Maximum Matching Width: new characterizations and a fast algorithm for dominating set

Size: px
Start display at page:

Download "Maximum Matching Width: new characterizations and a fast algorithm for dominating set"

Transcription

1 Maxmum Matcn Wdt: nw caractrzatons and a ast alortm or domnatn st 정지수 (KAIST) jont work wt Sv Hortmo Sætr and Jan Arn Tll (Unvrsty o Brn) 1st Koran Worksop on Grap Tory KAIST

2 Grap wdt paramtrs tr-wdt (Haln 1976, Robrtson and Symour 1984) branc-wdt (Robrtson and Symour 1991) carvn-wdt (Symour and Tomas 1994) clqu-wdt (Courcll and Olaru 2000) rank-wdt (Oum and Symour 2006) boolan-wdt (Bu-Xuan, Tll, Vatsll 2011) maxmum matcn-wdt (Vatsll 2012)

3 Tr-wdt tr-wdt (Haln 1976, Robrtson and Symour 1984) A masur o ow tr-lk t rap s. tr tr-lk Furs rom ttp://ptscool.mmuw.du.pl/slds/lc6.pd

4 Tr-wdt tr-wdt (Haln 1976, Robrtson and Symour 1984) A masur o ow tr-lk t rap s. tr tr-lk Furs rom ttp://ptscool.mmuw.du.pl/slds/lc6.pd

5 A tr-dcomposton o a rap GG s a par (TT, XX tt tt VV TT ) consstn o a tr TT and a amly XX tt tt VV TT o substs XX tt o VV GG, calld bas, satsyn t ollown tr condtons: 1. ac vrtx o GG s n at last on ba, 2. or ac d uuuu o GG, tr xsts a ba tat contans bot uu and vv, 3. XX and XX jj bot contan a vrtx vv, tn all bas XX kk n t pat btwn XX and XX jj contan vv as wll. a b c j d a j b j b b d b c d j

6 T wdt o a tr-dcomposton (TT, XX tt tt VV TT ) s max XX tt 1. T tr-wdt o a rap GG, dnotd by tttt(gg), s t mnmum wdt ovr all possbl tr-dcompostons o G. a b c d j a b j j b j b c d d b

7 Exampls tr-wdt 1 a orst no cycl tr-wdt 2 a srs-paralll rap no KK 4 mnor Furs rom wkpda

8 Exampls tr-wdt 1 a orst no cycl tr-wdt 2 a srs-paralll rap no KK 4 mnor T tr-wdt o a kk kk rd s kk. T tr-wdt o KK nn s nn rd KK 5

9 by Fdor V. Fomn (ttp://ptscool.mmuw.du.pl/s lds/lc6.pd)

10 Courcll s torm Torm (Courcll 1990) Lt PP b a proprty tat can b xprssd n MMMMMM 2 loc. I GG s a rap o boundd tr-wdt, tn tr xsts a lnar-tm alortm tat tsts wtr GG as proprty PP... 3-COLORING, HAMILTONICITY, SUBGRAPH ISOMORPHISM, MINOR TEST, VERTEX COVER, DOMINATING SET, INDEPENDENT SET, STEINER TREE, FEEDBACK VERTEX SET

11 Excludd Grd Torm T tr-wdt o a kk kk rd s kk. I a rap contans a lar rd as a mnor, tn ts tr-wdt s also lar. Torm (Robrtson, Symour, Tomas 1994) I t tr-wdt o a rap GG s at last 20 2kk5, tn GG as a kk kk rd as a mnor. Torm (Robrtson, Symour, Tomas 1994) I t tr-wdt o a planar rap GG s at last 6kk 4, tn GG as a kk kk rd as a mnor.

12 Grap wdt paramtrs tr-wdt (Haln 1976, Robrtson and Symour 1984) branc-wdt (Robrtson and Symour 1991) carvn-wdt (Symour and Tomas 1994) clqu-wdt (Courcll and Olaru 2000) rank-wdt (Oum and Symour 2006) boolan-wdt (Bu-Xuan, Tll, Vatsll 2011) maxmum matcn-wdt (Vatsll 2012)

13 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T valu o an d ㅅ o TT s t sz o t maxmum matcn o GG[ aa, bb,, cc, dd,,,, ]. a b c d a b c d a b c d ㅅ GG TT, LL GG[ aa, bb,, cc, dd,,,, ]

14 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T valu o an d ㅅ o TT s t sz o t maxmum matcn o GG[ aa, bb,, cc, dd,,,, ]. a b c d a b c d a b c d 2 GG TT, LL GG[ aa, bb,, cc, dd,,,, ]

15 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T wdt o a branc-dcomposton (TT, LL) s t maxmum valu amon all ds. T maxmum matcn-wdt (mm-wdt, mmmmmm(gg)) o a rap GG s t mnmum wdt ovr all possbl branc-dcompostons ovr VV(GG). a b c d GG a b c d T wdt o (TT, LL) s TT, LL

16 Wy mm-wdt? Torm (Vatsll 2012) For vry rap GG, mmmmmm GG tttt GG mmmmmm GG. A rap GG as boundd tr-wdt and only GG as boundd mm-wdt.

17 Wy mm-wdt? W want to solv Grap Problms cntly. A Domnatn St o a rap GG s a st DD o vrtcs suc tat NN DD DD = VV(GG). Wat s t mnmum sz o a domnatn st o GG?

18 Wy mm-wdt? Usn tr-wdt Torm (van Rooj, Bodlandr, Rossmant 2009) Mnmum Domnatn St Problm can b solvd n tm OO (3 tt ) wn a rap and ts tr-dcomposton o wdt tt s vn. Torm (Lokstanov, Marx, Saurab 2011) Mnmum Domnatn St Problm cannot b solvd n tm OO ((3 εε) tt ) wr tt s t tr-wdt o t vn rap.

19 Wy mm-wdt? Usn mm-wdt Torm (J., Sætr, Tll 2015+) Mnmum Domnatn St Problm can b solvd n tm OO (8 mm ) wn a rap and ts branc-dcomposton o mmwdt mm s vn.

20 Wy mm-wdt? Usn tr-wdt: OO (3 tt ) Usn mm-wdt: OO (8 mm ) Our alortm s astr wn 8 mm < 3 tt, tat s, mmmmmm GG < tttt GG. Not tat or vry rap GG, mmmmmm GG tttt GG mmmmmm GG.

21 Wy mm-wdt? Usn mm-wdt Torm (J., Sætr, Tll 2015+) Mnmum Domnatn St Problm can b solvd n tm OO (8 mm ) wn a rap and ts branc-dcomposton o mmwdt mm s vn. Proo das 1. Nw caractrzaton o raps o mm-wdt at most kk 2. Fast Subst Convoluton

22 Nw caractrzaton (tr-wdt) a b c a b j b c d j d b j b d j rap tr-dcomposton

23 Nw caractrzaton (tr-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as tr-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE(GG), tn TT and TT jj av at last on vrtx o TT n common, 2) or ac vrtx o TT, tr ar at most kk 1 subtrs contann t.

24 Nw caractrzaton (mm-wdt) a b c d a 1 1 b c d

25 Nw caractrzaton (mm-wdt) a a b c d 1 Torm (Kön 1931) 1 b c d For vry bpartt rap GG, t sz o a maxmum matcn s qual to t sz o a mnmum vrtx covr

26 Nw caractrzaton (mm-wdt) a b c d a a b c d b c d a b c d

27 Nw caractrzaton (mm-wdt) a b c d a b c d a b c d TT

28 Nw caractrzaton (mm-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as mm-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE GG, tn TT and TT jj av at last on vrtx o TT n common, 2) or ac d o TT, tr ar at most kk subtrs contann t.

29 Torm (J., Sætr, Tll 2015+) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as mm-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE GG, tn TT and TT jj av at last on vrtx o TT n common, 2) or ac d o TT, tr ar at most kk subtrs contann t.

30 Nw caractrzaton For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as tr-wdt (mm-wdt) at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE(GG), tn TT and TT jj av at last on vrtx o TT n common, 2) or ac vrtx (d) o TT, tr ar at most kk 1 (at most kk) subtrs contann t. Tank you

31 Nw caractrzaton (branc-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as branc-wdt at most kk and only tr ar a tr TT o max dr at most 3 and subtrs TT 1, TT 2,, TT nn suc tat 1) vv vv jj EE(GG) tn TT and TT jj av at last on d o TT n common, 2) or ac d o TT, tr ar at most kk subtrs usn t.

Graph width-parameters and algorithms

Graph width-parameters and algorithms Grph width-prmeters nd lgorithms Jisu Jeong (KAIST) joint work with Sigve Hortemo Sæther nd Jn Arne Telle (University of Bergen) 2015 KMS Annul Meeting 2015.10.24. YONSEI UNIVERSITY Grph width-prmeters

More information

a g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,...

a g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,... Warmup: Lt b 2 c 3 d 1 12 6 4 5 10 9 7 a 8 11 (i) Vriy tat G is connctd by ivin an xampl o a walk rom vrtx a to ac o t vrtics b. (ii) Wat is t sortst pat rom a to c? to? (iii) Wat is t lonst pat rom a

More information

Depth First Search. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong

Depth First Search. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong Dprtmnt o Computr Sn n Ennrn Cns Unvrsty o Hon Kon W v lry lrn rt rst sr (BFS). Toy, w wll suss ts sstr vrson : t pt rst sr (DFS) lortm. Our susson wll on n ous on rt rps, us t xtnson to unrt rps s strtorwr.

More information

Lecture 20: Minimum Spanning Trees (CLRS 23)

Lecture 20: Minimum Spanning Trees (CLRS 23) Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss

More information

The University of Sydney MATH 2009

The University of Sydney MATH 2009 T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n

More information

Minimum Spanning Trees

Minimum Spanning Trees Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst

More information

Weighted Graphs. Weighted graphs may be either directed or undirected.

Weighted Graphs. Weighted graphs may be either directed or undirected. 1 In mny ppltons, o rp s n ssot numrl vlu, ll wt. Usully, t wts r nonntv ntrs. Wt rps my tr rt or unrt. T wt o n s otn rrr to s t "ost" o t. In ppltons, t wt my msur o t lnt o rout, t pty o ln, t nry rqur

More information

Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra

Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets

More information

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2, 1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)

More information

Manipulator Dynamics (1) Read Chapter 6

Manipulator Dynamics (1) Read Chapter 6 Manipulator Dynamics (1) Read Capter 6 Wat is dynamics? Study te force (torque) required to cause te motion of robots just like engine power required to drive a automobile Most familiar formula: f = ma

More information

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk CMPS 2200 Fll 2017 Grps Crol Wnk Sls ourtsy o Crls Lsrson wt ns n tons y Crol Wnk 10/23/17 CMPS 2200 Intro. to Alortms 1 Grps Dnton. A rt rp (rp) G = (V, E) s n orr pr onsstn o st V o vrts (snulr: vrtx),

More information

Worksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra

Worksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation

More information

dy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c.

dy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c. AP CALCULUS BC SUMMER ASSIGNMENT DO NOT SHOW YOUR WORK ON THIS! Complt ts problms during t last two wks of August. SHOW ALL WORK. Know ow to do ALL of ts problms, so do tm wll. Itms markd wit a * dnot

More information

AP Calculus BC AP Exam Problems Chapters 1 3

AP Calculus BC AP Exam Problems Chapters 1 3 AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum

More information

" = Y(#,$) % R(r) = 1 4& % " = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V,

 = Y(#,$) % R(r) = 1 4& %  = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V, Recitation Problems: Week 4 1. Which of the following combinations of quantum numbers are allowed for an electron in a one-electron atom: n l m l m s 2 2 1! 3 1 0 -! 5 1 2! 4-1 0! 3 2 1 0 2 0 0 -! 7 2-2!

More information

Terms of Use. Copyright Embark on the Journey

Terms of Use. Copyright Embark on the Journey Terms of Use All rights reserved. No part of this packet may be reproduced, stored in a retrieval system, or transmitted in any form by any means - electronic, mechanical, photo-copies, recording, or otherwise

More information

ALPHABET. 0Letter Practice

ALPHABET. 0Letter Practice ALPHABET 0Ltt Pat Ltt Pat - Aa A A A A a a a Cl A s A a a A A Cl a s A a k Aa a Cut ut th ltt s a glu thm th ght st. Catal A Las a 2014 Lau Thms.msthmsstasus.m a a A a A A Ltt Pat - B B B B B Cl B s m

More information

CPS 616 W2017 MIDTERM SOLUTIONS 1

CPS 616 W2017 MIDTERM SOLUTIONS 1 CPS 616 W2017 MIDTERM SOLUTIONS 1 PART 1 20 MARKS - MULTIPLE CHOICE Instructions Plas ntr your answrs on t bubbl st wit your nam unlss you ar writin tis xam at t Tst Cntr, in wic cas you sould just circl

More information

A A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good.

A A A A A A A A A A A A. a a a a a a a a a a a a a a a. Apples taste amazingly good. Victorian Handwriting Sheet Aa A A A A A A A A A A A A Aa Aa Aa Aa Aa Aa Aa a a a a a a a a a a a a a a a Apples taste amazingly good. Apples taste amazingly good. Now make up a sentence of your own using

More information

7.3 The Jacobi and Gauss-Seidel Iterative Methods

7.3 The Jacobi and Gauss-Seidel Iterative Methods 7.3 The Jacobi and Gauss-Seidel Iterative Methods 1 The Jacobi Method Two assumptions made on Jacobi Method: 1.The system given by aa 11 xx 1 + aa 12 xx 2 + aa 1nn xx nn = bb 1 aa 21 xx 1 + aa 22 xx 2

More information

Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano

Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano RIGHT-ANGLE WEAVE Dv mons Mm t look o ts n rlt tt s ptvly p sn y Py Brnkmn Mttlno Dv your mons nto trnls o two or our olors. FCT-SCON0216_BNB66 2012 Klm Pulsn Co. Ts mtrl my not rprou n ny orm wtout prmsson

More information

10. EXTENDING TRACTABILITY

10. EXTENDING TRACTABILITY Coping with NP-compltnss 0. EXTENDING TRACTABILITY ining small vrtx covrs solving NP-har problms on trs circular arc covrings vrtx covr in bipartit graphs Q. Suppos I n to solv an NP-complt problm. What

More information

AP Calculus Multiple-Choice Question Collection

AP Calculus Multiple-Choice Question Collection AP Calculus Multipl-Coic Qustion Collction 985 998 . f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b

More information

Review for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa

Review for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 2 Chapter

More information

Outlines: Graphs Part-4. Applications of Depth-First Search. Directed Acyclic Graph (DAG) Generic scheduling problem.

Outlines: Graphs Part-4. Applications of Depth-First Search. Directed Acyclic Graph (DAG) Generic scheduling problem. Outlins: Graps Part-4 Applications o DFS Elmntary Grap Aloritms Topoloical Sort o Dirctd Acyclic Grap Stronly Connctd Componnts PART-4 1 2 Applications o Dpt-First Sarc Topoloical Sort: Usin dpt-irst sarc

More information

SECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems

SECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow

More information

TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES

TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note

More information

Chapter 22 : Electric potential

Chapter 22 : Electric potential Chapter 22 : Electric potential What is electric potential? How does it relate to potential energy? How does it relate to electric field? Some simple applications What does it mean when it says 1.5 Volts

More information

ON THE COMPLEXITY OF K-STEP AND K-HOP DOMINATING SETS IN GRAPHS

ON THE COMPLEXITY OF K-STEP AND K-HOP DOMINATING SETS IN GRAPHS MATEMATICA MONTISNIRI Vol XL (2017) MATEMATICS ON TE COMPLEXITY OF K-STEP AN K-OP OMINATIN SETS IN RAPS M FARAI JALALVAN AN N JAFARI RA partmnt of Mathmatcs Shahrood Unrsty of Tchnology Shahrood Iran Emals:

More information

Chapter 6. Heat capacity, enthalpy, & entropy

Chapter 6. Heat capacity, enthalpy, & entropy Chapter 6 Heat capacity, enthalpy, & entropy 1 6.1 Introduction In this lecture, we examine the heat capacity as a function of temperature, compute the enthalpy, entropy, and Gibbs free energy, as functions

More information

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our

More information

Lecture No. 5. For all weighted residual methods. For all (Bubnov) Galerkin methods. Summary of Conventional Galerkin Method

Lecture No. 5. For all weighted residual methods. For all (Bubnov) Galerkin methods. Summary of Conventional Galerkin Method Lecture No. 5 LL(uu) pp(xx) = 0 in ΩΩ SS EE (uu) = gg EE on ΓΓ EE SS NN (uu) = gg NN on ΓΓ NN For all weighted residual methods NN uu aaaaaa = uu BB + αα ii φφ ii For all (Bubnov) Galerkin methods ii=1

More information

Property Testing and Affine Invariance Part I Madhu Sudan Harvard University

Property Testing and Affine Invariance Part I Madhu Sudan Harvard University Property Testing and Affine Invariance Part I Madhu Sudan Harvard University December 29-30, 2015 IITB: Property Testing & Affine Invariance 1 of 31 Goals of these talks Part I Introduce Property Testing

More information

Lecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator

Lecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator Lecture No. 1 Introduction to Method of Weighted Residuals Solve the differential equation L (u) = p(x) in V where L is a differential operator with boundary conditions S(u) = g(x) on Γ where S is a differential

More information

Spanning Trees. BFS, DFS spanning tree Minimum spanning tree. March 28, 2018 Cinda Heeren / Geoffrey Tien 1

Spanning Trees. BFS, DFS spanning tree Minimum spanning tree. March 28, 2018 Cinda Heeren / Geoffrey Tien 1 Spnnn Trs BFS, DFS spnnn tr Mnmum spnnn tr Mr 28, 2018 Cn Hrn / Gory Tn 1 Dpt-rst sr Vsts vrts lon snl pt s r s t n o, n tn ktrks to t rst junton n rsums own notr pt Mr 28, 2018 Cn Hrn / Gory Tn 2 Dpt-rst

More information

Discrete Structures Lecture Solving Congruences. mathematician of the eighteenth century). Also, the equation gggggg(aa, bb) =

Discrete Structures Lecture Solving Congruences. mathematician of the eighteenth century). Also, the equation gggggg(aa, bb) = First Introduction Our goal is to solve equations having the form aaaa bb (mmmmmm mm). However, first we must discuss the last part of the previous section titled gcds as Linear Combinations THEOREM 6

More information

Uncertain Compression & Graph Coloring. Madhu Sudan Harvard

Uncertain Compression & Graph Coloring. Madhu Sudan Harvard Uncertain Compression & Graph Coloring Madhu Sudan Harvard Based on joint works with: (1) Adam Kalai (MSR), Sanjeev Khanna (U.Penn), Brendan Juba (WUStL) (2) Elad Haramaty (Harvard) (3) Badih Ghazi (MIT),

More information

(Minimum) Spanning Trees

(Minimum) Spanning Trees (Mnmum) Spnnn Trs Spnnn trs Kruskl's lortm Novmr 23, 2017 Cn Hrn / Gory Tn 1 Spnnn trs Gvn G = V, E, spnnn tr o G s onnt surp o G wt xtly V 1 s mnml sust o s tt onnts ll t vrts o G G = Spnnn trs Novmr

More information

2.11 That s So Derivative

2.11 That s So Derivative 2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point

More information

Classification of Components

Classification of Components Classification of Components Classification of SMD LEDs TLM.31../TLM.32../TLM.33../ + 3... series, Mini LEDs TLM.21... / TLM.23.. / TLM.2.. and 63 LEDs TLM.11../TLM1.. Light Intensity / Color Devices are

More information

General Strong Polarization

General Strong Polarization General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) December 4, 2017 IAS:

More information

10.4 The Cross Product

10.4 The Cross Product Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb

More information

Journal of Theoretical and Applied Information Technology 10 th January Vol. 47 No JATIT & LLS. All rights reserved.

Journal of Theoretical and Applied Information Technology 10 th January Vol. 47 No JATIT & LLS. All rights reserved. Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: 99-8645 www.att.org E-ISSN: 87-395 RESEARCH ON PROPERTIES OF E-PARTIAL DERIVATIVE OF LOGIC

More information

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)

Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a) Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5

More information

Gravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition

Gravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to

More information

Graph Search (6A) Young Won Lim 5/18/18

Graph Search (6A) Young Won Lim 5/18/18 Grp Sr (6A) Youn Won Lm Copyrt () 2015 2018 Youn W. Lm. Prmon rnt to opy, trut n/or moy t oumnt unr t trm o t GNU Fr Doumntton Ln, Vron 1.2 or ny ltr vron pul y t Fr Sotwr Founton; wt no Invrnt Ston, no

More information

Chem 263 Winter 2018 Problem Set #2 Due: February 16

Chem 263 Winter 2018 Problem Set #2 Due: February 16 Chem 263 Winter 2018 Problem Set #2 Due: February 16 1. Use size considerations to predict the crystal structures of PbF2, CoF2, and BeF2. Do your predictions agree with the actual structures of these

More information

COMP 250. Lecture 29. graph traversal. Nov. 15/16, 2017

COMP 250. Lecture 29. graph traversal. Nov. 15/16, 2017 COMP 250 Ltur 29 rp trvrsl Nov. 15/16, 2017 1 Toy Rursv rp trvrsl pt rst Non-rursv rp trvrsl pt rst rt rst 2 Hs up! Tr wr w mstks n t sls or S. 001 or toy s ltur. So you r ollown t ltur rorns n usn ts

More information

Secondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet

Secondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and

More information

Patterns of soiling in the Old Library Trinity College Dublin. Allyson Smith, Robbie Goodhue, Susie Bioletti

Patterns of soiling in the Old Library Trinity College Dublin. Allyson Smith, Robbie Goodhue, Susie Bioletti Patterns of soiling in the Old Library Trinity College Dublin Allyson Smith, Robbie Goodhue, Susie Bioletti Trinity College aerial view Old Library E N S W Old Library main (south) elevation Gallery Fagel

More information

MAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative

MAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative MAT 7 Tst Rviw (Spring ) Tst on April in PSA Sction.7 Implicit Drivativ Rmmbr: Equation of t tangnt lin troug t point ( ab, ) aving slop m is y b m( a ). dy Find t drivativ y d. y y. y y y. y 4. y sin(

More information

Exponential Functions

Exponential Functions Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.

More information

Introduction to Algorithms

Introduction to Algorithms Introduction to Algorithms 6.046J/18.401J LECTURE 14 Shortest Paths I Properties of shortest paths Dijkstra s algorithm Correctness Analysis Breadth-first search Prof. Charles E. Leiserson Paths in graphs

More information

Total Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are

Total Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Total Prm Graph M.Rav (a) Ramasubramaa 1, R.Kala 1 Dpt.of Mathmatcs, Sr Shakth Isttut of Egrg & Tchology, Combator 641 06. Dpt. of Mathmatcs,

More information

SECTION 7: STEADY-STATE ERROR. ESE 499 Feedback Control Systems

SECTION 7: STEADY-STATE ERROR. ESE 499 Feedback Control Systems SECTION 7: STEADY-STATE ERROR ESE 499 Feedback Control Systems 2 Introduction Steady-State Error Introduction 3 Consider a simple unity-feedback system The error is the difference between the reference

More information

Quantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.

Quantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of

More information

Big Bang Planck Era. This theory: cosmological model of the universe that is best supported by several aspects of scientific evidence and observation

Big Bang Planck Era. This theory: cosmological model of the universe that is best supported by several aspects of scientific evidence and observation Big Bang Planck Era Source: http://www.crystalinks.com/bigbang.html Source: http://www.odec.ca/index.htm This theory: cosmological model of the universe that is best supported by several aspects of scientific

More information

KENNESAW STATE UNIVERSITY ATHLETICS VISUAL IDENTITY

KENNESAW STATE UNIVERSITY ATHLETICS VISUAL IDENTITY KENNESAW STATE UNIVERSITY ATHLETICS VISUAL IDENTITY KENNESAW STATE DEPARTMENT OF ATHLETICS VISUAL IDENTITY GENERAL University Athletics Brand History...3 Referencing the Institution...3 Referencing the

More information

learning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms

learning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r

More information

Quantitative Screening of 46 Illicit Drugs in Urine using Exactive Ultrahigh Resolution and Accurate Mass system

Quantitative Screening of 46 Illicit Drugs in Urine using Exactive Ultrahigh Resolution and Accurate Mass system Quantitative Screening of 46 Illicit Drugs in Urine using Exactive Ultrahigh Resolution and Accurate Mass system Kevin Mchale Thermo Fisher Scientific, San Jose CA Presentation Overview Anabolic androgenic

More information

TSOKOS READING ACTIVITY Section 7-2: The Greenhouse Effect and Global Warming (8 points)

TSOKOS READING ACTIVITY Section 7-2: The Greenhouse Effect and Global Warming (8 points) IB PHYSICS Name: Period: Date: DEVIL PHYSICS BADDEST CLASS ON CAMPUS TSOKOS READING ACTIVITY Section 7-2: The Greenhouse Effect and Global Warming (8 points) 1. IB Assessment Statements for Topic 8.5.

More information

Chapter 4 Derivatives [ ] = ( ) ( )= + ( ) + + = ()= + ()+ Exercise 4.1. Review of Prerequisite Skills. 1. f. 6. d. 4. b. lim. x x. = lim = c.

Chapter 4 Derivatives [ ] = ( ) ( )= + ( ) + + = ()= + ()+ Exercise 4.1. Review of Prerequisite Skills. 1. f. 6. d. 4. b. lim. x x. = lim = c. Capter Derivatives Review of Prerequisite Skills. f. p p p 7 9 p p p Eercise.. i. ( a ) a ( b) a [ ] b a b ab b a. d. f. 9. c. + + ( ) ( + ) + ( + ) ( + ) ( + ) + + + + ( ) ( + ) + + ( ) ( ) ( + ) + 7

More information

Solar Photovoltaics & Energy Systems

Solar Photovoltaics & Energy Systems Solar Photovoltaics & Energy Systems Lecture 3. Solar energy conversion with band-gap materials ChE-600 Kevin Sivula, Spring 2014 The Müser Engine with a concentrator T s Q 1 = σσ CffT ss 4 + 1 Cff T pp

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function

More information

4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling.

4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling. Cptr 4 4 Intrvl Suln Gry Alortms Sls y Kvn Wyn Copyrt 005 Prson-Ason Wsly All rts rsrv Intrvl Suln Intrvl Suln: Gry Alortms Intrvl suln! Jo strts t s n nss t! Two os omptl ty on't ovrlp! Gol: n mxmum sust

More information

Math 95--Review Prealgebra--page 1

Math 95--Review Prealgebra--page 1 Math 95--Review Prealgebra--page 1 Name Date In order to do well in algebra, there are many different ideas from prealgebra that you MUST know. Some of the main ideas follow. If these ideas are not just

More information

Correlation in tree The (ferromagnetic) Ising model

Correlation in tree The (ferromagnetic) Ising model 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs.

More information

Supplemental Figures and Captions for Supplemental Tables

Supplemental Figures and Captions for Supplemental Tables Kuehn, S.C. and Negrini, R. M., 2010, A 250 k.y. record of Cascade arc pyroclastic volcanism from late Pleistocene lacustrine sediments near Summer Lake, Oregon, USA. Geosphere, Vol. 6, No. 4 (August),

More information

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Review for Exam3 12. 9. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University

More information

Introduction to Algorithms

Introduction to Algorithms Introduction to Algorithms 6.046J/18.401J LECTURE 17 Shortest Paths I Properties of shortest paths Dijkstra s algorithm Correctness Analysis Breadth-first search Prof. Erik Demaine November 14, 005 Copyright

More information

3-2-1 ANN Architecture

3-2-1 ANN Architecture ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.

More information

Math 31A Discussion Notes Week 4 October 20 and October 22, 2015

Math 31A Discussion Notes Week 4 October 20 and October 22, 2015 Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes

More information

CHAPTER 4. The First Law of Thermodynamics for Control Volumes

CHAPTER 4. The First Law of Thermodynamics for Control Volumes CHAPTER 4 T Frst Law of Trodynacs for Control olus CONSERATION OF MASS Consrvaton of ass: Mass, lk nrgy, s a consrvd proprty, and t cannot b cratd or dstroyd durng a procss. Closd systs: T ass of t syst

More information

Chapter Taylor Theorem Revisited

Chapter Taylor Theorem Revisited Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o

More information

Increased ionization during magnetron sputtering and its influence on the energy balance at the substrate

Increased ionization during magnetron sputtering and its influence on the energy balance at the substrate Institute of Experimental and Applied Physics XXII. Erfahrungsaustausch Oberflächentechnologie mit Plasma- und Ionenstrahlprozessen Mühlleithen, 10.-12. März, 2015 Increased ionization during magnetron

More information

ENGR 7181 LECTURE NOTES WEEK 3 Dr. Amir G. Aghdam Concordia University

ENGR 7181 LECTURE NOTES WEEK 3 Dr. Amir G. Aghdam Concordia University ENGR 78 LECTURE NOTES WEEK Dr. ir G. ga Concoria Univrity DT Equivalnt Tranfr Function for SSO Syt - So far w av tui DT quivalnt tat pac ol uing tp-invariant tranforation. n t ca of SSO yt on can u t following

More information

Math 171 Spring 2017 Final Exam. Problem Worth

Math 171 Spring 2017 Final Exam. Problem Worth Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:

More information

CMSC 451: Lecture 4 Bridges and 2-Edge Connectivity Thursday, Sep 7, 2017

CMSC 451: Lecture 4 Bridges and 2-Edge Connectivity Thursday, Sep 7, 2017 Rn: Not ovr n or rns. CMSC 451: Ltr 4 Brs n 2-E Conntvty Trsy, Sp 7, 2017 Hr-Orr Grp Conntvty: (T ollown mtrl ppls only to nrt rps!) Lt G = (V, E) n onnt nrt rp. W otn ssm tt or rps r onnt, t somtms t

More information

Week 3: Connected Subgraphs

Week 3: Connected Subgraphs Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y

More information

PHL424: Nuclear Shell Model. Indian Institute of Technology Ropar

PHL424: Nuclear Shell Model. Indian Institute of Technology Ropar PHL424: Nuclear Shell Model Themes and challenges in modern science Complexity out of simplicity Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few

More information

(1) Correspondence of the density matrix to traditional method

(1) Correspondence of the density matrix to traditional method (1) Correspondence of the density matrix to traditional method New method (with the density matrix) Traditional method (from thermal physics courses) ZZ = TTTT ρρ = EE ρρ EE = dddd xx ρρ xx ii FF = UU

More information

is an appropriate single phase forced convection heat transfer coefficient (e.g. Weisman), and h

is an appropriate single phase forced convection heat transfer coefficient (e.g. Weisman), and h For t BWR oprating paramtrs givn blow, comput and plot: a) T clad surfac tmpratur assuming t Jns-Lotts Corrlation b) T clad surfac tmpratur assuming t Tom Corrlation c) T clad surfac tmpratur assuming

More information

14- Hardening Soil Model with Small Strain Stiffness - PLAXIS

14- Hardening Soil Model with Small Strain Stiffness - PLAXIS 14- Hardening Soil Model with Small Strain Stiffness - PLAXIS This model is the Hardening Soil Model with Small Strain Stiffness as presented in PLAXIS. The model is developed using the user-defined material

More information

Introduction to Algorithms

Introduction to Algorithms Introduction to Algorithms 6.046J/8.40J/SMA550 Lecture 7 Prof. Erik Demaine Paths in graphs Consider a digraph G = (V, E) with edge-weight function w : E R. The weight of path p = v v L v k is defined

More information

General Strong Polarization

General Strong Polarization General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) May 1, 018 G.Tech:

More information

GG7: Beyond 2016 FROM PARIS TO ISE-SHIMA. The State of Climate Negotiations: What to Expect after COP 21. g20g7.com

GG7: Beyond 2016 FROM PARIS TO ISE-SHIMA. The State of Climate Negotiations: What to Expect after COP 21. g20g7.com 7: By 2016 A B Dy b B R INSIDE 7 ELOME: Sz Ab, P M J EONOMY T Ex EL D Ey D Sy ROM PARIS TO ISE-SHIMA T S N: Ex OP 21 A Pb by AT y I-S S ATOMPANYI Pb T 7 Mz VIP, D, D L 7: By 2016 A B Dy b B R ATOMPANYI

More information

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014 Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.

More information

Analyzing and improving the reproducibility of shots taken by a soccer robot

Analyzing and improving the reproducibility of shots taken by a soccer robot Analyzing and improving the reproducibility of shots taken by a soccer robot Overview Introduction Capacitor analysis Lever positioning analysis Validation Conclusion and recommendations Introduction Shots

More information

Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition

Work, Energy, and Power. Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition Work, Energy, and Power Chapter 6 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 With the knowledge we got so far, we can handle the situation on the left but not the one on the right.

More information

Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork and res u lts 2

Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork and res u lts 2 Internal Innovation @ C is c o 2 0 0 6 C i s c o S y s t e m s, I n c. A l l r i g h t s r e s e r v e d. C i s c o C o n f i d e n t i a l 1 Agenda Rationale for ETG S eek ing I d eas ETG fram ew ork

More information

Planar convex hulls (I)

Planar convex hulls (I) Covx Hu Covxty Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P A oyo P s ovx or y, P, t st s try P. Pr ovx us (I) Coutto Gotry [s 3250] Lur To Bowo Co ovx o-ovx 1 2 3 Covx Hu Covx Hu Covx Hu

More information

Shortest Paths in Graphs. Paths in graphs. Shortest paths CS 445. Alon Efrat Slides courtesy of Erik Demaine and Carola Wenk

Shortest Paths in Graphs. Paths in graphs. Shortest paths CS 445. Alon Efrat Slides courtesy of Erik Demaine and Carola Wenk S 445 Shortst Paths n Graphs lon frat Sls courtsy of rk man an arola Wnk Paths n raphs onsr a raph G = (V, ) wth -wht functon w : R. Th wht of path p = v v v k s fn to xampl: k = w ( p) = w( v, v + ).

More information

Winsome Winsome W Wins e ins e WUin ser some s Guide

Winsome Winsome W Wins e ins e WUin ser some s Guide Winsome Winsome Wins e Wins e U ser s Guide Winsome font faq HOW TO INSTALL YOUR FONT You will receive your files as a zipped folder. For instructions on how to unzip your folder, visit LauraWorthingtonType.com/faqs/.

More information

Heat, Work, and the First Law of Thermodynamics. Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition

Heat, Work, and the First Law of Thermodynamics. Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition Heat, Work, and the First Law of Thermodynamics Chapter 18 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Different ways to increase the internal energy of system: 2 Joule s apparatus

More information

MINI POST SERIES BALUSTRADE SYSTEM INSTALLATION GUIDE PRODUCT CODE: MPS-RP

MINI POST SERIES BALUSTRADE SYSTEM INSTALLATION GUIDE PRODUCT CODE: MPS-RP MN POST SRS LUSTR SYSTM NSTLLTON U PROUT O: MPS-RP 0 R0 WLL LN 0 RONT LVTON VW R0 N P 0 T RUR LOK LOT ON LSS. SLON SL TYP. OT SS 000 LSS T 0 00 SRS LSS WT 00/00 (0mm NRMNTS VLL) MX. 000 00-0 (ROMMN) 00

More information

Geometric Predicates P r og r a m s need t o t es t r ela t ive p os it ions of p oint s b a s ed on t heir coor d ina t es. S im p le exa m p les ( i

Geometric Predicates P r og r a m s need t o t es t r ela t ive p os it ions of p oint s b a s ed on t heir coor d ina t es. S im p le exa m p les ( i Automatic Generation of SS tag ed Geometric PP red icates Aleksandar Nanevski, G u y B lello c h and R o b ert H arp er PSCICO project h ttp: / / w w w. cs. cm u. ed u / ~ ps ci co Geometric Predicates

More information

Strongly connected components. Finding strongly-connected components

Strongly connected components. Finding strongly-connected components Stronly onnt omponnts Fnn stronly-onnt omponnts Tylr Moor stronly onnt omponnt s t mxml sust o rp wt rt pt twn ny two vrts SE 3353, SMU, Dlls, TX Ltur 9 Som sls rt y or pt rom Dr. Kvn Wyn. For mor normton

More information

Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall

Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s

More information

Free energy dependence along the coexistence curve

Free energy dependence along the coexistence curve Free energy dependence along the coexistence curve In a system where two phases (e.g. liquid and gas) are in equilibrium the Gibbs energy is G = GG l + GG gg, where GG l and GG gg are the Gibbs energies

More information

Rotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition

Rotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr

More information