0) 669 678 Content lit available at SciVere ScienceDirect Theoretical Computer Science journal homepage: www.elevier.com/locate/tc Optimal algorithm for online cheduling
More informationClustering Methods without Given Number of Clusters
Clutering Method without Given Number of Cluter Peng Xu, Fei Liu Introduction A we now, mean method i a very effective algorithm of clutering. It mot powerful feature i the calability and implicity. However,
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 22 Maximum Flow Applications Image segmentation Project selection Extensions to Max Flow Sofya Raskhodnikova 11/07/2016 S. Raskhodnikova; based on slides by E. Demaine,
More informationCS 580: Algorithm Design and Analysis. Jeremiah Blocki Purdue University Spring 2018
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 208 Midterm Exam Anticipate having midterm graded at this point Look for comments on Piazza Common Mistakes Average, Max,
More informationSparse Fault-Tolerant BFS Trees. Merav Parter and David Peleg Weizmann Institute Of Science BIU-CS Colloquium
Spare Fault-Tolerant BFS Tree Merav Parter and David Peleg Weizmann Intitute Of Science BIU-CS Colloquium 16-01-2014 v 5 Breadth Firt Search (BFS) Tree Unweighted graph G=(V,E), ource vertex V. Shortet-Path
More information- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs
LP-Duality ( Approximation Algorithms by V. Vazirani, Chapter 12) - Well-characterized problems, min-max relations, approximate certificates - LP problems in the standard form, primal and dual linear programs
More informationSource slideplayer.com/fundamentals of Analytical Chemistry, F.J. Holler, S.R.Crouch. Chapter 6: Random Errors in Chemical Analysis
Source lideplayer.com/fundamental of Analytical Chemitry, F.J. Holler, S.R.Crouch Chapter 6: Random Error in Chemical Analyi Random error are preent in every meaurement no matter how careful the experimenter.
More information7. NETWORK FLOW I. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley. Last updated on 11/22/17 6:11 AM
7. NETWORK FLOW I max-flow and min-cu problem Ford Fulkeron algorihm max-flow min-cu heorem capaciy-caling algorihm hore augmening pah blocking-flow algorihm imple uni-capaciy nework Lecure lide by Kevin
More informationLINEAR PROGRAMMING III
LINEAR PROGRAMMING III ellipsoid algorithm combinatorial optimization matrix games open problems Lecture slides by Kevin Wayne Last updated on 7/25/17 11:09 AM LINEAR PROGRAMMING III ellipsoid algorithm
More informationPreliminaries. Graphs. E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)}
Preliminaries Graphs G = (V, E), V : set of vertices E : set of edges (arcs) (Undirected) Graph : (i, j) = (j, i) (edges) 1 2 3 5 4 V = {1, 2, 3, 4, 5}, E = {(1, 3), (3, 2), (2, 4)} 1 Directed Graph (Digraph)
More informationTypes of Networks. Internet Telephone Cell Highways Rail Electrical Power Water Sewer Gas
Flow Networks Network Flows 2 Types of Networks Internet Telephone Cell Highways Rail Electrical Power Water Sewer Gas 3 Maximum Flow Problem How can we maximize the flow in a network from a source or
More informationParallel Graph Algorithms (con4nued)
Parallel Graph Algorithms (con4nued) MaxFlow A flow network G=(V,E): a directed graph, where each edge (u,v) E has a nonnega4ve capacity c(u,v)>=0. If (u,v) E, we assume that c(u,v)=0. Two dis4nct ver4ces
More informationNew bounds for Morse clusters
New bound for More cluter Tamá Vinkó Advanced Concept Team, European Space Agency, ESTEC Keplerlaan 1, 2201 AZ Noordwijk, The Netherland Tama.Vinko@ea.int and Arnold Neumaier Fakultät für Mathematik, Univerität
More informationMathematics for Decision Making: An Introduction. Lecture 13
Mathematics for Decision Making: An Introduction Lecture 13 Matthias Köppe UC Davis, Mathematics February 17, 2009 13 1 Reminder: Flows in networks General structure: Flows in networks In general, consider
More informationHybrid Control and Switched Systems. Lecture #6 Reachability
Hbrid Control and Switched Stem Lecture #6 Reachabilit João P. Hepanha Univerit of California at Santa Barbara Summar Review of previou lecture Reachabilit tranition tem reachabilit algorithm backward
More informationList coloring hypergraphs
Lit coloring hypergraph Penny Haxell Jacque Vertraete Department of Combinatoric and Optimization Univerity of Waterloo Waterloo, Ontario, Canada pehaxell@uwaterloo.ca Department of Mathematic Univerity
More informationPart V. Matchings. Matching. 19 Augmenting Paths for Matchings. 18 Bipartite Matching via Flows
Matching Input: undirected graph G = (V, E). M E is a matching if each node appears in at most one Part V edge in M. Maximum Matching: find a matching of maximum cardinality Matchings Ernst Mayr, Harald
More informationNetwork Flows made simple
Network Flows made simple A mild but rigorous introduction to Network Flows Periklis A. Papakonstantinou York University Network Flows is the last algorithmic paradigm we consider. Actually, we only address
More informationNetwork Flow. Data Structures and Algorithms Andrei Bulatov
Nework Flow Daa Srucure and Algorihm Andrei Bulao Algorihm Nework Flow 24-2 Flow Nework Think of a graph a yem of pipe We ue hi yem o pump waer from he ource o ink Eery pipe/edge ha limied capaciy Flow
More informationarxiv: v2 [math.co] 11 Sep 2017
The maximum number of clique in graph without long cycle Ruth Luo September 13, 017 arxiv:1701.0747v [math.co] 11 Sep 017 Abtract The Erdő Gallai Theorem tate that for k 3 every graph on n vertice with
More informationPreemptive scheduling on a small number of hierarchical machines
Available online at www.ciencedirect.com Information and Computation 06 (008) 60 619 www.elevier.com/locate/ic Preemptive cheduling on a mall number of hierarchical machine György Dóa a, Leah Eptein b,
More informationWe say that a flow is feasible for G (or just feasible if the graph and capacity function in question are obvious) if
CS787: Advanced Algorithms Lecture 4: Network Flow We devote this lecture to the network flow problem and the max-flow min-cut theorem. A number of other problems can be cast into a maximum flow or minimum
More informationMaximum flow problem
Maximum flow problem 7000 Network flows Network Directed graph G = (V, E) Source node s V, sink node t V Edge capacities: cap : E R 0 Flow: f : E R 0 satisfying 1. Flow conservation constraints e:target(e)=v
More information22 Max-Flow Algorithms
A process cannot be understood by stopping it. Understanding must move with the flow of the process, must join it and flow with it. The First Law of Mentat, in Frank Herbert s Dune (965) There s a difference
More information3/3/2015. Chapter 7. Network Flow. Maximum Flow and Minimum Cut. Minimum Cut Problem
// Chaper Nework Flow Maximum Flow and Minimum Cu Max flow and min cu. Two very rich algorihmic problem. Cornerone problem in combinaorial opimizaion. Beauiful mahemaical dualiy. Nonrivial applicaion /
More informationGraphs III - Network Flow
Graph III - Nework Flow Flow nework eup graph G=(V,E) edge capaciy w(u,v) 0 - if edge doe no exi, hen w(u,v)=0 pecial verice: ource verex ; ink verex - no edge ino and no edge ou of Aume every verex v
More informationMaximum Flow. Jie Wang. University of Massachusetts Lowell Department of Computer Science. J. Wang (UMass Lowell) Maximum Flow 1 / 27
Maximum Flow Jie Wang University of Massachusetts Lowell Department of Computer Science J. Wang (UMass Lowell) Maximum Flow 1 / 27 Flow Networks A flow network is a weighted digraph G = (V, E), where the
More informationMAXIMUM FLOW. introduction Ford-Fulkerson algorithm maxflow-mincut theorem
MAXIMUM FLOW inroducion Ford-Fulkeron algorihm maxflow-mincu heorem Mincu problem Inpu. An edge-weighed digraph, ource verex, and arge verex. each edge ha a poiive capaciy capaciy 9 10 4 15 15 10 5 8 10
More informationDiscrete Mathematics
Dicrete Mathematic 310 (010) 334 3333 Content lit available at ScienceDirect Dicrete Mathematic journal homepage: www.elevier.com/locate/dic Rank number of grid graph Hannah Alpert Department of Mathematic,
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 26 Computational Intractability Polynomial Time Reductions Sofya Raskhodnikova S. Raskhodnikova; based on slides by A. Smith and K. Wayne L26.1 What algorithms are
More informationParallel Graph Algorithms (con4nued)
Parallel Graph Algorithms (con4nued) MaxFlow A flow network G=(V,E): a directed graph, where each edge (u,v) E has a nonnega4ve capacity c(u,v)>=0. If (u,v) E, we assume that c(u,v)=0. Two dis4nct ver4ces
More information11.2 Stability. A gain element is an active device. One potential problem with every active circuit is its stability
5/7/2007 11_2 tability 1/2 112 tability eading Aignment: pp 542-548 A gain element i an active device One potential problem with every active circuit i it tability HO: TABIITY Jim tile The Univ of Kana
More information12. LOCAL SEARCH. gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria
12. LOCAL SEARCH gradient descent Metropolis algorithm Hopfield neural networks maximum cut Nash equilibria Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley h ttp://www.cs.princeton.edu/~wayne/kleinberg-tardos
More informationElectronic Theses and Dissertations
Eat Tenneee State Univerity Digital Common @ Eat Tenneee State Univerity Electronic Thee and Diertation Student Work 5-208 Vector Partition Jennifer French Eat Tenneee State Univerity Follow thi and additional
More informationPhysics 2212 G Quiz #2 Solutions Spring 2018
Phyic 2212 G Quiz #2 Solution Spring 2018 I. (16 point) A hollow inulating phere ha uniform volume charge denity ρ, inner radiu R, and outer radiu 3R. Find the magnitude of the electric field at a ditance
More informationMathematical Programs Linear Program (LP)
Mathematical Programs Linear Program (LP) Integer Program (IP) Can be efficiently solved e.g., by Ellipsoid Method Cannot be efficiently solved Cannot be efficiently solved assuming P NP Combinatorial
More informationChapter 11. Approximation Algorithms. Slides by Kevin Wayne Pearson-Addison Wesley. All rights reserved.
Chapter 11 Approximation Algorithms Slides by Kevin Wayne. Copyright @ 2005 Pearson-Addison Wesley. All rights reserved. 1 P and NP P: The family of problems that can be solved quickly in polynomial time.
More informationCSE 521: Design & Analysis of Algorithms I
CSE 52: Deign & Analyi of Algorihm I Nework Flow Paul Beame Biparie Maching Given: A biparie graph G=(V,E) M E i a maching in G iff no wo edge in M hare a verex Goal: Find a maching M in G of maximum poible
More informationSampling and the Discrete Fourier Transform
Sampling and the Dicrete Fourier Tranform Sampling Method Sampling i mot commonly done with two device, the ample-and-hold (S/H) and the analog-to-digital-converter (ADC) The S/H acquire a CT ignal at
More informationAlgorithms: COMP3121/3821/9101/9801
NEW SOUTH WALES Algorithms: COMP32/382/90/980 Aleks Ignjatović School of Computer Science and Engineering University of New South Wales LECTURE 7: MAXIMUM FLOW COMP32/382/90/980 / 24 Flow Networks A flow
More information