CMPS 6610/4610 Fall Flow Networks. Carola Wenk Slides adapted from slides by Charles Leiserson
|
|
- Benjamin Osborne
- 5 years ago
- Views:
Transcription
1 CMP 6610/4610 Fall 2016 Flow Nework Carola Wenk lide adaped rom lide by Charle Leieron
2 Max low and min c Fndamenal problem in combinaorial opimizaion Daliy beween max low and min c Many applicaion: Biparie maching Image egmenaion Airline chedling Nework reliabiliy rey deign Baeball eliminaion Gene ncion predicion 2
3 Flow nework Deiniion. A low nework i a direced graph G = V E wih wo diingihed erice: a orce and a ink. Each edge E ha a nonnegaie capaciy c. I E hen c = 0. We reqire ha i E hen E. 2 Example:
4 Flow nework Deiniion. A poiie low on G i a ncion : V V R aiying he ollowing: Capaciy conrain: For all V 0 c. Flow coneraion: For all V \{ } V V The ale o a low i he ne low o o he orce: V V 0 4
5 A low on a nework low capaciy 1:3 2:2 2:3 1:3 1:1 2:2 2:3 1:2 Flow coneraion like Kircho crren law: Flow ino i = 3. Flow o o i = 3. The ale o hi low i = 3. 5
6 The maximm-low problem Maximm-low problem: Gien a low nework G ind a low o maximm ale on G. 2:3 2:2 2:3 0:3 1:1 2:2 3:3 2:2 The ale o he maximm low i 4. 6
7 7 C Deiniion. A c T o a low nework G = V E i a pariion o V ch ha and T. I i a low on G hen he ne low acro he c i The capaciy o he c i T T T T c T c
8 C 2:3 0:3 2:2 1:1 2:3 T 2:2 2:2 3:3 T = = 4 ct = = 9 8
9 9 Anoher characerizaion o low ale Lemma. For any low and any c T we hae = T. } \{ T T T T T V V V V V V V V Proo: 0
10 10 Upper bond on he maximm low ale Theorem. The ale o any low i bonded rom aboe by he capaciy o any c: ct.. Proo. T c c T T T T T
11 Flow ino he ink 2:3 2:2 2:3 0:3 1:1 2:2 3:3 2:2 = {} V\{} = V\{} = 4 11
12 Reidal nework Deiniion. Le be a low on G = V E. The reidal nework G V E i he graph wih reidal capaciie c ie c = i E 0 oherwie Edge in E admi more low. Lemma. E 2 E. 12
13 Agmening pah Deiniion. Le p be a pah rom o in G. The reidal capaciy o p i c p min { c }. p I c p > 0 hen p i called an agmening pah in G wih repec o. The low ale can be increaed along an agmening pah p by c p. Ex.: G: 3:5 2:6 4:5 2:3 2:5 c p = 2 G : p
14 Agmening pah con. G: c p = 2 G : p 3:5 2: :5 2: :5 3 G: 3 5:5 2 4:6 1. 2:5 0:3 1 4:5 2 G :
15 Max-low min-c heorem Theorem. The ollowing are eqialen: 1. = c T or ome c T. min-c 2. i a maximm low. 3. admi no agmening pah. Proo. 1 2: ince c T or any c T he ampion ha c T implie ha i a maximm low. 2 3: I here wa an agmening pah he low ale cold be increaed conradicing he maximaliy o. 15
16 Proo conined 3 1: Deine = { V : here exi an agmening pah in G rom o } and le T = V \. ince admi no agmening pah here i no pah rom o in G. Hence and T and h T i a c. Conider any erice and T. pah in G T We m hae c = 0 ince i c > 0 hen no T a amed. Th = c i E ince c = c. And oherwie =0. mming oer all and T yield T = c T and ince = T he heorem ollow. 16
17 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G:
18 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 0:10 9 0:10 9 0:1 0:10 9 0:
19 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 0:10 9 0:10 9 0:1 0:10 9 0:
20 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 0:10 9 1:1 0:10 9 1:
21 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 0:10 9 1:1 0:10 9 1:
22 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 1:10 9 0:1 1:10 9 1:
23 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 1:10 9 1:10 9 0:1 1:10 9 1:
24 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Can be low: G: 2:10 9 1:10 9 1:1 1:10 9 2: billion ieraion on a graph wih 4 erice! 24
25 Ford-Flkeron max-low algorihm Algorihm: [ ] 0 or all E while an agmening pah p in G wr exi: agmen by c p Rnime: Le * be he ale o a maximm low and ame i i an inegral ale. The iniializaion ake O E ime There are a mo * ieraion o he loop Find an agmening pah wih DF in O V + E ime Each agmenaion ake O V ime O E * ime in oal 25
26 Edmond-Karp algorihm Edmond and Karp noiced ha many people implemenaion o Ford-Flkeron agmen along a breadh-ir agmening pah: a hore pah in G rom o where each edge wih poiie capaciy ha weigh 1. Thee implemenaion wold alway rn relaiely a. ince a breadh-ir agmening pah can be ond in O V + E ime heir analyi which proided he ir polynomial-ime bond on maximm low oce on bonding he nmber o low agmenaion. In independen work Dinic alo gae polynomial-ime bond. 26
27 Rnning ime o Edmond- Karp One can how ha he nmber o low agmenaion i.e. he nmber o ieraion o he while loop i O V E. Breadh-ir earch rn in O V + E ime All oher bookkeeping i O V per agmenaion. The Edmond-Karp maximm-low algorihm rn in O V E 2 ime. 27
28 Monooniciy lemma Lemma. Le = be he breadh-ir diance rom o in G. Dring he Edmond- Karp algorihm increae monoonically. Proo. ppoe ha i a low on G and agmenaion prodce a new low. Le =. We ll how ha by indcion on. For he bae cae = 0. For he indcie cae conider a breadh-ir pah in G. We m hae + 1 ince bpah o hore pah are hore pah. Cerainly E and now conider wo cae depending on wheher E. 28
29 Cae 1 Cae: E. We hae + 1 riangle ineqaliy + 1 indcion = breadh-ir pah and h monooniciy o i eablihed. 29
30 Cae 2 Cae: E. ince E he agmening pah p ha prodced rom m hae inclded. Moreoer p i a breadh-ir pah in G : p =. Th we hae 1 breadh-ir pah 1 indcion 2 breadh-ir pah < hereby eablihing monooniciy or hi cae oo. 30
31 Coning low agmenaion Theorem. The nmber o low agmenaion in he Edmond-Karp algorihm Ford-Flkeron wih breadh-ir agmening pah i O V E. Proo. Le p be an agmening pah and ppoe ha we hae c = c p or edge p. Then we ay ha i criical and i diappear rom he reidal graph aer low agmenaion. Example: G : c p = 2 31
32 Coning low agmenaion Theorem. The nmber o low agmenaion in he Edmond-Karp algorihm Ford-Flkeron wih breadh-ir agmening pah i O V E. Proo. Le p be an agmening pah and ppoe ha we hae c = c p or edge p. Then we ay ha i criical and i diappear rom he reidal graph aer low agmenaion. Example: G :
33 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: 33
34 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: = 5 = 6 34
35 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example: = 5 = 6 35
36 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:
37 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:
38 Coning low agmenaion conined The ir ime an edge i criical we hae = + 1 ince p i a breadh-ir pah. We m wai nil i on an agmening pah beore can be criical again. Le be he diance ncion when i on an agmening pah. Then we hae + 1 breadh-ir pah + 1 monooniciy + 2 breadh-ir pah. Example:
39 Rnning ime o Edmond- Karp Diance ar o nonnegaie neer decreae and are a mo V 1 nil he erex become nreachable. Th occr a a criical edge O V ime becae increae by a lea 2 beween occrrence. ince he reidal graph conain O E edge he nmber o low agmenaion i O V E. Corollary. The Edmond-Karp maximm-low algorihm rn in O V E 2 ime. Proo. Breadh-ir earch rn in O E ime and all oher bookkeeping i O V per agmenaion. 39
40 Be o dae The aympoically ae algorihm o dae or maximm low de o King Rao and Tarjan rn in O V E log E / V log V V ime. I we allow rnning ime a a ncion o edge weigh he ae algorihm or maximm low de o Goldberg and Rao rn in ime Omin{ V 2/3 E 1/2 } E log V 2 / E + 2 log C where C i he maximm capaciy o any edge in he graph. 40
Flow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001
CS 545 Flow Nework lon Efra Slide courey of Charle Leieron wih mall change by Carola Wenk Flow nework Definiion. flow nework i a direced graph G = (V, E) wih wo diinguihed verice: a ource and a ink. Each
More informationMaximum Flow 5/6/17 21:08. Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015
Maximm Flo 5/6/17 21:08 Preenaion for e ih he exbook, Algorihm Deign and Applicaion, by M. T. Goodrich and R. Tamaia, Wiley, 2015 Maximm Flo χ 4/6 4/7 1/9 2015 Goodrich and Tamaia Maximm Flo 1 Flo Neork
More informationMaximum Flow 3/3 4/6 1/1 4/7 3/3. s 3/5 1/9 1/1 3/5 2/2. 1/18/2005 4:03 AM Maximum Flow 1
Maximm Flo χ 4/6 4/7 1/9 8/2005 4:03 AM Maximm Flo 1 Oline and Reading Flo neork Flo ( 8.1.1) C ( 8.1.2) Maximm flo Agmening pah ( 8.2.1) Maximm flo and minimm c ( 8.2.1) Ford-Flkeron algorihm ( 8.2.2-8.2.3)
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 informationNetwork flows. The problem. c : V V! R + 0 [ f+1g. flow network G = (V, E, c), a source s and a sink t uv not in E implies c(u, v) = 0
Nework flow The problem Seing flow nework G = (V, E, c), a orce and a ink no in E implie c(, ) = 0 Flow from o capaciy conrain kew-ymmery flow-coneraion ale of he flow jfj = P 2V Find a maximm flow from
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 informationMain Reference: Sections in CLRS.
Maximum Flow Reied 09/09/200 Main Reference: Secion 26.-26. in CLRS. Inroducion Definiion Muli-Source Muli-Sink The Ford-Fulkeron Mehod Reidual Nework Augmening Pah The Max-Flow Min-Cu Theorem The Edmond-Karp
More informationFlow Networks Alon Efrat Slides courtesy of Charles Leiserson with small changes by Carola Wenk. Flow networks. Flow networks CS 445
CS 445 Flow Nework lon Efr Slide corey of Chrle Leieron wih mll chnge by Crol Wenk Flow nework Definiion. flow nework i direced grph G = (V, E) wih wo diingihed erice: orce nd ink. Ech edge (, ) E h nonnegie
More information20/20 20/20 0/5 0/5 20/20 20/20 5/5 0/5 0/5 5/5 0/20 25/30 20/20 30/30 20/20 0/5 5/5 20/20 0/5 0/5 15/20 15/25 20/20 10/10
Annoncemen CSEP Applied Algorihm Richard Anderon Lecre 9 Nework Flow Applicaion Reading for hi week 7.-7.. Nework flow applicaion Nex week: Chaper 8. NP-Compleene Final exam, March 8, 6:0 pm. A UW. hor
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 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 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 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 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 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 informationAlgorithms and Data Structures 2011/12 Week 9 Solutions (Tues 15th - Fri 18th Nov)
Algorihm and Daa Srucure 2011/ Week Soluion (Tue 15h - Fri 18h No) 1. Queion: e are gien 11/16 / 15/20 8/13 0/ 1/ / 11/1 / / To queion: (a) Find a pair of ube X, Y V uch ha f(x, Y) = f(v X, Y). (b) Find
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 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 information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
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 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 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 informationWarm Up. Correct order: s,u,v,y,x,w,t
Warm Up Rn Breadh Fir Search on hi graph aring from. Wha order are erice placed on he qee? When proceing a erex iner neighbor in alphabeical order. In a direced graph, BFS only follow an edge in he direcion
More informationNetwork Flows: Introduction & Maximum Flow
CSC 373 - lgorihm Deign, nalyi, and Complexiy Summer 2016 Lalla Mouaadid Nework Flow: Inroducion & Maximum Flow We now urn our aenion o anoher powerful algorihmic echnique: Local Search. In a local earch
More informationProblem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.
CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex
More informationFlow networks, flow, maximum flow. Some definitions. Edmonton. Saskatoon Winnipeg. Vancouver Regina. Calgary. 12/12 a.
Flow nework, flow, maximum flow Can inerpre direced graph a flow nework. Maerial coure hrough ome yem from ome ource o ome ink. Source produce maerial a ome eady rae, ink conume a ame rae. Example: waer
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 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 informationCS 473G Lecture 15: Max-Flow Algorithms and Applications Fall 2005
CS 473G Lecure 1: Max-Flow Algorihm and Applicaion Fall 200 1 Max-Flow Algorihm and Applicaion (November 1) 1.1 Recap Fix a direced graph G = (V, E) ha doe no conain boh an edge u v and i reveral v u,
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 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 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 informationCS4445/9544 Analysis of Algorithms II Solution for Assignment 1
Conider he following flow nework CS444/944 Analyi of Algorihm II Soluion for Aignmen (0 mark) In he following nework a minimum cu ha capaciy 0 Eiher prove ha hi aemen i rue, or how ha i i fale Uing he
More informationCSE 421 Introduction to Algorithms Winter The Network Flow Problem
CSE 42 Inroducion o Algorihm Winer 202 The Nework Flow Problem 2 The Nework Flow Problem 5 a 4 3 x 3 7 6 b 4 y 4 7 6 c 5 z How much uff can flow from o? 3 Sovie Rail Nework, 955 Reference: On he hiory
More informationNetwork Flows UPCOPENCOURSEWARE number 34414
Nework Flow UPCOPENCOURSEWARE number Topic : F.-Javier Heredia Thi work i licened under he Creaive Common Aribuion- NonCommercial-NoDeriv. Unpored Licene. To view a copy of hi licene, vii hp://creaivecommon.org/licene/by-nc-nd/./
More informationAdmin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t)
/0/ dmin lunch oday rading MX LOW PPLIION 0, pring avid Kauchak low graph/nework low nework direced, weighed graph (V, ) poiive edge weigh indicaing he capaciy (generally, aume ineger) conain a ingle ource
More informationAlgorithms. Algorithms 6.4 MAXIMUM FLOW
Algorihm ROBERT SEDGEWICK KEVIN WAYNE 6.4 MAXIMUM FLOW Algorihm F O U R T H E D I T I O N ROBERT SEDGEWICK KEVIN WAYNE hp://alg4.c.princeon.edu inroducion Ford Fulkeron algorihm maxflow mincu heorem analyi
More information4/12/12. Applications of the Maxflow Problem 7.5 Bipartite Matching. Bipartite Matching. Bipartite Matching. Bipartite matching: the flow network
// Applicaion of he Maxflow Problem. Biparie Maching Biparie Maching Biparie maching. Inpu: undireced, biparie graph = (, E). M E i a maching if each node appear in a mo one edge in M. Max maching: find
More information16 Max-Flow Algorithms and Applications
Algorihm A proce canno be underood by opping i. Underanding mu move wih he flow of he proce, mu join i and flow wih i. The Fir Law of Mena, in Frank Herber Dune (196) There a difference beween knowing
More informationIntroduction to Congestion Games
Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game
More information18 Extensions of Maximum Flow
Who are you?" aid Lunkwill, riing angrily from hi ea. Wha do you wan?" I am Majikhie!" announced he older one. And I demand ha I am Vroomfondel!" houed he younger one. Majikhie urned on Vroomfondel. I
More information26.1 Flow networks. f (u,v) = 0.
26 Maimum Flow Ju a we can model a road map a a direced graph in order o find he hore pah from one poin o anoher, we can alo inerpre a direced graph a a flow nework and ue i o anwer queion abou maerial
More informationCSC 364S Notes University of Toronto, Spring, The networks we will consider are directed graphs, where each edge has associated with it
CSC 36S Noe Univeriy of Torono, Spring, 2003 Flow Algorihm The nework we will conider are direced graph, where each edge ha aociaed wih i a nonnegaive capaciy. The inuiion i ha if edge (u; v) ha capaciy
More informationMaximum Flow. How do we transport the maximum amount data from source to sink? Some of these slides are adapted from Lecture Notes of Kevin Wayne.
Conen Conen. Maximum flow problem. Minimum cu problem. Max-flow min-cu heorem. Augmening pah algorihm. Capaciy-caling. Shore augmening pah. Chaper Maximum How do we ranpor he maximum amoun daa from ource
More informationFlow Networks. Ma/CS 6a. Class 14: Flow Exercises
0/0/206 Ma/CS 6a Cla 4: Flow Exercie Flow Nework A flow nework i a digraph G = V, E, ogeher wih a ource verex V, a ink verex V, and a capaciy funcion c: E N. Capaciy Source 7 a b c d e Sink 0/0/206 Flow
More information7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching
Chaper. Biparie Maching Nework Flow Slide by Kein Wayne. Copyrigh 00 Pearon-Addion Weley. All righ reered. Maching Biparie Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node
More informationAlgorithmic Discrete Mathematics 6. Exercise Sheet
Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap
More informationRandomized Perfect Bipartite Matching
Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for
More informationReminder: Flow Networks
0/0/204 Ma/CS 6a Cla 4: Variou (Flow) Execie Reminder: Flow Nework A flow nework i a digraph G = V, E, ogeher wih a ource verex V, a ink verex V, and a capaciy funcion c: E N. Capaciy Source 7 a b c d
More informationBasic Tools CMSC 641. Running Time. Problem. Problem. Algorithmic Design Paradigms. lg (n!) (lg n)! (lg n) lgn n.2
Baic Tool CMSC April, Review Aympoic Noaion Order of Growh Recurrence relaion Daa Srucure Li, Heap, Graph, Tree, Balanced Tree, Hah Table Advanced daa rucure: Binomial Heap, Fibonacci Heap Soring Mehod
More informationintroduction Ford-Fulkerson algorithm
Algorihm ROBERT SEDGEWICK KEVIN WAYNE. MAXIMUM FLOW. MAXIMUM FLOW inroducion inroducion Ford-Fulkeron algorihm Ford-Fulkeron algorihm Algorihm F O U R T H E D I T I O N maxflow-mincu heorem analyi of running
More informationThey were originally developed for network problem [Dantzig, Ford, Fulkerson 1956]
6. Inroducion... 6. The primal-dual algorihmn... 6 6. Remark on he primal-dual algorihmn... 7 6. A primal-dual algorihmn for he hore pah problem... 8... 9 6.6 A primal-dual algorihmn for he weighed maching
More informationLecture 16 (Momentum and Impulse, Collisions and Conservation of Momentum) Physics Spring 2017 Douglas Fields
Lecure 16 (Momenum and Impulse, Collisions and Conservaion o Momenum) Physics 160-02 Spring 2017 Douglas Fields Newon s Laws & Energy The work-energy heorem is relaed o Newon s 2 nd Law W KE 1 2 1 2 F
More informationNetwork Flow Applications
Hopial problem Neork Flo Applicaion Injured people: n Hopial: k Each peron need o be brough o a hopial no more han 30 minue aay Each hopial rea no more han n/k" people Gien n, k, and informaion abou people
More informationintroduction Ford-Fulkerson algorithm
Algorihm ROBERT SEDGEWICK KEVIN WAYNE. MAXIMUM FLOW. MAXIMUM FLOW inroducion inroducion Ford-Fulkeron algorihm Ford-Fulkeron algorihm Algorihm F O U R T H E D I T I O N maxflow-mincu heorem analyi of running
More informationPlease Complete Course Survey. CMPSCI 311: Introduction to Algorithms. Approximation Algorithms. Coping With NP-Completeness. Greedy Vertex Cover
Pleae Complee Coure Survey CMPSCI : Inroducion o Algorihm Dealing wih NP-Compleene Dan Sheldon hp: //owl.oi.uma.edu/parner/coureevalsurvey/uma/ Univeriy of Maachue Slide Adaped from Kevin Wayne La Compiled:
More informationCOMPETITIVE LOCAL ROUTING WITH CONSTRAINTS
COMPETITIVE LOCAL ROUTING WITH CONSTRAINTS Proenji Boe, Rolf Fagerberg, André van Renen, and Sander Verdoncho Abrac. Le P be a e of n verice in he plane and S a e of non-croing line egmen beween verice
More informationOn the Submodularity of Influence in Social Networks
On he ubmodulariy o Inluence in ocial Neworks Elchanan Mossel & ebasien Roch OC07 peaker: Xinran He Xinranhe1990@gmail.com ocial Nework ocial nework as a graph Nodes represen indiiduals. Edges are social
More information23 Maximum Flows and Minimum Cuts
A proce canno be underood by opping i. Underanding mu move wih he flow of he proce, mu join i and flow wih i. The Fir Law of Mena, in Frank Herber Dune (196) Conrary o expecaion, flow uually happen no
More informationOn Line Supplement to Strategic Customers in a Transportation Station When is it Optimal to Wait? A. Manou, A. Economou, and F.
On Line Spplemen o Sraegic Comer in a Tranporaion Saion When i i Opimal o Wai? A. Mano, A. Economo, and F. Karaemen 11. Appendix In hi Appendix, we provide ome echnical analic proof for he main rel of
More informationMaximum Flow in Planar Graphs
Maximum Flow in Planar Graph Planar Graph and i Dual Dualiy i defined for direced planar graph a well Minimum - cu in undireced planar graph An - cu (undireced graph) An - cu The dual o he cu Cu/Cycle
More informationCS Lunch This Week. Special Talk This Week. Soviet Rail Network, Flow Networks. Slides20 - Network Flow Intro.key - December 5, 2016
CS Lunch This Week Panel on Sudying Engineering a MHC Wednesday, December, : Kendade Special Talk This Week Learning o Exrac Local Evens from he Web John Foley, UMass Thursday, December, :, Carr Sovie
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 informationToday s topics. CSE 421 Algorithms. Problem Reduction Examples. Problem Reduction. Undirected Network Flow. Bipartite Matching. Problem Reductions
Today opic CSE Algorihm Richard Anderon Lecure Nework Flow Applicaion Prolem Reducion Undireced Flow o Flow Biparie Maching Dijoin Pah Prolem Circulaion Loweround conrain on flow Survey deign Prolem Reducion
More information16 Max-Flow Algorithms
A process canno be undersood by sopping i. Undersanding mus move wih he flow of he process, mus join i and flow wih i. The Firs Law of Mena, in Frank Herber s Dune (196) There s a difference beween knowing
More informationCompetitive Routing in the Half-θ 6 -Graph
Compeiie Roing in he Half-θ 6 -Graph Proenji Boe Rolf Fagererg André an Renen Sander Verdoncho Arac We preen a deerminiic local roing cheme ha i garaneed o find a pah eeen any pair of erice in a halfθ
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationI. OBJECTIVE OF THE EXPERIMENT.
I. OBJECTIVE OF THE EXPERIMENT. Swissmero raels a high speeds hrough a unnel a low pressure. I will hereore undergo ricion ha can be due o: ) Viscosiy o gas (c. "Viscosiy o gas" eperimen) ) The air in
More informationME 425: Aerodynamics
ME 45: Aerodnamics Dr. A.B.M. Toiqe Hasan Proessor Deparmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET, Dhaka Lecre-7 Fndamenals so Aerodnamics oiqehasan.be.ac.bd oiqehasan@me.be.ac.bd
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 informationEnergy Problems 9/3/2009. W F d mgh m s 196J 200J. Understanding. Understanding. Understanding. W F d. sin 30
9/3/009 nderanding Energy Proble Copare he work done on an objec o a.0 kg a) In liing an objec 0.0 b) Puhing i up a rap inclined a 30 0 o he ae inal heigh 30 0 puhing 0.0 liing nderanding Copare he work
More information7.5 Bipartite Matching. Chapter 7. Network Flow. Matching. Bipartite Matching
Chaper. Biparie Maching Nework Flow Slide by Kevin Wayne. Copyrigh PearonAddion Weley. All righ reerved. Maching Biparie Maching Maching. Inpu: undireced graph G = (V, E). M E i a maching if each node
More information2. VECTORS. R Vectors are denoted by bold-face characters such as R, V, etc. The magnitude of a vector, such as R, is denoted as R, R, V
ME 352 VETS 2. VETS Vecor algebra form he mahemaical foundaion for kinemaic and dnamic. Geomer of moion i a he hear of boh he kinemaic and dnamic of mechanical em. Vecor anali i he imehonored ool for decribing
More informationPHYSICS 151 Notes for Online Lecture #4
PHYSICS 5 Noe for Online Lecure #4 Acceleraion The ga pedal in a car i alo called an acceleraor becaue preing i allow you o change your elociy. Acceleraion i how fa he elociy change. So if you ar fro re
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 informationIntroduction to SLE Lecture Notes
Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will
More informationChapter 5. Localization. 5.1 Localization of categories
Chaper 5 Localizaion Consider a caegory C and a amily o morphisms in C. The aim o localizaion is o ind a new caegory C and a uncor Q : C C which sends he morphisms belonging o o isomorphisms in C, (Q,
More informationarxiv: v1 [cs.cg] 21 Mar 2013
On he rech facor of he Thea-4 graph Lui Barba Proenji Boe Jean-Lou De Carufel André van Renen Sander Verdoncho arxiv:1303.5473v1 [c.cg] 21 Mar 2013 Abrac In hi paper we how ha he θ-graph wih 4 cone ha
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More informationApril 3, The maximum flow problem. See class notes on website.
5.05 April, 007 The maximum flow problem See cla noe on webie. Quoe of he day You ge he maxx for he minimum a TJ Maxx. -- ad for a clohing ore Thi wa he mo unkinde cu of all -- Shakepeare in Juliu Caear
More informationMath 221: Mathematical Notation
Mah 221: Mahemaical Noaion Purpose: One goal in any course is o properly use he language o ha subjec. These noaions summarize some o he major conceps and more diicul opics o he uni. Typing hem helps you
More informationFord Fulkerson algorithm max-flow min-cut theorem. max-flow min-cut theorem capacity-scaling algorithm
7. NETWORK FLOW I 7. NETWORK FLOW I max-flow and min-cu problem max-flow and min-cu problem Ford Fulkeron algorihm Ford Fulkeron algorihm max-flow min-cu heorem max-flow min-cu heorem capaciy-caling algorihm
More informationDifferential Geometry: Revisiting Curvatures
Differenial Geomery: Reisiing Curaures Curaure and Graphs Recall: hus, up o a roaion in he x-y plane, we hae: f 1 ( x, y) x y he alues 1 and are he principal curaures a p and he corresponding direcions
More informationPage 1 o 13 1. The brighes sar in he nigh sky is α Canis Majoris, also known as Sirius. I lies 8.8 ligh-years away. Express his disance in meers. ( ligh-year is he disance coered by ligh in one year. Ligh
More informationCS261: A Second Course in Algorithms Lecture #1: Course Goals and Introduction to Maximum Flow
CS61: A Second Coure in Algorihm Lecure #1: Coure Goal and Inroducion o Maximum Flo Tim Roughgarden January 5, 016 1 Coure Goal CS61 ha o major coure goal, and he coure pli roughly in half along hee line.
More information, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max
ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen
More informationMIXED BOUNDARY VALUE PROBLEM FOR A QUARTER-PLANE WITH A ROBIN CONDITION
Jornal o Science, Ilamic Repblic o Iran 3: 65-69 Naional Cener For Scieniic Reearch, ISSN 6-4 MIXED OUNDRY VLUE PROLEM FOR QURTER-PLNE WITH ROIN CONDITION ghili * Deparmen o Mahemaic, Facl o Science, Gilan
More informationBrock University Physics 1P21/1P91 Fall 2013 Dr. D Agostino. Solutions for Tutorial 3: Chapter 2, Motion in One Dimension
Brock Uniersiy Physics 1P21/1P91 Fall 2013 Dr. D Agosino Soluions for Tuorial 3: Chaper 2, Moion in One Dimension The goals of his uorial are: undersand posiion-ime graphs, elociy-ime graphs, and heir
More informationWrap up: Weighted, directed graph shortest path Minimum Spanning Tree. Feb 25, 2019 CSCI211 - Sprenkle
Objecive Wrap up: Weighed, direced graph hore pah Minimum Spanning Tree eb, 1 SI - Sprenkle 1 Review Wha are greedy algorihm? Wha i our emplae for olving hem? Review he la problem we were working on: Single-ource,
More informationHYPOTHESIS TESTING. four steps. 1. State the hypothesis and the criterion. 2. Compute the test statistic. 3. Compute the p-value. 4.
Inrodcion o Saisics in Psychology PSY Professor Greg Francis Lecre 24 Hypohesis esing for correlaions Is here a correlaion beween homework and exam grades? for seps. Sae he hypohesis and he crierion 2.
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
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 informationTopics in Combinatorial Optimization May 11, Lecture 22
8.997 Topics in Combinaorial Opimizaion May, 004 Lecure Lecurer: Michel X. Goemans Scribe: Alanha Newman Muliflows an Disjoin Pahs Le G = (V,E) be a graph an le s,,s,,...s, V be erminals. Our goal is o
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationGreedy algorithms. Shortest paths in weighted graphs. Tyler Moore. Shortest-paths problem. Shortest path applications.
Shortet-path problem Greedy algorithm Shortet path in weighted graph Problem. Gien a digraph G = (V, E), edge length e 0, orce V, and detination t V, find the hortet directed path from to t. Tyler Moore
More information6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson
6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen
More informationStar coloring of sparse graphs
Sar coloring of pare graph Yeha B Daniel W. Cranon Mickaël Monaier André Rapad Weifan Wang Abrac A proper coloring of he erice of a graph i called a ar coloring if he nion of eery o color clae indce a
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More information