Ford-Fulkerson Algorithm for Maximum Flow
|
|
- Morgan Short
- 6 years ago
- Views:
Transcription
1 Ford-Fulkerson Algorihm for Maximum Flow 1. Assign an iniial flow f ij (for insance, f ij =0) for all edges.label s by Ø. Mark he oher verices "unlabeled.". Find a labeled verex i ha has no ye been scanned. Scan i as follows For every unlabeled adjacen verex j, (a or b or c) a) if C ij > f ij and f ij 0 compue Δ ij = C ij f ij and Δ j where Δij if i = 1 Δ j = min( Δi, Δij) if i > 1 Label j wih a forward label (i +, f ij ) b) if C ij > f ij and f ij <0 (opposie direcion) Δ j =min (Δ j, f ij ) Label j wih a backward label (i, Δ j ) c) if C ij = f ij No operaion. If no unlabeled j exiss STOP. ) Repea sep unil is reached. [This gives a flow augmening pah P: s > ] If i is impossible o reach hen STOP. ). Backrack he pah P, using he labels. )Using P, augmen he exising flow by Δ, Se f = f + Δ,. 7) Remove all labels from verices,, n. Go o Sep. Example: Find he maximum flow from s o in he following graph. 0,0 10,0 11,0,0 7,0 Soluion 1) C 1 =0, C =11, C =1, C =, C 1 =10, C =7, C =, C =, f 1 =f =f =f =f 1 =f =f =f =0,,0 1,0,0 ) verex 1 (s) is labeled Ø,,,,, are unlabeled ) Scan 1. i=1,adjacen labels and. [ j= and j=] C 1 =0. f 1 =0. (perform a) For verex j= Δ 1 = C 1 f 1 = 0 0=0 Δ = Δ 1 =0. L = {1 +, 0} For verex j= C 1 =0. f 1 =0. (perform a) Δ 1 = C 1 f 1 = 10 0=10 Δ = Δ 1 =10. L = {1 +, 10} Scan. i=,adjacen label, and. [ j= and j=] For verex j= C =11. f =0. Δ = C f = 11 0 = 11 Δ = min (Δ, Δ ) = min(0, 11) =11 L = { +, 11} For verex j=, f <0 (perform b) Δ = min( Δ f ) = min(0,0) =0 L = {, 0} Scan. i=,adjacen labels, and. [ j= ] (j= and j= are already labelled) C =1. f =0. Δ = C f = 1 0 = 1 Δ = min (Δ, Δ ) = min(11, 1) =11 L = { +, 11} Since verex is Δ = Now all verices are all labeled Find he pah L = { +, 11} L = { +, 11} L = {1 +, 0} L1 Thus one augmening pah i--- Add Δ =11 o his pah f 1(new) = f 1(old) + Δ f 1 = f =0 +11=11 f =0 +11=11 Remove all he labels. Sar scanning
2 1) C 1 =0, C =11, C =1, C =, C 1 =10, C =7, C =, C =, f 1 =f =f =11, f =f 1 =f =f =f =0, ) verex 1 (s) is labeled Ø,,,,, are unlabeled ) Scan 1. i=1,adjacen labels and. [ j= and j=] C 1 =0. f 1 =11. (perform a) For verex j= Δ 1 = C 1 f 1 = 0 11=9 Δ = Δ 1 =9. L = {1 +, 9} For verex j= C 1 =10. f 1 =0. Δ 1 = C 1 f 1 = 10 0=10 Δ = Δ 1 =10. L = {1 +, 10} Scan. i=, Adjacen label, and. [ j= and j=] For verex j=, C =11. f =11. C = f No acion. For verex j=, f <0 (perform b) Δ = min( Δ f ) = min(9,0) =0 L = {, 0} Scan. 0,11 10,0,0 7,0 i= no labeled no acion. Scan. i= Adjacen label,. [ j=] C =7. f =0. Δ = C f = 7 0 = 7 Δ = min (Δ, Δ ) = min(10, 7) =10 L = { +, 10} Scan. i= Adjacen labels,,, [ j=, j=] (j= j= are already labelled) C =. f =0. Δ = C f = 0 = Δ = min (Δ, Δ ) = min(10, ) = L = { +, },0 1,11,0 Since verex is Δ = Now all verices are all labeled Find he pah L = { +, } L = { +, 10} L = {1 +, 0} L1 Thus one augmening pah i--- Add Δ = o his pah f 1(new) = f 1(old) + Δ f 1 = 0 + f =0 += f =0 += Remove all he labels. Sar scanning ) verex 1 (s) is labeled Ø,,,,, are unlabeled ) Scan 1. i=1,adjacen labels and. [ j= and j=] C 1 =0. f 1 =11. (perform a) For verex j= Δ 1 = C 1 f 1 = 0 11=9 Δ = Δ 1 =9. L = {1 +, 9} For verex j= C 1 =10. f 1 =. Δ 1 = C 1 f 1 = 10 =7 Δ = Δ 1 =7. L = {1 +, 7} Scan. i=, Adjacen label, and. [ j= and j=] For verex j=, C =11. f =11. C = f No acion. For verex j=, f <0 (perform b) Δ = min( Δ f ) = min(9,0) =0 L = {, 0} Scan. 0,11 10,,0 7, i= no labeled no acion. Scan. i= Adjacen label,. [ j=] C =7. f =0. Δ = C f = 7 0 = 7,0 1,11,
3 Δ = min (Δ, Δ ) = min(10, 7) =10 L = { +, 10} Scan. i= Adjacen labels,,, [ j=, j=] (j= j= are already labelled) C =. f =0. Δ = C f = 0 = Δ = min (Δ, Δ ) = min(10, ) = L = { +, } Soluion 1) C 1 =0, C =11, C =1, C =, C 1 =10, C Scan. i=,adjacen label, and. [ j= ] (j= and j= are already scanned) C =1. f =0. Δ = C f = 1 0 = 1 Δ = min (Δ, Δ ) = min(11, 1) =11 L = { +, 11} Since L is For verex j=, f <0 (perform b) Δ = min( Δ f ) = min(0,0) =0 L = {, 0} vv Δ = Δ 1 =10. L = {1 +, 10} Scan and. Second Number: given flow (f i,j ) S: source : arge Pah: sequence of edges in a diagraph Flow augmening pah: Pahs from S o. Examples: Pah 1=(1---) Pah =(1--- ) Pah =(1----) Forward edge:if he direcion of pah is he same as he direcion of edge i is called forward edge. Backward edge:if he direcion of pah is he opposie of he direcion of edge i is called forward edge. Pah 1: 1-, -, - all forward edges Pah : 1-, -, - forward edges, backward edge C ij =he capaciy of edge from i o j f ij =The value of curren flow from i o j. ij =possible addiional flow from edge i o j. ij = C ij - f ij 1 =0-=1, =11-8=, =1-=7, 1 =10-=, =7-=, =-=0, =- =. Maximum Flow: Maximum possible flow from s o Kirchof s rule: Incoming flow=ougoing flow Example: for verex,, incoming flow. 8:ougoing flow.. +=8 Possible addiional flow in pah 1 We can increase maximum flow by because he edge, allows only. No addiional flow is possible in pah, because =0, addiional flow is possible in pah. 0,8 10, Δ =11,,0 7, Ford-Fulkerson Algorihm for Maximum Flow ALGORفTHM FORD-FULKERSON [G = (V, E), verices l (= s),, n (= ), edges (;', j), Cy] This algorihm compues he maximum flow in a nework G wih source s, sink (, aý capaciies Cy > O of he edges (;', j). INPUT: n, s = l, = n, edges (;', j) of G, Cy OUTPUT: Maximum flow f in G 1. Assign an iniial flow f y (for insance, f y = O for ali edges), compue f.. Label s by 0. Mark he oher verices "unlabeled." J. l'lllu a laü^l^u v^/ha/a. ilul lýuo llü J^l Uüülý a^^mý^u. 0^0.11 For every unlabeled adjacen verex j, if Cy > f\p compue 1,11,
4 Ay = Cy - fy and A, and label7 wih a "fonvard label" (i" 1 ', A,); or if f^ > O, compue A, == min(a,, f,,) and label j by a "backward label" (;~, Aj). If no such j exiss hen OUTPUT f. Sop [^" ; r/îe ma xim um flo w.} Els e con in ue (h a is, go o Se p ).. Repea Sep unil is reached. A if ( = l ' ü im in (A,, A y) if ; > l [This gives a flow augmening pah P: s >.} If i is impossible o reach hen OUTPUT f. Sop [f is he ma xim um flo w.} Els e con in ue (h a is, go o Se p ).. Backrack he pah P, using he labels.. Using P, augmen he exising flow by a(. Se f = f + a(. 7. Remove ali labels from verices,, n. Go o Sep. End FORD-FULKERSON A Nework is a diagraph in which each edge has assigned o i a capaciy (maximum flow) 0, 10, 11,8, 7, Graphs and Combinaorial Opimizaion 1.7 Ford-Fulkerson Algorihm for Maximum Flow, 1,,
5 Flow augmening pahs, as discussed in he las secion, are used as he basic ool ini Ford-Fulkerson algorihm in Table 1.8 in which a given flow (for insance, zero flofl ali edges) is increased unil i is maximum. The algorihm accomplishes he increase a sepwise consrucion of flow augmening pahs, one a a ime, unil no furher sý pahs can be consruced, which happens precisely when he flow is maximum. Table 1.8 Ford-Fulkerson Algorihm for Maximum Flow ALGORفTHM FORD-FULKERSON [G = (V, E), verices l (= s),, n (= ), edges (;', j), Cy] This algorihm compues he maximum flow in a nework G wih source s, sink (, aý capaciies Cy > O of he edges (;', j). INPUT: n, s = l, = n, edges (;', j) of G, Cy OUTPUT: Maximum flow f in G 1. Assign an iniial flow f y (for insance, f y = O for ali edges), compue f.. Label s by 0. Mark he oher verices "unlabeled." J. l'lllu a laü^l^u v^/ha/a. ilul lýuo llü J^l Uüülý a^^mý^u. 0^0.11 For every unlabeled adjacen verex j, if Cy > f\p compue A if ( = l 'ü Ay = Cy - fy and A, imin (A,, Ay) if ; > l and label7 wih a "fonvard label" (i" 1 ', A,); or if f^ > O, compue A, == min(a,, f,,) and label j by a "backward label" (;~, Aj). If no such j exiss hen OUTPUT f. Sop [^" ; r/îe maximum flow.} Else coninue (ha is, go o Sep ).. Repea Sep unil is reached. [This gives a flow augmening pah P: s >.} If i is impossible o reach hen OUTPUT f. Sop [f is he maximum flow.} Else coninue (ha is, go o Sep ).. Backrack he pah P, using he labels.. Using P, augmen he exising flow by a(. Se f = f + a(. 7. Remove ali labels from verices,, n. Go o Sep. End FORD-FULKERSON Graphs and Combinaorial Opimizaion Chap. Table1.9
Maximum 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018
MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren
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 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 informationHidden Markov Models. Adapted from. Dr Catherine Sweeney-Reed s slides
Hidden Markov Models Adaped from Dr Caherine Sweeney-Reed s slides Summary Inroducion Descripion Cenral in HMM modelling Exensions Demonsraion Specificaion of an HMM Descripion N - number of saes Q = {q
More informationMEI Mechanics 1 General motion. Section 1: Using calculus
Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy
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 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 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 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 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 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. 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 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 informationa 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s)
Name: Dae: Kinemaics Review (Honors. Physics) Complee he following on a separae shee of paper o be urned in on he day of he es. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. 1. The graph below describes he
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 informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
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 informationRandom Walk with Anti-Correlated Steps
Random Walk wih Ani-Correlaed Seps John Noga Dirk Wagner 2 Absrac We conjecure he expeced value of random walks wih ani-correlaed seps o be exacly. We suppor his conjecure wih 2 plausibiliy argumens and
More informationk-remainder Cordial Graphs
Journal of Algorihms and Compuaion journal homepage: hp://jac.u.ac.ir k-remainder Cordial Graphs R. Ponraj 1, K. Annahurai and R. Kala 3 1 Deparmen of Mahemaics, Sri Paramakalyani College, Alwarkurichi
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 informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationComments on Window-Constrained Scheduling
Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
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 informationMA 366 Review - Test # 1
MA 366 Review - Tes # 1 Fall 5 () Resuls from Calculus: differeniaion formulas, implici differeniaion, Chain Rule; inegraion formulas, inegraion b pars, parial fracions, oher inegraion echniques. (1) Order
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
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 informationt dt t SCLP Bellman (1953) CLP (Dantzig, Tyndall, Grinold, Perold, Anstreicher 60's-80's) Anderson (1978) SCLP
Coninuous Linear Programming. Separaed Coninuous Linear Programming Bellman (1953) max c () u() d H () u () + Gsusds (,) () a () u (), < < CLP (Danzig, yndall, Grinold, Perold, Ansreicher 6's-8's) Anderson
More informationA Hop Constrained Min-Sum Arborescence with Outage Costs
A Hop Consrained Min-Sum Arborescence wih Ouage Coss Rakesh Kawara Minnesoa Sae Universiy, Mankao, MN 56001 Email: Kawara@mnsu.edu Absrac The hop consrained min-sum arborescence wih ouage coss problem
More informationWaveform Transmission Method, A New Waveform-relaxation Based Algorithm. to Solve Ordinary Differential Equations in Parallel
Waveform Transmission Mehod, A New Waveform-relaxaion Based Algorihm o Solve Ordinary Differenial Equaions in Parallel Fei Wei Huazhong Yang Deparmen of Elecronic Engineering, Tsinghua Universiy, Beijing,
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 information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
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 informationLecture 23: I. Data Dependence II. Dependence Testing: Formulation III. Dependence Testers IV. Loop Parallelization V.
Lecure 23: Array Dependence Analysis & Parallelizaion I. Daa Dependence II. Dependence Tesing: Formulaion III. Dependence Tesers IV. Loop Parallelizaion V. Loop Inerchange [ALSU 11.6, 11.7.8] Phillip B.
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
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 information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationAppendix to Online l 1 -Dictionary Learning with Application to Novel Document Detection
Appendix o Online l -Dicionary Learning wih Applicaion o Novel Documen Deecion Shiva Prasad Kasiviswanahan Huahua Wang Arindam Banerjee Prem Melville A Background abou ADMM In his secion, we give a brief
More informationY. Xiang, Learning Bayesian Networks 1
Learning Bayesian Neworks Objecives Acquisiion of BNs Technical conex of BN learning Crierion of sound srucure learning BN srucure learning in 2 seps BN CPT esimaion Reference R.E. Neapolian: Learning
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 informationHidden Markov Models
Hidden Markov Models Probabilisic reasoning over ime So far, we ve mosly deal wih episodic environmens Excepions: games wih muliple moves, planning In paricular, he Bayesian neworks we ve seen so far describe
More informationDifferential Geometry: Numerical Integration and Surface Flow
Differenial Geomery: Numerical Inegraion and Surface Flow [Implici Fairing of Irregular Meshes using Diffusion and Curaure Flow. Desbrun e al., 1999] Energy Minimizaion Recall: We hae been considering
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 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 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 informationMA Study Guide #1
MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()
More informationLecture 9: September 25
0-725: Opimizaion Fall 202 Lecure 9: Sepember 25 Lecurer: Geoff Gordon/Ryan Tibshirani Scribes: Xuezhi Wang, Subhodeep Moira, Abhimanu Kumar Noe: LaTeX emplae couresy of UC Berkeley EECS dep. Disclaimer:
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationRL Lecture 7: Eligibility Traces. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1
RL Lecure 7: Eligibiliy Traces R. S. Suon and A. G. Baro: Reinforcemen Learning: An Inroducion 1 N-sep TD Predicion Idea: Look farher ino he fuure when you do TD backup (1, 2, 3,, n seps) R. S. Suon and
More informationPhys1112: DC and RC circuits
Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.
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 informationParticle Swarm Optimization Combining Diversification and Intensification for Nonlinear Integer Programming Problems
Paricle Swarm Opimizaion Combining Diversificaion and Inensificaion for Nonlinear Ineger Programming Problems Takeshi Masui, Masaoshi Sakawa, Kosuke Kao and Koichi Masumoo Hiroshima Universiy 1-4-1, Kagamiyama,
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 informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationComputer-Aided Analysis of Electronic Circuits Course Notes 3
Gheorghe Asachi Technical Universiy of Iasi Faculy of Elecronics, Telecommunicaions and Informaion Technologies Compuer-Aided Analysis of Elecronic Circuis Course Noes 3 Bachelor: Telecommunicaion Technologies
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 informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationApplication of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing
Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology
More informationPhysics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008
Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly
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 informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
More informationCOMPUTING SZEGED AND SCHULTZ INDICES OF HAC C C [ p, q ] NANOTUBE BY GAP PROGRAM
Diges Journal of Nanomaerials and Biosrucures, Vol. 4, No. 1, March 009, p. 67-7 COMPUTING SZEGED AND SCHULTZ INDICES OF HAC C C [ p, q ] 5 6 7 NANOTUBE BY GAP PROGRAM A. Iranmanesh *, Y. Alizadeh Deparmen
More informationOptimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms
1 Opimal Embedding of Funcions for In-Nework Compuaion: Complexiy Analysis and Algorihms Pooja Vyavahare, Nuan Limaye, and D. Manjunah, Senior Member, IEEE arxiv:1401.2518v3 [cs.dc] 14 Jul 2015 Absrac
More informationx i v x t a dx dt t x
Physics 3A: Basic Physics I Shoup - Miderm Useful Equaions A y A sin A A A y an A y A A = A i + A y j + A z k A * B = A B cos(θ) A B = A B sin(θ) A * B = A B + A y B y + A z B z A B = (A y B z A z B y
More informationMachine Learning 4771
ony Jebara, Columbia Universiy achine Learning 4771 Insrucor: ony Jebara ony Jebara, Columbia Universiy opic 20 Hs wih Evidence H Collec H Evaluae H Disribue H Decode H Parameer Learning via JA & E ony
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 informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationUniversity of Cyprus Biomedical Imaging and Applied Optics. Appendix. DC Circuits Capacitors and Inductors AC Circuits Operational Amplifiers
Universiy of Cyprus Biomedical Imaging and Applied Opics Appendix DC Circuis Capaciors and Inducors AC Circuis Operaional Amplifiers Circui Elemens An elecrical circui consiss of circui elemens such as
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 informationOptima and Equilibria for Traffic Flow on a Network
Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1 A Traffic
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More information1. Kinematics I: Position and Velocity
1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his
More informationProblem Set #1. i z. the complex propagation constant. For the characteristic impedance:
Problem Se # Problem : a) Using phasor noaion, calculae he volage and curren waves on a ransmission line by solving he wave equaion Assume ha R, L,, G are all non-zero and independen of frequency From
More informationi L = VT L (16.34) 918a i D v OUT i L v C V - S 1 FIGURE A switched power supply circuit with diode and a switch.
16.4.3 A SWITHED POWER SUPPY USINGA DIODE In his example, we will analyze he behavior of he diodebased swiched power supply circui shown in Figure 16.15. Noice ha his circui is similar o ha in Figure 12.41,
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationUST/DME: A Clock Tree Router for General Skew Constraints
UST/DME: A Clock Tree Rouer for General Skew Consrains C.-W. Alber Tsao Ulima Inerconnec Technology, Inc. Cheng-Kok Koh School of ECE, Purdue Universiy (Suppored in par by NSF) 4// Ouline of Talk Inroducion
More informationAn Introduction to Constraint Based Scheduling
Lesson 1: An Inroducion o Consrain Based Scheduling Michele Lombardi DEIS, Universiy of Bologna Ouline Course ouline Lecure 1: An Inroducion o Consrain Based Scheduling Lecure
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 informationTHE MATRIX-TREE THEOREM
THE MATRIX-TREE THEOREM 1 The Marix-Tree Theorem. The Marix-Tree Theorem is a formula for he number of spanning rees of a graph in erms of he deerminan of a cerain marix. We begin wih he necessary graph-heoreical
More informationFarr High School NATIONAL 5 PHYSICS. Unit 3 Dynamics and Space. Exam Questions
Farr High School NATIONAL 5 PHYSICS Uni Dynamics and Space Exam Quesions VELOCITY AND DISPLACEMENT D B D 4 E 5 B 6 E 7 E 8 C VELOCITY TIME GRAPHS (a) I is acceleraing Speeding up (NOT going down he flume
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
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 informationDecentralized Stochastic Control with Partial History Sharing: A Common Information Approach
1 Decenralized Sochasic Conrol wih Parial Hisory Sharing: A Common Informaion Approach Ashuosh Nayyar, Adiya Mahajan and Demoshenis Tenekezis arxiv:1209.1695v1 [cs.sy] 8 Sep 2012 Absrac A general model
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 informationGiambattista, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76
Giambaisa, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76 9. Sraeg Le be direced along he +x-axis and le be 60.0 CCW from Find he magniude of 6.0 B 60.0 4.0 A x 15. (a) Sraeg Since he angle
More information