A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata

Transcription

1 A Simple Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata HIROSHI UMEO Uiversity of Osaka Electro-Commuicatio KINUO NISHIDE Uiversity of Osaka Electro-Commuicatio KEISUKE KUBO Uiversity of Osaka Electro-Commuicatio The firig squad sychroizatio problem (FSSP) o cellular automata has bee studied extesively for more tha forty years, ad a rich variety of sychroizatio algorithms have bee proposed for ot oly oe-dimesioal arrays but two-dimesioal arrays. I the preset paper, we propose a simple recursive-halvig based optimum-time sychroizatio algorithm that ca sychroize ay rectagle arrays of size m with a geeral at oe corer i m + + max(m,) 3 steps. The algorithm is a atural expasio of the well-kow FSSP algorithms proposed by Balzer [1967], Gerke [1987], ad Waksma [1966] ad it ca be easily expaded to three-dimesioal arrays, eve to multi-dimesioal arrays with a geeral at ay positio of the array. The algorithm proposed is isotropic cocerig the side-legths of multi-dimesioal arrays ad its algorithmic correctess is trasparet ad easily verified. 1 Itroductio We study a sychroizatio problem that gives a fiite-state protocol for sychroizig large-scale cellular automata. The sychroizatio i cellular automata has bee kow as a firig squad sychroizatio problem (FSSP) sice its developmet, i which it was origially proposed by J. Myhill i Moore [1964] to sychroize all parts of self-reproducig cellular automata. The problem has bee studied extesively for more tha forty years [1-23], ad a rich variety of sychroizatio algorithms have bee proposed for ot oly oe-dimesioal (1D) arrays but two-dimesioal (2D) arrays. The 1D FSSP is described as follows: give a oe-dimesioal array of idetical cellular automata, icludig a geeral at oe ed that is activated at time t = 0, we wat to desig the automata such that, at some future time, all the cells will simultaeously ad, for the first time, eter a special firig state. Some questios may arise: How ca we sychroize multi-dimesioal arrays? Ca we expad those 2D FSSP algorithms proposed so far to 3D arrays, or more geerally to multi-dimesioal arrays? Ca we geeralize those 2D FSSP algorithms to geeralized oes, i which a iitial geeral is located at ay positio of the array? What is a lower boud of time steps eeded for sychroizig multi-dimesioal arrays with a geeral at oe corer? What is a lower boud for the geeralized case? E. Formeti (Ed.): AUTOMATA ad JAC 2012 cofereces EPTCS 90, 2012, pp , doi: /eptcs c H. Umeo, K. Nishide & K. Kubo This work is licesed uder the Creative Commos Attributio Licese.

2 152 A Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata I the preset paper, we attempt to aswer these questios by proposig a ew, simple recursivehalvig based sychroizatio algorithm for 2D rectagle cellular automata. The algorithm ca sychroize ay 2D rectagle array of size m with a geeral at oe corer i m + + max(m,) 3 steps. A implemetatio i terms of local trasitio rules is also give o a 2D cellular automato, ot oly for the array with a geeral at oe corer but a geeralized case. The algorithms proposed i this paper are iterestig i the followig view poits. The 2D algorithm proposed is isotropic with respect to shape of a give rectagle array, i.e. o eed to cotrol the FSSP algorithm for loger-tha-wide ad wider-tha-log iput rectagles. The algorithm proposed ca be easily expaded to 3D arrays, eve to multi-dimesioal arrays. The algorithm is a atural geeralizatio of the well-kow 1D FSSP algorithms developed by Waksma [1966], Balzer [1967] ad Gerke [1987]. It gives us a ew view poit of those classical 1D FSSP algorithms based o recursive-halvig. The algorithm ca be expaded to a geeralized FSSP solutio, where a iitial geeral is at a arbitrary positio of a give array. The algorithm ca be geeralized to a optimum-time geeralized FSSP solutio. I Sectio 2 we give a descriptio of the 2D FSSP ad review some basic results o 2D FSSP algorithms. Sectio 3 defies the recursive-halvig markig o 1D arrays ad gives some prelimiary lemmas for the costructio of 2D FSSP algorithms. I Sectios 4, 5, ad 6 we preset a ew 2D FSSP algorithm based o the recursive-halvig markig ad several multi-dimesioal expasios. Two implemetatios i terms of 2D cellular automata are also preseted for the optimum-time FSSP algorithms. Most of the descriptios of the multi-dimesioal FSSP algorithms are based o the 2D FSSP algorithms. Some expaded ad geeralized theorems for multi-dimesioal arrays are give without proofs. 2 Firig Squad Sychroizatio Problem o Two-Dimesioal Arrays Figure 1 shows a fiite two-dimesioal (2D) cellular automato cosistig of m cells. Each cell is a idetical (except the border cells) fiite-state automato. The array operates i lock-step mode i such a way that the ext state of each cell (except border cells) is determied by both its ow preset state ad the preset states of its orth, south, east ad west eighbors. All cells (soldiers), except the orth-west corer cell (geeral), are iitially i the quiescet state at time t = 0 with the property that the ext state of a quiescet cell with quiescet eighbors is the quiescet state agai. At time t = 0, the orth-west corer cell C 11 is i the fire-whe-ready state, which is the iitiatio sigal for the array. The firig squad sychroizatio problem is to determie a descriptio (state set ad ext-state fuctio) for cells that esures all cells eter the fire state at exactly the same time ad for the first time. The tricky part of the problem is that the same kid of soldier havig a fixed umber of states must be sychroized, regardless of the size m of the array. The set of states ad ext state fuctio must be idepedet of m ad. The problem was first solved by J. McCarthy ad M. Misky who preseted a 3-step algorithm for 1D cellular array of legth. I 1962, the first optimum-time, i.e. (2 2)-step, sychroizatio algorithm was preseted by Goto [1962], with each cell havig several thousads of states. Waksma [1966] preseted a 16-state optimum-time sychroizatio algorithm. Afterward, Balzer [1967] ad Gerke [1987] developed a eight-state algorithm ad a seve-state sychroizatio algorithm, respectively, thus decreasig the umber of states required for the sychroizatio. Mazoyer [1987] developed

3 H. Umeo, K. Nishide & K. Kubo C11 C12 C13 C14 C1 C21 C22 C23 C24 C2 C31 C32 C33 C34 C3 m Cm1 Cm2 Cm3 Cm4 Cm Figure 1: A two-dimesioal (2D) cellular automato. a six-state sychroizatio algorithm which, at preset, is the algorithm havig the fewest states for 1D arrays. O the other had, several sychroizatio algorithms o 2D arrays have bee proposed by Beyer [1969], Grasselli [1975], Shiahr [1974], Szweriski [1982], Schmid [2003], Schmid ad Worsch [2004], Umeo, Maeda, Hisaoka ad Teraoka [2006], ad Umeo ad Uchio [2008]. It has bee show idepedetly by Beyer [1969] ad Shiahr [1974] that there exists o 2D cellular automato that ca sychroize ay 2D array of size m i less tha m + + max(m,) 3 steps. I additio they first proposed a optimum-time sychroizatio algorithm that ca sychroize ay 2D array of size m i optimum m + + max(m, ) 3 steps. Shiahr [1974] gave a 28-state implemetatio. Umeo, Hisaoka ad Akiguchi [2005] preseted a ew 12-state sychroizatio algorithm operatig i optimum-step, realizig a smallest solutio to the rectagle sychroizatio problem at preset. As for the time optimality of the 2D FSSP algorithms, the followig theorems have bee show. Theorem 1 Beyer [1969], Shiahr [1974] There exists o cellular automato that ca sychroize ay 2D array of size m i less tha m + + max(m,) 3 steps, where the geeral is located at oe corer of the array. Theorem 2 Shiahr [1974], Umeo, Hisaoka, ad Akiguchi [2005] There exists a cellular automato that ca sychroize ay 2D array of size m at exactly m + + max(m,) 3 steps, where the geeral is located at oe corer of the array. 3 Recursive-Halvig Markig I this sectio, we develop a markig schema for 1D arrays referred to as recursive-halvig markig. The markig schema prits a special mark o cells i a cellular space defied by the recursive-halvig markig. The markig itself is based o a 1D FSSP sychroizatio algorithm. It will be effectively used for costructig multi-dimesioal FSSP algorithms operatig i optimum-time. Let S be a 1D cellular space cosistig of cells C i, C i+1,, C j, deoted by [i j], where j > i. Let S deote the umber of cells i S, that is S = j i + 1. A ceter cell(s) C x of S is defied by

4 154 A Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata x = { (i + j)/2 S : odd (i + j 1)/2,(i + j + 1)/2 S : eve. (1) The recursive-halvig markig for a give cellular space S = [1] is defied as follows: Recursive-Halvig Markig: RHM Algorithm RHM(S) begi if S 2 the if S is odd the mark a ceter cell C x i S; S L := [1x]; S R := [x]; RHM L (S L ); RHM R (S R ); else mark ceter cells Cx ad C x+1 i S; S L := [1x]; S R := [x + 1]; RHM L (S L ); RHM R (S R ); ed Left-Side Recursive-Halvig Markig: RHM L Algorithm RHM L (S) begi while S > 2 do if S is odd the mark a ceter cell C x i S; S L := [1x]; RHM L (S L ); else mark ceter cells Cx ad C x+1 i S; S L := [1x]; RHM L (S L ); ed Right-Side Recursive-Halvig Markig: RHM R Algorithm RHM R (S) begi while S > 2 do if S is odd the mark a ceter cell C x i S; S R := [x]; RHM R (S R ); else mark ceter cells Cx ad C x+1 i S; S R := [x + 1]; RHM R (S R ); ed For example, we cosider a cellular space S = [115] cosistig of 15 cells. The first ceter cell is C 8, the the secod oe is C 4, C 5 ad C 11, C 12, ad the last oe is C 2, C 3, C 13, C 14, respectively. I case S = [117], we get C 9, C 5, C 13, C 3, C 15, ad C 2, C 16 after four iteratios. Figure 2 (left) shows a space-time diagram for the markig. At time t = 0, the leftmost cell C 1 geerates a ifiite set of sigals w 1,w 2,,w k,.., each propagatig i the right directio at 1/(2 k 1) speed,

5 H. Umeo, K. Nishide & K. Kubo G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 G B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 2 G O B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 3 G O K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 4 G O K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 5 G O O C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 6 G E O Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 7 G E O Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 8 G E O O C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 9 G E K O Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 10 G E K O Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 11 G E K O O C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 12 G E K C O Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 13 G E E C O Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 14 G E E C O O C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 15 G E E C K O Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 16 G E E Q K O Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 17 G E E Q K O O C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 18 G E E Q K C O Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 19 G E E Q C C O Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 20 G E E E C C O O C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 21 G E K E C C K O Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 22 G E K E C Q K O Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 23 G E K E Q Q K O O C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 24 G E K E Q Q K C O Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 25 G E K E Q Q C C O Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 26 G E K E Q C C C O O C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 27 G E K E E C C C K O Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 28 G E K C E C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 29 G E E C E C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 30 G E E C E Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 31 G E E C E Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 32 G E E C E Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 33 G E E C E Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 34 G E E C E E C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 35 G E E C K E C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 36 G E E Q K E C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 37 G E E Q K E Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 38 G E E Q K E Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 39 G E E Q K E Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 40 G E E Q K E Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 41 G E E Q K E E C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 42 G E E Q K C E C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 43 G E E Q C C E C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 44 G E E E C C E Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 45 G E K E C C E Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 46 G E K E C C E Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 47 G E K E C C E Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 48 G E K E C C E E C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 49 G E K E C C K E C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 50 G E K E C Q K E C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 51 G E K E Q Q K E Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 52 G E K E Q Q K E Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 53 G E K E Q Q K E Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 54 G E K E Q Q K E Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 55 G E K E Q Q K E E C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 56 G E K E Q Q K C E C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q 57 G E K E Q Q C C E C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q 58 G E K E Q C C C E Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q 59 G E K E E C C C E Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q 60 G E K C E C C C E Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q 61 G E E C E C C C E Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q 62 G E E C E C C C E E C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q 63 G E E C E C C C K E C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q 64 G E E C E C C Q K E C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q 65 G E E C E C Q Q K E Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q 66 G E E C E Q Q Q K E Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q 67 G E E C E Q Q Q K E Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q 68 G E E C E Q Q Q K E Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q 69 G E E C E Q Q Q K E E C C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q 70 G E E C E Q Q Q K C E C C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C G 71 G E E C E Q Q Q C C E C Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B G 72 G E E C E Q Q C C C E Q Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D G 73 G E E C E Q C C C C E Q Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K E G 74 G E E C E E C C C C E Q Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K E G 75 G E E C K E C C C C E Q C C C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D E E G 76 G E E Q K E C C C C E E C C C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q E E G 77 G E E Q K E C C C C K E C C Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q E E G 78 G E E Q K E C C C Q K E C Q Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D E E E G 79 G E E Q K E C C Q Q K E Q Q Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q E K E G 80 G E E Q K E C Q Q Q K E Q Q Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q E K E G 81 G E E Q K E Q Q Q Q K E Q Q C C C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D E E K E G 82 G E E Q K E Q Q Q Q K E Q C C C C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q E D K E G 83 G E E Q K E Q Q Q Q K E E C C C Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q E D E E G 84 G E E Q K E Q Q Q Q K C E C C Q Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D E E D E E G 85 G E E Q K E Q Q Q Q C C E C Q Q Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q E K D E E G 86 G E E Q K E Q Q Q C C C E Q Q Q Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q E K Q E E G 87 G E E Q K E Q Q C C C C E Q Q Q C C C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D E E K Q E E G 88 G E E Q K E Q C C C C C E Q Q C C C C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q Q E D K Q E E G 89 G E E Q K E E C C C C C E Q C C C C Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D D Q E D D Q E E G 90 G E E Q K C E C C C C C E E C C C Q Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q Q D E E D D E E E G 91 G E E Q C C E C C C C C K E C C Q Q Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D D Q Q E K D D E K E G 92 G E E E C C E C C C C Q K E C Q Q Q Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q Q D D Q E K Q D E K E G 93 G E K E C C E C C C Q Q K E Q Q Q Q C C C C Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D D Q Q D E E K Q Q E K E G 94 G E K E C C E C C Q Q Q K E Q Q Q C C C C Q Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q Q D D Q Q E D K Q Q E K E G 95 G E K E C C E C Q Q Q Q K E Q Q C C C C Q Q Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D D Q Q D D Q E D D Q Q E K E G 96 G E K E C C E Q Q Q Q Q K E Q C C C C Q Q Q Q C C C C Q Q Q K C O Q Q C C Q Q C C Q Q C B D K Q D D Q Q D D Q Q D D Q Q D E E D D D Q E K E G 97 G E K E C C E Q Q Q Q Q K E E C C C Q Q Q Q C C C C Q Q Q Q C C O Q C C Q Q C C Q Q C B K D D Q Q D D Q Q D D Q Q D D Q Q E K D D D E E K E G 98 G E K E C C E Q Q Q Q Q K C E C C Q Q Q Q C C C C Q Q Q Q C C C O O C Q Q C C Q Q C B D K Q D D Q Q D D Q Q D D Q Q D D Q E K Q D D E D K E G 99 G E K E C C E Q Q Q Q Q C C E C Q Q Q Q C C C C Q Q Q Q C C C C K O Q Q C C Q Q C B K D D Q Q D D Q Q D D Q Q D D Q Q D E E K Q Q D E D E E G 100 G E K E C C E Q Q Q Q C C C E Q Q Q Q C C C C Q Q Q Q C C C C Q K O Q C C Q Q C B D K Q D D Q Q D D Q Q D D Q Q D D Q Q E D K Q Q Q E D E E G 101 G E K E C C E Q Q Q C C C C E Q Q Q C C C C Q Q Q Q C C C C Q Q K O O C Q Q C B K D D Q Q D D Q Q D D Q Q D D Q Q D D Q E D D Q Q Q E D E E G 102 G E K E C C E Q Q C C C C C E Q Q C C C C Q Q Q Q C C C C Q Q Q K C O Q Q C B D K Q D D Q Q D D Q Q D D Q Q D D Q Q D E E D D D Q Q E D E E G 103 G E K E C C E Q C C C C C C E Q C C C C Q Q Q Q C C C C Q Q Q Q C C O Q C B K D D Q Q D D Q Q D D Q Q D D Q Q D D Q Q E K D D D D Q E D E E G 104 G E K E C C E E C C C C C C E E C C C Q Q Q Q C C C C Q Q Q Q C C C O O B D K Q D D Q Q D D Q Q D D Q Q D D Q Q D D Q E K Q D D D E E D E E G 105 G E K E C C K E C C C C C C K E C C Q Q Q Q C C C C Q Q Q Q C C C C K M K D D Q Q D D Q Q D D Q Q D D Q Q D D Q Q D E E K Q Q D D E K D E E G 106 G E K E C Q K E C C C C C Q K E C Q Q Q Q C C C C Q Q Q Q C C C C Q A M A Q D D Q Q D D Q Q D D Q Q D D Q Q D D Q Q E D K Q Q Q D E K Q E E G 107 G E K E Q Q K E C C C C Q Q K E Q Q Q Q C C C C Q Q Q Q C C C C Q A C M D A Q D D Q Q D D Q Q D D Q Q D D Q Q D D Q E D D Q Q Q Q E K Q E E G 108 G E K E Q Q K E C C C Q Q Q K E Q Q Q C C C C Q Q Q Q C C C C Q A C Q M Q D A Q D D Q Q D D Q Q D D Q Q D D Q Q D E E D D D Q Q Q E K Q E E G 109 G E K E Q Q K E C C Q Q Q Q K E Q Q C C C C Q Q Q Q C C C C Q A C Q Q M Q Q D A Q D D Q Q D D Q Q D D Q Q D D Q Q E K D D D D Q Q E K Q E E G 110 G E K E Q Q K E C Q Q Q Q Q K E Q C C C C Q Q Q Q C C C C Q A C Q Q Q M Q Q Q D A Q D D Q Q D D Q Q D D Q Q D D Q E K Q D D D D Q E K Q E E G 111 G E K E Q Q K E Q Q Q Q Q Q K E E C C C Q Q Q Q C C C C Q A C Q Q Q Q M Q Q Q Q D A Q D D Q Q D D Q Q D D Q Q D E E K Q Q D D D E E K Q E E G 112 G E K E Q Q K E Q Q Q Q Q Q K C E C C Q Q Q Q C C C C Q A C Q Q Q Q Q M Q Q Q Q Q D A Q D D Q Q D D Q Q D D Q Q E D K Q Q Q D D E D K Q E E G 113 G E K E Q Q K E Q Q Q Q Q Q C C E C Q Q Q Q C C C C Q A C Q Q Q Q Q Q M Q Q Q Q Q Q D A Q D D Q Q D D Q Q D D Q E D D Q Q Q Q D E D D Q E E G 114 G E K E Q Q K E Q Q Q Q Q C C C E Q Q Q Q C C C C Q A C Q Q Q Q Q Q Q M Q Q Q Q Q Q Q D A Q D D Q Q D D Q Q D E E D D D Q Q Q Q E D D E E E G 115 G E K E Q Q K E Q Q Q Q C C C C E Q Q Q C C C C Q A C Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q D A Q D D Q Q D D Q Q E K D D D D Q Q Q E D D E K E G 116 G E K E Q Q K E Q Q Q C C C C C E Q Q C C C C Q A C Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q D A Q D D Q Q D D Q E K Q D D D D Q Q E D D E K E G 117 G E K E Q Q K E Q Q C C C C C C E Q C C C C Q A C Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q D A Q D D Q Q D E E K Q Q D D D D Q E D D E K E G 118 G E K E Q Q K E Q C C C C C C C E E C C C Q A C Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q D A Q D D Q Q E D K Q Q Q D D D E E D D E K E G 119 G E K E Q Q K E E C C C C C C C K E C C Q A C Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q D A Q D D Q E D D Q Q Q Q D D E K D D E K E G 120 G E K E Q Q K C E C C C C C C Q K E C Q A C Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q D A Q D E E D D D Q Q Q Q D E K Q D E K E G 121 G E K E Q Q C C E C C C C C Q Q K E Q A C Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q D A Q E K D D D D Q Q Q Q E K Q Q E K E G 122 G E K E Q C C C E C C C C Q Q Q K E A C Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q D A E K Q D D D D Q Q Q E K Q Q E K E G 123 G E K E E C C C E C C C Q Q Q Q K U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U K Q Q D D D D Q Q E K Q Q E K E G 124 G E K C E C C C E C C Q Q Q Q Q A U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U A Q Q Q D D D D Q E K Q Q E K E G 125 G E E C E C C C E C Q Q Q Q Q A C U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U D A Q Q Q D D D E E K Q Q E K E G 126 G E E C E C C C E Q Q Q Q Q A C Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q D A Q Q Q D D E D K Q Q E K E G 127 G E E C E C C C E Q Q Q Q A C Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q D A Q Q Q D E D D Q Q E K E G 128 G E E C E C C C E Q Q Q A C Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q D A Q Q Q E D D D Q E K E G 129 G E E C E C C C E Q Q A C Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q D A Q Q E D D D E E K E G 130 G E E C E C C C E Q A C Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q D A Q E D D D E D K E G 131 G E E C E C C C E A C Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q D A E D D D E D E E G 132 G E E C E C C C U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U D D D E D E E G 133 G E E C E C C B U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U B D D E D E E G 134 G E E C E C B C U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U D B D E D E E G 135 G E E C E B C Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q D B E D E E G 136 G E E C I Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q I D E E G 137 G E E B I Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q I B E E G 138 G E I Q I Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q I Q I E G 139 G I I Q I Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q I Q I I G 140 I I I Q I Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q M Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U U Q Q Q Q Q Q Q U U Q Q Q I Q I I I G Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 G B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 2 G O B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 3 G O K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 4 G O K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 5 G O O C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 6 G E O Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 7 G E O Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 8 G E O O C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 9 G E K O Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 10 G E K O Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 11 G E K O O C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 12 G E K C O Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 13 G E E C O Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 14 G E E C O O C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 15 G E E C K O Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 16 G E E Q K O Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 17 G E E Q K O O C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 18 G E E Q K C O Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 19 G E E Q C C O Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 20 G E E E C C O O C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 21 G E K E C C K O Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 22 G E K E C Q K O Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 23 G E K E Q Q K O O C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 24 G E K E Q Q K C O Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 25 G E K E Q Q C C O Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 26 G E K E Q C C C O O C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 27 G E K E E C C C K O Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q Q Q 28 G E K C E C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q Q Q 29 G E E C E C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q Q Q 30 G E E C E Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q Q Q 31 G E E C E Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q Q Q 32 G E E C E Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q Q Q 33 G E E C E Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q Q Q 34 G E E C E E C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q Q Q 35 G E E C K E C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q Q Q 36 G E E Q K E C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q Q Q 37 G E E Q K E Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q Q Q 38 G E E Q K E Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q Q Q 39 G E E Q K E Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K B Q Q 40 G E E Q K E Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q K C B Q 41 G E E Q K E E C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C K G 42 G E E Q K C E C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q A G 43 G E E Q C C E C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q A D G 44 G E E E C C E Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q A K E G 45 G E K E C C E Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q A D K E G 46 G E K E C C E Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q Q C C Q Q C C Q A K D E E G 47 G E K E C C E Q C C C C Q Q K O O C Q Q C C Q Q C C Q Q C C Q Q C C Q A D K Q E E G 48 G E K E C C E E C C C Q Q Q K C O Q Q C C Q Q C C Q Q C C Q Q C C Q A K D D Q E E G 49 G E K E C C K E C C Q Q Q Q C C O Q C C Q Q C C Q Q C C Q Q C C Q A D K Q D E E E G 50 G E K E C Q K E C Q Q Q Q C C C O O C Q Q C C Q Q C C Q Q C C Q A K D D Q Q E K E G 51 G E K E Q Q K E Q Q Q Q C C C C K O Q Q C C Q Q C C Q Q C C Q A D K Q D D Q E K E G 52 G E K E Q Q K E Q Q Q C C C C Q K O Q C C Q Q C C Q Q C C Q A K D D Q Q D E E K E G 53 G E K E Q Q K E Q Q C C C C Q Q K O O C Q Q C C Q Q C C Q A D K Q D D Q Q E D K E G 54 G E K E Q Q K E Q C C C C Q Q Q K C O Q Q C C Q Q C C Q A K D D Q Q D D Q E D E E G 55 G E K E Q Q K E E C C C Q Q Q Q C C O Q C C Q Q C C Q A D K Q D D Q Q D E E D E E G 56 G E K E Q Q K C E C C Q Q Q Q C C C O O C Q Q C C Q A K D D Q Q D D Q Q E K D E E G 57 G E K E Q Q C C E C Q Q Q Q C C C C K O Q Q C C Q A D K Q D D Q Q D D Q E K Q E E G 58 G E K E Q C C C E Q Q Q Q C C C C Q K O Q C C Q A K D D Q Q D D Q Q D E E K Q E E G 59 G E K E E C C C E Q Q Q C C C C Q Q K O O C Q A D K Q D D Q Q D D Q Q E D K Q E E G 60 G E K C E C C C E Q Q C C C C Q Q Q K C O Q A K D D Q Q D D Q Q D D Q E D D Q E E G 61 G E E C E C C C E Q C C C C Q Q Q Q C C O A D K Q D D Q Q D D Q Q D E E D D E E E G 62 G E E C E C C C E E C C C Q Q Q Q C C C W W D D Q Q D D Q Q D D Q Q E K D D E K E G 63 G E E C E C C C K E C C Q Q Q Q C C C B W W B D D Q Q D D Q Q D D Q E K Q D E K E G 64 G E E C E C C Q K E C Q Q Q Q C C C B C W W D B D D Q Q D D Q Q D E E K Q Q E K E G 65 G E E C E C Q Q K E Q Q Q Q C C C B C Q W W Q D B D D Q Q D D Q Q E D K Q Q E K E G 66 G E E C E Q Q Q K E Q Q Q C C C B C Q Q W W Q Q D B D D Q Q D D Q E D D Q Q E K E G 67 G E E C E Q Q Q K E Q Q C C C B C Q Q Q W W Q Q Q D B D D Q Q D E E D D D Q E K E G 68 G E E C E Q Q Q K E Q C C C B C Q Q Q Q W W Q Q Q Q D B D D Q Q E K D D D E E K E G 69 G E E C E Q Q Q K E E C C B C Q Q Q Q Q W W Q Q Q Q Q D B D D Q E K Q D D E D K E G 70 G E E C E Q Q Q K C E C B C Q Q Q Q Q Q W W Q Q Q Q Q Q D B D E E K Q Q D E D E E G 71 G E E C E Q Q Q C C E B C Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q D B E D K Q Q Q E D E E G 72 G E E C E Q Q C C C I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I D D Q Q Q E D E E G 73 G E E C E Q C C C B I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I B D D Q Q E D E E G 74 G E E C E E C C B C I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I D B D D Q E D E E G 75 G E E C K E C B C Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q D B D E E D E E G 76 G E E Q K E B C Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q D B E K D E E G 77 G E E Q K I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I K Q E E G 78 G E E Q A I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I A Q E E G 79 G E E A C I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I D A E E G 80 G E U U Q I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I Q U U E G 81 G I U U Q I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I Q U U I G 82 I I U U Q I Q Q Q Q I Q Q Q Q Q Q Q Q Q W W Q Q Q Q Q Q Q Q Q I Q Q Q Q I Q U U I I /7 t = 0 t = 2-2 w1 w3 w2 w1 w2 w3 1/7 5 w4 t = -1 t = Figure 2: Space-time diagram for recursive-halvig markig o 1D array of legth (left) ad some sapshots for the markig o 42 (middle) ad 71 (right) cells, respectively. where k = 1,2,3,,. The -speed sigal w 1 arrives at C at time t = 1. The, the rightmost cell C also emits a ifiite set of sigals w 1,w 2,,w k,.., each propagatig i the left directio at 1/(2 k 1) speed, where k = 1,2,3,,. The readers ca fid that each crossig of two sigals, show i Fig. 2 (left), eables the markig at middle poits defied by the recursive-halvig. A fiite state realizatio for geeratig the ifiite set of sigals above is a well-kow techique employed i Balzer [1967], Gerke [1987], ad Waksma [1966] for the implemetatios of the optimum-time sychroizatio algorithms o 1D arrays. We have developed a simple implemetatio of the recursive-halvig markig o a 13-state, 314-rule cellular automato. I Fig. 2 (middle ad right) we preset several sapshots for the markig o 42 ad 71 cells, respectively. Thus we have: Lemma 3 There exists a 1D 13-state, 314-rule cellular automato that ca prit the recursive-halvig markig i ay cellular space of legth i 2 2 steps. A optimum-time complexity 2 2 eeded for sychroizig cellular space of legth i the classical WBG-type (Waksma [1966], Balzer [1967], ad Gerke [1987]) FSSP algorithms ca be iterpreted as follows: Let S be a cellular space of legth = , where 1 1. The first ceter mark i S is prited o cell C 1 +1 at time t 1D ceter = 3 1. Additioal 1 steps are required for the markigs thereafter,

6 156 A Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata yieldig a fial sychroizatio at time t 1D opt = = 4 1 = 2 2. I the case = 2 1, where 1 1, the first ceter mark is prited simultaeously o cells C 1 ad C 1 +1 at time t 1D ceter = Additioal 1 1 steps are required for the markig ad sychroizatio thereafter, yieldig the fial sychroizatio at time t 1D opt = = = 2 2. t 1D ceter = { 3 1 S = , S = 2 1. Thus, additioal t 1D syc steps are required for the sychroizatio for a cellular space with the recursive-halvig marks: { 1 S = , t 1D syc = (3) 1 1 S = 2 1. (2) 1 t = 0 t = tsyc G G I I Q I Q Q Q M2 Q Q Q I Q I I G 0 1 G I I Q I Q Q Q2 M2 C2 Q Q I Q I I G 2 G I I Q I Q Q2 Q2 M2 C2 C2 Q I Q I I G 3 G I I Q I Q2 L2 Q2 M2 C2 R2 C2 I Q I I G 4 G I I Q M2 A2 L2 Q2 M2 C2 R2 B2 M2 Q I I G 5 G I I Q2 M2 B2 Q2 Q2 M2 C2 C2 A2 M2 C2 I I G 6 G I M2 Q2 M2 C2 M2 Q2 M2 C2 M2 Q2 M2 C2 M2 I G 7 G M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 G 8 F F F F F F F F F F F F F F F F F G U U U U Q Q U U Q Q Q Q Q Q M2 M2 Q Q Q Q Q Q U U Q Q U U U U G 1 G U U U U Q Q U U Q Q Q Q Q Q2 M2 M2 C2 Q Q Q Q Q U U Q Q U U U U G 2 G U U U U Q Q U U Q Q Q Q Q2 Q2 M2 M2 C2 C2 Q Q Q Q U U Q Q U U U U G 3 G U U U U Q Q U U Q Q Q Q2 L2 Q2 M2 M2 C2 R2 C2 Q Q Q U U Q Q U U U U G 4 G U U U U Q Q U U Q Q Q2 A2 L2 Q2 M2 M2 C2 R2 B2 C2 Q Q U U Q Q U U U U G 5 G U U U U Q Q U U Q Q2 L2 A2 Q2 Q2 M2 M2 C2 C2 B2 R2 C2 Q U U Q Q U U U U G 6 G U U U U Q Q U U Q2 A2 L2 L2 Q2 Q2 M2 M2 C2 C2 R2 R2 B2 C2 U U Q Q U U U U G 7 G U U U U Q Q U MM L2 A2 A2 L2 Q2 Q2 M2 M2 C2 C2 R2 B2 B2 R2 MM U Q Q U U U U G 8 G U U U U Q Q M2 M2 L2 L2 A2 Q2 Q2 Q2 M2 M2 C2 C2 C2 B2 R2 R2 M2 M2 Q Q U U U U G 9 G U U U U Q Q2 M2 M2 C2 L2 L2 Q2 L2 Q2 M2 M2 C2 R2 C2 R2 R2 Q2 M2 M2 C2 Q U U U U G 10 G U U U U Q2 Q2 M2 M2 C2 C2 L2 Q2 L2 Q2 M2 M2 C2 R2 C2 R2 Q2 Q2 M2 M2 C2 C2 U U U U G 11 G U U U MM L2 Q2 M2 M2 C2 R2 C2 Q2 L2 Q2 M2 M2 C2 R2 C2 Q2 L2 Q2 M2 M2 C2 R2 MM U U U G 12 G U U M2 M2 L2 Q2 M2 M2 C2 R2 M2 M2 L2 Q2 M2 M2 C2 R2 M2 M2 L2 Q2 M2 M2 C2 R2 M2 M2 U U G 13 G U MM M2 M2 C2 Q2 M2 M2 C2 Q2 M2 M2 C2 Q2 M2 M2 C2 Q2 M2 M2 C2 Q2 M2 M2 C2 Q2 M2 M2 MM U G 14 G M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 G 15 F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Figure 3: Space-time diagram for sychroizig a cellular space with recursive-halvig markig (left) ad some sapshots for the sychroizatio o 17 (middle) ad 32 (right) cells, respectively. I this way, it ca be easily see that ay cellular space of legth with the recursive-halvig markig iitially with a geeral o a ceter cell or two geerals o adjacet ceter cells ca be sychroized i /2 1 optimum-steps. I Fig. 3, we illustrate a space-time diagram for sychroizig a cellular space with recursive-halvig markig (left) ad some sapshots for the sychroizatio o 17 (middle) ad 32 (right) cells, respectively. Thus we have: Lemma 4 Ay 1D cellular space S of legth with the recursive-halvig markig iitially with a geeral(s) o a ceter cell(s) i S ca be sychroized i /2 1 optimum-steps. As was see, the first markig of ceter cell(s) plays a importat role. We prit a special mark for the first ceter cell(s) of a give cellular space. O the other had, for the ceter cells geerated thereafter are marked with a differet symbol from the first oe. G Step 0 Step 1 Step 3 Step 2 Figure 4: Sychroizatio schema for 2D cellular automato.

7 H. Umeo, K. Nishide & K. Kubo A Optimum-Time 2D FSSP Algorithm A Overview of the Algorithm A 1 We assume that a iitial geeral G is o the orth-west corer cell C 11 of a give array of size m. The algorithm cosists of three phases: a markig phase, a pre-sychroizatio phase ad a fial sychroizatio phase. A overview of the 2D sychroizatio algorithm A 1 is as follows: Step 1. Start the recursive-halvig markig for cells o each row ad colum, fid a ceter cell(s) of the give array, ad geerate a ew geeral(s) o the ceter cell(s). Note that a crossig(s) of the ceter colum(s) with the ceter row(s) is a ceter cell(s) of the array. Step 2. Pre-sychroize the ceter colum(s) usig Lemma 4, which is iitiated by the geeral i Step 1. Every cell o the ceter colum(s) acts as a geeral at the followig Step 3. Step 3. Sychroize each row usig Lemma 4, iitiated by the geeral geerated i Step 2. This yields the fial sychroizatio of the array. G 1 1 i t = 0 1 t = 0 1 m t = 0 1 t = i-1 1/7 t = /7 1/7 t = m-1 t = 2-2 t = +1 +i-2 5 1/7 1/7 t = 2m+ 1-2 t = 2m+ 1-2 t = 2m+ 1-2 t = 2m+-3 t = 2m+-3 t = m t = 2m /7 Figure 5: Space-time diagrams for the sychroizatio algorithm o the 1st, ith, ad mth rows of a loger-tha-wide rectagle array of size m, respectively. Figure 4 illustrates the sychroizatio schema for 2D cellular automato. We assume that m =

8 158 A Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata 2m 1 + 1, = , where m 1, 1 1. The algorithm operates as follows: 1. At time t = 0 a iitial geeral o the orth-west corer emits a -speed sigal alog the first row ad colum to prit recursive-halvig marks. Oce a ceter mark is prited, it is copied to the adjacet row ad colum. At time t = 3m 1 + 1, a ceter mark of the ceter colum of the array is marked, ad the ceter mark of the ceter row is marked at time t = m 1. The ceter of the array is marked at time t = t 2D ceter = max(3m 1 + 1,3 1 + m 1 ). 2. Usig Lemma 4, the ceter colum will be sychroized with a tetative pre-firig state at time t = t 2D ceter + m Oce the ceter colum could be sychroized with the pre-firig state, the the cell C i,1 +1 iitiates the sychroizatio for the ith row for each i such that 1 i m. Usig Lemma 4, for ay i,1 i m, the ith row will be sychroized at time t = t 2D ceter + m = max(3m 1 + 1,3 1 +m 1 )+m = max(2m 1 +1,2 1 +1)+2m = m ++max(m,) 3. Thus, the array ca be sychroized at time t = m + + max(m, ) 3 i optimum-steps. G 1 1 i m t = 0 1 t = 0 1 t = 0 1 t = i-1 t = m-1 1/7 t = +1-1 t = 2-2 t = m t = m /7 t = +1+i-2 t = m t = m t = m /7 1/7 t = m+2-3 1/7 5 1/7 Figure 6: Space-diagrams of the sychroizatio algorithm o the 1st, ith, ad mth rows of a widertha-log rectagle array of size m, respectively. Note that the sigal propagatio for the recursive-halvig markig ad the wake-up sigal for the sychroizatio are made by the same speed. Thus, the sychroizatio ca be performed successfully for each colum ad the array ca be sychroized i optimum steps. Figures 5 ad 6 illustrate a space-time diagram for the recursive-halvig markig ad the sychroizatio operatios o the 1st, ith, ad mth row of a loger-tha-wide ad wider-tha-log array, respectively. Oe ca see that

9 H. Umeo, K. Nishide & K. Kubo 159 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 1 1G2G Q Q Q Q Q Q Q Q Q Q Q 1 1G2G 1B Q Q Q Q Q Q Q Q Q Q 1 1G2G 1O 1B Q Q Q Q Q Q Q Q Q 1 1G2G 1O 1K 1B Q Q Q Q Q Q Q Q 1 1G2G 1O 1K 1C 1B Q Q Q Q Q Q Q 1 1G2G 1O 1O 1C 1K 1B Q Q Q Q Q Q 1 1G2G 1E 1O Q 1K 1C 1B Q Q Q Q Q 2 Q Q Q Q Q Q Q Q Q Q Q Q 2 2B Q Q Q Q Q Q Q Q Q Q Q 2 2O Q Q Q Q Q Q Q Q Q Q Q 2 2O Q Q Q Q Q Q Q Q Q Q Q 2 2O Q Q Q Q Q Q Q Q Q Q Q 2 2O Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q Q 3 Q Q Q Q Q Q Q Q Q Q Q Q 3 Q Q Q Q Q Q Q Q Q Q Q Q 3 2B Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2O Q Q Q Q Q Q Q Q Q Q Q 3 2O Q Q Q Q Q Q Q Q Q Q Q 4 Q Q Q Q Q Q Q Q Q Q Q Q 4 Q Q Q Q Q Q Q Q Q Q Q Q 4 Q Q Q Q Q Q Q Q Q Q Q Q 4 2B Q Q Q Q Q Q Q Q Q Q Q 4 2C Q Q Q Q Q Q Q Q Q Q Q 4 2C Q Q Q Q Q Q Q Q Q Q Q 4 Q Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 2B Q Q Q Q Q Q Q Q Q Q Q 5 2K Q Q Q Q Q Q Q Q Q Q Q 5 2K Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 6 2B Q Q Q Q Q Q Q Q Q Q Q 6 2C Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 7 2B Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q t = 7 t = 8 t = 9 t = 10 t = 11 t = 12 t = G2G 1E 1O Q 1C 1C 1K 1B Q Q Q Q 1 1G2G 1E 1O 1O 1C Q 1K 1C 1B Q Q Q 1 1G2G 1E 1K 1O Q Q 1C 1C 1K 1B Q Q 1 1G2G 1E 1K 1O Q 1C 1C Q 1K 1C 1B Q 1 1G2G 1E 1K 1O 1O 1C Q Q 1C 1C 1K 1G2G 1 1G2G 1E 1K 1C 1O Q Q 1C 1C Q 1A 1G2G 1 1G2G 1E 1E 1C 1O Q 1C 1C Q 1A 1D 1G2G 2 2E Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q Q 2 2E Q Q Q Q Q Q Q Q Q Q 1G2x 2 2E Q Q Q Q Q Q Q Q Q Q 1G2x 3 2O Q Q Q Q Q Q Q Q Q Q Q 3 2O Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2K Q Q Q Q Q Q Q Q Q Q Q 3 2E Q Q Q Q Q Q Q Q Q Q 1G2x 4 Q Q Q Q Q Q Q Q Q Q Q Q 4 2O Q Q Q Q Q Q Q Q Q Q Q 4 2O Q Q Q Q Q Q Q Q Q Q Q 4 2O Q Q Q Q Q Q Q Q Q Q Q 4 2O Q Q Q Q Q Q Q Q Q Q Q 4 2C Q Q Q Q Q Q Q Q Q Q Q 4 2B Q Q Q Q Q Q Q Q Q Q Q 5 2C Q Q Q Q Q Q Q Q Q Q Q 5 2C Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 Q Q Q Q Q Q Q Q Q Q Q Q 5 2O Q Q Q Q Q Q Q Q Q Q Q 5 2M Q Q Q Q Q Q Q Q Q Q Q 5 2x 2M Q Q Q Q Q Q Q Q Q Q 6 2C Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 6 2C Q Q Q Q Q Q Q Q Q Q Q 6 2B Q Q Q Q Q Q Q Q Q Q Q 6 2D Q Q Q Q Q Q Q Q Q Q Q 6 2B Q Q Q Q Q Q Q Q Q Q Q 7 2K Q Q Q Q Q Q Q Q Q Q Q 7 2K Q Q Q Q Q Q Q Q Q Q Q 7 2C Q Q Q Q Q Q Q Q Q Q Q 7 2B Q Q Q Q Q Q Q Q Q Q Q 7 2K Q Q Q Q Q Q Q Q Q Q Q 7 2K Q Q Q Q Q Q Q Q Q Q Q 7 2E Q Q Q Q Q Q Q Q Q Q Q 8 2B Q Q Q Q Q Q Q Q Q Q Q 8 2C Q Q Q Q Q Q Q Q Q Q Q 8 2B Q Q Q Q Q Q Q Q Q Q Q 8 2D Q Q Q Q Q Q Q Q Q Q Q 8 2E Q Q Q Q Q Q Q Q Q Q Q 8 2E Q Q Q Q Q Q Q Q Q Q Q 8 2E Q Q Q Q Q Q Q Q Q Q Q Q Q Q 9 1G2G Q Q Q Q Q Q Q Q Q Q Q 9 1G2G1x2G Q Q Q Q Q Q Q Q Q Q 9 1G2G1x2G1x2G Q Q Q Q Q Q Q Q Q 9 1G2G1x2G1x2G1x2G Q Q Q Q Q Q Q Q 9 1G2G1x2G1x2G1x2G1x2G Q Q Q Q Q Q Q 9 1G2G1x2G1x2G1x2G1x2G1x2G Q Q Q Q Q Q t = 14 t = 15 t = 16 t = 17 t = 18 t = 19 t = G2G 1E 1E 1C 1O 1O 1C Q 1A 1K 1E 1G2G 1 1G2G 1E 1E 1C 1K 1O Q 1A 1D 1K 1E 1G2G 1 1G2G 1E 1E Q 1K 1O 1A 1K 1D 1E 1E 1G2G 1 1G2G 1E 1E Q 1K 1W 1M 1K Q 1E 1E 1G2G 1 1G2G 1E 1E Q 1A 1x 1x 1A Q 1E 1E 1G2G 1 1G2G 1E 1E 1A 1x 1x 1x 1x 1A 1E 1E 1G2G 1 1G2G 1E 1I 1U 1x 1x 1x 1x 1U 1I 1E 1G2G 2 2E Q Q Q Q Q Q Q Q Q Q 1G2x 2 2I Q Q Q Q Q Q Q Q Q Q 1G2x 2 2x 2I Q Q Q Q Q Q Q Q Q 1G2x 2 2x 2I 2I Q Q Q Q Q Q Q Q 1G2x 2 2x 2I 2I 2I Q 1W 1M Q Q Q Q 1G2x 2 2x 2I 2I 2I 2I 1W 1M Q Q Q Q 1G2x 2 2x 2I 2I 2I 2I 1W2I 1M Q Q Q Q 1G2x 3 2I Q Q Q Q Q Q Q Q Q Q 1G2x 3 2x 2I Q Q Q Q Q Q Q Q Q 1G2x 3 2x 2I 2I Q Q Q Q Q Q Q Q 1G2x 3 2x 2I 2I 2I Q Q Q Q Q Q Q 1G2x 3 2x 2I 2I 2I 2I Q Q Q Q Q Q 1G2x 3 2x 2I 2I 2I 2I 1W2I 1M Q Q Q Q 1G2x 3 2x 2I 2I 2I 2I 1W2I1M2I Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q Q Q Q Q Q Q 1G2x 4 2x Q Q Q Q 1W 1M Q Q Q Q 1G2x 5 2x 2M 2M Q Q Q Q Q Q Q Q Q 5 2x 2M 2M 2M Q Q Q Q Q Q Q 1G2x 5 2x 2M 2M 2M 2M Q Q Q Q Q Q 1G2x 5 2x 2M 2M 2M 2M 2M Q Q Q Q Q 1G2x 5 2x 2M 2M 2M 2M 2M 2M Q Q Q Q 1G2x 5 2x 2M 2M 2M 2M 2M 2M 2M Q Q Q 1G2x 5 2x 2M 2M 2M 2M 2M 2M 2M 2M Q Q 1G2x 6 2x Q Q Q Q Q Q Q Q Q Q Q 6 2x Q Q Q Q Q Q Q Q Q Q Q 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 7 2I Q Q Q Q Q Q Q Q Q Q Q 7 2x 2I Q Q Q Q Q Q Q Q Q Q 7 2x 2I 2I Q Q Q Q Q Q Q Q Q 7 2x 2I 2I 2I Q Q Q Q Q Q Q 1G2x 7 2x 2I 2I 2I 2I Q Q Q Q Q Q 1G2x 7 2x 2I 2I 2I 2I 2I Q Q Q Q Q 1G2x 7 2x 2I 2I 2I 2I 2I 2I Q Q Q Q 1G2x 8 2E Q Q Q Q Q Q Q Q Q Q Q 8 2I Q Q Q Q Q Q Q Q Q Q Q 8 2x 2I Q Q Q Q Q Q Q Q Q Q 8 2x 2I 2I Q Q Q Q Q Q Q Q Q 8 2x 2I 2I 2I Q Q Q Q Q Q Q 1G2x 8 2x 2I 2I 2I 2I Q Q Q Q Q Q 1G2x 8 2x 2I 2I 2I 2I 2I Q Q Q Q Q 1G2x 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G Q Q Q Q Q 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G Q Q Q Q 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G Q Q Q 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G Q Q 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G Q 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G t = 21 t = 22 t = 23 t = 24 t = 25 t = 26 t = G2G 1I 1x 1x 1x 1x 1x 1x 1x 1x 1I 1G2G 1 1G2G 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1G2G 1 1G2G 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1G2G 1 1G2G 1x 1x 1x 1x 1x 1x 1x 1x 1x 1x 1G2G 1 1G2G 1x 1x 1x 1x M+ M+ 1x 1x 1x 1x 1G2G 1 1G2G 1x 1x 1x Q+ M+ M+ C+ 1x 1x 1x 1G2G 1 1G2G 1x 1x MM+ Q+ M+ M+ C+ MM+ 1x 1x 1G2G 2 2x 2I 1I2I 1U2I 2I 1W2I1M2I Q 1U 1I Q 1G2x 2 2x 1I2I 1I2I 1U2I 2I 1W2I1M2I 2I 1U 1I 1I 1G2x 2 2x 1I2I 1I2I 1U2I 2I 1W2I1M2I 2I 1U2I 1I 1I 1G2x 2 2x 1I2I 1I2I 1U2I 2I M- M- 2I 1U2I 1I2I 1I 1G2x 2 2x 1I2I 1I2I 1U2I 2I M+ M+ 2I 1U2I 1I2I 1I2I 1G2x 2 2x 1I2I 1I2I 1U2I Q+ M+ M+ C+ 1U2I 1I2I 1I2I 1G2x 2 2x 1I2I 1I2I MM+ Q+ M+ M+ C+ MM+ 1I2I 1I2I 1G2x 3 2x 2I 2I 2I 2I 1W2I1M2I 2I Q Q Q 1G2x 3 2x 2I 1I2I 1U2I 2I 1W2I1M2I 2I 1U2I 1I Q 1G2x 3 2x 1I2I 1I2I 1U2I 2I M- M- 2I 1U2I 1I2I 1I 1G2x 3 2x 1I2I 1I2I 1U2I 2I M- M- 2I 1U2I 1I2I 1I2I 1G2x 3 2x 1I2I 1I2I 1U2I 2I M+ M+ 2I 1U2I 1I2I 1I2I 1G2x 3 2x 1I2I 1I2I 1U2I Q+ M+ M+ C+ 1U2I 1I2I 1I2I 1G2x 3 2x 1I2I 1I2I MM+ Q+ M+ M+ C+ MM+ 1I2I 1I2I 1G2x 4 2x Q Q Q Q 1W 1M Q Q Q Q 1G2x 4 2x Q Q Q Q C- C- Q Q Q Q 1G2x 4 2x Q 1I 1U Q C- C- Q 1U 1I Q 1G2x 4 2x 1I 1I 1U Q M- M- Q 1U 1I 1I 1G2x 4 2x 1I 1I 1U Q M+ M+ Q 1U 1I 1I 1G2x 4 2x 1I 1I 1U Q+ M+ M+ C+ 1U 1I 1I 1G2x 4 2x 1I 1I MM+ Q+ M+ M+ C+ MM+ 1I 1I 1G2x 5 2x 2M 2M 2M 2M M^ M^ 2M 2M 2M Q 1G2x 5 2x 2M 2M 2M 2M M- M- 2M 2M 2M 2M 1G2x 5 2x 2M 2M 2M 2M M- M- 2M 2M 2M 2M 1G2x 5 2x 2M 1I2M1U2M2M M- M- 2M1U2M1I2M 2M 1G2x 5 2x 1I2M1I2M1U2M2M M+ M+ 2M1U2M1I2M1I2M1G2x 5 2x 1I2M1I2M1U2MQ+ M+ M+ C+1U2M1I2M1I2M1G2x 5 2x 1I2M1I2MMM+ Q+ M+ M+ C+ MM+1I2M1I2M1G2x 6 2x Q Q Q Q Q Q Q Q Q Q 1G2x 6 2x Q Q Q Q Q- Q- Q Q Q Q 1G2x 6 2x Q Q Q Q Q- Q- Q Q Q Q 1G2x 6 2x Q Q Q Q M- M- Q Q Q Q 1G2x 6 2x Q 1I 1U Q M+ M+ Q 1U 1I Q 1G2x 6 2x 1I 1I 1U Q+ M+ M+ C+ 1U 1I 1I 1G2x 6 2x 1I 1I MM+ Q+ M+ M+ C+ MM+ 1I 1I 1G2x 7 2x 2I 2I 2I 2I 2I 2I 2I Q Q Q 1G2x 7 2x 2I 2I 2I 2I 2I 2I 2I 2I Q Q 1G2x 7 2x 2I 2I 2I 2I M- M- 2I 2I 2I Q 1G2x 7 2x 2I 2I 2I 2I M- M- 2I 2I 2I 2I 1G2x 7 2x 2I 2I 2I 2I M+ M+ 2I 2I 2I 2I 1G2x 7 2x 2I 1I2I 1U2I Q+ M+ M+ C+ 1U2I 1I2I 2I 1G2x 7 2x 1I2I 1I2I MM+ Q+ M+ M+ C+ MM+ 1I2I 1I2I 1G2x 8 2x 2I 2I 2I 2I 2I 2I Q Q Q Q 1G2x 8 2x 2I 2I 2I 2I 2I 2I 2I Q Q Q 1G2x 8 2x 2I 2I 2I 2I 2I 2I 2I 2I Q Q 1G2x 8 2x 2I 2I 2I 2I M- M- 2I 2I 2I Q 1G2x 8 2x 2I 2I 2I 2I M+ M+ 2I 2I 2I 2I 1G2x 8 2x 2I 2I 2I Q+ M+ M+ C+ 2I 2I 2I 1G2x 8 2x 2I 1I2I MM+ Q+ M+ M+ C+ MM+ 1I2I 2I 1G2x 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G1x2G M+ M+ 1x2G1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G1x2G Q+ M+ M+ C+ 1x2G1x2G1x2G1G2G 9 1G2G1x2G1x2G Q+ Q+ M+ M+ C+ C+ 1x2G1x2G1G2G t = 28 t = 29 t = G2G 1x M+ M+ Q+ M+ M+ C+ M+ M+ 1x 1G2G 1 1G2G M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2G 1 F F F F F F F F F F F F 2 2x 1I2I M+ M+ Q+ M+ M+ C+ M+ M+ 1I2I 1G2x 2 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 2 F F F F F F F F F F F F 3 2x 1I2I M+ M+ Q+ M+ M+ C+ M+ M+ 1I2I 1G2x 3 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 3 F F F F F F F F F F F F 4 2x 1I M+ M+ Q+ M+ M+ C+ M+ M+ 1I 1G2x 4 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 4 F F F F F F F F F F F F 5 2x 1I2M M+ M+ Q+ M+ M+ C+ M+ M+ 1I2M1G2x 5 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 5 F F F F F F F F F F F F 6 2x 1I M+ M+ Q+ M+ M+ C+ M+ M+ 1I 1G2x 6 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 6 F F F F F F F F F F F F 7 2x 1I2I M+ M+ Q+ M+ M+ C+ M+ M+ 1I2I 1G2x 7 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 7 F F F F F F F F F F F F 8 2x 1I2I M+ M+ Q+ M+ M+ C+ M+ M+ 1I2I 1G2x 8 2x M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2x 8 F F F F F F F F F F F F 9 1G2G1x2G M+ M+ Q+ M+ M+ C+ M+ M+ 1x2G1G2G 9 1G2G M+ M+ M+ M+ M+ M+ M+ M+ M+ M+ 1G2G 9 F F F F F F F F F F F F Figure 7: Sapshots for sychroizatio o a 9 12 array. each markig operatio has bee fiished before the arrival of the first wake-up sigal for the sychroizatio. The algorithm operates i optimum-steps i a similar way for the rectagles such as case 1: m = 2m 1 + 1, = 2 1, case 2: m = 2m 1, = , ad case 3: m = 2m 1, = 2 1. Thus, we ca establish the followig theorem. Theorem 5 The sychroizatio algorithm A 1 ca sychroize ay m rectagular array i optimum m + + max(m,) 3 steps. We have implemeted the algorithm A 1 o a 2D cellular automato havig 60 states ad local rules. I Figures 7 ad 8 we preset some sapshots of the sychroizatio processes of the algorithm o 9 12 ad 12 9 arrays, respectively. 5 Expasio to Multi-Dimesioal Arrays 5.1 Three-Dimesioal Arrays I this sectio, we show that there exists o algorithm that ca sychroize ay 3D array of size m l with a geeral at a arbitrary corer i less tha m + + l + max(m,,l) 4 steps. Theorem 6 The miimum time i which the firig squad sychroizatio could occur is o earlier tha m + + l + max(m,,l) 4 for ay 3D array of size m l with a geeral at a arbitrary corer cell. Proof. The proof is made by cotradictio. Without loss of geerality, we assume that l m. It is assumed that there is a cellular automato M that ca sychroize a array of size m 0 0 l 0 (l 0 m 0 0 ) at step t = t 0 such that: