A 4-state solution to the Firing Squad Synchronization Problem based on hybrid rule 60 and 102 cellular automata
|
|
- Lorena Waters
- 5 years ago
- Views:
Transcription
1 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, Chnghun, , Chin; 2. Collg o sin, ChngChun Univrsity o Sin n Thnology, ChngChun, ,Chin) 1 Astrt: In this ppr, w prsnt 4-stt solution to th Firing Squ Synhroniztion Prolm (FSSP) s on hyri rul 60/102 Cllulr Automt(CA),This solution solvs th prolm on th lin o lngth 2 n with two gnrls. Prvious work on FSSP or 4-stt systms ous mostly on linr llulr utomt, whr synhronizs n ininit numr o lins ut not ll possil lins. W giv tim-optiml solutions to synhroniz n ininit numr o lins y rul 60 n rul 102 rsptivly, n onstrut hyri rul 60 n 102 stts trnsition tl. Compr to th known solutions o llulr utomt, th hyri CA wy is simplr n str, th miniml tim is (n-1) stp. Kywors: Prlll lgorithms, llulr utomt, Firing Squ Synhroniztion, Rul 60, Rul Introution Th Firing Squ Synhroniztion Prolm (FSSP), is on o th st stui prolms or llulr utomt. This prolm in whih ntwork o intil lls(init utomt) work synhronously t isrt tim stps. Figur 1 shows init on-imnsionl llulr rry onsisting o n lls, All lls (solirs) xpt th lt n ll (gnrl), r initilly in th quisnt stt t tim t=0. Th nxt stt is trmin y oth its own stt n th prsnt stts o its right n lt nighors. At som utur tim, ll o th lls will simultnously n, or th irst tim, ntr spil iring stt. * Corrsponing uthor. E-mil: lsl@nnu.u.n, Tl.:
2 Fig. 1. A on-imnsionl llulr utomton. Th FSSP prolm hs n stui or long tim ([1]). Minsky n MCrthy ([2] )onstrut solution synhronizing n lls in 3n stps using ivi-n-onqur mtho. Atr tht Wksmn ([3]) n Blzr ([4]) sign nw lgorithm whih n synhroniz in miniml tim in T (n) =(2n stps, n hv stt omplxity o 16 n 8 stts rsptivly. Mzoyr ([5]) show 6-stt miniml-tim solution,ut still unknown whthr iv-stt ull solution xists,yunès ([6, 7]), Sttl n Simon ( [8]) n Umo ( [9]) sign non miniml-tim solutions with w stts. Umo ([10,11]) n in lss thn 6-stt solutions to synhroniz n ininit numr. Whn it oms to our-stt spil solution, som popl try to in solutions to synhroniz n ininit numr ut not ll lins. Blzr prov tht th miniml tim 4-stt solution os not xist, n tht Blzr s rsult shows tht thr is no 4-stt solution whih synhronizs vry lin in miniml-tim. Now th 4-stt solutions r ll uilt using som lmntry lgr, n mostly s on linr llulr utomt with rul 60 n 150, Jn-Bptist ([12]) prsnt on o ths solutions in only 4-stt whih synhronizs vry lin whos lngth is powr o 2. In this ppr, w qust 4-stt solution to lin o 2 n FSSP with two gnrls s on hyri rul 60/102 Cllulr Automt. 2. Prliminry For th our-stt FSSP solution, Yunès sign n lgorithm y Wolrm Rul 60 or 2 n ll rry to mt n ininit numr o lins. Our onstrution is lso s on Wolrm s linr llulr utomt using rul 60 n rul 102.Hr r som si FSSP initions n lmntry onpts. 2.1 Dinitions Thr is llulr utomt A(Q,δ) whr Q is init st, ll th stts st o A, nδis trnsition untion rom Q 3 ->Q. ) A llulr utomt S is oupl A whih is n pplition rom Z in Q. A onigurtion C volvs to nothr onigurtion C* so tht C*(Z)=δ(C(z-1),C(z),C(z+1)) W n in C*=Δ(C)s glol trnsition untion. So th initil onigurtion o llulr utomt is C0 (t th tim o t=0), th onigurtion o tim t is C t =Δ t (C). ) At lst our istinguish stts long to Q. ) Stt Q is th quisnt stt. It stisisδ(q,q,q)=q,δ(q,q,!)=q, δ(!,q,q)=q ) Stt * is th ounry stt. It stisis: q q Q, (q,*,q ) * 1, ) Stt G is th gnrl stt n th stt F is th Fir stt, suh tht, strting rom th
3 initil onigurtion in y: ( ) z 0, C[ n]( z)! () z n 1, C[ n]( z)! ()C[n](1) G () z {2,3...,n},C[n](z) Q ) Th volution o th onigurtion C[n] is suh tht, or synhroniztion tim t(n): () z N, t {1,...,t(n) 1},C[n] (z) F () z {1,...,n},C[n] t(n) (z) F 2.2 Som FSSP lmntry onpts ) Synhroniztion Tim[3] [4] [9] [10] [13]. Th solution to th iring squ synhroniztion prolm n shown: Synhroniztion o n lls in lss thn 2n-2 stps is impossil, n Synhroniztion o n lls in xtly 2n-2stps is possil. So th miniml synhroniztion tim is 2n-2 stps. ) Th numr o stts[4] [9] [10]. To sign trnsition tls, thr is t lst thr stts: th stt o quisnt, th stt o gnrl n th stt o iring. Bsis, w lso sign ounry stt. It is shown tht thr xists no thr-stt solution n no our-stt symmtri solution on rings. Blzr prov tht thr is no our-stt ull solution or this prolm. A miniml tim solution o six stts ws introu y Jqus Mzoyr in 1987[13]. ) Th numr o trnsition ruls[9] [10]. Any k-stt trnsition tl or th synhroniztion hs t most (k-1)k 2 ntris in (k-1) mtris o siz k*k. Th numr o trnsition ruls rlts th omplxity o synhroniztion lgorithms. t 3. Yunès s Rul 60 lgorithm Yunès[12] sign our stts solution to hiv this synhroniztion, th stt gnrl (g) is , th quisnt stt is, th stt x 1 is, n th stt o iring is, th symol $ rprsnts th ounry stts. Th rul 60 volution igur is ig 2:
4 Fig. 2. Th Rul 60 Cllulr Auotmt I w ol th psl s tringl moulo2, w n gt ollow igur 3: Fig. 3. Th Rul 60 s solution. Th gry olor stts rprsnt th stt o gnrl, th whit olor stts rprsnt th stt o quisnt, th lk olor stts rprsnt th stt o 1 x, n th rk r olor stts rprsnt th stt o iring. Mnwhil, Yunès rwrit th trnsition untion into lgri orm: (mo $ (mo $ (mo (mo
5 4. Rul 102 lgorithm Bs on th work o Yunès, w sign our-stt solution y rul 102, in this wy th gnrl is lot in th right n o ll rry. Th rul 102 volution pross n sn in ig. 4: Fig.4. th Rul 102 Cllulr Automt n=16 I w ol th psl s tringl o rul 102, w will gt ig. 5: Fig.5. th Rul 102 s solution n=8 Similrly, w n lso rwrit th trnsition untion into this lgri orm: (mo $ (mo $ (mo (mo
6 An th rul 102 trnsition tl n got s Tl 1.: Tl 1. Th Rul 102 trnsition tl 5. Th hyri solution o Rul 60 n Rul 102 Bs on th work o ormr txt, w omin th Rul 60 n 102 to onstrut solution or 2 gnrls tht r t lt n right o th lin o 2 n FSSP rsptivly, w lso hv our stts: th stt o gnrl (th gry olor) ; th stt o quisnt (th whit olor), th stt o x 1 (th lk olor stts), th stt o iring(th rk r olor), Hr, w giv snpshot or synhroniztion oprtion or this lgorithm on ight lls: Fig.6. th Rul 102 n Rul 60 s solution n=8 W n rw th onlusion sly tht this lgorithm n short th synhroniztion tim or 2-gnrl, w in th stt o gnrl 1 is G, th stt o gnrl 2 is H, th stt o quisnt is Q, th stt o iring is F, th ounry stt is *, th trnsition tls r s ollows:
7 Tl 2. Th Rul 102 n Rul 60 trnsition tls In trms o th trnsition tls w n gt th stts hng tls s ollows: Tl 2. Th Rul 102 n Rul 60 stts trnsition tls 6. Conlusion An xistn or non-xistn o our-stt iring squ synhroniztion protool hs n long-stning, mous opn prolm or long tim. In this ppr, w hv prsnt 4-stt solution to th iring squ whih synhronizs th lin o lngth 2 n in (n-1) stp. Bs on th rsult o th Rul 60 n rul 102 to FSSP, w omin with th Rul 60 n Rul 102 to giv solution or 2 gnrls tht r t lt n right o th lin o 2 n FSSP rsptivly, in th n w lso gt stts hng tls. W think this is vry promising pproh or th srh o solutions sign or non miniml-tim solutions with w stts. Rrns [1] Y. Nishitni n N. Hon, Th Firing Squ Synhroniztion Prolm or Grphs, Thortil Computr Sin 14 (1981), [2]Minsky, M.: Computtion: Finit n Ininit Mhins. Prnti-Hll, Englwoo Clis (1967). [3] A. Wksmn, An optimum solution to th iring squ synhroniztion prolm, Inormtion n Control 9 (1966)
8 [4]Blzr, R.: An 8-Stt Miniml Tim Solution to th Firing Squ Synhroniztion Prolm. Inormtion n Control 10 (1967) [5]Mzoyr, J.: A Six-Stt Miniml-Tim Solution to th Firing Squ Synhroniztion Prolm. Thortil Computr Sin 50(1987) [6]Yun`s, J.-B.: Svn Stts Solutions to th Firing Squ Synhroniztion Prolm. Thortil Computr Sin 127( (1994) [7]Yun`s, J.-B.: An Intrinsilly non Miniml-Tim Minsky-lik 6-Stts Solution to th Firing Squ Synhroniztion Prolm. RAIRO ITA/TIA 42(1) (2008) [8] A. Sttl, J. Simon, Non-miniml tim solutions or th iring squ synhroniztion prolm, Thnil Rport 97-08, Univrsity o Chigo, [9] H. Umo, M. M, K. Hongyo, A sign o symmtril six stt 3n-stps iring squ synhroniztion lgorithms n thir implmnttions, in: Proings o ACRI 2006, in: Ltur Nots in Computr Sin, 4173(2006) [10] H. Umo, T. Yngihr, A smllst iv-stt solution to th iring squ synhroniztion prolm, in: J. Durn-Los,M.Mrgnstrn (Es.), Proings, 5th Intrntionl Conrn on Mhins, Computtions n Univrslity, MCU 2007 Orléns, Frn, in: Ltur Nots in Computr Sin, 4664(2007) [11] H. Umo, N. Kmikw, A 4-stts solution to th iring squ s on Wolrm s rul 150, Privt ommunition. [12]Jn-Bptist,Yunès, A 4-stts lgri solution to linr llulr utomt Synhroniztion, Inormtion Prossing Lttrs 107 (2008) [13] J. Mzoyr, A six-stt miniml-tim solution to th iring squ synhroniztion prolm, Thortil Computr Sin 50(1987)
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)
More informationOutline. 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
More information(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,
More informationCycles 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
More information1 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 informationAlgorithmic 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
More informationAn 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
More informationProperties 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
More informationGraph 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
More informationCSC 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, =
More informationConstructive 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
More informationCOMPLEXITY 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
More informationCS 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
More informationPaths. 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. {, } {, } {, } {, } {, } {,
More informationWhy 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
More informationMath 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
More informationProblem 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
More informationb. 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
More informationCSE 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
More informationThe 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
More informationFSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *
CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if
More informationSimilarity 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
More informationPlanar 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
More informationOutline. 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
More informationCS 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
More informationQUESTIONS 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
More information12. 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}
More informationGraphs. 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
More information, 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
More informationGarnir 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,
More informationCSE 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
More informationV={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
More informationECE 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
More informationV={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
More informationUsing 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
More informationGraphs. 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
More informationPresent 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
More informationDEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM
Fr Est Journl o Mthtil Sins (FJMS) Volu 6 Nur Pgs 8- Pulish Onlin: Sptr This ppr is vill onlin t http://pphjo/journls/jsht Pushp Pulishing Hous DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF MATRICES
More information# 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
More information0.1. 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
More informationCS200: 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
More informationCS61B 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:
More informationCSE 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
More informationDesigning 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
More informationModule 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
More informationSteinberg 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
More information12/3/12. 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
More information5/9/13. 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
More informationS 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
More informationXML 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
More informationCSE303 - 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
More informationCOMP108 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
More informationSection 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
More informationCS 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
More informationExam 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
More informationLast time: introduced our first computational model the DFA.
Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody
More informationAnalysis 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
More informationQUESTIONS 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
More informationGREEDY 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
More informationSection 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
More informationModule 2 Motion Instructions
Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr
More informationNumbering 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
More informationSeven-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
More informationa 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
More informationECE 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
More informationSolutions 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 ):
More informationAnnouncements. 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
More informationNefertiti. Echoes of. Regal components evoke visions of the past MULTIPLE STITCHES. designed by Helena Tang-Lim
MULTIPLE STITCHES Nrtiti Ehos o Rgl omponnts vok visions o th pst sign y Hln Tng-Lim Us vrity o stiths to rt this rgl yt wrl sign. Prt sping llows squr s to mk roun omponnts tht rp utiully. FCT-SC-030617-07
More informationEE1000 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
More informationRegister 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,
More informationOutline. 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
More informationConstruction 11: Book I, Proposition 42
Th Visul Construtions of Euli Constrution #11 73 Constrution 11: Book I, Proposition 42 To onstrut, in givn rtilinl ngl, prlllogrm qul to givn tringl. Not: Equl hr mns qul in r. 74 Constrution # 11 Th
More informationMore 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 ({,,,,}, {{,}, {,},
More informationIntegration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals
Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion
More informationMultipoint 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
More informationDUET 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
More informationA Low Noise and Reliable CMOS I/O Buffer for Mixed Low Voltage Applications
Proings of th 6th WSEAS Intrntionl Confrn on Miroltronis, Nnoltronis, Optoltronis, Istnul, Turky, My 27-29, 27 32 A Low Nois n Rlil CMOS I/O Buffr for Mix Low Voltg Applitions HWANG-CHERNG CHOW n YOU-GANG
More informationFormal 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
More informationTrees 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
More informationNew challenges on Independent Gate FinFET Transistor Network Generation
Nw hllngs on Inpnnt Gt FinFET Trnsistor Ntwork Gnrtion Viniius N. Possni, Anré I. Ris, Rnto P. Ris, Flip S. Mrqus, Lomr S. Ros Junior Thnology Dvlopmnt Cntr, Frl Univrsity o Plots, Plots, Brzil Institut
More informationSOLVED EXAMPLES. be the foci of an ellipse with eccentricity e. For any point P on the ellipse, prove that. tan
LOCUS 58 SOLVED EXAMPLES Empl Lt F n F th foci of n llips with ccntricit. For n point P on th llips, prov tht tn PF F tn PF F Assum th llips to, n lt P th point (, sin ). P(, sin ) F F F = (-, 0) F = (,
More informationSolutions 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. ()
More informationResearch Article On the Genus of the Zero-Divisor Graph of Z n
Intrntionl Journl o Comintoris, Artil ID 7, pgs http://x.oi.org/.1/14/7 Rsrh Artil On th Gnus o th Zro-Divisor Grph o Z n Huong Su 1 n Piling Li 2 1 Shool o Mthmtil Sins, Gungxi Thrs Eution Univrsity,
More informationMAT3707. 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
More informationAn Efficient FPGA Implementation of the Advanced Encryption Standard Algorithm G. Mohan 1 K. Rambabu 2
IJSRD - Intrntionl Journl or Sintii Rsrh & Dvlopmnt Vol., Issu, ISSN (onlin): - An Eiint FPGA Implmnttion o th Avn Enryption Stnr Algorithm G. Mohn K. Rmu M.Th Assistnt Prossor Dprtmnt o Eltronis n ommunition
More information(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
More informationTURFGRASS DISEASE RESEARCH REPORT J. M. Vargas, Jr. and R. Detweiler Department of Botany and Plant Pathology Michigan State University
I TURFGRASS DISEASE RESEARCH REPORT 9 J. M. Vrgs, Jr. n R. Dtwilr Dprtmnt f Btny n Plnt Pthlgy Mihign Stt Univrsity. Snw Ml Th 9 snw ml fungii vlutin trils wr nut t th Byn Highln Rsrt, Hrr Springs, Mihign
More informationlearning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms
rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r
More informationWeighted 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 ) >.
More informationInstructions 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
More informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationLayout Decomposition for Triple Patterning Lithography
Lyout Domposition or Tripl Pttrning Lithogrphy Bi Yu, Kun Yun, Boyng Zhng, Duo Ding, Dvi Z. Pn ECE Dpt. Univrsity o Txs t Austin, Austin, TX USA 7871 Cn Dsign Systms, In., Sn Jos, CA USA 9514 Emil: {i,
More informationPolygons POLYGONS.
Polgons PLYGNS www.mthltis.o.uk ow os it work? Solutions Polgons Pg qustions Polgons Polgon Not polgon Polgon Not polgon Polgon Not polgon Polgon Not polgon f g h Polgon Not polgon Polgon Not polgon Polgon
More informationarxiv: v1 [cs.ar] 11 Feb 2014
Lyout Domposition or Tripl Pttrning Lithogrphy Bi Yu, Kun Yun, Boyng Zhng, Duo Ding, Dvi Z. Pn ECE Dpt. Univrsity o Txs t Austin, Austin, TX USA 7871 Cn Dsign Systms, In., Sn Jos, CA USA 9514 Emil: {i,
More informationAnnouncements. 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
More informationGraph 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
More informationNP-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
More informationDecimals DECIMALS.
Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths
More informationLimits Indeterminate Forms and L Hospital s Rule
Limits Indtrmint Forms nd L Hospitl s Rul I Indtrmint Form o th Tp W hv prviousl studid its with th indtrmint orm s shown in th ollowin mpls: Empl : Empl : tn [Not: W us th ivn it ] Empl : 8 h 8 [Not:
More informationWalk 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
More information