CIT 596 Theory of Computation 1. Graphs and Digraphs


 Rosamond Preston
 1 years ago
 Views:
Transcription
1 CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege set of the grph, often enote y just E, whih is possily empty set of elements lle eges, suh tht eh ege e in E is ssigne n unorere pir (u, v) of verties, lle the en verties of e. Sometimes, it is onvenient to enote (u, v) y simply uv, or equivlently, vu.
2 CIT 596 Theory of Computtion 2 Consier the grph G = (V, E) suh tht V = {,,,, e} n E = {e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8 }, where e 1 (, ) e 2 (, ) e 3 (, ) e 4 (, ) e 5 (, ) e 6 (, e) e 7 (, e) e 8 (, e)
3 CIT 596 Theory of Computtion 3 A grph is often represente y igrm in whih verties re rwn s irles n eges s line or urve segments joining the irles representing the en verties of the ege. e 8 e 1 e e 7 e 2 e 5 e 6 e 3 e 4
4 CIT 596 Theory of Computtion 4 Verties re lso lle points, noes, or just ots. If e is n ege with en verties u n v then e is si to join u n v. Note tht the efinition of grph llows the possiility of the ege e hving ietil en verties, i.e., it is possile to hve vertex u joine to itself y n ege suh n ege is lle loop. If two (or more) eges hve the sme en verties then eges re lle prllel. A grph is lle simple is it hs no loops n no prllel eges.
5 CIT 596 Theory of Computtion 5 For n exmple of simple grph, onsier the grph G = (V, E) suh tht V = {,,, } n E = {e 1, e 2, e 3, e 4 }, where e 1 (, ) e 2 (, ) e 3 (, ) e 4 (, ) Some uthors use the term multigrph for grphs with loops n prllel eges, n reserve the term grph for simple grphs only. Sine we will el with grphs with loops very often, it is more onvenient not to mke this istintion.
6 CIT 596 Theory of Computtion 6 A pitoril representtion of the simple grph G in the previous slie: e 1 e 2 e 3 e 4
7 CIT 596 Theory of Computtion 7 The numer of verties in G is lle the orer of G. The numer of eges in G is lle the size of G. Two verties u n v of grph G re si to e jent if uv E(G). If uv E(G) then we sy tht u n v re nonjent verties. An ege e of grph G is si to e inient with or inient to the vertex v if v is n en vertex of e. In this se, we lso sy tht v is inient with or inient to e. Two eges e n f whih re inient with ommon vertex v re si to e jent.
8 CIT 596 Theory of Computtion 8 Let v e vertex of the grph G. The egree (v) of v is the numer of eges of G inient to v, ounting eh loop twie, i.e., it is the numer of times v is n en vertex of n ege. e For exmple, () = 1, () = 3, () = 3, () = 3, n (e) = 4 in the grph ove.
9 CIT 596 Theory of Computtion 9 The First Theorem of Grph Theory. For ny grph G with n e eges n n v verties v 1,..., v nv, we hve tht Chek y yourself: n v i=1 (v i ) = 2 n e. e
10 CIT 596 Theory of Computtion 10 A wlk in grph G is finite sequene W = v 0 e 1 v 1 e 2 v 2... v k 1 e k v k whose terms re lterntely verties n eges suh tht, for 1 i k, the ege e i hs en verties v i 1 n v i. We sy tht the ove wlk is v 0 v k wlk or wlk from v 0 to v k. The integer k, the numer of eges of the wlk, is lle the length of W. A trivil wlk is one ontining no eges.
11 CIT 596 Theory of Computtion 11 The sequene e 2 e 8 f e 6 e 3 e 4 e e 4 is wlk of length 6 in the grph elow. e 1 e 2 e 3 e 8 e 4 e 7 e 6 e 5 f e In simple grph, wlk is etermine y the sequene of verties only: f e.
12 CIT 596 Theory of Computtion 12 Given two verties u n v of grph G, u v wlk is lle lose or open epening on whether u = v or u v. If the eges e 1, e 2,..., e k of the wlk v 0 e 1 v 1 e 2 v 2... v k 1 e k v k re istint then W is lle tril. If the verties v 0, v 1,..., v k of the wlk v 0 e 1 v 1 e 2 v 2... v k 1 e k v k re istint then W is lle pth. For the grph in the previous slie, f e is f tril, ut not pth! The sequene f e is f e pth.
13 CIT 596 Theory of Computtion 13 A vertex u is si to e onnete to vertex v in grph G if there is pth in G from u to v. A grph G is onnete if every two verties of G re onnete; otherwise, G is isonnete. M N Connete Disonnete
14 CIT 596 Theory of Computtion 14 A nontrivil lose tril C = v 1 v 2... v n v 1 in grph G is lle yle if the verties v 2... v n re ll istint. A yle of length k, i.e., yle with k eges, is lle kyle. For exmple, f is 3yle in the grph elow: e 1 e 2 e 3 e 8 e 4 e 7 e 6 e 5 f e
15 CIT 596 Theory of Computtion 15 A grph G is si to e yli if it ontins no yles. A grph G is lle tree if it is onnete n yli. e The verties of egree (t most) 1 in tree re lle the leves of the tree.
16 CIT 596 Theory of Computtion 16 A irete grph (or simply igrph) D = (V (D), A(D)) onsists of two finite sets: V (D), the vertex set of the igrph, often enote y just V, whih is nonempty set of elements lle verties, n A(D), the r set of the igrph, often enote y just A, whih is possily empty set of elements lle rs, suh tht eh r in A is ssigne (orere) pir (u, v) of verties. If is n r in D with ssoite orere pir of verties (u, v), then is si to join u to v, u is lle the initil vertex of, n v is lle the terminl vertex of.
17 CIT 596 Theory of Computtion 17 For exmple, onsier the igrph D = (V (D), A(D)) suh tht V (D) = {,,,, e} n A(D) = {e 1, e 2, e 3, e 4, e 5, e 6 }, suh tht e 1 (, ), e 2 (, ), e 3 (, ), e 3 (, e), e 4 (e, ), n e 6 (e, e). e 6 e e 1 e 5 e 4 e 2 e 3
18 CIT 596 Theory of Computtion 18 Let D e igrph. Then irete wlk in D is finite sequene W = v 0 1 v 1... k v k, whose terms re lterntely verties n rs suh tht for 1 i k, the initil vertex of the r i is v i 1 n its terminl vertex is v i. The numer k of rs is the length of W. The wlk W given in the efinition ove is si to e v 0 v k irete wlk or irete wlk from v 0 to v k. There re similr efinitions for irete trils, irete pths, n irete yles.
19 CIT 596 Theory of Computtion 19 For exmple, onsier the igrph D elow: e 6 e e 1 e 5 e 4 e 2 e 3 The sequene e 1 e 3 e 4 e is e irete wlk, e irete tril, n e irete pth in D.
20 CIT 596 Theory of Computtion 20 A vertex v of the igrph D is si to e rehle from vertex u if there is irete pth from u to v. Given ny igrph D = (V (D), A(D)), we n otin grph G = (V (G), E(G)) from D s follows: Let V (G) = V (D) n E(G) = {e (, ) (, ) A(D)}. The grph G is the unerlying grph of D. A igrph D is si to e wekly onnete (or simply onnete) if its unerlying grph is onnete. A igrph D is si to e strongly onnete (or ionnete) if for ny pir of verties u n v in D there is irete pth from u to v.
21 CIT 596 Theory of Computtion 21 e 6 e 6 e e e 1 e 5 e 4 e 5 e 1 e 4 e 2 e 2 e 3 e 3 A igrph Its unerlying grph Is the ove irete grph onnete? If so, is it strongly onnete?
22 CIT 596 Theory of Computtion 22 Let v e vertex in igrph D. The inegree i(v) of v is the numer of rs of D tht hve v s the terminl vertex, i.e., the numer of rs tht go to to v. Similrly, the outegree o(v) of v is the numer of rs of D tht hve v s the initil vertex, i.e., the numer of rs tht go out of v. The First Theorem of Digrph Theory. Let D e igrph with n verties n q rs. If v 1,..., v n is the set of verties of D, then we hve tht n i=1 i(v i ) = n i=1 Cn you figure out the proof for this one? o(v i ) = q.
Solutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More informationMidTerm Examination  Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours
MiTerm Exmintion  Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into
More informationGraph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or MultiGrph We llow loops n multiple eges. G = (V, E.ψ)
More informationCounting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n onetoone n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
More informationProportions: A ratio is the quotient of two numbers. For example, 2 3
Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)
More informationLecture 11 Binary Decision Diagrams (BDDs)
C 474A/57A ComputerAie Logi Design Leture Binry Deision Digrms (BDDs) C 474/575 Susn Lyseky o 3 Boolen Logi untions Representtions untion n e represente in ierent wys ruth tle, eqution, Kmp, iruit, et
More informationLogic, Set Theory and Computability [M. Coppenbarger]
14 Orer (Hnout) Definition 711: A reltion is qusiorering (or preorer) if it is reflexive n trnsitive. A quisiorering tht is symmetri is n equivlene reltion. A qusiorering tht is ntisymmetri is n orer
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationConnectivity in Graphs. CS311H: Discrete Mathematics. Graph Theory II. Example. Paths. Connectedness. Example
Connetiit in Grphs CSH: Disrete Mthemtis Grph Theor II Instrtor: Işıl Dillig Tpil qestion: Is it possile to get from some noe to nother noe? Emple: Trin netork if there is pth from to, possile to tke trin
More informationGraph Algorithms. Vertex set = { a,b,c,d } Edge set = { {a,c}, {b,c}, {c,d}, {b,d}} Figure 1: An example for a simple graph
Inin Institute of Informtion Tehnology Design n Mnufturing, Knheepurm, Chenni 00, Ini An Autonomous Institute uner MHRD, Govt of Ini http://www.iiitm..in COM 0T Design n Anlysis of Algorithms Leture Notes
More informationCS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
More informationDiscrete Structures, Test 2 Monday, March 28, 2016 SOLUTIONS, VERSION α
Disrete Strutures, Test 2 Mondy, Mrh 28, 2016 SOLUTIONS, VERSION α α 1. (18 pts) Short nswer. Put your nswer in the ox. No prtil redit. () Consider the reltion R on {,,, d with mtrix digrph of R.. Drw
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More informationAnswers and Solutions to (Some Even Numbered) Suggested Exercises in Chapter 11 of Grimaldi s Discrete and Combinatorial Mathematics
Answers n Solutions to (Some Even Numere) Suggeste Exercises in Chpter 11 o Grimli s Discrete n Comintoril Mthemtics Section 11.1 11.1.4. κ(g) = 2. Let V e = {v : v hs even numer o 1 s} n V o = {v : v
More informationarxiv: v2 [math.co] 31 Oct 2016
On exlue minors of onnetivity 2 for the lss of frme mtrois rxiv:1502.06896v2 [mth.co] 31 Ot 2016 Mtt DeVos Dryl Funk Irene Pivotto Astrt We investigte the set of exlue minors of onnetivity 2 for the lss
More informationI 3 2 = I I 4 = 2A
ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents
More informationLecture 4: Graph Theory and the FourColor Theorem
CCS Disrete II Professor: Pri Brtlett Leture 4: Grph Theory n the FourColor Theorem Week 4 UCSB 2015 Through the rest of this lss, we re going to refer frequently to things lle grphs! If you hen t seen
More informationPROPERTIES OF TRIANGLES
PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.
More informationComparing the Preimage and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) nglengle Similrity Theorem (.2) SideSideSide Similrity
More informationAlgebra 2 Semester 1 Practice Final
Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationLecture 2: Cayley Graphs
Mth 137B Professor: Pri Brtlett Leture 2: Cyley Grphs Week 3 UCSB 2014 (Relevnt soure mteril: Setion VIII.1 of Bollos s Moern Grph Theory; 3.7 of Gosil n Royle s Algeri Grph Theory; vrious ppers I ve re
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationSection 2.3. Matrix Inverses
Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue
More informationOn a Class of Planar Graphs with StraightLine Grid Drawings on Linear Area
Journl of Grph Algorithms n Applitions http://jg.info/ vol. 13, no. 2, pp. 153 177 (2009) On Clss of Plnr Grphs with StrightLine Gri Drwings on Liner Are M. Rezul Krim 1,2 M. Siur Rhmn 1 1 Deprtment of
More informationOn the Spectra of Bipartite Directed Subgraphs of K 4
On the Spetr of Biprtite Direte Sugrphs of K 4 R. C. Bunge, 1 S. I. ElZnti, 1, H. J. Fry, 1 K. S. Kruss, 2 D. P. Roerts, 3 C. A. Sullivn, 4 A. A. Unsiker, 5 N. E. Witt 6 1 Illinois Stte University, Norml,
More informationSolutions to Problem Set #1
CSE 233 Spring, 2016 Solutions to Prolem Set #1 1. The movie tse onsists of the following two reltions movie: title, iretor, tor sheule: theter, title The first reltion provies titles, iretors, n tors
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationF / x everywhere in some domain containing R. Then, + ). (10.4.1)
0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More information18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2106
8. Problem Set Due Wenesy, Ot., t : p.m. in  Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the oneelement, linerly epenent sets forme from these. (b) Wht re the twoelement, linerly
More informationFree groups, Lecture 2, part 1
Free groups, Lecture 2, prt 1 Olg Khrlmpovich NYC, Sep. 2 1 / 22 Theorem Every sugroup H F of free group F is free. Given finite numer of genertors of H we cn compute its sis. 2 / 22 Schreir s grph The
More informationGeometry of the Circle  Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.
Geometry of the irle  hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion
More information22: Union Find. CS 473u  Algorithms  Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u  Algorithms  Spring 2005 April 14, 2005 1 UnionFin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x)  rete set tht ontins the single element x. 2. Fin(x)
More informationGraph widthparameters and algorithms
Grph widthprmeters 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 widthprmeters
More informationEigenvectors and Eigenvalues
MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.
More informationThe DOACROSS statement
The DOACROSS sttement Is prllel loop similr to DOALL, ut it llows proueronsumer type of synhroniztion. Synhroniztion is llowe from lower to higher itertions sine it is ssume tht lower itertions re selete
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationSTRAND I: Geometry and Trigonometry. UNIT 32 Angles, Circles and Tangents: Student Text Contents. Section Compass Bearings
ME Jmi: STR I UIT 32 ngles, irles n Tngents: Stuent Tet ontents STR I: Geometry n Trigonometry Unit 32 ngles, irles n Tngents Stuent Tet ontents Setion 32.1 ompss erings 32.2 ngles n irles 1 32.3 ngles
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higherdimensionl nlogues) with the definite integrls
More informationGraph States EPIT Mehdi Mhalla (Calgary, Canada) Simon Perdrix (Grenoble, France)
Grph Sttes EPIT 2005 Mehdi Mhll (Clgry, Cnd) Simon Perdrix (Grenole, Frne) simon.perdrix@img.fr Grph Stte: Introdution A grphsed representtion of the entnglement of some (lrge) quntum stte. Verties: quits
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationCS 573 Automata Theory and Formal Languages
Nondeterminism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Nondeterministi Finite Automton with moves (NFA) An NFA is tuple
More informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationCHEM1611 Answers to Problem Sheet 9
CEM1611 Answers to Prolem Sheet 9 1. Tutomers re struturl isomers whih re relte y migrtion of hyrogen tom n the exhnge of single on n jent oule on. Compoun Tutomer 2 2 2 2 2 2 2 2 2 2 2 2. () Whih pir
More informationTrigonometry and Constructive Geometry
Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationAnalysis of Temporal Interactions with Link Streams and Stream Graphs
Anlysis of Temporl Intertions with n Strem Grphs, Tiphine Vir, Clémene Mgnien http:// ltpy@ LIP6 CNRS n Soronne Université Pris, Frne 1/23 intertions over time 0 2 4 6 8,,, n for 10 time units time 2/23
More informationCommon intervals of genomes. Mathieu Raffinot CNRS LIAFA
Common intervls of genomes Mthieu Rffinot CNRS LIF Context: omprtive genomis. set of genomes prtilly/totlly nnotte Informtive group of genes or omins? Ex: COG tse Mny iffiulties! iology Wht re two similr
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More informationVidyalankar S.E. Sem. III [CMPN] Discrete Structures Prelim Question Paper Solution
S.E. Sem. III [CMPN] Discrete Structures Prelim Question Pper Solution 1. () (i) Disjoint set wo sets re si to be isjoint if they hve no elements in common. Exmple : A = {0, 4, 7, 9} n B = {3, 17, 15}
More informationCompression of Palindromes and Regularity.
Compression of Plinromes n Regulrity. Kyoko ShikishimTsuji Center for Lierl Arts Eution n Reserh Tenri University 1 Introution In [1], property of likstrem t t view of tse is isusse n it is shown tht
More informationComputing data with spreadsheets. Enter the following into the corresponding cells: A1: n B1: triangle C1: sqrt
Computing dt with spredsheets Exmple: Computing tringulr numers nd their squre roots. Rell, we showed 1 ` 2 ` `n npn ` 1q{2. Enter the following into the orresponding ells: A1: n B1: tringle C1: sqrt A2:
More informationCS 360 Exam 2 Fall 2014 Name
CS 360 Exm 2 Fll 2014 Nme 1. The lsses shown elow efine singlylinke list n stk. Write three ifferent O(n)time versions of the reverse_print metho s speifie elow. Eh version of the metho shoul output
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationLecture 08: Feb. 08, 2019
4CS46:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 09765840 Volume 6, Numer (04), pp. 89 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationParticle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap
Prtile Physis Mihelms Term 2011 Prof Mrk Thomson g X g X g g Hnout 3 : Intertion y Prtile Exhnge n QED Prof. M.A. Thomson Mihelms 2011 101 Rep Working towrs proper lultion of ey n sttering proesses lnitilly
More informationDirected acyclic graphs with the unique dipath property
Direte yli grphs with the unique ipth property JenClue Bermon, Mihel Cosnr, Stéphne érennes To ite this version: JenClue Bermon, Mihel Cosnr, Stéphne érennes. Direte yli grphs with the unique ipth property.
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationCoding Techniques. Manjunatha. P. Professor Dept. of ECE. June 28, J.N.N. College of Engineering, Shimoga.
Coing Tehniques Mnjunth. P mnjup.jnne@gmil.om Professor Dept. of ECE J.N.N. College of Engineering, Shimog June 8, 3 Overview Convolutionl Enoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8 Overview
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationRecent advances in analysis of evolutionary transpositions
Reent vnes in nlysis of evolutionry trnspositions Mx Alekseyev Deprtment of Mthemtis / Computtionl Biology Institute George Wshington University 2015 Genome Rerrngements Mouse X hromosome unknown nestor
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationFormal Languages and Automata
Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt ChunMing Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University
More informationAPPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More informationExercise sheet 6: Solutions
Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd
More informationNondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
More informationCS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014
S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown
More informationTotal score: /100 points
Points misse: Stuent's Nme: Totl sore: /100 points Est Tennessee Stte University Deprtment of Computer n Informtion Sienes CSCI 2710 (Trnoff) Disrete Strutures TEST 2 for Fll Semester, 2004 Re this efore
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationMaximum size of a minimum watching system and the graphs achieving the bound
Mximum size of minimum wthing system n the grphs hieving the oun Tille mximum un système e ontrôle minimum et les grphes tteignnt l orne Dvi Auger Irène Chron Olivier Hury Antoine Lostein 00D0 Mrs 00 Déprtement
More informationy = c 2 MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark) is...
. Liner Equtions in Two Vriles C h p t e r t G l n e. Generl form of liner eqution in two vriles is x + y + 0, where 0. When we onsier system of two liner equtions in two vriles, then suh equtions re lle
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationNonDeterministic Finite Automata. Fall 2018 Costas Busch  RPI 1
NonDeterministic Finite Automt Fll 2018 Costs Busch  RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch  RPI 2 Nondeterministic Finite Automton (NFA) Alphbet
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationSEMIEXCIRCLE OF QUADRILATERAL
JP Journl of Mthemtil Sienes Volume 5, Issue &, 05, Pges  05 Ishn Pulishing House This pper is ville online t http://wwwiphsiom SEMIEXCIRCLE OF QUADRILATERAL MASHADI, SRI GEMAWATI, HASRIATI AND HESY
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the solled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationThe master ring problem
2005 Interntionl Conferene on Anlysis of Algorithms DMTCS pro. AD, 2005, 287 296 The mster ring problem Hs Shhni 1 n Lis Zhng 2 1 Computer Siene Dept., Tehnion, Hif 32000, Isrel. 2 Bell Lbs, Luent Tehnologies,
More informationMCH T 111 Handout Triangle Review Page 1 of 3
Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:
More informationRegular expressions, Finite Automata, transition graphs are all the same!!
CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104W11 1
More informationEuler and Hamilton Paths
Euler an Hamilton Paths The town of Königserg, Prussia (now know as Kaliningra an part of the Russian repuli), was ivie into four setion y ranhes of the Pregel River. These four setions C A D B Figure:
More informationSystem Validation (IN4387) November 2, 2012, 14:0017:00
System Vlidtion (IN4387) Novemer 2, 2012, 14:0017:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise
More informationC. C^mpenu, K. Slom, S. Yu upper boun of mn. So our result is tight only for incomplete DF's. For restricte vlues of m n n we present exmples of DF's
Journl of utomt, Lnguges n Combintorics u (v) w, x{y c OttovonGuerickeUniversitt Mgeburg Tight lower boun for the stte complexity of shue of regulr lnguges Cezr C^mpenu, Ki Slom Computing n Informtion
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationNecessary and sucient conditions for some two. Abstract. Further we show that the necessary conditions for the existence of an OD(44 s 1 s 2 )
Neessry n suient onitions for some two vrile orthogonl esigns in orer 44 C. Koukouvinos, M. Mitrouli y, n Jennifer Seerry z Deite to Professor Anne Penfol Street Astrt We give new lgorithm whih llows us
More informationInspiration and formalism
Inspirtion n formlism Answers Skills hek P(, ) Q(, ) PQ + ( ) PQ A(, ) (, ) grient ( ) + Eerise A opposite sies of regulr hegon re equl n prllel A ED i FC n ED ii AD, DA, E, E n FC No, sies of pentgon
More informationLESSON 11: TRIANGLE FORMULAE
. THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.
More informationCS261: A Second Course in Algorithms Lecture #5: MinimumCost Bipartite Matching
CS261: A Seon Course in Algorithms Leture #5: MinimumCost Biprtite Mthing Tim Roughgren Jnury 19, 2016 1 Preliminries Figure 1: Exmple of iprtite grph. The eges {, } n {, } onstitute mthing. Lst leture
More information