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

Size: px
Start display at page:

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

Transcription

1 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 Commnts n Rrns Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 / 0 Introution Computtion o Min-Cost Spnnin Trs Min-Cost Spnnin Trs Costs o Spnnin Trs in Wiht Grphs Motivtion: Givn st o sits (rprsnt y vrtis o rph), onnt ths ll s hply s possil (usin onntions rprsnt y th s o wiht rph). Gols or Toy: prsnttion o th initions n to ormlly in prolm motivt y th ov prsnttion o n lorithm (Prim s ) or solvin th prolm Rll tht i G = (V, E) is onnt, unirt rph, thn spnnin tr o G is surph Ĝ = ( V, Ê) suh tht V = V (so Ĝ inlus ll th vrtis in G) Ĝ is tr Suppos now tht G = (V, E) is onnt wiht rph with wiht untion w : E N, n tht G = (V, E ) is spnnin tr o G Th ost o G, w(g ), is th sum o th wihts o th s in G, tht is, w(g ) = E w(). Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 3 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 4 / 0

2 Min-Cost Spnnin Trs Min-Cost Spnnin Trs Suppos G is wiht rph with wihts s shown low. Th ost o th ollowin spnnin tr, G = (V, E ), is Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 5 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 6 / 0 Min-Cost Spnnin Trs Min-Cost Spnnin Trs Minimum-Cost Spnnin Trs Th ost o th ollowin spnnin tr, G = (V, E ), is 6. Suppos (G, w) is wiht rph. 5 A surph G o G is minimum-ost spnnin tr o (G, w) i th ollowin proprtis r stisi. G is spnnin tr o G. 3 w(g ) w(g ) or vry spnnin tr G o G. : In th prvious xmpl, G is lrly not minimum-ost spnnin tr, us G is spnnin tr o G suh tht w(g ) > w(g ). It n shown tht G is minimum-ost spnnin tr o (G, w). Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 7 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 8 / 0

3 Gnrl Constrution Builin Minimum-Cost Spnnin Tr Gnrl Constrution Builin Minimum-Cost Spnnin Tr To onstrut minimum-ost spnnin tr o G = (V, E): Strt with Ĝ = ( V, Ê), whr V V n Ê =. Not: Ĝ is surph o som minimum-ost spnnin tr o (G, w). Rptly vrtis (i nssry) n s nsurin tht Ĝ is still surph o minimum-ost spnnin tr s you o so. Continu oin this until V = V n Ê = V (so tht Ĝ is spnnin tr o Ĝ). Aitionl Nots: This n on in svrl irnt wys, n thr r t lst two irnt lorithms tht us this pproh to solv this prolm. Th lorithm to prsnt hr ins with V = {s} or som vrtx s V, n mks sur tht Ĝ is lwys tr. As rsult, this lorithm is struturlly vry similr to Dijkstr s to omput minimum-ost pths (whih w hv lry isuss in lss). Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 9 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 0 / 0 Spiition o Rquirmnts Dt Struturs Pr-Conition G = (V, E) is onnt rph with wiht untion w Post-Conition: π is untion π : V V {NIL} I Ê = {(π(v), v) v V n π(v) NIL} thn (V, Ê) is minimum-ost spnnin tr or G Th rph G = (V, E) n its wiht untion hv not n hn Th lorithm (to prsnt nxt) will us priority quu to stor inormtion out wihts o s tht r in onsir or inlusion Th priority quu will MinHp: th ntry with th smllst priority will t th top o th hp Eh no in th priority quu will stor vrtx in G n th wiht o n inint to this vrtx Th wiht will us s th vrtx s priority An rry-s rprsnttion o th priority quu will us A son rry will us to lot h ntry o th priority quu or ivn no in onstnt tim Not: Th t struturs will, thror, look vry muh lik th t struturs us y Dijkstr s lorithm. Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 / 0

4 Psuoo Psuoo, Continu MST-Prim(G, w, s) or v V o olour[v] = whit [v] = + π[v] = NIL n or Initiliz n mpty priority quu Q olour[s] = ry [s] = 0 s with priority 0 to Q whil (Q is not mpty) o (u, ) = xtrt-min(q) {Not: = [u]} or h v Aj[u] o i (olour[v] == whit) thn [v] = w((u, v)) olour[v] = ry; π[v] = u v with priority [v] to Q ls i (olour[v] == ry) thn Upt inormtion out v (Shown on nxt sli) n i n or olour[u] = lk n whil rturn π Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 3 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 4 / 0 Psuoo, Conlu Uptin Inormtion Aout v i (w((u, v)) < [v]) thn ol = [v] [v] = w((u, v)) π[v] = u Us Drs-Priority to rpl (v, ol) in Q with (v, [v]) n i Stp 7: Extrt-Min (rturns (, )) olor lk on! E. on MST (totl ost is 8): 0 π - Q: (mpty) {(π(), ), (π(), ), (π(), ), (π(), ), (π( ), ), (π(), )} = {(, ), (, ), (, ), (, ), (, ), (, )} Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 5 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 6 / 0

5 Trmintion n Eiiny Trmintion n Eiiny Trmintion n Eiiny Trmintion n Eiiny (ont.) Clim: I MST-Prim is xut on wiht unirt rph G = (V, E) thn th lorithm trmints tr prormin O(( V + E ) lo V ) stps in th worst s. Proo. This is virtully intil to th proo o th orrsponin rsult or Dijkstr s lorithm (to omput minimum-ost pths). Th numr o oprtions on th priority quu, n th numr o oprtions tht o not involv this t strutur, r h in Θ( V + E ) in th worst s (y th rumnt tht hs n ppli to th lst thr lorithms onsir). Proo (ontinu). Sin th siz o th priority quu nvr xs V n sin th only oprtions on th priority quu us r insrtions, rss o ky vlus, n xtrtions o th minimum (top priority) lmnt, th ost o h oprtion on th t strutur is in O(lo V ). It ollows immitly tht th totl numr o stps is in O(( V + E ) lo V ), s lim. O( V lo V + E ) usin Fioni hp (mortiz) Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 7 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 8 / 0 Aitionl Commnts n Rrns Aitionl Commnts Rrns Aitionl Commnts n Rrns On Gry s Prim s lorithm is n xmpl o ry lorithm: A lol optimiztion prolm (inin minimum-ost spnnin tr) is solv y mkin squn o lol ry hois (y xtnin tr with s whos wihts r s smll s possil). Provin orrtnss o ry lorithms is otn hllnin. In, ry huristis r otn inorrt. On th othr hn, whn thy r orrt, ry lorithms r rquntly simplr n mor iint thn othr lorithms or th sm omputtion. S CPSC 43 or mor out ry lorithms! Furthr Rin n Jv Co: Introution to s, Chptr 3 Chptr 3 inlus Prim s lorithm lon with nothr ry lorithm or this prolm (Kruskl s lorithm). Dt Struturs & s in Jv, Chptr 4 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 9 / 0 Mik Joson (Univrsity o Clry) Computr Sin 33 Ltur #34 0 / 0

### 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)

### 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

### 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

### 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, =

### 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

### 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

### 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

### 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

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

CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 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

### 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

### 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

### 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

### GREEDY TECHNIQUE. Greedy method vs. Dynamic programming method:

Dinition: GREEDY TECHNIQUE Gry thniqu is gnrl lgorithm sign strtgy, uilt on ollowing lmnts: onigurtions: irnt hois, vlus to in ojtiv untion: som onigurtions to ithr mximiz or minimiz Th mtho: Applil to

### , 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

### 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

### 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

### 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

### 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

### 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:

### 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

### 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

### 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

### 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

### 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

### 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. {, } {, } {, } {, } {, } {,

### 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:

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am

16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)

### 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 ) >.

### 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

### Outline. Binary Tree

Outlin Similrity Srh Th Binry Brnh Distn Nikolus Austn nikolus.ustn@s..t Dpt. o Computr Sins Univrsity o Slzur http://rsrh.uni-slzur.t 1 Binry Brnh Distn Binry Rprsnttion o Tr Binry Brnhs Lowr Boun or

### 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

### 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

### 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

### 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

### 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

### 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

### A 4-state solution to the Firing Squad Synchronization Problem based on hybrid rule 60 and 102 cellular automata

A 4-stt solution to th Firing Squ Synhroniztion Prolm s on hyri rul 60 n 102 llulr utomt LI Ning 1, LIANG Shi-li 1*, CUI Shung 1, XU Mi-ling 1, ZHANG Ling 2 (1. Dprtmnt o Physis, Northst Norml Univrsity,

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### Computational Biology, Phylogenetic Trees. Consensus methods

Computtionl Biology, Phylognti Trs Consnsus mthos Asgr Bruun & Bo Simonsn Th 16th of Jnury 2008 Dprtmnt of Computr Sin Th univrsity of Copnhgn 0 Motivtion Givn olltion of Trs Τ = { T 0,..., T n } W wnt

### # 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

### 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

### Applications: The problem has several applications, for example, to compute periods of maximum net expenses for a design department.

A Gntl Introution to Aloritms: Prt III Contnts o Prt I: 1. Mr: (to mr two sort lists into sinl sort list.). Bul Sort 3. Mr Sort: 4. T Bi-O, Bi-Θ, Bi-Ω nottions: symptoti ouns Contnts o Prt II: 5. Bsis

### Minimum Spanning Trees

Mnmum Spnnng Trs Spnnng Tr A tr (.., connctd, cyclc grph) whch contns ll th vrtcs of th grph Mnmum Spnnng Tr Spnnng tr wth th mnmum sum of wghts 1 1 Spnnng forst If grph s not connctd, thn thr s spnnng

### Steinberg s Conjecture is false

Stinrg s Conjtur is als arxiv:1604.05108v2 [math.co] 19 Apr 2016 Vinnt Cohn-Aa Mihal Hig Danil Král Zhntao Li Estan Salgao Astrat Stinrg onjtur in 1976 that vry planar graph with no yls o lngth our or

### 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. ()

### Organization. Dominators. Control-flow graphs 8/30/2010. Dominators, control-dependence. Dominator relation of CFGs

Orniztion Domintors, ontrol-pnn n SSA orm Domintor rltion o CFGs postomintor rltion Domintor tr Computin omintor rltion n tr Dtlow lorithm Lnur n Trjn lorithm Control-pnn rltion SSA orm Control-low rphs

### Multipoint Alternate Marking method for passive and hybrid performance monitoring

Multipoint Altrnt Mrkin mtho or pssiv n hyri prormn monitorin rt-iool-ippm-multipoint-lt-mrk-00 Pru, Jul 2017, IETF 99 Giuspp Fiool (Tlom Itli) Muro Coilio (Tlom Itli) Amo Spio (Politnio i Torino) Riro

### RAM Model. I/O Model. Real Machine Example: Nehalem : Algorithms in the Real World 4/9/13

4//3 RAM Mol 5-853: Algorithms in th Rl Worl Lolity I: Ch-wr lgorithms Introution Sorting List rnking B-trs Bur trs Stnr thortil mol or nlyzing lgorithms: Ininit mmory siz Uniorm ss ost Evlut n lgorithm

### 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

### Designing A Concrete Arch Bridge

This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr

### 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

### 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

### 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

### 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

### (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,

### 14 Shortest Paths (November 8)

CS G Ltur : Shortt Pth Fll 5 Shortt Pth (Novmr ). Introution Givn wight irt grph G = (V, E, w) with two pil vrti, our n trgt t, w wnt to in th hortt irt pth rom to t. In othr wor, w wnt to in th pth p

### 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

### 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

### 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

### 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 =

### Graph Algorithms and Combinatorial Optimization Presenters: Benjamin Ferrell and K. Alex Mills May 7th, 2014

Grp Aloritms n Comintoril Optimiztion Dr. R. Cnrskrn Prsntrs: Bnjmin Frrll n K. Alx Mills My 7t, 0 Mtroi Intrstion In ts ltur nots, w mk us o som unonvntionl nottion or st union n irn to kp tins lnr. In

### 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}

### Yehuda Lindell Bar-Ilan University

Wintr Shool on Sur Computtion n iiny Br-Iln Unirsity, Isrl 3//2-/2/2 Br Iln Unirsity Dpt. o Computr Sin Yhu Linll Br-Iln Unirsity Br Iln Unirsity Dpt. o Computr Sin Protool or nrl sur to-prty omputtion

### 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

### 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 ):

### Using the Printable Sticker Function. Using the Edit Screen. Computer. Tablet. ScanNCutCanvas

SnNCutCnvs Using th Printl Stikr Funtion On-o--kin stikrs n sily rt y using your inkjt printr n th Dirt Cut untion o th SnNCut mhin. For inormtion on si oprtions o th SnNCutCnvs, rr to th Hlp. To viw th

### 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

### (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

### 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,

### Properties of Hexagonal Tile local and XYZ-local Series

1 Proprtis o Hxgonl Til lol n XYZ-lol Sris Jy Arhm 1, Anith P. 2, Drsnmik K. S. 3 1 Dprtmnt o Bsi Sin n Humnitis, Rjgiri Shool o Enginring n, Thnology, Kkkn, Ernkulm, Krl, Ini. jyjos1977@gmil.om 2 Dprtmnt

### Register Allocation. Register Allocation. Principle Phases. Principle Phases. Example: Build. Spills 11/14/2012

Rgistr Allotion W now r l to o rgistr llotion on our intrfrn grph. W wnt to l with two typs of onstrints: 1. Two vlus r liv t ovrlpping points (intrfrn grph) 2. A vlu must or must not in prtiulr rhitturl

### 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

### DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Leslie Rogalski

Dut with Dimons Brlt DUET WITH DIAMONDS COLOR SHIFTING BRACELET By Lsli Roglski Photo y Anrw Wirth Supruo DUETS TM from BSmith rt olor shifting fft tht mks your work tk on lif of its own s you mov! This