# CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018

Size: px
Start display at page:

Download "CSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018"

Transcription

1 CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14,

2 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs it n hv, in trms of x? Now, suppos w hv iffrnt simpl, unirt grph with y gs. Wht is th mximum numr of vrtis it n hv, in trms of y? 2

3 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? A simpl grph is grph tht hs no slf-loops n no prlll gs. Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs it n hv, in trms of x? Eh vrtx n onnt to x 1 othr vrtis, so x(x 1). Now, suppos w hv iffrnt simpl, unirt grph with y gs. Wht is th mximum numr of vrtis it n hv, in trms of y? Infinit: just kp ing nos with no gs tth. 2

4 Wrmup: Som follow-up qustions: Suppos w hv simpl, unirt grph with x nos. Wht is th mximum numr of gs it n hv? Wht if th grph is not simpl? Now, suppos w hv iffrnt simpl, unirt grph with y gs. Wht is th mximum numr of vrtis it n hv? Wht if th grph is not simpl? 3

5 Wrmup: Som follow-up qustions: Suppos w hv simpl, unirt grph with x nos. Wht is th mximum numr of gs it n hv? Wht if th grph is not simpl? If th grph is simpl, th mx numr of gs is xtly hlf x(x 1) of wht it woul if th grph wr irt. So,. 2 If th grph is not simpl, it s infinit: ssuming x > 0, w n just kp ing mor n mor slf-loops. Not tht if x = 0, thr n t ny gs t ll. Now, suppos w hv iffrnt simpl, unirt grph with y gs. Wht is th mximum numr of vrtis it n hv? Wht if th grph is not simpl? Eithr wy, it s still infinit, for th sm rsons givn prviously. 3

6 Summry Wht i w lrn? In grphs with no rstritions, numr of gs n numr of vrtis r inpnnt. 4

7 Summry Wht i w lrn? In grphs with no rstritions, numr of gs n numr of vrtis r inpnnt. In simpl grphs, if w know V is som fix vlu, w lso know E O ( V 2), for oth irt n unirt grphs. 4

8 Summry Wht i w lrn? In grphs with no rstritions, numr of gs n numr of vrtis r inpnnt. In simpl grphs, if w know V is som fix vlu, w lso know E O ( V 2), for oth irt n unirt grphs. Dns grph If E Θ ( V 2), w sy th grph is ns. To put it nothr wy, ns grphs hv lots of gs 4

9 Summry Wht i w lrn? In grphs with no rstritions, numr of gs n numr of vrtis r inpnnt. In simpl grphs, if w know V is som fix vlu, w lso know E O ( V 2), for oth irt n unirt grphs. Dns grph If E Θ ( V 2), w sy th grph is ns. To put it nothr wy, ns grphs hv lots of gs Sprs grph If E O ( V ), w su th grph is sprs. To put it nothr wy, sprs grphs hv fw gs. 4

10 How o w rprsnt grphs in o? So, how o w tully rprsnt grphs in o? 5

11 How o w rprsnt grphs in o? So, how o w tully rprsnt grphs in o? Two ommon pprohs, with iffrnt troffs: Ajny mtrix Ajny list 5

12 Ajny mtrix Cor i: Assign h no numr from 0 to V 1 Crt V V nst rry of oolns or ints If (x, y) E, thn nstarry[x][y] == tru 6

13 Ajny mtrix Cor i: Assign h no numr from 0 to V 1 Crt V V nst rry of oolns or ints If (x, y) E, thn nstarry[x][y] == tru 6

14 Ajny list Wht is th worst-s runtim to: Gt out-gs: Gt in-gs: Di if n g xists: Insrt n g: Dlt n g: How muh sp o w us? Is this ttr for sprs or ns grphs? Cn w hnl slf-loops n prlll gs? 7

15 Ajny list Wht is th worst-s runtim to: Gt out-gs: O ( V ) Gt in-gs: O ( V ) Di if n g xists: O (1) Insrt n g: O (1) Dlt n g: O (1) How muh sp o w us? O ( V 2) Is this ttr for sprs or ns grphs? Dns ons Cn w hnl slf-loops n prlll gs? Slf-loops ys, prlll gs, not sily 7

16 Ajny list Cor i: Assign h no numr from 0 to V 1 Crt n rry of siz V Eh lmnt in th rry stors its out gs in list or st 8

17 Ajny list Cor i: Assign h no numr from 0 to V 1 Crt n rry of siz V Eh lmnt in th rry stors its out gs in list or st On highr lvl: rprsnt s IDitionry<Vrtx, Egs>. 8

18 Ajny list Cor i: Assign h no numr from 0 to V 1 Crt n rry of siz V Eh lmnt in th rry stors its out gs in list or st On highr lvl: rprsnt s IDitionry<Vrtx, Egs>. 8

19 Ajny list W n stor gs using ithr sts or lists. Answr ths qustions for oth. Wht is th worst-s runtim to: Gt out-gs: Gt in-gs: Di if n g xists: Insrt n g: Dlt n g: How muh sp o w us? Is this ttr for sprs or ns grphs? Cn w hnl slf-loops n prlll gs? 9

20 Whih o w pik? So whih o w pik? 10

21 Whih o w pik? So whih o w pik? Osrvtions: Most grphs r sprs If w implmnt jny lists using sts, w n gt omprl worst-s prformn 10

22 Whih o w pik? So whih o w pik? Osrvtions: Most grphs r sprs If w implmnt jny lists using sts, w n gt omprl worst-s prformn So y fult, pik jny lists. 10

23 Wlks n pths Wlk A wlk is list of vrtis v 0, v 1, v 2,..., v n whr if i is som int whr 0 i < v n, vry pir (v i, v i+1 ) E is tru. Mor intuitivly, wlk is on ontinous lin following th gs. 11

24 Wlks n pths Wlk A wlk is list of vrtis v 0, v 1, v 2,..., v n whr if i is som int whr 0 i < v n, vry pir (v i, v i+1 ) E is tru. Mor intuitivly, wlk is on ontinous lin following th gs. Pth A pth is wlk tht nvr visits th sm vrtx twi. 11

25 Wlks n pths Wlk A wlk is list of vrtis v 0, v 1, v 2,..., v n whr if i is som int whr 0 i < v n, vry pir (v i, v i+1 ) E is tru. Mor intuitivly, wlk is on ontinous lin following th gs. Pth A pth is wlk tht nvr visits th sm vrtx twi. Pth or wlk? Pth or wlk? 11

26 Wlks n pths Wlk A wlk is list of vrtis v 0, v 1, v 2,..., v n whr if i is som int whr 0 i < v n, vry pir (v i, v i+1 ) E is tru. Mor intuitivly, wlk is on ontinous lin following th gs. Pth A pth is wlk tht nvr visits th sm vrtx twi. Wlk Pth 11

27 Connt omponnts Connt grph A grph is onnt if vry vrtx is onnt to vry othr vrtx vi som pth. E.g.: if w pik up th grph n shk it, nothing flis off. 12

28 Connt omponnts Connt grph A grph is onnt if vry vrtx is onnt to vry othr vrtx vi som pth. E.g.: if w pik up th grph n shk it, nothing flis off. Connt or not onnt? Connt or not onnt? 12

29 Connt omponnts Connt grph A grph is onnt if vry vrtx is onnt to vry othr vrtx vi som pth. E.g.: if w pik up th grph n shk it, nothing flis off. Connt Not onnt Connt omponnt A onnt omponnt of grph is ny sugrph (prt of grph) whr ll vrtis r onnt to h othr. Not: A onnt grph hs only on onnt omponnt. 12

30 Trs vs grphs Is this grph or tr? f g 13

31 Trs vs grphs Is this grph or tr? f g Both! 13

32 Trs vs grphs Is this grph or tr? f g Both! Tr A tr is onnt n yli grph. 13

33 Trs vs grphs Is this grph or tr? Is this th sm thing? f Both! g g f Tr A tr is onnt n yli grph. 13

34 Trs vs grphs Is this grph or tr? Is this th sm thing? f g g f Both! Ys! (If is th root...) Tr A tr is onnt n yli grph. 13

35 Trs vs grphs Is this grph or tr? Is this th sm thing? f g g f Both! Ys! (If is th root...) Tr A tr is onnt n yli grph. Root tr A root tr is tr whr w ll on spil no th root. 13

36 Dtting if grph is onnt Qustion: How n w tll if grph is onnt or not? 14

37 Dtting if grph is onnt Qustion: How n w tll if grph is onnt or not? I: Lt s just fin out! Pik no n s if thr s pth to vry othr no! 14

38 Brth-first srh (BFS) Brth-first trvrsl, or i: 1. Pik som no n mrk it (or sv it in st, t...) 15

39 Brth-first srh (BFS) Brth-first trvrsl, or i: 1. Pik som no n mrk it (or sv it in st, t...) 2. Exmin h nighor n visit h on (not: sv th ons w hvn t visit yt in som t strutur, lik quu?) 15

40 Brth-first srh (BFS) Brth-first trvrsl, or i: 1. Pik som no n mrk it (or sv it in st, t...) 2. Exmin h nighor n visit h on (not: sv th ons w hvn t visit yt in som t strutur, lik quu?) 3. Dquu som no from th t strutur. Go to stp 1. 15

41 Brth-first srh (BFS) Brth-first trvrsl, or i: 1. Pik som no n mrk it (or sv it in st, t...) 2. Exmin h nighor n visit h on (not: sv th ons w hvn t visit yt in som t strutur, lik quu?) 3. Dquu som no from th t strutur. Go to stp Kp going until th t strutur is mpty. 15

42 Brth-first srh (BFS) Brth-first trvrsl, or i: 1. Pik som no n mrk it (or sv it in st, t...) 2. Exmin h nighor n visit h on (not: sv th ons w hvn t visit yt in som t strutur, lik quu?) 3. Dquu som no from th t strutur. Go to stp Kp going until th t strutur is mpty. Psuoo, vrsion 1: srh(v): visit = mpty st quu.nquu(v) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) 15

43 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:, 16

44 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu: 16

45 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:,, 16

46 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:, 16

47 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:,,,, 16

48 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:,,, 16

49 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:,,, f, g, 16

50 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:,, f, g, 16

51 Brth-first srh (BFS) xmpl srh(v): quu.nquu(v) f whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): quu.nquu(w) i g j h Currnt no: Quu:, f, g, Wht wnt wrong? 16

52 A rokn trvrsl Prolm: W r r-visiting nos w lry visit! 17

53 A rokn trvrsl Prolm: W r r-visiting nos w lry visit! A fix: Kp trk of nos w v lry visit in st! 17

54 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:, Visit:, 18

55 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu: Visit:, 18

56 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:,, Visit:,,, 18

57 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:, Visit:,,, 18

58 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:,,, Visit:,,,,, 18

59 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:,, Visit:,,,,, 18

60 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:,, f, g, Visit:,,,,, f, g, 18

61 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu:, f, g, Visit:,,,,, f, g, 18

62 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu: f, g, Visit:,,,,, f, g, 18

63 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: Quu: f, g, h, Visit:,,,,, f, g, h, 18

64 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: f Quu: g, h, Visit:,,,,, f, g, h, 18

65 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: g Quu: h, Visit:,,,,, f, g, h, 18

66 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: g Quu: h, i, Visit:,,,,, f, g, h, i, 18

67 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: h Quu: i, Visit:,,,,, f, g, h, i, 18

68 Brth-first srh (BFS) xmpl srh(v): visit = mpty st f quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) i g h j Currnt no: i Quu: Visit:,,,,, f, g, h, i, 18

69 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

70 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

71 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

72 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

73 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

74 An intrsting proprty... Not: W visit th nos in rings mintin grully growing frontir of nos. f i g j h 19

75 BFS Psuoo srh(v): visit = mpty st quu.nquu(v) visit.(urr) whil (quu is not mpty): urr = quu.quu() for (w : v.nighors()): if (w not in visit): quu.nquu(w) visit.(urr) 20

76 BFS nlysis Qustions: Wht is th worst-s runtim? (Lt V th numr of vrtis, lt E th numr of gs) Wht is th worst-s mount of mmory us? 21

77 BFS nlysis Qustions: Wht is th worst-s runtim? (Lt V th numr of vrtis, lt E th numr of gs) W visit h vrtx on, n h g on, so O ( V + E ). Wht is th worst-s mount of mmory us? Whtvr th lrgst horizon siz is. In th worst s, th horizon will ontin V 1 nos, so O ( V ). Not: O ( V + E ) is lso ll grph linr. 21

78 Othr pplitions of BFS Dsri how you woul us or moify BFS to solv th following: Dtrmin if som grph is lso tr. Print out ll th lmnts in tr lvl y lvl. Fin th shortst pth from on no to nothr. 22

### CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp

CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140

### V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous

### V={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}

s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn

### CS 461, Lecture 17. Today s Outline. Example Run

Prim s Algorithm CS 461, Ltur 17 Jr Si Univrsity o Nw Mxio In Prim s lgorithm, th st A mintin y th lgorithm orms singl tr. Th tr strts rom n ritrry root vrtx n grows until it spns ll th vrtis in V At h

### CS200: Graphs. Graphs. Directed Graphs. Graphs/Networks Around Us. What can this represent? Sometimes we want to represent directionality:

CS2: Grphs Prihr Ch. 4 Rosn Ch. Grphs A olltion of nos n gs Wht n this rprsnt? n A omputr ntwork n Astrtion of mp n Soil ntwork CS2 - Hsh Tls 2 Dirt Grphs Grphs/Ntworks Aroun Us A olltion of nos n irt

### CS 241 Analysis of Algorithms

CS 241 Anlysis o Algorithms Prossor Eri Aron Ltur T Th 9:00m Ltur Mting Lotion: OLB 205 Businss HW6 u lry HW7 out tr Thnksgiving Ring: Ch. 22.1-22.3 1 Grphs (S S. B.4) Grphs ommonly rprsnt onntions mong

### Graphs. CSC 1300 Discrete Structures Villanova University. Villanova CSC Dr Papalaskari

Grphs CSC 1300 Disrt Struturs Villnov Univrsity Grphs Grphs r isrt struturs onsis?ng of vr?s n gs tht onnt ths vr?s. Grphs n us to mol: omputr systms/ntworks mthm?l rl?ons logi iruit lyout jos/prosss f

### Paths. Connectivity. Euler and Hamilton Paths. Planar graphs.

Pths.. Eulr n Hmilton Pths.. Pth D. A pth rom s to t is squn o gs {x 0, x 1 }, {x 1, x 2 },... {x n 1, x n }, whr x 0 = s, n x n = t. D. Th lngth o pth is th numr o gs in it. {, } {, } {, } {, } {, } {,

### CSC Design and Analysis of Algorithms. Example: Change-Making Problem

CSC 801- Dsign n Anlysis of Algorithms Ltur 11 Gry Thniqu Exmpl: Chng-Mking Prolm Givn unlimit mounts of oins of nomintions 1 > > m, giv hng for mount n with th lst numr of oins Exmpl: 1 = 25, 2 =10, =

### Graphs. Graphs. Graphs: Basic Terminology. Directed Graphs. Dr Papalaskari 1

CSC 00 Disrt Struturs : Introuon to Grph Thory Grphs Grphs CSC 00 Disrt Struturs Villnov Univrsity Grphs r isrt struturs onsisng o vrs n gs tht onnt ths vrs. Grphs n us to mol: omputr systms/ntworks mthml

### An undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V

Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon

### Exam 1 Solution. CS 542 Advanced Data Structures and Algorithms 2/14/2013

CS Avn Dt Struturs n Algorithms Exm Solution Jon Turnr //. ( points) Suppos you r givn grph G=(V,E) with g wights w() n minimum spnning tr T o G. Now, suppos nw g {u,v} is to G. Dsri (in wors) mtho or

### CSE 373: AVL trees. Warmup: Warmup. Interlude: Exploring the balance invariant. AVL Trees: Invariants. AVL tree invariants review

rmup CSE 7: AVL trs rmup: ht is n invrint? Mihl L Friy, Jn 9, 0 ht r th AVL tr invrints, xtly? Disuss with your nighor. AVL Trs: Invrints Intrlu: Exploring th ln invrint Cor i: xtr invrint to BSTs tht

### CS61B Lecture #33. Administrivia: Autograder will run this evening. Today s Readings: Graph Structures: DSIJ, Chapter 12

Aministrivi: CS61B Ltur #33 Autogrr will run this vning. Toy s Rings: Grph Struturs: DSIJ, Chptr 12 Lst moifi: W Nov 8 00:39:28 2017 CS61B: Ltur #33 1 Why Grphs? For xprssing non-hirrhilly rlt itms Exmpls:

### Outline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example

Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)

### Graph Isomorphism. Graphs - II. Cayley s Formula. Planar Graphs. Outline. Is K 5 planar? The number of labeled trees on n nodes is n n-2

Grt Thortil Is In Computr Sin Vitor Amhik CS 15-251 Ltur 9 Grphs - II Crngi Mllon Univrsity Grph Isomorphism finition. Two simpl grphs G n H r isomorphi G H if thr is vrtx ijtion V H ->V G tht prsrvs jny

### 12/3/12. Outline. Part 10. Graphs. Circuits. Euler paths/circuits. Euler s bridge problem (Bridges of Konigsberg Problem)

12/3/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 Ciruits Cyl 2 Eulr

### 5/9/13. Part 10. Graphs. Outline. Circuits. Introduction Terminology Implementing Graphs

Prt 10. Grphs CS 200 Algorithms n Dt Struturs 1 Introution Trminology Implmnting Grphs Outlin Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 2 Ciruits Cyl A spil yl

### N=4 L=4. Our first non-linear data structure! A graph G consists of two sets G = {V, E} A set of V vertices, or nodes f

lulu jwtt pnlton sin towr ounrs hpl lpp lu Our irst non-linr t strutur! rph G onsists o two sts G = {V, E} st o V vrtis, or nos st o E s, rltionships twn nos surph G onsists o sust o th vrtis n s o G jnt

### Outline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs

Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl

### Announcements. Not graphs. These are Graphs. Applications of Graphs. Graph Definitions. Graphs & Graph Algorithms. A6 released today: Risk

Grphs & Grph Algorithms Ltur CS Spring 6 Announmnts A6 rls toy: Risk Strt signing with your prtnr sp Prlim usy Not grphs hs r Grphs K 5 K, =...not th kin w mn, nywy Applitions o Grphs Communition ntworks

### Outline. Circuits. Euler paths/circuits 4/25/12. Part 10. Graphs. Euler s bridge problem (Bridges of Konigsberg Problem)

4/25/12 Outlin Prt 10. Grphs CS 200 Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits 1 2 Eulr s rig prolm

### b. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?

MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth

### Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.

Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right

### Cycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!

Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik

### COMP108 Algorithmic Foundations

Grdy mthods Prudn Wong http://www.s.liv..uk/~pwong/thing/omp108/01617 Coin Chng Prolm Suppos w hv 3 typs of oins 10p 0p 50p Minimum numr of oins to mk 0.8, 1.0, 1.? Grdy mthod Lrning outoms Undrstnd wht

### Module graph.py. 1 Introduction. 2 Graph basics. 3 Module graph.py. 3.1 Objects. CS 231 Naomi Nishimura

Moul grph.py CS 231 Nomi Nishimur 1 Introution Just lik th Python list n th Python itionry provi wys of storing, ssing, n moifying t, grph n viw s wy of storing, ssing, n moifying t. Bus Python os not

### , each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management

nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o

### 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

### 5/7/13. Part 10. Graphs. Theorem Theorem Graphs Describing Precedence. Outline. Theorem 10-1: The Handshaking Theorem

Thorm 10-1: Th Hnshkin Thorm Lt G=(V,E) n unirt rph. Thn Prt 10. Grphs CS 200 Alorithms n Dt Struturs v V (v) = 2 E How mny s r thr in rph with 10 vrtis h of r six? 10 * 6 /2= 30 1 Thorm 10-2 An unirt

### Section 10.4 Connectivity (up to paths and isomorphism, not including)

Toy w will isuss two stions: Stion 10.3 Rprsnting Grphs n Grph Isomorphism Stion 10.4 Conntivity (up to pths n isomorphism, not inluing) 1 10.3 Rprsnting Grphs n Grph Isomorphism Whn w r working on n lgorithm

### Weighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths

Dt Strutur LECTURE Shortt pth lgorithm Proprti of hortt pth Bllmn-For lgorithm Dijktr lgorithm Chptr in th txtook (pp ). Wight grph -- rminr A wight grph i grph in whih g hv wight (ot) w(v i, v j ) >.

### ECE 407 Computer Aided Design for Electronic Systems. Circuit Modeling and Basic Graph Concepts/Algorithms. Instructor: Maria K. Michael.

0 Computr i Dsign or Eltroni Systms Ciruit Moling n si Grph Conptslgorithms Instrutor: Mri K. Mihl MKM - Ovrviw hviorl vs. Struturl mols Extrnl vs. Intrnl rprsnttions Funtionl moling t Logi lvl Struturl

### CSI35 Chapter 11 Review

1. Which of th grphs r trs? c f c g f c x y f z p q r 1 1. Which of th grphs r trs? c f c g f c x y f z p q r . Answr th qustions out th following tr 1) Which vrtx is th root of c th tr? ) wht is th hight

### Why the Junction Tree Algorithm? The Junction Tree Algorithm. Clique Potential Representation. Overview. Chris Williams 1.

Why th Juntion Tr lgorithm? Th Juntion Tr lgorithm hris Willims 1 Shool of Informtis, Univrsity of Einurgh Otor 2009 Th JT is gnrl-purpos lgorithm for omputing (onitionl) mrginls on grphs. It os this y

### Announcements. These are Graphs. This is not a Graph. Graph Definitions. Applications of Graphs. Graphs & Graph Algorithms

Grphs & Grph Algorithms Ltur CS Fll 5 Announmnts Upoming tlk h Mny Crrs o Computr Sintist Or how Computr Sin gr mpowrs you to o muh mor thn o Dn Huttnlohr, Prossor in th Dprtmnt o Computr Sin n Johnson

### Planar Upward Drawings

C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th

### The University of Sydney MATH2969/2069. Graph Theory Tutorial 5 (Week 12) Solutions 2008

Th Univrsity o Syny MATH2969/2069 Grph Thory Tutoril 5 (Wk 12) Solutions 2008 1. (i) Lt G th isonnt plnr grph shown. Drw its ul G, n th ul o th ul (G ). (ii) Show tht i G is isonnt plnr grph, thn G is

### Algorithmic and NP-Completeness Aspects of a Total Lict Domination Number of a Graph

Intrntionl J.Mth. Comin. Vol.1(2014), 80-86 Algorithmi n NP-Compltnss Aspts of Totl Lit Domintion Numr of Grph Girish.V.R. (PES Institut of Thnology(South Cmpus), Bnglor, Krntk Stt, Ini) P.Ush (Dprtmnt

### 10/30/12. Today. CS/ENGRD 2110 Object- Oriented Programming and Data Structures Fall 2012 Doug James. DFS algorithm. Reachability Algorithms

0/0/ CS/ENGRD 0 Ojt- Orint Prormmin n Dt Strutur Fll 0 Dou Jm Ltur 9: DFS, BFS & Shortt Pth Toy Rhility Dpth-Firt Srh Brth-Firt Srh Shortt Pth Unwiht rph Wiht rph Dijktr lorithm Rhility Alorithm Dpth Firt

### 0.1. Exercise 1: the distances between four points in a graph

Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 pg 1 Mth 707 Spring 2017 (Drij Grinrg): mitrm 3 u: W, 3 My 2017, in lss or y mil (grinr@umn.u) or lss S th wsit or rlvnt mtril. Rsults provn in th nots, or in

### ECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS

C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h

### Constructive Geometric Constraint Solving

Construtiv Gomtri Constrint Solving Antoni Soto i Rir Dprtmnt Llngutgs i Sistms Inormàtis Univrsitt Politèni Ctluny Brlon, Sptmr 2002 CGCS p.1/37 Prliminris CGCS p.2/37 Gomtri onstrint prolm C 2 D L BC

### Garnir Polynomial and their Properties

Univrsity of Cliforni, Dvis Dprtmnt of Mthmtis Grnir Polynomil n thir Proprtis Author: Yu Wng Suprvisor: Prof. Gorsky Eugny My 8, 07 Grnir Polynomil n thir Proprtis Yu Wng mil: uywng@uvis.u. In this ppr,

### QUESTIONS BEGIN HERE!

Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt of Computr n Informtion Sins CSCI 710 (Trnoff) Disrt Struturs TEST for Fll Smstr, 00 R this for strtin! This tst is los ook

### 1 Introduction to Modulo 7 Arithmetic

1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w

### More Foundations. Undirected Graphs. Degree. A Theorem. Graphs, Products, & Relations

Mr Funtins Grphs, Pruts, & Rltins Unirt Grphs An unirt grph is pir f 1. A st f ns 2. A st f gs (whr n g is st f tw ns*) Friy, Sptmr 2, 2011 Ring: Sipsr 0.2 ginning f 0.4; Stughtn 1.1.5 ({,,,,}, {{,}, {,},

### Improving Union. Implementation. Union-by-size Code. Union-by-Size Find Analysis. Path Compression! Improving Find find(e)

POW CSE 36: Dt Struturs Top #10 T Dynm (Equvln) Duo: Unon-y-Sz & Pt Comprsson Wk!! Luk MDowll Summr Qurtr 003 M! ZING Wt s Goo Mz? Mz Construton lortm Gvn: ollton o rooms V Conntons twn t rooms (ntlly

### Instructions for Section 1

Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks

### 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),

### CS 103 BFS Alorithm. Mark Redekopp

CS 3 BFS Aloritm Mrk Rkopp Brt-First Sr (BFS) HIGHLIGHTED ALGORITHM 3 Pt Plnnin W'v sn BFS in t ontxt o inin t sortst pt trou mz? S?? 4 Pt Plnnin W xplor t 4 niors s on irtion 3 3 3 S 3 3 3 3 3 F I you

### CSE303 - Introduction to the Theory of Computing Sample Solutions for Exercises on Finite Automata

CSE303 - Introduction to th Thory of Computing Smpl Solutions for Exrciss on Finit Automt Exrcis 2.1.1 A dtrministic finit utomton M ccpts th mpty string (i.., L(M)) if nd only if its initil stt is finl

### QUESTIONS BEGIN HERE!

Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook

### Minimum Spanning Trees

Minimum Spnning Trs Minimum Spnning Trs Problm A town hs st of houss nd st of rods A rod conncts nd only houss A rod conncting houss u nd v hs rpir cost w(u, v) Gol: Rpir nough (nd no mor) rods such tht:

### Walk Like a Mathematician Learning Task:

Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics

### 12. Traffic engineering

lt2.ppt S-38. Introution to Tltrffi Thory Spring 200 2 Topology Pths A tlommunition ntwork onsists of nos n links Lt N not th st of nos in with n Lt J not th st of nos in with j N = {,,,,} J = {,2,3,,2}

### (2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely

. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,

### Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1

Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):

### a b c cat CAT A B C Aa Bb Cc cat cat Lesson 1 (Part 1) Verbal lesson: Capital Letters Make The Same Sound Lesson 1 (Part 1) continued...

Progrssiv Printing T.M. CPITLS g 4½+ Th sy, fun (n FR!) wy to tch cpitl lttrs. ook : C o - For Kinrgrtn or First Gr (not for pr-school). - Tchs tht cpitl lttrs mk th sm souns s th littl lttrs. - Tchs th

### CS September 2018

Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o

### Numbering Boundary Nodes

Numring Bounry Nos Lh MBri Empori Stt Univrsity August 10, 2001 1 Introution Th purpos of this ppr is to xplor how numring ltril rsistor ntworks ffts thir rspons mtrix, Λ. Morovr, wht n lrn from Λ out

### CMSC 451: Lecture 2 Graph Basics Thursday, Aug 31, 2017

Dv Mount CMSC 45: Ltur Grph Bsis Thursy, Au, 07 Rin: Chpt. in KT (Klinr n Tros) n Chpt. in DBV (Dsupt, Ppimitriou, n Vzirni). Som o our trminoloy irs rom our txt. Grphs n Dirphs: A rph G = (V, E) is strutur

### COMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS

OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson

### Chapter 9. Graphs. 9.1 Graphs

Chptr 9 Grphs Grphs r vry gnrl lss of ojt, us to formliz wi vrity of prtil prolms in omputr sin. In this hptr, w ll s th sis of (finit) unirt grphs, inluing grph isomorphism, onntivity, n grph oloring.

### Similarity Search. The Binary Branch Distance. Nikolaus Augsten.

Similrity Srh Th Binry Brnh Distn Nikolus Augstn nikolus.ugstn@sg..t Dpt. of Computr Sins Univrsity of Slzurg http://rsrh.uni-slzurg.t Vrsion Jnury 11, 2017 Wintrsmstr 2016/2017 Augstn (Univ. Slzurg) Similrity

### NP-Completeness. CS3230 (Algorithm) Traveling Salesperson Problem. What s the Big Deal? Given a Problem. What s the Big Deal? What s the Big Deal?

NP-Compltnss 1. Polynomil tim lgorithm 2. Polynomil tim rution 3.P vs NP 4.NP-ompltnss (som slis y P.T. Um Univrsity o Txs t Dlls r us) Trvling Slsprson Prolm Fin minimum lngth tour tht visits h ity on

### 1. Determine whether or not the following binary relations are equivalence relations. Be sure to justify your answers.

Mth 0 Exm - Prti Prolm Solutions. Dtrmin whthr or not th ollowing inry rltions r quivln rltions. B sur to justiy your nswrs. () {(0,0),(0,),(0,),(,),(,),(,),(,),(,0),(,),(,),(,0),(,),(.)} on th st A =

### Problem solving by search

Prolm solving y srh Tomáš voo Dprtmnt o Cyrntis, Vision or Roots n Autonomous ystms Mrh 5, 208 / 3 Outlin rh prolm. tt sp grphs. rh trs. trtgis, whih tr rnhs to hoos? trtgy/algorithm proprtis? Progrmming

### Chapter 18. Minimum Spanning Trees Minimum Spanning Trees. a d. a d. a d. f c

Chptr 8 Minimum Spnning Trs In this hptr w ovr importnt grph prolm, Minimum Spnning Trs (MST). Th MST o n unirt, wight grph is tr tht spns th grph whil minimizing th totl wight o th gs in th tr. W irst

### Page 1. Question 19.1b Electric Charge II Question 19.2a Conductors I. ConcepTest Clicker Questions Chapter 19. Physics, 4 th Edition James S.

ConTst Clikr ustions Chtr 19 Physis, 4 th Eition Jms S. Wlkr ustion 19.1 Two hrg blls r rlling h othr s thy hng from th iling. Wht n you sy bout thir hrgs? Eltri Chrg I on is ositiv, th othr is ngtiv both

### EE1000 Project 4 Digital Volt Meter

Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s

### MAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017

MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT

### Greedy Algorithms, Activity Selection, Minimum Spanning Trees Scribes: Logan Short (2015), Virginia Date: May 18, 2016

Ltur 4 Gry Algorithms, Ativity Sltion, Minimum Spnning Trs Sris: Logn Short (5), Virgini Dt: My, Gry Algorithms Suppos w wnt to solv prolm, n w r l to om up with som rursiv ormultion o th prolm tht woul

### Round 7: Graphs (part I)

Roun 7: Grphs (prt I) Tommi Junttil Alto Univrsity Shool o Sin Dprtmnt o Computr Sin CS-A40 Dt Struturs n Alorithms Autumn 207 Tommi Junttil (Alto Univrsity) Roun 7 CS-A40 / Autumn 207 / 55 Topis: Grphs

### S i m p l i f y i n g A l g e b r a SIMPLIFYING ALGEBRA.

S i m p l i y i n g A l g r SIMPLIFYING ALGEBRA www.mthltis.o.nz Simpliying SIMPLIFYING Algr ALGEBRA Algr is mthmtis with mor thn just numrs. Numrs hv ix vlu, ut lgr introus vrils whos vlus n hng. Ths

### Aquauno Video 6 Plus Page 1

Connt th timr to th tp. Aquuno Vio 6 Plus Pg 1 Usr mnul 3 lik! For Aquuno Vio 6 (p/n): 8456 For Aquuno Vio 6 Plus (p/n): 8413 Opn th timr unit y prssing th two uttons on th sis, n fit 9V lklin ttry. Whn

### # 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0

### Scientific Programming. Graphs

Sintifi Progrmming Grphs Alrto Montrsor Univrsità i Trnto 08//07 This work is lins unr Crtiv Commons Attriution-ShrAlik 4.0 Intrntionl Lins. Tl of ontnts Introution Exmpls Dfinitions Spifition Rprsnttions

### XML and Databases. Outline. Recall: Top-Down Evaluation of Simple Paths. Recall: Top-Down Evaluation of Simple Paths. Sebastian Maneth NICTA and UNSW

Smll Pth Quiz ML n Dtss Cn you giv n xprssion tht rturns th lst / irst ourrn o h istint pri lmnt? Ltur 8 Strming Evlution: how muh mmory o you n? Sstin Mnth NICTA n UNSW

### Seven-Segment Display Driver

7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on

### INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

### Graph Contraction and Connectivity

Chptr 17 Grph Contrtion n Conntivity So r w hv mostly ovr thniqus or solving prolms on grphs tht wr vlop in th ontxt o squntil lgorithms. Som o thm r sy to prllliz whil othrs r not. For xmpl, w sw tht

### CS553 Lecture Register Allocation I 3

Low-Lvl Issus Last ltur Intrproural analysis Toay Start low-lvl issus Rgistr alloation Latr Mor rgistr alloation Instrution shuling CS553 Ltur Rgistr Alloation I 2 Rgistr Alloation Prolm Assign an unoun

### WORKSHOP 6 BRIDGE TRUSS

WORKSHOP 6 BRIDGE TRUSS WS6-2 Workshop Ojtivs Lrn to msh lin gomtry to gnrt CBAR lmnts Bom fmilir with stting up th CBAR orinttion vtor n stion proprtis Lrn to st up multipl lo ss Lrn to viw th iffrnt

### Section 3: Antiderivatives of Formulas

Chptr Th Intgrl Appli Clculus 96 Sction : Antirivtivs of Formuls Now w cn put th is of rs n ntirivtivs togthr to gt wy of vluting finit intgrls tht is ct n oftn sy. To vlut finit intgrl f(t) t, w cn fin

### (a) v 1. v a. v i. v s. (b)

Outlin RETIMING Struturl optimiztion mthods. Gionni D Mihli Stnford Unirsity Rtiming. { Modling. { Rtiming for minimum dly. { Rtiming for minimum r. Synhronous Logi Ntwork Synhronous Logi Ntwork Synhronous

### Solutions to Homework 5

Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()

### Analysis of Algorithms - Elementary graphs algorithms -

Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsarch Cntr) anras.rmahl@mh.s Autumn 004 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring

### Graph Theory. Vertices. Vertices are also known as nodes, points and (in social networks) as actors, agents or players.

Stphn P. Borgtti Grph Thory A lthough grph thory is on o th youngr rnhs o mthmtis, it is unmntl to numr o ppli ils, inluing oprtions rsrh, omputr sin, n soil ntwork nlysis. In this hptr w isuss th si onpts

### Plan. I Gale-Shapley Running Time. I Graphs. I Motivation and definitions I Graph traversal: BFS and DFS

Pln CS : Grphs: BFS n DFS Dn Shlon Gl-Shply Running Tim Grphs Motivtion n finitions Grph trvrsl: BFS n DFS Frury, 0 Running Tim of Gl-Shply? nitilly ll ollgs n stunts r fr whil som ollg is fr n hsn t m

### A Simple Code Generator. Code generation Algorithm. Register and Address Descriptors. Example 3/31/2008. Code Generation

A Simpl Co Gnrtor Co Gnrtion Chptr 8 II Gnrt o for singl si lok How to us rgistrs? In most mhin rhitturs, som or ll of th oprnsmust in rgistrs Rgistrs mk goo tmporris Hol vlus tht r omput in on si lok

### Present state Next state Q + M N

Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I

### Formal Concept Analysis

Forml Conpt Anlysis Conpt intnts s losd sts Closur Systms nd Implitions 4 Closur Systms 0.06.005 Nxt-Closur ws dvlopd y B. Gntr (984). Lt M = {,..., n}. A M is ltilly smllr thn B M, if B A if th smllst

### 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

### Trees as operads. Lecture A formalism of trees

Ltur 2 rs s oprs In this ltur, w introu onvnint tgoris o trs tht will us or th inition o nroil sts. hs tgoris r gnrliztions o th simpliil tgory us to in simpliil sts. First w onsir th s o plnr trs n thn

### Analysis for Balloon Modeling Structure based on Graph Theory

Anlysis for lloon Moling Strutur bs on Grph Thory Abstrt Mshiro Ur* Msshi Ym** Mmoru no** Shiny Miyzki** Tkmi Ysu* *Grut Shool of Informtion Sin, Ngoy Univrsity **Shool of Informtion Sin n Thnology, hukyo

### Register Allocation. How to assign variables to finitely many registers? What to do when it can t be done? How to do so efficiently?

Rgistr Allotion Rgistr Allotion How to ssign vrils to initly mny rgistrs? Wht to o whn it n t on? How to o so iintly? Mony, Jun 3, 13 Mmory Wll Disprity twn CPU sp n mmory ss sp improvmnt Mony, Jun 3,

### Analysis of Algorithms - Elementary graphs algorithms -

Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsach Cntr) anras.rmahl@mh.s Autumn 00 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring