Admin 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)
|
|
- Myra Holland
- 5 years ago
- Views:
Transcription
1 /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 V wih no incoming edge conain a ingle ink/arge V wih no ougoing edge every verex i on a pah from o low conrain in-flow = ou-flow for every verex (excep, ) flow along an edge canno exceed he edge capaciy flow are poiive 0 0 0
2 /0/ Max flow problem Nework flow properie iven a flow nework: wha i he maximum flow we can end from o ha mee he flow conrain? 0 If one of hee i rue hen all are rue (i.e. each implie he he oher): f i a maximum flow f (reidual graph) ha no pah from o f = minimum capaciy cu 0 0 ord-ulkeron pplicaion: biparie graph maching ord-ulkeron(,, ) flow = 0 for all edge f = reidualraph() while a imple pah exi from o in f end a much flow along he pah a poible f = reidualraph() reurn flow iparie graph a graph where every verex can be pariioned ino wo e X and Y uch ha all edge connec a verex u X and a verex v Y
3 /0/ pplicaion: biparie graph maching pplicaion: biparie graph maching maching M i a ube of edge uch ha each node occur a mo once in M maching M i a ube of edge uch ha each node occur a mo once in M maching pplicaion: biparie graph maching pplicaion: biparie graph maching maching M i a ube of edge uch ha each node occur a mo once in M maching M i a ube of edge uch ha each node occur a mo once in M maching no a maching
4 /0/ pplicaion: biparie graph maching pplicaion: biparie graph maching maching can be hough of a pairing he verice iparie maching problem: find he large maching in a biparie graph Where migh hi problem come up? - deparmen ha n coure and m faculy - very inrucor can each ome of he coure - Wha coure hould each peron each? - nyime we wan o mach n hing wih m, bu no all hing can mach pplicaion: biparie graph maching pplicaion: biparie graph maching iparie maching problem: find he large maching in a biparie graph eup a a flow problem: idea? - greedy? - dynamic programming?
5 /0/ pplicaion: biparie graph maching pplicaion: biparie graph maching eup a a flow problem: edge weigh? eup a a flow problem: all edge weigh are pplicaion: biparie graph maching pplicaion: biparie graph maching eup a a flow problem: afer we find he flow, how do we find he maching? eup a a flow problem: mach hoe node wih flow beween hem
6 /0/ pplicaion: biparie graph maching pplicaion: biparie graph maching I i correc? ume i no here i a beer maching becaue of how we eup he graph flow = # of mache herefore, he beer maching would have a higher flow conradicion (max-flow algorihm find maximal!) Run-ime? o o build he flow? O() n each exiing edge ge a capaciy of n inroduce V new edge (o and from and ) n V i O() (for non-degenerae biparie maching problem) Max-flow calculaion? aic ord-ulkeron: O(max-flow * ) dmund-karp: O(V ) Preflow-puh: O(V ) pplicaion: biparie graph maching pplicaion: biparie graph maching Run-ime? o o build he flow? O() n each exiing edge ge a capaciy of n inroduce V new edge (o and from and ) n V i O() (for non-degenerae biparie maching problem) Max-flow calculaion? aic ord-ulkeron: O(max-flow * ) n max-flow = O(V) n O(V ) iparie maching problem: find he large maching in a biparie graph - deparmen ha n coure and m faculy - very inrucor can each ome of he coure - Wha coure hould each peron each? - ach faculy can each a mo coure a emeer? hange he edge weigh (repreening faculy) o
7 /0/ urvey eign urvey eign eign a urvey wih he following requiremen: eign urvey aking n conumer abou m produc an only urvey conumer abou a produc if hey own i Queion conumer abou a mo q produc ach produc hould be urveyed a mo ime Maximize he number of urvey/queion aked How can we do hi? each conumer can anwer a mo q queion q q q q conumer c c c produc capaciy edge if conumer owned produc p p each produc can be queioned abou a mo ime p c urvey deign I i correc? ach of he commen above he flow graph mach he problem conrain max-flow find he maximum maching, given he problem conrain dge ijoin Pah wo pah are edge-dijoin if hey have no edge in common Wha i he run-ime? aic ord-ulkeron: O(max-flow * ) dmund-karp: O(V ) Preflow-puh: O(V )
8 /0/ dge ijoin Pah dge ijoin Pah Problem wo pah are edge-dijoin if hey have no edge in common iven a direced graph = (V, ) and wo node and, find he max number of edge-dijoin pah from o Why migh hi be ueful? dge ijoin Pah Problem dge ijoin Pah iven a direced graph = (V, ) and wo node and, find he max number of edge-dijoin pah from o lgorihm idea? Why migh hi be ueful? edge are unique reource (e.g. communicaion, ranporaion, ec.) how many concurren (non-conflicing) pah do we have from o 8
9 /0/ dge ijoin Pah Max flow formulaion: aign uni capaciy o every edge dge ijoin Pah Max flow formulaion: aign uni capaciy o every edge Wha doe he max flow repreen? Why? - max-flow = maximum number of dijoin pah - correcne: - each edge can have a mo flow =, o can only be ravered once - herefore, each uni ou of repreen a eparae pah o Max-flow variaion Wha if we have muliple ource and muliple ink (e.g. he Ruian rain problem ha muliple ink)? Max-flow variaion reae a new ource and ink and connec up wih infinie capaciie capaciy nework capaciy nework 9
10 /0/ Max-flow variaion Max-flow variaion Verex capaciie: in addiion o having edge capaciie we can alo reric he amoun of flow hrough each verex Verex capaciie: in addiion o having edge capaciie we can alo reric he amoun of flow hrough each verex 0/ 0 0/ 0/ /0 0/ 0 0/0 Wha i he max-flow now? uni Max-flow variaion Max-flow variaion Verex capaciie: in addiion o having edge capaciie we can alo reric he amoun of flow hrough each verex How can we olve hi problem? or each verex v - creae a new node v - creae an edge wih he verex capaciy from v o v - move all ougoing edge from v o v an you now prove i correc? 0
11 /0/ Max-flow variaion Max-flow variaion Proof:. how ha if a oluion exi in he original graph, hen a oluion exi in he modified graph. how ha if a oluion exi in he modified graph, hen a oluion exi in he original graph Proof: we know ha he verex conrain are aified n no incoming flow can exceed he verex capaciy ince we have a ingle edge wih ha capaciy from v o v we can obain he oluion, by collaping each v and v back o he original v node n in-flow = ou-flow ince here i only a ingle edge from v o v n becaue here i only a ingle edge from v o v and all he in edge go in o v and ou o v, hey can be viewed a a ingle node in he original graph More problem: maximum independen pah wo pah are independen if hey have no verice in common More problem: maximum independen pah wo pah are independen if hey have no verice in common
12 /0/ More problem: maximum independen pah ind he maximum number of independen pah Idea? maximum independen pah Max flow formulaion: - aign uni capaciy o every edge (hough any value would work) - aign uni capaciy o every verex ame idea a he maximum edge-dijoin pah, bu now we alo conrain he verice More problem: wirele nework he campu ha hired you o eup he wirele nework here are currenly m wirele aion poiioned a variou (x,y) coordinae on campu he range of each of hee aion i r (i.e. he ignal goe a mo diance r) ny paricular wirele aion can only ho k people conneced You ve calculae he n mo popular locaion on campu and have heir (x,y) coordinae ould he curren nework uppor n differen people rying o connec a each of he n mo popular locaion (i.e. one peron per locaion)? Prove correcne and ae run-ime noher maching problem n people node and m aion node if di(p i,w j ) < r hen add an edge from pi o wj wih weigh (where di i euclidean diance) add edge -> p i wih weigh add edge w j -> wih weigh k add edge if di(p i, w j ) < r p w k p w m n - olve for max-flow - check if flow = m k
13 /0/ orrecne If here i flow from a peron node o a wirele node hen ha peron i aached o ha wirele node if di(pi,wj) < r hen add an edge from pi o wj wih weigh (where di i euclidean diance) only people able o connec o node could have flow add edge -> pi wih weigh each peron can only connec o one wirele node add edge wj -> wih weigh L a mo L people can connec o a wirele node If flow = m, hen every peron i conneced o a node Runime = O(mn): every peron i wihin range of every node V = m + n + max-flow = O(m), ha a mo m ou-flow O(max-flow * ) = O(m n): ord-ulkeron O(V ) = O((m+n)m n ): dmund-karp O(V ) = O((m+n) ): preflow-puh varian
Problem 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationSelfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos
Selfih Rouing Tim Roughgarden Cornell Univeriy Include join work wih Éva Tardo 1 Which roue would you chooe? Example: one uni of raffic (e.g., car) wan o go from o delay = 1 hour (no congeion effec) long
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 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 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 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 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 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 informationStationary Distribution. Design and Analysis of Algorithms Andrei Bulatov
Saionary Disribuion Design and Analysis of Algorihms Andrei Bulaov Algorihms Markov Chains 34-2 Classificaion of Saes k By P we denoe he (i,j)-enry of i, j Sae is accessible from sae if 0 for some k 0
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 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 informationDiscussion Session 2 Constant Acceleration/Relative Motion Week 03
PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll
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 informationAverage Case Lower Bounds for Monotone Switching Networks
Average Cae Lower Bound for Monoone Swiching Nework Yuval Filmu, Toniann Piai, Rober Robere, Sephen Cook Deparmen of Compuer Science Univeriy of Torono Monoone Compuaion (Refreher) Monoone circui were
More informationCMPS 6610/4610 Fall Flow Networks. Carola Wenk Slides adapted from slides by Charles Leiserson
CMP 6610/4610 Fall 2016 Flow Nework Carola Wenk lide adaped rom lide by Charle Leieron Max low and min c Fndamenal problem in combinaorial opimizaion Daliy beween max low and min c Many applicaion: 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 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 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 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 informationSolutions for Assignment 2
Faculy of rs and Science Universiy of Torono CSC 358 - Inroducion o Compuer Neworks, Winer 218 Soluions for ssignmen 2 Quesion 1 (2 Poins): Go-ack n RQ In his quesion, we review how Go-ack n RQ can be
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 informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
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 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 informationWe just finished the Erdős-Stone Theorem, and ex(n, F ) (1 1/(χ(F ) 1)) ( n
Lecure 3 - Kövari-Sós-Turán Theorem Jacques Versraëe jacques@ucsd.edu We jus finished he Erdős-Sone Theorem, and ex(n, F ) ( /(χ(f ) )) ( n 2). So we have asympoics when χ(f ) 3 bu no when χ(f ) = 2 i.e.
More informationNotes on cointegration of real interest rates and real exchange rates. ρ (2)
Noe on coinegraion of real inere rae and real exchange rae Charle ngel, Univeriy of Wiconin Le me ar wih he obervaion ha while he lieraure (mo prominenly Meee and Rogoff (988) and dion and Paul (993))
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More information18.03SC Unit 3 Practice Exam and Solutions
Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care
More informationLecture 26. Lucas and Stokey: Optimal Monetary and Fiscal Policy in an Economy without Capital (JME 1983) t t
Lecure 6. Luca and Sokey: Opimal Moneary and Fical Policy in an Economy wihou Capial (JME 983. A argued in Kydland and Preco (JPE 977, Opimal governmen policy i likely o be ime inconien. Fiher (JEDC 98
More information7. NETWORK FLOW II. Soviet rail network (1950s) Max-flow and min-cut applications. "Free world" goal. Cut supplies (if cold war turns into real war).
Sovie rail nework (9). NETWORK FLOW II "Free world" goal. Cu upplie (if cold war urn ino real war). Lecure lide by Kevin Wayne Copyrigh Pearon-Addion Weley Copyrigh Kevin Wayne hp://www.c.princeon.edu/~wayne/kleinberg-ardo
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 informationGraph Theory: Network Flow
Univeriy of Wahingon Mah 336 Term Paper Graph Theory: Nework Flow Auhor: Ellio Broard Advier: Dr. Jame Morrow 3 June 200 Conen Inroducion...................................... 2 2 Terminology......................................
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 informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationPerformance Comparison of LCMV-based Space-time 2D Array and Ambiguity Problem
Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 Performance Comparion of LCMV-baed pace-ime 2D Arra and Ambigui Problem 2 o uan Chang and Jin hinghia Deparmen of Communicaion
More informationBipartite Matching. Matching. Bipartite Matching. Maxflow Formulation
Mching Inpu: undireced grph G = (V, E). Biprie Mching Inpu: undireced, biprie grph G = (, E).. Mching Ern Myr, Hrld äcke Biprie Mching Inpu: undireced, biprie grph G = (, E). Mflow Formulion Inpu: undireced,
More informationMaximum Network Lifetime in Wireless Sensor Networks with Adjustable Sensing Ranges
1 Maximum Nework Lifeime in Wirele Senor Nework wih Adjuable Sening Range Mihaela Cardei, Jie Wu, Mingming Lu, and Mohammad O. Pervaiz Abrac Thi paper addree he arge coverage problem in wirele enor nework
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
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 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 informationPhysics 240: Worksheet 16 Name
Phyic 4: Workhee 16 Nae Non-unifor circular oion Each of hee proble involve non-unifor circular oion wih a conan α. (1) Obain each of he equaion of oion for non-unifor circular oion under a conan acceleraion,
More informationLinear Algebra Primer
Linear Algebra rimer And a video dicuion of linear algebra from EE263 i here (lecure 3 and 4): hp://ee.anford.edu/coure/ee263 lide from Sanford CS3 Ouline Vecor and marice Baic Mari Operaion Deerminan,
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationHow to Solve System Dynamic s Problems
How o Solve Sye Dynaic Proble A ye dynaic proble involve wo or ore bodie (objec) under he influence of everal exernal force. The objec ay uliaely re, ove wih conan velociy, conan acceleraion or oe cobinaion
More informationPHYSICS Solving Equations
Sepember 20, 2013 PHYSIS Solving Equaion Sepember 2013 www.njcl.org Solving for a Variable Our goal i o be able o olve any equaion for any variable ha appear in i. Le' look a a imple equaion fir. The variable
More informationAnalyze patterns and relationships. 3. Generate two numerical patterns using AC
envision ah 2.0 5h Grade ah Curriculum Quarer 1 Quarer 2 Quarer 3 Quarer 4 andards: =ajor =upporing =Addiional Firs 30 Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 andards: Operaions and Algebraic Thinking
More informationMathcad Lecture #8 In-class Worksheet Curve Fitting and Interpolation
Mahcad Lecure #8 In-class Workshee Curve Fiing and Inerpolaion A he end of his lecure, you will be able o: explain he difference beween curve fiing and inerpolaion decide wheher curve fiing or inerpolaion
More information15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel
15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he
More informationFishing limits and the Logistic Equation. 1
Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationGeometric Path Problems with Violations
Click here o view linked Reference 1 1 1 0 1 0 1 0 1 0 1 Geomeric Pah Problem wih Violaion Anil Mahehwari 1, Subha C. Nandy, Drimi Paanayak, Saanka Roy and Michiel Smid 1 1 School of Compuer Science, Carleon
More informationElectrical and current self-induction
Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deparmen of Civil and Environmenal Engineering 1.731 Waer Reource Syem Lecure 17 River Bain Planning Screening Model Nov. 7 2006 River Bain Planning River bain planning
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationOptimal State-Feedback Control Under Sparsity and Delay Constraints
Opimal Sae-Feedback Conrol Under Spariy and Delay Conrain Andrew Lamperki Lauren Leard 2 3 rd IFAC Workhop on Diribued Eimaion and Conrol in Neworked Syem NecSy pp. 24 29, 22 Abrac Thi paper preen he oluion
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 informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationd = ½(v o + v f) t distance = ½ (initial velocity + final velocity) time
BULLSEYE Lab Name: ANSWER KEY Dae: Pre-AP Physics Lab Projecile Moion Weigh = 1 DIRECTIONS: Follow he insrucions below, build he ramp, ake your measuremens, and use your measuremens o make he calculaions
More information