Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Size: px
Start display at page:

Download "Luby s Alg. for Maximal Independent Sets using Pairwise Independence"

Transcription

1 Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, Maxmal Independent Sets For a graph G = (V, E), an ndependent set s a set S V whch contans no edges of G,.e., for all (u, v) E ether u S and/or v S. The ndependent set S s a maxmal ndependent set f for all v V, ether v S or N(v) S where N(v) denotes the neghbors of v. It s easy to fnd a maxmal ndependent set. For example, the followng algorthm works:. I =, V = V.. Whle (V ) do (a) Choose any v V. (b) Set I = I v. (c) Set V = V \ (v N(v)). 3. Output I. Our focus s fndng an ndependent set usng a parallel algorthm. The dea s that n every round we fnd a set S whch s an ndependent set. Then we add S to our current ndependent set I, and we remove S N(S) from the current graph V. If S N(S) s a constant fracton of V, then we wll only need O(log V ) rounds. We wll nstead ensure that by removng S N(S) from the graph, we remove a constant fracton of the edges. To choose S n parallel, each vertex v ndependently adds themselves to S wth a well chosen probablty p(v). We want to avod addng adjacent vertces to S. Hence, we wll prefer to add low degree vertces. But, f for some edge (u, v), both endponts were added to S, then we keep the hgher degree vertex. Here s the algorthm: The Algorthm Problem : Gven a graph fnd a maxmal ndependent set.. I =, G = G.. Whle (G s not the empty graph) do (a) Choose a random set of vertces S G by selectng each vertex v ndependently wth probablty /(d(v)). 8-

2 8- Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence (b) For every edge (u, v) E(G ) f both endponts are n S then remove the vertex of lower degree from S (Break tes arbtrarly). Call ths new set S. (c) I = I S. G = G \ (S N(S )),.e., G s the nduced subgraph on V \ (S N(S )) where V s the prevous vertex set. 3. output I Fg : The algorthm Correctness : We see that at each stage the set S that s added s an ndependent set. Moreover snce we remove, at each stage, S N(S ) the set I remans an ndependent set. Also note that all the vertces removed from G at a partcular stage are ether vertces n I or neghbours of some vertex n I. So the algorthm always outputs a maxmal ndependent set. We also note that t can be easly parallelzed on a CREW PRAM. 8. Expected Runnng Tme In ths secton we bound the expected runnng tme of the algorthm and n the next secton we derandomze t. Let G j = (V j, E j ) denote the graph after stage j. Man Lemma : For some c <, E ( E j E j ) < c E j. Hence, n expectaton, only O(log m) rounds wll be requred, where m = E 0. We say vertex v s BAD f more than /3 of the neghbors of v are of hgher degree than v. We say an edge s BAD f both of ts endponts are bad; otherwse the edge s GOOD. The key clams are that at least half the edges are GOOD, and each GOOD edge s deleted wth a constant probablty. The man lemma then follows mmedately. Lemma 8. At least half the edges are GOOD. Proof: Denote the set of bad edges by E B. We wll defne f : E B ( E ) so that for all e e E B, f(e ) f(e ) =. Ths proves E B E /, and we re done. The functon f s defned as follows. For each (u, v) E, drect t to the hgher degree vertex. Break tes as n the algorthm. Now, suppose (u, v) E B, and s drected towards v. Snce v s BAD, t has at least twce as many edges out as n. Hence we can par two edges out of v wth every edge nto v. Ths gves our mappng. Lemma 8. If v s GOOD then Pr ( N(v) S ) α, where α := ( e /6 ). Proof: Defne L(v) := {w N(v) d(w) d(v)}.

3 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence 8-3 By defnton, L(v) d(v) 3 f v s a GOOD vertex. Pr ( N(v) S ) = Pr ( N(v) S = ) = Pr ( w S ) w N(v) w L(v) = w L(v) w L(v) Pr ( w S ) ( ) d(w) ( ) d(v) exp( L(v) /d(v)) exp( /6), usng full ndependence Note, the above lemma s usng full ndependence n ts proof. Lemma 8.3 Pr ( w / S w S ) /. Proof: Let H(w) = N(w) \ L(w) = {z N(w) : d(z) > d(w)}. Pr ( w / S w S ) = Pr ( H(w) S w S ) Pr ( z S w S ) = = = Pr ( z S, w S ) Pr ( w S ) Pr ( z S ) Pr ( w S ) Pr ( w S ) Pr ( z S ) d(z) d(v) usng parwse ndependence. Lemma 8.4 If v s GOOD then Pr ( v N(S ) ) α

4 8-4 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Proof: Let V G denote the GOOD vertces. We have Pr ( v N(S ) v V G ) = Pr ( N(v) S v V G ) = Pr ( N(v) S N(v) S, v V G ) Pr ( N(v) S v V G ) Pr ( w S w N(v) S, v V G ) Pr ( N(v) S v V G ) (/)(α) = α Corollary 8.5 If v s GOOD then the probablty that v gets deleted s at least α. Corollary 8.6 If an edge e s GOOD then the probablty that t gets deleted s at least α. Proof: Pr ( e = (u, v) E j \ E j ) Pr ( v gets deleted ). We now re-state the man lemma : Man Lemma : E ( E j E j ) E j ( α/). Proof: E ( E j E j ) = Pr ( e gets deleted ) e E j E j α GOOD edges E j ( α/). The constant α s approxmately Thus, E ( E j ) E 0 ( α )j m exp( jα/) <, for j > α log m. Therefore, the expected number of rounds requred s 4m = O(log m). 8.3 Derandomzng MIS The only step where we use full ndependence s n Lemma 8. for lower boundng the probablty that a GOOD vertex gets pcked.the argument we used was essentally the followng : Lemma 8.7 Let X, n be {0, } random varables and p := Pr ( X = ). ndependent then ( n ) n Pr X > 0 ( p ) If the X are fully

5 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence 8-5 Here s the correspondng bound f the varables are parwse ndependent Lemma 8.8 Let X, n be {0, } random varables and p := Pr ( X = ). If the X are parwse ndependent then ( n ) Pr X > 0 mn{, p } Proof: Suppose p /. Then we have the followng, (the condton p / wll only come nto some algebra at the end) ( ) Pr X > 0 Pr ( X = ) Pr ( X =, X j = ) j = p p p j j ( ) p p = ( ) p p p when p. If p > /, then we restrct our ndex of summaton to a set S [n] such that / S p, and the same basc argument works. Note, f p >, there always must exst a subset S [n] where / S p Usng the constructon descrbed earler we can now derandomze to get an algorthm that runs n O(mn/α) tme (ths s asymptotcally as good as the sequental algorthm). The advantage of ths method s that t can be easly parallelzed to gve an NC algorthm (usng O(m) processors). 8.4 Hstory and Open Questons The k wse ndependence derandomzaton approach was developed n [CG], [ABI] and [L85]. The maxmal ndependence problem was frst shown to be n NC n [KW].They showed that MIS n s NC 4. Subsequently, mprovements on ths were found by [ABI] and [L85]. The verson descrbed n these notes s the latter. The queston of whether MIS s n NC s stll open. References [] N Alon, L. Baba, A. Ita, A fast and smple randomzed parallel algorthm for the Maxmal Independent Set Problem, Journal of Algorthms, Vol. 7, 986, pp [] B. Chor, O. Goldrech, On the power of two-pont samplng, Journal of Complexty, Vol. 5, 989, pp

6 8-6 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence [3] R.M. Karp, A. Wgderson, A fast parallel algorthm for the maxmal ndependent set problem, Proc. 6th ACM Symposum on Theory of Computng, 984, pp [4] M. Luby, A smple parallel algorthm for the maxmal ndependent set problem, Proc, 7th ACM Symposum on Theory of Computng, 985, pp. -0.

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Generalizing the degree sequence problem

Generalizing the degree sequence problem Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance

) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6

NPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6 PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has

More information

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Lecture 7 March 20, 2002

Lecture 7 March 20, 2002 MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng

More information

Graph Calculus: Scalable Shortest Path Analytics for Large Social Graphs through Core Net

Graph Calculus: Scalable Shortest Path Analytics for Large Social Graphs through Core Net Graph Calculus: Scalable Shortest Path Analytcs for Large Socal Graphs through Core Net Lxn Fu Department of Computer Scence Unversty of North Carolna at Greensboro Greensboro, NC, U.S.A. lfu@uncg.edu

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

More information

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

More information

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

On Lockett pairs and Lockett conjecture for π-soluble Fitting classes On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs

More information

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1 (4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

A Constant Factor Approximation for the Single Sink Edge Installation Problem

A Constant Factor Approximation for the Single Sink Edge Installation Problem A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buy-at-bulk network

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

An Empirical Study of Two MIS Algorithms

An Empirical Study of Two MIS Algorithms An Empirical Study of Two MIS Algorithms Email: Tushar Bisht and Kishore Kothapalli International Institute of Information Technology, Hyderabad Hyderabad, Andhra Pradesh, India 32. tushar.bisht@research.iiit.ac.in,

More information

Availability-Based Path Selection and Network Vulnerability Assessment

Availability-Based Path Selection and Network Vulnerability Assessment Avalablty-Based Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl

More information

Chapter 7: Answers to Questions and Problems

Chapter 7: Answers to Questions and Problems 19. Based on the nformaton contaned n Table 7-3 of the text, the food and apparel ndustres are most compettve and therefore probably represent the best match for the expertse of these managers. Chapter

More information

Two-phase algorithms for the parametric shortest path problem

Two-phase algorithms for the parametric shortest path problem Two-phase algorthms for the parametrc shortest path problem Sourav Chakraborty 1, Eldar Fscher 1, Oded Lachsh 2, and Raphael Yuster 3 1 Department of Computer Scence, Technon, Hafa 32000, Israel {eldar,sourav}@cs.technon.ac.l

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

Efficient Reinforcement Learning in Factored MDPs

Efficient Reinforcement Learning in Factored MDPs Effcent Renforcement Learnng n Factored MDPs Mchael Kearns AT&T Labs mkearns@research.att.com Daphne Koller Stanford Unversty koller@cs.stanford.edu Abstract We present a provably effcent and near-optmal

More information

Approximation algorithms for allocation problems: Improving the factor of 1 1/e

Approximation algorithms for allocation problems: Improving the factor of 1 1/e Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e Urel Fege Mcrosoft Research Redmond, WA 98052 urfege@mcrosoft.com Jan Vondrák Prnceton Unversty Prnceton, NJ 08540 jvondrak@gmal.com

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars

More information

Ring structure of splines on triangulations

Ring structure of splines on triangulations www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

Loop Parallelization

Loop Parallelization - - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Quantization Effects in Digital Filters

Quantization Effects in Digital Filters Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value

More information

General Auction Mechanism for Search Advertising

General Auction Mechanism for Search Advertising General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

More information

To Fill or not to Fill: The Gas Station Problem

To Fill or not to Fill: The Gas Station Problem To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Bandwdth Packng E. G. Coman, Jr. and A. L. Stolyar Bell Labs, Lucent Technologes Murray Hll, NJ 07974 fegc,stolyarg@research.bell-labs.com Abstract We model a server that allocates varyng amounts of bandwdth

More information

Complete Fairness in Secure Two-Party Computation

Complete Fairness in Secure Two-Party Computation Complete Farness n Secure Two-Party Computaton S. Dov Gordon Carmt Hazay Jonathan Katz Yehuda Lndell Abstract In the settng of secure two-party computaton, two mutually dstrustng partes wsh to compute

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM

GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 -NORM BARRIOT Jean-Perre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jean-perre.barrot@cnes.fr 1/Introducton The

More information

Solutions to the exam in SF2862, June 2009

Solutions to the exam in SF2862, June 2009 Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Addendum to: Importing Skill-Biased Technology

Addendum to: Importing Skill-Biased Technology Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

ESCI 341 Atmospheric Thermodynamics Lesson 9 Entropy

ESCI 341 Atmospheric Thermodynamics Lesson 9 Entropy ESCI 341 Atmosherc hermodynamcs Lesson 9 Entroy References: An Introducton to Atmosherc hermodynamcs, sons Physcal Chemstry (4 th edton), Levne hermodynamcs and an Introducton to hermostatstcs, Callen

More information

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004 OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

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

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

The Full-Wave Rectifier

The Full-Wave Rectifier 9/3/2005 The Full Wae ectfer.doc /0 The Full-Wae ectfer Consder the followng juncton dode crcut: s (t) Power Lne s (t) 2 Note that we are usng a transformer n ths crcut. The job of ths transformer s to

More information

The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values

The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values Lecture 7: Expectaton Values The materal n ths lecture covers the followng n Atkns. 11.5 The nformton of a wavefuncton (d) superpostons and expectaton values Lecture on-lne Expectaton Values (PDF) Expectaton

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Period and Deadline Selection for Schedulability in Real-Time Systems

Period and Deadline Selection for Schedulability in Real-Time Systems Perod and Deadlne Selecton for Schedulablty n Real-Tme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng

More information

An ACO Algorithm for. the Graph Coloring Problem

An ACO Algorithm for. the Graph Coloring Problem Int. J. Contemp. Math. Scences, Vol. 3, 2008, no. 6, 293-304 An ACO Algorthm for the Graph Colorng Problem Ehsan Salar and Kourosh Eshgh Department of Industral Engneerng Sharf Unversty of Technology,

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.

INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton

More information

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures

Minimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng

More information

Stable Distributions, Pseudorandom Generators, Embeddings, and Data Stream Computation

Stable Distributions, Pseudorandom Generators, Embeddings, and Data Stream Computation Stable Dstrbutons, Pseudorandom Generators, Embeddngs, and Data Stream Computaton PIOTR INDYK MIT, Cambrdge, Massachusetts Abstract. In ths artcle, we show several results obtaned by combnng the use of

More information

Proactive Secret Sharing Or: How to Cope With Perpetual Leakage

Proactive Secret Sharing Or: How to Cope With Perpetual Leakage Proactve Secret Sharng Or: How to Cope Wth Perpetual Leakage Paper by Amr Herzberg Stanslaw Jareck Hugo Krawczyk Mot Yung Presentaton by Davd Zage What s Secret Sharng Basc Idea ((2, 2)-threshold scheme):

More information

INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS

INVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMA-HDR NETWORKS 21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech-2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

PAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of Illinois-Urbana Champaign

PAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of Illinois-Urbana Champaign PAS: A Packet Accountng System to Lmt the Effects of DoS & DDoS Debsh Fesehaye & Klara Naherstedt Unversty of Illnos-Urbana Champagn DoS and DDoS DDoS attacks are ncreasng threats to our dgtal world. Exstng

More information

Stochastic Games on a Multiple Access Channel

Stochastic Games on a Multiple Access Channel Stochastc Games on a Multple Access Channel Prashant N and Vnod Sharma Department of Electrcal Communcaton Engneerng Indan Insttute of Scence, Bangalore 560012, Inda Emal: prashant2406@gmal.com, vnod@ece.sc.ernet.n

More information

On some special nonlevel annuities and yield rates for annuities

On some special nonlevel annuities and yield rates for annuities On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

More information

The Power of Slightly More than One Sample in Randomized Load Balancing

The Power of Slightly More than One Sample in Randomized Load Balancing The Power of Slghtly More than One Sample n Randomzed oad Balancng e Yng, R. Srkant and Xaohan Kang Abstract In many computng and networkng applcatons, arrvng tasks have to be routed to one of many servers,

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Secure Network Coding Over the Integers

Secure Network Coding Over the Integers Secure Network Codng Over the Integers Rosaro Gennaro Jonathan Katz Hugo Krawczyk Tal Rabn Abstract Network codng has receved sgnfcant attenton n the networkng communty for ts potental to ncrease throughput

More information

Embedding lattices in the Kleene degrees

Embedding lattices in the Kleene degrees F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded

More information

Week 6 Market Failure due to Externalities

Week 6 Market Failure due to Externalities Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to

More information

A Lyapunov Optimization Approach to Repeated Stochastic Games

A Lyapunov Optimization Approach to Repeated Stochastic Games PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://www-bcf.usc.edu/

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

More information

Relay Secrecy in Wireless Networks with Eavesdropper

Relay Secrecy in Wireless Networks with Eavesdropper Relay Secrecy n Wreless Networks wth Eavesdropper Parvathnathan Venktasubramanam, Tng He and Lang Tong School of Electrcal and Computer Engneerng Cornell Unversty, Ithaca, NY 14853 Emal : {pv45, th255,

More information

The Geometry of Online Packing Linear Programs

The Geometry of Online Packing Linear Programs The Geometry of Onlne Packng Lnear Programs Marco Molnaro R. Rav Abstract We consder packng lnear programs wth m rows where all constrant coeffcents are n the unt nterval. In the onlne model, we know the

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information