Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Max Flow. Lecture 4. Optimization on graphs. C25 Optimization Hilary 2013 A. Zisserman. Max-flow & min-cut. The augmented path algorithm"

Transcription

1 Lecture 4 C5 Optimization Hilary 03 A. Zisserman Optimization on graphs Max-flow & min-cut The augmented path algorithm Optimization for binary image graphs Applications Max Flow Given: a weighted directed graph with two distinguished nodes: source s, sink (destination) t B reminder Interpret edge weights (all positive) as capacities Goal: Find maximum flow from s to t Flow does not exceed capacity in any edge Flow at every vertex satisfies equilibrium [ flow in equals flow out ] e.g. oil flowing through pipes, internet routing

2 Example 3 cont. B reminder Slide: Robert Sedgewick and Kevin Wayne Example cont. B reminder

3 Max flow example 3 S T max flow is unique = but there may be multiple paths (solutions) that achieve it Matlab LP function linprog for max-flow >> f = [ -; -; 0; 0; 0 ]; >> A = [ e e e 3 e 4 e ]; >> b = [; ; ; 3; ]; >> Aeq = [ >> beq = [ 0; 0 ]; >> lb = zeros(5,); ]; >> x = linprog( f, A, b, Aeq, beq, lb ); 3 e e 4 S e 3 T e e 5 >> Optimization terminated. >> x x = >> flow = - x * f flow =.0

4 Another example 9 / 5/9 / v v 5 4 v v 3/5 3 4/4 Max Flow = 7 The st-mincut Problem An st-cut (S,T) divides the nodes between source and sink 9 The cost of the cut is the sum of costs of all edges going from S to T v v 5 4 The st-min-cut is the cut with lowest cost Each node is either assigned to the source S or sink T The cost of the edge (i, j) is taken if (i S) and (j T) Slides from Pushmeet Kohli

5 The st-mincut Problem An st-cut (S,T) divides the nodes between source and sink 9 The cost of the cut is the sum of costs of all edges going from S to T v v 5 4 The st-min-cut is the cut with lowest cost Each node is either assigned to the source S or sink T = 6 The cost of the edge (i, j) is taken if (i S) and (j T) The st-mincut Problem An st-cut (S,T) divides the nodes between source and sink 9 The cost of the cut is the sum of costs of all edges going from S to T v v 5 4 The st-min-cut is the cut with lowest cost Each node is either assigned to the source S or sink T = 7 The cost of the edge (i, j) is taken if (i S) and (j T)

6 Min-cut\Max-flow Theorem 9 v v 5 4 In every network, the maximum flow equals the cost of the st-mincut Max flow = min cut = 7 Next: the augmented path algorithm for computing the max-flow/min-cut Maxflow Algorithms v v 5 Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

7 Maxflow Algorithms v v 5 Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity Maxflow Algorithms - 9 v v 5- Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

8 Maxflow Algorithms 0 9 v v 3 Flow = 4 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity Maxflow Algorithms 0 9 v v 3 Flow = 4 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

9 Maxflow Algorithms 0 9 v v 3 Flow = 4 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity Maxflow Algorithms 0 5 v v 3 Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

10 Maxflow Algorithms 0 5 v v 3 Flow = 6 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity Maxflow Algorithms 0 5 v v 3 Flow = 6 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

11 Maxflow Algorithms v v Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity Maxflow Algorithms v v Flow = Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path 3. Repeat until no path can be found Algorithms assume non-negative capacity

12 Maxflow Algorithms Flow = 7 Min cut = v v 3 0 v v = 7 Image Graphs many loops very large number of nodes (variables) millions dynamic programming can t be used Consider binary graphs (h = ) pixels

13 Example: noise removal in an image N (i) i j n is the number of pixels in the image, e.g. n = M Graph structure: each vertex is connected to four neighbours noisy data z i = x i + w i where x i {0, } w i N(0, σ ) and φ(x i,x j ) = ( 0 ifxi = x j ifx i 6= x j. 0 i f(x) = X i (z i x i ) +dφ(x i,x i ) Graph cut algorithms Binary optimization: each variable x has one of two possible values f(x) = nx {m i (x i )+ X φ i (x i,x j )} i= j N (i) x i {0, } N (i) is the neighbourhood of node i i The cost function f(x) issub-modularif φ(0, 0) + φ(, ) <= φ(0, ) + φ(, 0) If f(x) is sub-modular, then it can be optimized by the Min-Cut algorithm Complexity of minimization: exhaustive search O( n ) min cut O(n 3 )

14 d = 0 original original plus noise Min x d = 60 original original plus noise Min x n 0 K

15 Optimization using graph cuts Stage : map the cost function f(x) onto a flow network so that a cut of the network corresponds to the cost f(x) Stage : compute the min-cut of the network using an augmented path algorithm Map f(x) onto network flow Construct a network so that a cut corresponds to an assignment of x i m () D Label m () C B Label 0 m (0) m A (0) s n-links w pq = 0 a cut x x t = f(x) = nx {m i (x i )+ X φ i (x i,x j )} i= j N (i)

16 m () D m () C B m (0) m A (0) x x For unary terms only: m () x m (0) (0) m () m (0) () x (0) m () m () cut x =,x = x x m (0) m (0) f(x) =m () + m () () Now, include pair wise term D m () m () C B m (0) m (0) A (0) m () + C - A m () + D - C B+C-A-D x x x x m (0) m (0) x () x 0 A C B D = A + B+C D-C 0 0 -A-D + + C-A C-A 0 D-C 0 0 add C-A if x = add D-C if x = Sub-modular constraint: flows must be positive. So, B+C-A-D >= 0

17 m () m (0) C D A B x x (0) m () + C - A m () + D - C cut B+C-A-D x x x =,x = m (0) m (0) () f(x) =m () + m () + D A m () m (0) C D A B x x (0) m () + C - A m () + D - C cut x =0,x = B+C-A-D x x f(x) = m (0) + m () m (0) m (0) () +D C + B + C A D = m (0) + m () + B A

18 Summary: optimization using graph cuts Stage : map the cost function f(x) onto a flow network so that a cut of the network corresponds to the cost f(x) Stage : compute the min-cut of the network using an augmented path algorithm Applications Optimization of binary image graph using graph-cuts:. Image cut-out and editing. Image quilting 3. Interactive Digital Photo-montage

19 . Image cut-out by binary segmentation Object - white, Background - green/grey Graph G = (V,E) Each vertex corresponds to a pixel Edges define a 4-neighbourhood grid graph Assign a label to each vertex from L = {obj,bkg}

20 Object - white, Background - green/grey Cost of a labelling f : V L Cost of label obj low Graph G = (V,E) Per Vertex Cost Cost of label bkg high Object - white, Background - green/grey Cost of a labelling f : V L Graph G = (V,E) Per Vertex Cost Cost of label obj high UNARY COST Cost of label bkg low

21 Object - white, Background - green/grey Cost of a labelling f : V L Graph G = (V,E) Per Edge Cost Cost of same label low Cost of different labels high Object - white, Background - green/grey Cost of a labelling f : V L Graph G = (V,E) Per Edge Cost Cost of different labels low PAIRWISE COST

22 Object - white, Background - green/grey Graph G = (V,E) Problem: Find the labelling with minimum cost f* N (i) i j f(x) = nx {m i (x i )+ X φ i (x i,x j )} i= j N (i) x i = for foreground pixels, x i = 0 for background m i (x i ) is likelihood that pixel at i is foreground (if x i =),orbackground (if x i = 0 ), e.g. using colour histogram of seed regions φ(x i,x j ) penalizes a change of state: φ i (x i,x j ) = ( 0 ifxi = x j γe β(i i I j ) ifx i 6= x j.

23 Application: foreground/background image segmentation foreground Seed Pixels Background Seed Pixels use seed pixels to learn colour distribution Image editing Available in Microsoft Office

24 Image Quilting Example: Texture Synthesis Goal of Texture Synthesis: create new samples of a given texture Many applications: virtual environments, hole-filling, texturing surfaces

25 block Input texture B B B B B B Random placement of blocks Neighboring blocks constrained by overlap Minimal error boundary cut Algorithm Pick size of block and size of overlap Synthesize blocks in raster order Search input texture for block that satisfies overlap constraints (above and left) Paste new block into resulting texture > use graph cuts to compute minimal error boundary cut Efros & Freeman 00, Kwatra et al. 003

26 Minimal error boundary overlapping blocks vertical boundary _ = overlap error min. error boundary

27 Interactive Digital Photomontage

28 Agarwala et al. 004 Use graph-cuts to quilt images

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

More information

Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem

Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem Lecture 6 Network Flow I 6. Overview In these next two lectures we are going to talk about an important algorithmic problem called the Network Flow Problem. Network flow is important because it can be

More information

Introduction to Segmentation

Introduction to Segmentation Lecture 2: Introduction to Segmentation Jonathan Krause 1 Goal Goal: Identify groups of pixels that go together image credit: Steve Seitz, Kristen Grauman 2 Types of Segmentation Semantic Segmentation:

More information

Proximal mapping via network optimization

Proximal mapping via network optimization L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:

More information

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

Lecture 2: The SVM classifier

Lecture 2: The SVM classifier Lecture 2: The SVM classifier C19 Machine Learning Hilary 2015 A. Zisserman Review of linear classifiers Linear separability Perceptron Support Vector Machine (SVM) classifier Wide margin Cost function

More information

Dynamic Programming and Graph Algorithms in Computer Vision

Dynamic Programming and Graph Algorithms in Computer Vision Dynamic Programming and Graph Algorithms in Computer Vision Pedro F. Felzenszwalb and Ramin Zabih Abstract Optimization is a powerful paradigm for expressing and solving problems in a wide range of areas,

More information

Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

More information

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

More information

Approximation Algorithms

Approximation Algorithms Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

More information

Algorithm Design and Analysis

Algorithm Design and Analysis Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;

More information

Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images

Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images Proceedings of Internation Conference on Computer Vision, Vancouver, Canada, July 2001 vol.i, p.105 Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images Yuri Y. Boykov

More information

Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

More information

Class One: Degree Sequences

Class One: Degree Sequences Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

More information

Social Media Mining. Graph Essentials

Social Media Mining. Graph Essentials Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

More information

Distributed Computing over Communication Networks: Topology. (with an excursion to P2P)

Distributed Computing over Communication Networks: Topology. (with an excursion to P2P) Distributed Computing over Communication Networks: Topology (with an excursion to P2P) Some administrative comments... There will be a Skript for this part of the lecture. (Same as slides, except for today...

More information

Artificial Intelligence Methods (G52AIM)

Artificial Intelligence Methods (G52AIM) Artificial Intelligence Methods (G52AIM) Dr Rong Qu rxq@cs.nott.ac.uk Constructive Heuristic Methods Constructive Heuristics method Start from an empty solution Repeatedly, extend the current solution

More information

Flow and Activity Analysis

Flow and Activity Analysis Facility Location, Layout, and Flow and Activity Analysis Primary activity relationships Organizational relationships» Span of control and reporting hierarchy Flow relationships» Flow of materials, people,

More information

An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision

An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision In IEEE Transactions on PAMI, Vol. 26, No. 9, pp. 1124-1137, Sept. 2004 p.1 An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision Yuri Boykov and Vladimir Kolmogorov

More information

Impact Of Interference On Multi-hop Wireless Network Performance

Impact Of Interference On Multi-hop Wireless Network Performance Impact Of Interference On Multi-hop Wireless Network Performance Kamal Jain Jitendra Padhye Venkata N. Padmanabhan Lili Qiu Microsoft Research One Microsoft Way, Redmond, WA 98052. {kamalj, padhye, padmanab,

More information

Lecture 11: Graphical Models for Inference

Lecture 11: Graphical Models for Inference Lecture 11: Graphical Models for Inference So far we have seen two graphical models that are used for inference - the Bayesian network and the Join tree. These two both represent the same joint probability

More information

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ The Binary Blocking Flow Algorithm Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ Theory vs. Practice In theory, there is no difference between theory and practice.

More information

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010 CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

More information

Segmentation. Lecture 12. Many slides from: S. Lazebnik, K. Grauman and P. Kumar

Segmentation. Lecture 12. Many slides from: S. Lazebnik, K. Grauman and P. Kumar Segmentation Lecture 12 Many slides from: S. Lazebnik, K. Grauman and P. Kumar Image Segmentation Image segmentation The goals of segmentation Group together similar-looking pixels for efficiency of further

More information

Automatic Reconstruction of Parametric Building Models from Indoor Point Clouds. CAD/Graphics 2015

Automatic Reconstruction of Parametric Building Models from Indoor Point Clouds. CAD/Graphics 2015 Automatic Reconstruction of Parametric Building Models from Indoor Point Clouds Sebastian Ochmann Richard Vock Raoul Wessel Reinhard Klein University of Bonn, Germany CAD/Graphics 2015 Motivation Digital

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling

Minimum cost maximum flow, Minimum cost circulation, Cost/Capacity scaling 6.854 Advanced Algorithms Lecture 16: 10/11/2006 Lecturer: David Karger Scribe: Kermin Fleming and Chris Crutchfield, based on notes by Wendy Chu and Tudor Leu Minimum cost maximum flow, Minimum cost circulation,

More information

Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

More information

Dipartimento di Informatica Università del Piemonte Orientale A. Avogadro Via Bellini 25/G, 15100 Alessandria

Dipartimento di Informatica Università del Piemonte Orientale A. Avogadro Via Bellini 25/G, 15100 Alessandria Dipartimento di Informatica Università del Piemonte Orientale A. Avogadro Via Bellini 25/G, 15100 Alessandria http://www.di.unipmn.it Reliability and QoS Analysis of the Italian GARR network Authors: Andrea

More information

Data Structures and Algorithms Written Examination

Data Structures and Algorithms Written Examination Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where

More information

Linear Programming Problems

Linear Programming Problems Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number

More information

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology

Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Solving NP Hard problems in practice lessons from Computer Vision and Computational Biology Yair Weiss School of Computer Science and Engineering The Hebrew University of Jerusalem www.cs.huji.ac.il/ yweiss

More information

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/

The Binary Blocking Flow Algorithm. Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ The Binary Blocking Flow Algorithm Andrew V. Goldberg Microsoft Research Silicon Valley www.research.microsoft.com/ goldberg/ Why this Max-Flow Talk? The result: O(min(n 2/3, m 1/2 )mlog(n 2 /m)log(u))

More information

Social and Technological Network Analysis. Lecture 3: Centrality Measures. Dr. Cecilia Mascolo (some material from Lada Adamic s lectures)

Social and Technological Network Analysis. Lecture 3: Centrality Measures. Dr. Cecilia Mascolo (some material from Lada Adamic s lectures) Social and Technological Network Analysis Lecture 3: Centrality Measures Dr. Cecilia Mascolo (some material from Lada Adamic s lectures) In This Lecture We will introduce the concept of centrality and

More information

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II. Matrix Algorithms DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

More information

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

More information

A Practical Scheme for Wireless Network Operation

A Practical Scheme for Wireless Network Operation A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.

Linear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued. Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.

More information

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm. Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

More information

SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM

SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the

More information

Euler Paths and Euler Circuits

Euler Paths and Euler Circuits Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

More information

A Network Flow Approach in Cloud Computing

A Network Flow Approach in Cloud Computing 1 A Network Flow Approach in Cloud Computing Soheil Feizi, Amy Zhang, Muriel Médard RLE at MIT Abstract In this paper, by using network flow principles, we propose algorithms to address various challenges

More information

Why? A central concept in Computer Science. Algorithms are ubiquitous.

Why? A central concept in Computer Science. Algorithms are ubiquitous. Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

More information

Lecture 7: NP-Complete Problems

Lecture 7: NP-Complete Problems IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

More information

NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness

NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

Image Segmentation and Registration

Image Segmentation and Registration Image Segmentation and Registration Dr. Christine Tanner (tanner@vision.ee.ethz.ch) Computer Vision Laboratory, ETH Zürich Dr. Verena Kaynig, Machine Learning Laboratory, ETH Zürich Outline Segmentation

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

Chapter 10: Network Flow Programming

Chapter 10: Network Flow Programming Chapter 10: Network Flow Programming Linear programming, that amazingly useful technique, is about to resurface: many network problems are actually just special forms of linear programs! This includes,

More information

Lecture 4: Thresholding

Lecture 4: Thresholding Lecture 4: Thresholding c Bryan S. Morse, Brigham Young University, 1998 2000 Last modified on Wednesday, January 12, 2000 at 10:00 AM. Reading SH&B, Section 5.1 4.1 Introduction Segmentation involves

More information

A network flow algorithm for reconstructing. binary images from discrete X-rays

A network flow algorithm for reconstructing. binary images from discrete X-rays A network flow algorithm for reconstructing binary images from discrete X-rays Kees Joost Batenburg Leiden University and CWI, The Netherlands kbatenbu@math.leidenuniv.nl Abstract We present a new algorithm

More information

Natural Neighbour Interpolation

Natural Neighbour Interpolation Natural Neighbour Interpolation DThe Natural Neighbour method is a geometric estimation technique that uses natural neighbourhood regions generated around each point in the data set. The method is particularly

More information

Texture Unmapping Combining Parametric and Non- Parametric Techniques for Image Reconstruction

Texture Unmapping Combining Parametric and Non- Parametric Techniques for Image Reconstruction Texture Unmapping Combining Parametric and Non- Parametric Techniques for Image Reconstruction S. Mills and T.P. Pridmore School of Computer Science & Information Technology University of Nottingham Jubilee

More information

Step 3: Go to Column C. Use the function AVERAGE to calculate the mean values of n = 5. Column C is the column of the means.

Step 3: Go to Column C. Use the function AVERAGE to calculate the mean values of n = 5. Column C is the column of the means. EXAMPLES - SAMPLING DISTRIBUTION EXCEL INSTRUCTIONS This exercise illustrates the process of the sampling distribution as stated in the Central Limit Theorem. Enter the actual data in Column A in MICROSOFT

More information

Scheduling Shop Scheduling. Tim Nieberg

Scheduling Shop Scheduling. Tim Nieberg Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations

More information

Layout Design Post Woods Apartment Complex

Layout Design Post Woods Apartment Complex Layout Design Some of these slides are courtesy of Professor Paul Griffin Post Woods partment Complex tlanta Zoo Lenox Mall, tlanta Hospital floor plan Guildcrest Homes Factory floor plan 6 Furniture parts

More information

BOUNDARY EDGE DOMINATION IN GRAPHS

BOUNDARY EDGE DOMINATION IN GRAPHS BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

Seminar. Path planning using Voronoi diagrams and B-Splines. Stefano Martina stefano.martina@stud.unifi.it

Seminar. Path planning using Voronoi diagrams and B-Splines. Stefano Martina stefano.martina@stud.unifi.it Seminar Path planning using Voronoi diagrams and B-Splines Stefano Martina stefano.martina@stud.unifi.it 23 may 2016 This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24

Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divide-and-conquer algorithms

More information

Binary Image Analysis

Binary Image Analysis Binary Image Analysis Segmentation produces homogenous regions each region has uniform gray-level each region is a binary image (0: background, 1: object or the reverse) more intensity values for overlapping

More information

Graph Cuts in Vision and Graphics: Theories and Applications

Graph Cuts in Vision and Graphics: Theories and Applications 100 Math. Models of C.Vision: The Handbook, edts. Paragios, Chen, Faugeras Graph Cuts in Vision and Graphics: Theories and Applications Yuri Boykov and Olga Veksler Computer Science, The University of

More information

Graph Mining and Social Network Analysis

Graph Mining and Social Network Analysis Graph Mining and Social Network Analysis Data Mining and Text Mining (UIC 583 @ Politecnico di Milano) References Jiawei Han and Micheline Kamber, "Data Mining: Concepts and Techniques", The Morgan Kaufmann

More information

A Numerical Study on the Wiretap Network with a Simple Network Topology

A Numerical Study on the Wiretap Network with a Simple Network Topology A Numerical Study on the Wiretap Network with a Simple Network Topology Fan Cheng and Vincent Tan Department of Electrical and Computer Engineering National University of Singapore Mathematical Tools of

More information

Social Media Mining. Network Measures

Social Media Mining. Network Measures Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

More information

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm. Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

More information

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved.

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved. 7.5 SYSTEMS OF INEQUALITIES Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities

More information

Bandwidth Allocation in a Network Virtualization Environment

Bandwidth Allocation in a Network Virtualization Environment Bandwidth Allocation in a Network Virtualization Environment Juan Felipe Botero jfbotero@entel.upc.edu Xavier Hesselbach xavierh@entel.upc.edu Department of Telematics Technical University of Catalonia

More information

Linear programming and reductions

Linear programming and reductions Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence,

More information

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

More information

LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING. ----Changsheng Liu 10-30-2014

LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING. ----Changsheng Liu 10-30-2014 LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING ----Changsheng Liu 10-30-2014 Agenda Semi Supervised Learning Topics in Semi Supervised Learning Label Propagation Local and global consistency Graph

More information

Arrangements And Duality

Arrangements And Duality Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

More information

Robert Collins CSE598G. More on Mean-shift. R.Collins, CSE, PSU CSE598G Spring 2006

Robert Collins CSE598G. More on Mean-shift. R.Collins, CSE, PSU CSE598G Spring 2006 More on Mean-shift R.Collins, CSE, PSU Spring 2006 Recall: Kernel Density Estimation Given a set of data samples x i ; i=1...n Convolve with a kernel function H to generate a smooth function f(x) Equivalent

More information

Computational Approach for Assessment of Critical Infrastructure in Network Systems

Computational Approach for Assessment of Critical Infrastructure in Network Systems Computational Approach for Assessment of Critical Infrastructure in Network Systems EMIL KELEVEDJIEV 1 Institute of Mathematics and Informatics Bulgarian Academy of Sciences ABSTRACT. Methods of computational

More information

Graph Cuts for Image Segmentation

Graph Cuts for Image Segmentation Graph Cuts for Image Segmentation Ph.D. Seminar Report Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Meghshyam G. Prasad Roll No: 124058001 under the guidance

More information

Chap 4 The Simplex Method

Chap 4 The Simplex Method The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

More information

Equilibrium computation: Part 1

Equilibrium computation: Part 1 Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium

More information

SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis

SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 17, 2015 Outline

More information

Environmental Remote Sensing GEOG 2021

Environmental Remote Sensing GEOG 2021 Environmental Remote Sensing GEOG 2021 Lecture 4 Image classification 2 Purpose categorising data data abstraction / simplification data interpretation mapping for land cover mapping use land cover class

More information

Network Design with Coverage Costs

Network Design with Coverage Costs Network Design with Coverage Costs Siddharth Barman 1 Shuchi Chawla 2 Seeun William Umboh 2 1 Caltech 2 University of Wisconsin-Madison APPROX-RANDOM 2014 Motivation Physical Flow vs Data Flow vs. Commodity

More information

Algorithms and Data Structures

Algorithms and Data Structures Algorithms and Data Structures CMPSC 465 LECTURES 20-21 Priority Queues and Binary Heaps Adam Smith S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 1 Trees Rooted Tree: collection

More information

Signature Segmentation from Machine Printed Documents using Conditional Random Field

Signature Segmentation from Machine Printed Documents using Conditional Random Field 2011 International Conference on Document Analysis and Recognition Signature Segmentation from Machine Printed Documents using Conditional Random Field Ranju Mandal Computer Vision and Pattern Recognition

More information

Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

More information

NP-Hardness Results Related to PPAD

NP-Hardness Results Related to PPAD NP-Hardness Results Related to PPAD Chuangyin Dang Dept. of Manufacturing Engineering & Engineering Management City University of Hong Kong Kowloon, Hong Kong SAR, China E-Mail: mecdang@cityu.edu.hk Yinyu

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Introduction to Deep Learning Variational Inference, Mean Field Theory

Introduction to Deep Learning Variational Inference, Mean Field Theory Introduction to Deep Learning Variational Inference, Mean Field Theory 1 Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Center for Visual Computing Ecole Centrale Paris Galen Group INRIA-Saclay Lecture 3: recap

More information

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

More information

Capacity of Inter-Cloud Layer-2 Virtual Networking!

Capacity of Inter-Cloud Layer-2 Virtual Networking! Capacity of Inter-Cloud Layer-2 Virtual Networking! Yufeng Xin, Ilya Baldin, Chris Heermann, Anirban Mandal, and Paul Ruth!! Renci, University of North Carolina at Chapel Hill, NC, USA! yxin@renci.org!

More information

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004 Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

More information

Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections

Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections Blind Deconvolution of Barcodes via Dictionary Analysis and Wiener Filter of Barcode Subsections Maximilian Hung, Bohyun B. Kim, Xiling Zhang August 17, 2013 Abstract While current systems already provide

More information

Pulser Gating: A Clock Gating of Pulsed-Latch Circuits. Sangmin Kim, Inhak Han, Seungwhun Paik, and Youngsoo Shin Dept.

Pulser Gating: A Clock Gating of Pulsed-Latch Circuits. Sangmin Kim, Inhak Han, Seungwhun Paik, and Youngsoo Shin Dept. Pulser Gating: A Clock Gating of Pulsed-Latch Circuits Sangmin Kim, Inhak Han, Seungwhun Paik, and Youngsoo Shin Dept. of EE, KAIST Introduction Pulsed-latch circuits Clock gating synthesis Pulser gating

More information

Decentralized Utility-based Sensor Network Design

Decentralized Utility-based Sensor Network Design Decentralized Utility-based Sensor Network Design Narayanan Sadagopan and Bhaskar Krishnamachari University of Southern California, Los Angeles, CA 90089-0781, USA narayans@cs.usc.edu, bkrishna@usc.edu

More information

Machine vision systems - 2

Machine vision systems - 2 Machine vision systems Problem definition Image acquisition Image segmentation Connected component analysis Machine vision systems - 1 Problem definition Design a vision system to see a flat world Page

More information

On the effect of forwarding table size on SDN network utilization

On the effect of forwarding table size on SDN network utilization IBM Haifa Research Lab On the effect of forwarding table size on SDN network utilization Rami Cohen IBM Haifa Research Lab Liane Lewin Eytan Yahoo Research, Haifa Seffi Naor CS Technion, Israel Danny Raz

More information

Lecture 17 : Equivalence and Order Relations DRAFT

Lecture 17 : Equivalence and Order Relations DRAFT CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

More information

The Line Connectivity Problem

The Line Connectivity Problem Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany RALF BORNDÖRFER MARIKA NEUMANN MARC E. PFETSCH The Line Connectivity Problem Supported by the DFG Research

More information