REIHE INFORMATIK 7/95. Extremal Codes for Speed-Up of Distributed Parallel Arbitration

Transcription

1 REIHE INFORMATIK 7/95 Extremal Codes for Speed-Up of Dstrbuted Parallel Arbtrato M. Makhaok, V. Cheravsky, R. Mäer, K.-H. Noffz Uverstät Mahem Semargebäude A5 D Mahem 1

2 EXTREMAL CODES FOR SPEED-UP OF DISTRIBUTED PARALLEL ARBITRATION M. Makhaok 1), V. Cheravsky 1), R. Mäer 2,3), K.-H. Noffz 2) 1) Isttute of Egeerg Cyberetcs, Academy of Sceces, Surgaov St. 6, Msk, Republc Belarus 2) Lehrstuhl für Iformatk V, Uverstät Mahem, A5, D Mahem, Germay 3) Iterdszpläres Zetrum für Wsseschaftlches Reche, Uverstät Hedelberg, Im Neuehemer Feld 368, D Hedelberg, Germay Idex terms: Buses, arbtrato prortes, extremal codes, dstrbuted parallel arbtrato, Futurebus+ Abstract. Ths paper descrbes a method that allows the speed up of parallel processes dstrbuted arbtrato schemes as used Futurebus+. It s based o specal arbtrato codes that decrease the maxmal arbtrato tme to a specfed value. Such codes ca be appled wth few, f ay, mor chages of the hardware. The geeral structure of these codes s gve. Itroducto. Oe mportat compoet of every multprocessor bus s the arbtrato system. May moder multprocessor buses use arbtrato systems prcpally based o the same dstrbuted parallel arbtrato scheme [1]. I ths scheme, the tme for acqurg bus mastershp depeds o the maxmal arbtrato tme requred. Ths tme s determed by the set of arbtrato prortes avalable [2-4]. Such a set wll be called a arbtrato code or smply a code below. The maxmal arbtrato tme s reduced, e.g., by usg bomal codes as dscussed [4] where these codes were troduced. From the dscusso t follows that the achevable speed-up depeds o the umber of arbtrato prortes the bomal code,.e., o ts capacty. However, t has ot bee cosdered whether the bomal codes have the maxmal possble capacty,.e., whether there are other codes wth equal or greater capacty that ca reduce the maxmal arbtrato tme to the same or a lower lmt. I the preset paper we derve the geeral structure of the codes of maxmal capacty that we wll call extremal codes. It wll be show that there exst dfferet extremal codes wth bomal codes as oe example oly. Arbtrato process. We cosder a dstrbuted parallel arbtrato process for a sglebus multprocessor system. To each processor a uque arbtrato prorty s assged that s a m-bt bary word. It wll be called a arbtrato word or smply a word below. If oe or more processors request bus mastershp, a arbtrato process s started. At Ths work has bee supported by the Deutsche Forschugsgemeschaft (DFG) uder grats Ma 1150/8-1 ad 436-WER ( /117/0). 2

3 ths tme each processor decdes whether t partcpates the arbtrato process or ot. The objectve of the arbtrato process s to fd the processor of hghest prorty amog all partcpatg oes. Arbtrato s doe a decetralzed way. Each processor has ts ow local arbtrato crcut whch s coected to m commo arbtrato les. O these les the logcal OR fucto s formed of the correspodg outputs of the local arbtrato crcuts. I Fg. 1 ths crcut s show wth outputs postve logc correspodg to expressos (1). I a real mplemetato egatve logc would be used,.e., the outputs would be verted so that the OR fucto could be realzed as a wred-or o opecollector les. It s mportat to dstgush betwee the state of a arbtrato le ad the state of the correspodg output of a arbtrato crcut. Such a output ca be chaged by a sgle processor by assertos or wthdrawals of bts of ts arbtrato word. Ths may or may ot fluece the state of the arbtrato le. Usually the arbtrato crcuts are realzed as combatoral logc ad the OR fuctos are computed as a wred-or usg ope-collector drvers. The output sgals deped o the word assged to the processor ad o the sgals curretly formed o the arbtrato les. Ay chage of the output sgals of a processor - assertos as well as wthdrawals - wll be called a swtch of the processor. If a processor s ot partcpatg the process t does ot assert ts output sgals,.e., keeps them the "zero" state. Whe the arbtrato process starts, all partcpatg processors assert all bts of ther words oto the arbtrato les. Smultaeously they compare ther ow bt values wth the curret state of the correspodg les. If a le s set to a logc "oe" ad the correspodg bt of the processor s word s zero, the processor swtches. Ths meas t determes the most sgfcat bt that s ot equal to the state of the correspodg arbtrato le ad wthdraws ths ad all ts less sgfcat bts from the les. If the codto s o loger true the processor swtches back to the prevous state. Let all processors have the same speed. The the arbtrato process for the j-th partcpatg processor ca be descrbed by the followg expressos 1. b 1,j (t+τ)=s(t)a 1,j, b 2,j (t+τ)=s(t)a 2,j (a 1,j Β 1 (t)), b 3,j (t+τ)=s(t)a 3,j (a 1,j Β 1 (t))(a 2,j Β 2 (t)), 1 If the local arbtrato crcuts of the processors takg part the arbtrato process have dfferet delays τ for swtchg ther sgals o the arbtarto les t s ecessary to take to accout results from [6] to mmze the maxmal arbtrato tme. 3

4 ... (1) b m,j (t+τ)=s(t)a m,j r=1,...,m-1 (a r,j Β r (t)), B (t)=v j P b,j (t), =1,...,m, where j P, P s the set of processors partcpatg the process, t s the curret tme (- <t< ), a,j s bt of the word whch s assged to processor j, (a,j {0,1}, a 1,j s the most sgfcat bt), b,j (t) s the output sgal asserted by processor j oto arbtrato le, B (t) s the sgal formed o arbtrato le, τ s the delay troduced by ay processor j, (0<τ< ), to assert ts sgals oto the arbtrato les or to wthdraw them from there, s(t) s the start sgal (s(t)=0 f t<0, ad s(t)=1 otherwse), m s the umber of bts the arbtrato words,.e., ther wdth. The frst m equaltes (1) express the dvdual output sgals of all processors. The last oe descrbes the curret state of the ope collector arbtrato les whch form the logc OR fucto of the sgals asserted. Durato of the arbtrato process. The arbtrato process s a sequece of swtches. From expressos (1) t follows that every processor ca execute at most m swtches oe arbtrato process. The followg example shows that m swtches ca deed occur. Let m=4, let three processors partcpate the process, ad let them have the words, 01, ad The processors eed a frst swtch to assert these umbers oto the arbtrato les. After the frst swtch, the les are the state The secod swtch s eeded to wthdraw the thrd bt of word, the forth bt of word 01, ad the last three bts of word The arbtrato les wll therefore be the state 00 after the secod swtch. Now the frst ad the secod processors assert ther last three bts aga. Thus, after the thrd swtch, the arbtrato les wll be the state 11. Fally, the wthdrawal of the last bt by the secod processor represets the fourth swtch. The word wll appear o the arbtrato les ad wll be stable. By comparg ths word wth ts ow arbtrato word each processor decdes ow f t has the maxmal prorty. The durato of the arbtrato process s proportoal to the umber of swtches executed. Sce we cosder a stuato such as Futurebus+, ether the delays of other modules or the modules partcpatg the process or ther arbtrato words are kow. Because the durato of the process caot be determed dyamcally as a 4

5 fucto of the partcpats ad ther delays, every module has always to assume that the maxmal umber of swtches wll occur ad that they wll be executed by the slowest modules,.e., the worst case. We, therefore, ca assume wthout loss of geeralty that the delays are detcal. Thus to get a correct result of the process for ay combato of m-bt arbtrato words, the partcpatg processors have to wat the tme m τ before they make ther decso. However, the maxmal umber of swtches ca be cosderably smaller tha m as show below. Deote by K the umber of words code K,.e., ts capacty. 1. If a code of wdth m has a capacty K <2 m, t s possble to reduce the maxmal umber of swtches that ca occur f ay combato of words from K wll partcpate the process. Cosder, e.g., a code of wdth 4 ad of capacty 15. If the words ad 01 are cluded the code K ad both partcpate the process, the worst case t wll requre four swtches as show above. However, f we exclude from K ether (the oly word possbly requrg four swtches) or 01 (the oly word creatg ths kd of coflct wth ), the the arbtrato process wll requre at most three swtches for all combatos of the 15 remag 4-bt arbtrato words. I ths case a speed-up of 25% s obtaed. 2. If a code of wdth m has the capacty K =2 m, the we caot leave out ay word because all of them are use. However, such a possblty exsts, f we use a addtoal arbtrato le,.e., a (m+1)-bt code stead of a m-bt code. Now we ca tur to the frst case ad - by approprate selecto of the words - reduce the maxmal umber of swtches by a factor of approxmately 2 as show [4]. Therefore, the followg problem arses. If the wdth of a code ad ts capacty are gve, how to select ts words from the set of all possble m-bt bary words so that the maxmal umber of swtches s mmzed. Extremal codes. To compare the umber of swtches dfferet arbtrato codes we troduce the followg otatos. a*b s the word that s the cocateato of words a ad b, a () s the word that s the cocateato of words a, {a*k} s the code formed by the left cocateato of word a wth every word from code K, {a* }={ *a}={a}, X (m) s the code cosstg of all possble words of wdth m. For example, 0 (3) = 000, X (2) ={11,,01,00}; ad f a=, b= 11, K={111,1} the a*b= 11, a (2) =, a*k={111,1}. Defto 1. We defe the depth d(k) of a arbtrato code K as the maxmal umber of swtches that ca occur f ay subset Z of K wll take part a arbtrato process. 5

6 Defto 2. We defe a code of wdth m ad depth as the extremal code Ex m f there s o code of the same wdth ad depth, ad greater capacty. Usg a code of smaller depth results a faster arbtrato process. Accordg to expressos (1) the depth ca vary from 1 to m. Therefore, our task s to fd for every depth the code K of maxmal capacty. Let us frst focus o the specal classes of extremal codes of depth =0,1,m. The structures of these codes are defed the followg Lemmas. Lemma 1. The code Ex 0 m has the capacty 1 ad cossts of the sgle zero word Ex 0 m={0 (m) }. (2) The process starts at t=0. For t<0 the arbtrato les are set to zero. Thus, the arbtrato process of the sgle word 0 (m) does ot requre ay swtches. Ay other word geerates at least oe swtch durg whch ts o-zero bt values are asserted oto the arbtrato les. Lemma 2. The code Ex 1 m has the capacty (m+1) ad s costructed by ay bt permutato appled to all words of,...,m 0 (m-) *1 (). (3) Proof. Let a 1 = a 1,1...a m,1, a 2 = a 1,2...a m,2, wth a 1 >a 2, be two words of a code Ex 1 m. If a,1 =0 ( [1,m]) the a,2 =0 also has to be true. Otherwse, the arbtrato process of {a 1,a 2 } could requre two swtches, the asserto of a 1 ad a 2 ad at least the wthdrawal of the bt a,2 =1. Ths s mpossble for a code of a depth equal to 1. Therefore, w a1 >w a2 f w a1 ad w a2 are the umber of o-zero bts the words a 1 ad a 2. Let us arrage all words of the code creasg order ad prove that ther umber s s equal to m+1. If w j s the umber of o-zero bts the word a j, the the expresso 0 w a1 <w a2 <...<w as m s vald. Obvously, the umber of o-zero bts the word a j s (j-1) ad the maxmal possble value of s s m+1. These codtos are oly satsfed for the codes Ex 1 m defed Lemma 2. For them the sgals B (t) are settled after the frst asserto of the maxmal word,.e., after the frst swtch. Fg. 2 shows examples of codes Ex 1 4 wth the correspodg permutato of the words defed by (3). Lemma 3. The code Ex m m has the capacty 2 m ad cossts of all possble m-bt words Ex m m=x (m). (4) 6

7 The code X (m) has the maxmal possble capacty accordg to ts defto ad ca cause a sequece of m swtches but ot more, as t was show [3,4]. Theorem. The capacty of ay extremal code Ex m s gve by the equato Ex m =Σ,..., C m, m 1, m 0 (5) where C m are the bomal coeffcets. Proof. Equalty (5) s obvously true for the cases =0,1,m that have bee cosdered the prevous Lemmas. We wll prove (5) for m>>1 by a double ducto wth parameters ad m. The hypothess s that (5) s true for the codes Ex m-1 ad Ex -1, Ex,..., Ex m-2. From ths we wll derve that equato (5) holds true for the code Ex m. Ths ducto starts from the cases =0,1,m cofrmed by the Lemmas, ad proves the Theorem up to ay gve m ad the progresso Ex 2 2,Ex 0 m Ex 2 3 Ex Ex 2 m; Ex 3 3,Ex 1 m Ex 3 4 Ex Ex 3 m;... Ex,Ex m Ex +1 Ex Ex m. Bass. Assume that the followg equatos are true. Ex m-1 =Σ,..., C m-1, (6) Ex k-1 =Σ,..., C k-1, k=,(+1),...,(m-1). (7) Iducto. We ted to prove (5) for the code Ex m for m>>1 assumg the truth of the bass. Frst we costruct a certa code R m ad cosder ts depth ad capacty. The code R m= j=-1,...,m E j (8) s the uo of the followg subcodes (Fg. 3a) E -1 ={1 (m-+1) *X (-1) }, E k ={1 (m-k) *0*Ex k-1}, k=,(+1),...,(m-1), E m ={0*Ex m-1}. 7

8 Obvously E p E q = for ay par of subcodes (p,q [-1,m]). Thus, the capacty of the code R m s the sum of the capactes of all subcodes R m = E -1 + E + E E m-1 + E m = = X (-1) + Ex -1 + Ex + + Ex m-2 + Ex m-1. Takg to accout the expressos (6) ad (7), assumed to be true as the bass, we ca derve (see Appedx) R m =2-1 +Σ,..., C m-1+σ k=,...,m-1 Σ,..., C k-1=σ,..., C m. (9) Below t wll be show that the depth of R m s. All words of E -1, E k, ad E m cosst of two parts, a detcal leadg part, ad a tralg part whch s dfferet for every word (Fg. 3a). Thus f oly words of oe of the subcodes E -1, E k, or E m partcpate the arbtrato, swtches happe oly the secod part. Therefore, the process takes o more tha -1,, or swtches correspodgly. Below we show that f words from dfferet subcodes partcpate the process the t caot have more tha swtches. Cosder the case whe words from E m ad E m-1 are partcpatg a arbtrato process smultaeously. The words from E m-1 are headed by the bts 1, ad the words of E m are headed by 0. Thus, after the frst swtch the logcal OR of the words of E m ad E m-1 appears o the bus. After the secod swtch all words of E m wll be wthdraw ad oly the words of E m-1 are left. I the worst case addtoal swtches are requred to determe the maxmum word amog them. The same argumet apples to ay other combato of subcodes from R m whch proves that the depth of R m s. The code R m has depth, wdth m, ad the capacty defed by (5). Let us show that there exsts o other code of the same depth ad wdth whch has a capacty greater tha R m,.e., R m s the extremal code. Assume the cotrary,.e., that a code S m of m-bt words wth d(s m)= ad S m > R m exsts. I order to costruct the code S m a form smlar to R m let us troduce the followg masks L -1 ={1 (m-+1) *X (-1) }, L k ={1 (m-k) *0*X (k-1) }, k=,(+1),...,(m-1), L m ={0*X (m-1) }. The uo of these masks s the code cosstg of all m-bt bary words X (m). So the code S m s a uo of subcodes (Fg. 3b): 8

9 where S m= j=-1,...,m T j T -1 =L -1 S m={1 (m-+1) *T -1 } T k =L k S m={1 (m-k) *0*T k-1 }, k=,(+1),...,(m-1) T m =L m S m={0*t m-1 }. As before T p T q = ; p,q [-1,m]; for ay par of subcodes ad therefore S m = T -1 + T + T T m-1 + T m. To estmate S m, cosder the capactes of the dvdual subcodes. Sce subcode T -1 cossts of (-1)-bt words ts capacty does ot exceed 2-1, so T -1 = T = X(-1). The depth of the subcode T m-1 caot be greater tha. Otherwse the depth of T m ad therefore the depth of S m would exceed. Sce T m-1 cossts of (m-1)-bt words ts capacty has the upper lmt Ex m-1. For T m we therefore have T m = T m-1 Ex m-1. Sce the capacty of T -1 caot exceed that of E -1, ad sce the capacty of T m caot exceed that of E m, the equalty T k > E k has to hold true at least for oe k [,m-1] to meet the assumpto S m > R m. We defe s as the mmal value of the dex k for whch the equalty T k > E k holds. Sce T s = T s-1 > E s ad E s = Ex s-1, the depth d(t s )=d(t s-1 ) s greater tha (). Let us form the code D s+1 =T s+1 T s+2... T m. Cosder ths code the bts of colum (m-s+1) (see Fg. 3b). If D s+1 there exsts a word a j wth a o-zero bt (ms+1), the the arbtrato process for the code Z=T s {a j } ca requre more tha swtches. They are: the asserto of the words of Z (the frst swtch), the wthdrawal of the o-zero bt (m-s+1) of the word a j (the secod swtch), ad the settlemet of the maxmal word of the code T s-1 (a sequece of more tha () swtches). Ths s mpossble for the code S m of depth. So, all words of D s+1 as well as the words of T s must also have zeros colum (m-s+1). Let us form the code D* s by elmato of ths colum (m-s+1) from the code D s =T s D s+1. Obvously the depth d(d* s ) s stll equal to d(d s ). Furthermore, the depth d(d s ) as well as the depth of ay other subcode of S m caot be greater tha. The we have D* s < Ex m-1 for the capacty of the code D* s of (m-1)-bt words. Therefore, D s = D* s < Ex m-1. Now we have upper bouds for the subcodes of S m. Takg to accout the bass (6) ad (7), we ca fally estmate S m as S m = T -1 + T + T T s-1 + D s X (-1) + Ex -1 + Ex + Ex s-2 + Ex m-1 = =2-1 +Σ k=,...,s-1 Σ,..., C k-1+σ,..., C m-1 9

10 After comparg ths expresso wth (9), we drectly obta S m <Σ =1,...,m C m= R m () sce s<m. So the assumpto S m > R m leads to the cotradcto () whch proves that a code wth capacty greater tha R m caot exst. Let us prove ow that ay code S m of depth whch s dfferet from R m has a smaller capacty. Assume cotrast that S m = R m ad S m R m. If T k > Ex k-1 or d(t k)>() holds true for some k [,m-1] the the prevous argumet brgs us back to () whch cotradcts the assumpto. O the other had, T k < E k-1 for all k=,...,m-1 drectly yelds S m < R m. The oly alteratve left s T k = E k ad d(t k ) () for all k=,...,m-1. Ths s possble oly f T k-1 =Ex k-1 ad T k =E k. I the same way we have T m =E m, T -1 =E -1, ad therefore S m=r m. So, o code of depth ad wdth m ca have a hgher capacty tha determed by (5). Q.E.D. We proved that the capacty of the extremal codes s determed by expresso (5) for all m ad. Some values are gve Tab. 1. I [4] t was show that the capacty of the bomal codes of wdth m ad depth s also determed by the same expresso. Therefore, the bomal codes are extremal. Costructo of the Extremal Codes. Durg the proof of (5) t has bee show that ay code R m costructed accordace wth (8) has the depth ad the maxmum capacty. It also has bee prove that ay code dfferet from R m has a lower capacty. I vew of ths, expresso (8) represets the geeral structure of the extremal codes for ay ( 0,1,m). We ca, therefore, rewrte (8) as Ex m={0*ex m-1} {1 (m-+1) *X (-1) } =1,...,m- {1 () *0*Ex m--1}, 0,1,m. (11) Usg expressos (2)-(4) ad (11) the extremal codes of ay depth ad wdth ca be costructed. For eve ( 0,m) expresso (11) allows to costruct the code Ex m from the codes of depth 0, whch are determed uquely by Lemma 1. So for eve oly oe extremal code exsts. Sce the bomal code wth the same parameters m ad has the capacty gve by (5) t s ths extremal code. If s odd ( 1,m) the the codes Ex m are costructed from the codes of depth 1, whch are ot uque (Lemma 2). So t s possble to costruct dfferet extremal codes of the same odd. Here the bomal codes are oly oe example amog them. I Fg. 4 some examples of codes Ex 5 are show. The code Ex 3 5, e.g., s formed accordace wth (11) as

11 {0*Ex 3 4} {1 (3) *X (2) } {1 (2) *0*Ex 1 2} {1*0*Ex 1 3} where Ex 3 4={0*Ex 3 3} {11*X (2) } {*Ex 1 2} s also gve by (11). It s easy to determe the umber γ m of dfferet codes Ex m. From Lemmas 1-3 we have γ 0 m=γ m m=1 ad γ 1 m=m!. For other follows drectly from (11). γ m=γ m-1 γ m-2 γ m-3... γ -1 for m>>1 It allows to calculate γ m recursvely. Some values of γ m are gve Tab. 2. Applcato. Havg derved the expressos for the capactes of extremal codes we ca calculate the achevable speed-up. Cosder a system that uses a code K of wdth m, K <2 m. If the arbtrato words of ths code are chose wthout takg to accout ts depth, the the chaces are hgh to get a depth equal to m. Sce the system uses ot all possble bary words of wdth m, t s possble to replace the orgal code K by a extremal code order to reduce the maxmal umber of swtches. To acheve the maxmal speed-up oe has to fd a code Ex m of capacty Ex m K ad mmal depth. Now f K s replaced by Ex m, the maxmal umber of swtches s reduced from m to. Ths results a speed-up of the arbtrato process by factor of m/. Obvously the maxmal speed-up s s obtaed f the depth of the code K s mmal ad the umber of processors p s maxmal. For arbtrato the mmal s equal to 1. Here the maxmal umber of processors s the wdth of the arbtrato words plus 1. Thus, the hghest speed-up s obtaed wth a relatvely small umber of processors. Nowadays ths s the most relevat case. Cosder, e.g., a Futurebus+ system wth e processors. Sce Futurebus+ uses a prorty word of wdth m=8, t s possble to acheve a speed-up s=8 by applyg the extremal code of depth =1. For a hgher umber of processors the achevable speed-up decreases because codes Ex m of creasg depth are requred to assg arbtrato words to all processors. Tab. 3 shows the achevable speed-up for a system where all arbtrato words are used. I ths case the speed-up s obtaed by provdg addtoal arbtrato les. Ths allows to apply extremal codes of hgher wdth ad smaller depth. Note that the arbtrato tme ca be reduced by a factor of 2 by addg a sgle le oly [4]. Hgher speedups ca oly be obtaed usg addtoal arbtrato les. Note that after the maxmal word s fally asserted oto the arbtrato les swtches the local arbtrato crcuts of processors ca stll occur. Cosder, e.g., a arbtrato betwee the two words { 1111, 11 }. The arbtrato process takes oly oe swtch to assert the word After ths momet the processor havg the word 11 wth- 11

12 draws ts least sgfcat o-zero bt. Depedg o the techology used ths wthdrawal mght or mght ot fluece the arbtrato le whch s already set to a logc "oe". If, e.g., ope collector les are used to mplemet wred OR logc, the wthdrawal of a sgal from a le the logc "oe" state may result a "zero" gltch [5]. It s possble to avod ths gltch by usg the code =1,...,m {1 () *0 (m-) } as the Ex 1 m code. I ths case the code Ex m s uquely defed for ay m ad. For eve t cocdes wth the bomal code. Coclusos. I ths paper we derved the structure of codes whch reduce the maxmal arbtrato tme to a defed lmt. We obtaed a recursve formula that allows oe to costruct codes of hghest capacty for all possble umbers of swtches. No hgher speed-up ca be acheved by usg ay other code. It was show that, some cases, a cosderable umber of extremal codes exst. Bomal codes are oly oe represetatve of these codes. The recursve formula (11) that gves us the geeral structure of extremal codes s qute complcated. However, oe has to use t oly oce whe the arbtrato system s set up. For most arbtrato systems the codes ca be appled drectly wthout chages of the hardware. APPENDIX. Here we preset the detaled dervato of (9). The propertes m 1) C m = 2 m, m 0, m 2) = +1 C = Cm+1, m 0, ) C m-1 + Cm-1 = C m, m-2 0 are used below. For ay m ad (m>>1) we ca derve m j=-1 =2-1 + C j + m-2 j= C m-1 C j - =2-1 + j= m-2 j=-1 C j + C m-1 C j + C m-1 =2-1 + =2-1 + m-2 j= m-2 ( C j - j=0 j= j C j- j= C j + C j )+ 1, 2) C m-1 = C m-1 = 12

13 = C m j=0 2 j + =o C m-1 = ) + C m-1+cm-1+1= C m-1 C m-1 = (Cm Cm-1 )+ 3) ) = C m +Cm-1 +Cm = C m +Cm+1= C m C m - C m 0 +1=. Refereces. [1] Borrll P. "A Comparso of 32-bt Buses", IEEE Mcro, Dec. 1985, pp [2] Taub D.M. "Corrected Settlg Tme of the Dstrbuted Parallel Arbter", IEE Proc. E, Vol. 139, No. 4, 1992, pp [3] Taub D.M. "Improved Cotrol Acqusto Scheme for the IEEE 896 Futurebus", IEEE Mcro, Vol. 7, No. 3, 1987, pp [4] Makhaok M., Cheravsky V., Mäer R., Stucky O.: "Bomal Codg Dstrbuted Arbtrato Systems", Electr. Lett., Vol. 25, No. 20, 1989, pp [5] Gustavso D.B., Theus J.: Wre-OR Logc o Trasmsso Les; IEEE Mcro, Vol. 3, No. 3, 1983, pp [6] Makhaok M., Mäer R.: "Sychrozato of Parallel Processes Dstrbuted Systems; Bougè L., Cosard M., Robert Y., Trystram D. (Eds.): Proc. CONPAR 92 - VAPP V, Lyos, Frace; Lect. Notes Comp. Sc. 634, Sprger, Berl (1992) pp

14 Captos. Fg. 1. Local arbtrato crcut. Fg. 2. Examples of extremal codes Ex 1 4. Fg. 3. The structure of the sets R m ad S m. Fg. 4. Examples of extremal codes Ex 5. Tab. 1. The capacty Ex m of extremal codes of depth ad wdth m. Tab. 2. The umber γ m of extremal codes of depth ad wdth m. Tab. 3. The speed-up s for dfferet word wdths m ad umber of processors p=2 m acheved by addg dm arbtrato les. 14

15 d m Table 1. m Table 2. m p dm Table 3. 15

16 s(t) b 1,j a 1,j B 1 b 2,j a 2,j B 2 b 3,j a 3,j B 3 b 4,j a 4,j B 4 wo/lost Fg. 1 16

17 ( )( 1243 )( 1324 ) ( 4321 ) Fg. 2 E -1 E E +1 E m-1 E m X (-1) Ex Ex Ex m Ex.. m-1 0 a) R m Fg. 3 T -1 T T s D s (m-s+1)-th colum b) S m T T -1 T s-1 D s 17

18 X (3) Ex Ex X (2) 1 Ex 2 1 Ex 3 X (2) 1 Ex 2 3 Ex 3 Ex Ex 5 5 Ex 5 4 Ex 5 3 Ex 5 2 Ex 5 1 Fg. 4 18