High-Speed Computation of the Kleene Star in Max-Plus Algebra Using a Cell Broadband Engine

Transcription

1 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING High-Speed Computatio of the Kleee Star i Max-Plus Algebra Usig a Cell Broadbad Egie HIROYUKI GOTO ad TAKAHIRO ICHIGE Departmet of Maagemet ad Iformatio Systems Egieerig Nagaoa Uiversity of Techology Kamitomioamachi, Nagaoa, Niigata JAPAN {hgoto@js, s83354@ics}.agaoaut.ac.jp Abstract: - This research addresses a high-speed computatio method for the Kleee Star of the weighted adjacecy matrix i max-plus algebraic system. We focus o systems whose precedece costraits are represeted by a directed acyclic graph (DAG), ad implemet o a Cell Broadbad Egie (CBE) processor. Usig the implemetatio o a Soy Playstatio3 (PS3) equipped with a CBE processor, we attempt to achieve a speedup by usig two approaches: parallelizatio ad SIMDizatio (Sigle Istructio, Multiple Data). We foud that the SIMDizatio is effective regardless of the system s size ad the umber of processor cores used, ad that the scalability of usig multiple cores is remarable especially for systems with large umber of odes. Key-Words: - Directed acyclic graph, max-plus algebra, parallel processig, SIMD; Cell Broadbad Egie 1 Itroductio This research aims to develop a fast computatio method for the trasitio matrix i a state-equatio i max-plus algebra [1]. For the computatio, we use a processor called the Cell Broadbad Egie [2], [3] which has istructios for vector operatios ad parallel processig. We assume that the precedece relatioships of the system are described by a Directed Acyclic Graph. The behavior of this class ca be represeted by a pair of simple equatios i max-plus algebra, referred to as the state ad output equatios. Sice this is similar to the represetatio i moder cotrol theory, a umber of research developmets i moder cotrol theory have bee applied to various ids of schedulig problems [4], [5]. The bottleec of computig the equatios is i the trasitio matrix of the state equatio. The matrix is obtaied by applyig the Kleee Star [1] operator to the adjacecy matrix. For efficiet computatio, algorithms with O( ( + m)) time complexity have bee proposed [6], [7], where ad m represet the system size ad the umber of precedece costraits, respectively. I particular, the algorithm proposed i [7] has the followig attractive features: Partitioig ca be carried out by arbitrary colum. The resultig matrix is obtaied row by row. However, the paper oly presets a implemetatio usig a sigle processor, thereby the above advatages are ot cosidered. As such, this research addresses a implemetatio usig SIMD istructios [8] ad multiple cores i parallel, both of which are available o the CBE. We mae experimets o a Soy Playstatio3 equipped with a CBE. 2 Mathematical Prelimiaries 2.1 Max-plus algebra Defie a field R max R { }, where R represets the real field. Operators for additio ad multiplicatio are defied as: x y = max( x, y), x y = x + y. Deote the uit elemets for these by ε ( = ) ad e (= ), respectively. The priority of is higher tha, ad it is abbreviated if o cofusio is liely to arise. For a operatio o multiple umbers, if m, = m x = max( xm, xm + 1, L, x ). For matrices, [] represets the ( i, j) th elemet of matrix. If m l, Y R max ad Z R max, [ Y] = [ ] [Y ], [ Z] = = 1[ ] i [ Z] j. Let the uit matrices for operators ad be deoted by ε ad e, respectively. ε is a matrix, all elemets of which ISSN: ISBN:

2 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING are ε, whereas e is a matrix, all diagoal elemets of which are e ad all off-diagoal elemets are ε. 2.2 State equatio ad Kleee star Terms for maufacturig systems are used to facilitate uderstadig. Let the umber of facilities be, ad the job umber be. The the processig completio times of job i all facilities is deoted by x () ad the costraits regardig their miimum values by u (). Uder these otatios, the state equatio is expressed as follows [1]: x( ) = A [ x( 1) u( )], (1) where x () ad u () are also referred to as the state ad iput vectors, respectively. A is ow as the system or trasitio matrix. With the help of Eq. (1), the earliest times of evets for job ca be obtaied. Let the list of precedig facilities of facility i ( 1 i ) be P (i), ad the processig times of job be d (). Suppose the precedece costraits with respect to the processig order relatio are give by the followig adjacecy matrix: [ F ] = { e : j P( i), ε P( i)}. I terms of graph theory, facilities ad precedece costraits correspod to odes ad arcs, respectively. It is ow that the system matrix A ca be represeted as A = P, where P = diag[ d( )] ad = F P. is also referred to as a weighted adjacecy matrix. Operator is called the Kleee Star. With a calculatio of the Kleee Star, reachabilities betwee two arbitrary facilities ca be idetified ad propagatio times of evets be calculated. Let the weighted adjacecy matrix be deoted simply by. If the precedece costraits of the system are represeted by a DAG, the Kleee Star is calculated as: s 1 l s 1 = = e L, (2) l= s 1 s where ε, = ε ( 1 s ), ad s depeds o the precedece costraits of the system. i Eq. (2) has the followig properties: [ ] τ = ε e : if ode i is reachable from ode j, : if ode i is ot reachable from ode j, : if i = j, where τ is the sum of the weights of arcs from ode j to ode i. If there are multiple paths betwee two odes, it gives the maximum value. 3 Implemetatio o a CBE 3.1 Structure of the CBE We briefly overview of the structure of the CBE istalled o a PS3. I the CBE processor, there is a all-purpose processor called a PPE (PowerPC Processor Elemet), ad eight processors called SPEs (Syergetic Processor Elemets). Of these SPEs, the umber available i Liux is six. The PPE is a 64-bit processor with a PowerPC architecture, that drives the operatig system, as well as beig i cotrol of the SPEs. O the other had, the SPE is a processor with 128-bit registers that supports SIMD operatios, which allow multiple elemets to be calculated with a sigle istructio. I a operatio o 32-bit float variables, four elemets ca be calculated. For the mai memory, the processor is equipped with 256MB of fast memory, called DR (etreme Data Rate) DRAM. Each SPE has 256KB iteral RAM, ow as the LS (Local Store) which ca be accessed from the correspodig SPE very quicly. Data trasfer betwee the LS ad DR is accomplished by DMA trasfer through a iterface called the MFC (Memory Flow Cotroller). 3.2 Outlie of the algorithm Let the weighted adjacecy matrix be. First, we carry out a topological sort [9]. If ode j is located upstream of ode i physically, we deote this by j p i. Moreover, we deote the idex of ode i after the topological sort by î. A topological sort is a method for sortig the odes such that: if j p i, ˆ j < iˆ. If we adopt the DFS (Depth First Search) algorithm, the time complexity is O( + m), where ad m represet the umber of odes ad arcs, respectively. Note that the result is ot uique ad depeds o the implemetatio. Next, we calculate the values of from the topologically upstream odes to the dowstream odes. First, the wor matrix Z is iitialized as Z e. For the i th ode i topological order, let the ISSN: ISBN:

3 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING origial ode umber be deoted by i. Moreover, let the collectio of succeedig odes of ode ~ ~ l be S (l ). The, we update the wor matrix Z accordig to the followig procedure: [ Z] i j [ Z] i j [ ] ~ [ Z]~, il l j ~ for all i S(l ) ad j ( 1 j ). (3) This is repeated for all ~ l ( 1 l ) by icremetig l. After this procedure, the values of are stored i Z. 3.3 Iteral represetatio of matrices As the iteral represetatio of the matrices, this research adopts the method proposed i [1]. We use four-byte float variables for represetig values. The elemets of vectors ad matrices are stored i a oe-dimesioal array, ad those for matrices are i a row-major order. This structure is stadard i several laguages such as C, but differs slightly i respect of the followig: A uit bloc is defied as 16 bytes, which is equal to the size of the registers i the SPE. Ay o-square matrix is exteded to a square matrix. The umber of rows ad colums are rouded up to a multiple of 4. The exteded verbose elemets are iitialized to ε. The iteral structure of the m matrix is illustrated i Fig. 1, where x y represets the value of x rouded-up to a multiple of y. To represet a colum vector, we prepare a array for a row vector with the same size, ad distiguish its type by attachig a attributio variable. With regards the secod feature above, extedig rows ad colums is aalogous to addig dummy odes, which does ot participate i ay calculatios of the target odes. 1 bloc 4 b = ( m ) / 4 blocs ε m 4 b rows Fig. 1. Iteral structure of a matrix. 3.4 Allocatio of SPEs As metioed earlier, each SPE o the CBE istalled o a PS3 has 256KB memory as the LS, which meas that all the elemets of a large-scale matrix caot be hadled by a sigle SPE at oe time. Thus, this research divides the problem ito several blocs. We partitio the wor matrix depicted i Fig. 1 ito several vertically log matrices, where each partitio is assiged to oe SPE. Let the umber of odes ad SPEs allocated be ad p ( 6), respectively. For the maximum umber of colums that a sigle SPE ca store i the available LS, let the value trucated to a multiple of 4 be s. We eforce each SPE to process for 4 = / p colums. If > s, the SPEs have to compute the resultig partial matrices step by step. 3.5 Details of the algorithm We cosider a method for creatig a list of arcs for a adjacecy matrix, amely a adjacecy list. This is required as pre-processig for the topological sort. The set of succeedig odes S ( j) for ode j is obtaied as follows: S ( j) { i [ ] ε}. (4) = First, we partitio the iput matrix ito vertically log bloc matrices as show i Fig. 2. Each of these is allocated to oe SPE, where is rouded dow to a multiple of 4 to comply with the SIMDizatio. For each partitioed bloc matrix, we ispect whether or ot [ ] = ε holds for all 1 j, ad repeat this for all i ( 1 i ). This process is equivalet to ispectig whether or ot i is icluded i the set S ( j) i Eq. (4). The we store the ispectio results i a array of usiged short type. Let us deote this array by U hereafter. I the first elemet of the i th row of U, the umber of elemets of S ( j) is stored, ad the cotets of S ( j) are stored i the succeedig elemets. After the ispectio process for all rows, the resultig matrix is trasferred from the SPE to the PPE. O the other had, the PPE waits util all trasfers of the resultig matrices from the SPEs have bee completed. Subsequetly, we carry out a topological sort based o the DFS algorithm. We have built a experimetal implemetatio usig oly a PPE ad ISSN: ISBN:

4 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING measured the computatio time thereof. Sice the time for the sort is much smaller tha the time for computig, we perform the topological sort usig oly the PPE. We deote the array storig the result of the sort by T. Next, for all sets ( i, j) that satisfy [ ] ε, we create lists of the succeedig odes S ( j) for ode j ad the weights of arcs [ ]. These are created sequetially from topologically upstream odes to dowstream odes with respect to j. Let the lists be deoted by N, L ad W, where the first two lists ad the latter oe are arrays of usiged short ad float types, respectively. N ad L hold a list of the umber of succeedig odes ad the correspodig ode umbers, respectively. The former list is created from the first elemets of each row i U whilst the latter is from the succeedig elemets. This is performed i the topological order specified by T. Moreover, cosider creatig W that holds the weights of the arcs. The weights are obtaied from [ ], where i ad j represet the precedig ad succeedig ode umber give by the elemet of T ad L, respectively. For the performace of creatig the above four lists, we have built a experimetal implemetatio usig oly a PPE. Sice the computatio time for creatig these is much smaller tha the overall computatio, we calculate this part usig the PPE oly. The above lists are passed to all SPEs for use i calculatig the partial matrices of. The, the partial matrix of is computed based o Eq. (3). O the SPE side, we dyamically allocate a data area for storig the partial matrix show i Fig. 2, ad iitialize the elemets to be a partial matrix of the uit matrix e. Next, we update the values of the partial matrix based o Eq. (3), where j is executed oly for the allocated colums. Moreover, if > s p, the update is carried out for s colums at oe time, ad this is repeated multiple times. 4 rows colums SPE 1 SPE 2 SPE p Fig. 2. Allocatios to multiple SPEs. 3.6 SIMDizatio We use multiple SPEs i the followig two procedures: Geeratig a adjacecy list from the weighted adjacecy matrix, equivalet to the set of succeedig odes S. Calculatig a partial matrix of from lists T, N, L ad W. With respect to the first issue, cosider the partial matrix of the weighted adjacecy matrix show i Fig. 2. I each SPE, the values of [ ] ( 1 j ) are ispected at every trasfer of a row from the PPE. If a value is aythig other tha ε, we icremet the ( j,1) th elemet of the array i U, ad the apped the ode umber i to the last positio of the correspodig row. We ca utilize SIMD istructios for the ispectio irrespective of whether or ot the elemets are ε. With respect to the secod issue, every SPE has the same lists T, N, L ad W, ad each calculates the values of the assiged colums of. Accordig to Eq. (3), the values are obtaied sequetially from the topologically upstream odes to the dowstream odes. The essece of this procedure cosists of a simple operatio i which the values of the i th row multiplied by a costat are added to other rows. Thus, a SIMDizatio of this procedure is straightforward. Moreover, we pay attetio to the feature that, at the start of processig the i th row, the correspodig row will ot be chaged ay more. This meas that the values of the i th row of are already fixed before processig the i th row. Accordigly, if we issue a trasfer commad to the PPE i o-blocig mode before processig the i th row, we ca reduce the computatio time. ISSN: ISBN:

5 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING 3.7 Overall flow We cofirm the overall flow of the proposed method. Fig. 3 gives a flowchart of the overall process of calculatig after the weighted adjacecy matrix has bee give, ad defies the roles of the PPE ad SPEs. I the calculatio of, each SPE is ivoed twice. The first time is to create a list of succeedig odes ad to retur the result to the PPE. O the PPE side, barrier sychroizatio is performed whilst receivig the results from all the SPEs. The secod ivocatio is to calculate a partial matrix of, with the result retured to the PPE row by row i o-blocig mode for every process of oe row. O the PPE side, barrier sychroizatio is oce agai performed to receive the results from all the SPEs. Sice the required size of worig memory differs betwee the first ad secod ivocatios, the maximum umber of colums that a SPE ca hadle at oe time, s, may differ. give Start determie, p partitio to,,l 1 2 barrier topological sort T create lists N LW determie, p barrier retur ed PPE SPE1 SPE2 1 2 adjacecy list calculate wait for a request 1 U 1 U 2 wait for the ext request 2 Fig. 3. Roles of the PPE ad SPEs i computig. 4 Performace Evaluatio The speedup effect of SIMDizatio ad parallelizatio is examied. The executio eviromet is a Soy Playstatio III, ruig Fedora Core 1. We use gcc as the compiler ad the CELL Software Developmet Kit (SDK) Versio.3.1. The followig commads are used for compilig programs for the PPE ad SPEs: PPE: ppu-gcc: -O3 m64 maltivec mabi=altivec SPE: spu-gcc: -O3 We reserve 128KB of memory i the LS of a SPE for the data area, which is shared by the partial matrix ad the lists for succeedig odes. For the maximum size of lists L ad W, we limit the maximum elemets to 496. Regardig the system s size, we cosider cases where the umber of odes are = 1, 2, 4, 1 ad 2. For the adjacecy matrix F, we attach the costrait i j with a probability of.5 for all pairs of ( i, j) that satisfy ip j. This meas that the umber of arcs is m ( 1) / 4 +. The, we sort the idexes of odes radomly ad use the resultig matrix as the adjacecy matrix F. For the diagoal matrix P that represets the weights of odes, suppose the diagoal elemets obey the ormal uiform distributio [,1]. We measure the computatio time of calculatig from the time whe ( = FP ) is give. This measuremet is executed te times, ad the average time is adopted. Tables 1 ad 2 show the computatio times of the SIMDized ad o-simdized fuctios, respectively. The uit used i all cases is microsecods. The first colums represet the umber of SPEs allocated, equivalet to p. Regardig p = 1, the computatio times icrease 3 i proportio to roughly. This would because the umber of arcs is m ( 1) / 4 + ad thus the computatio time complexity is O( 3 ). I respect to the effect of usig multiple SPEs, the computatio times are reduced as p icreases. We ow examie speedup effects of the help of usig multiple SPEs. By defiig a idex for evaluatig scalability as: Computatio time usig a SPE / Computatio time usig p SPEs, ISSN: ISBN:

6 Proceedigs of the 9th WSEAS Iteratioal Coferece o APPLICATIONS of COMPUTER ENGINEERING the idex is calculated for all cases based o Table 1. Figure 4 depicts the speedup effects of usig multiple SPEs. For large scale systems = 1 ad = 2, the performace curves icrease almost liearly, ad the scalability idices for p = 6 are 4.54 ad 5.2, respectively. Thus, the proposed algorithm would be eough scalable for large scale systems. For smaller-scale systems 4, the curves icrease but ot that remarable compared with the large-scale oes. I additio, they are liely to saturate aroud 4 p 7. This would because the relevat overheads regardig the cotrol of the SPEs ad the DMA trasfers become relatively large. Table 1. Computatio times of the SIMDized fuctio. SPE s ,732 39, ,816 6,957, ,769 24,47 356,795 3,64, ,213 19, ,142 2,513, ,954 14,446 27,17 1,949, ,748 14, ,75 1,627, ,678 14, ,21 1,385,228 Table 2. Computatio times of the o-simdized fuctio. SPE s ,351 33, ,486 3,234,94 27,269, ,847 13, ,751 1,652,351 13,972,26 3 1,382 9,288 1,715 1,121,28 9,42, ,127 7,454 68,69 894,86 7,135, ,97 69,394 74,23 5,836, ,464 69, ,397 4,843,476 Speedup effect =1 =2 =4 =1 = Number of SPEs Fig. 4. Speedup effects of the SIMDized algorithm. 5 Coclusio This research has ivestigated the speedup effect of calculatig the Kleee Star for a weighted adjacecy matrix of a DAG type i max-plus algebraic system. By implemetig a program adapted for a Cell Broadbad Egie processor, the overall results of the umerical experimets idicate that the computatio time ca be reduced if we use multiple SPEs ad perform SIMDizatio. Refereces: [1] B. Heidergott, G. J. Olsder, ad L. Woude, Max Plus at Wor: Modelig ad Aalysis of Sychroized Systems, Priceto Uiversity Press, New Jersey, 26. [2] J. Kahle, M. Day, H. Hofstee, C. Johs, T. Maeurer, ad D. Shippy, Itroductio to The Cell Multiprocessor, IBM Joural of Research ad Developmet, Vol.49, No.4/5, 25, pp [3] M. Scarpio, Programmig The Cell Processor: for Games, Graphics, ad Computatio, Pretice Hall, New Yor, 28. [4] G. Schullerus, V. Krebs, B. De Schutter, ad T. Boom, Iput Sigal Desig for Idetificatio of Max-Plus-Liear-Systems, Automatica, Vol.42, No.6, 26, pp [5] R. Goverde, Railway Timetable Stability Aalysis Usig Max-Plus System Theory, Trasportatio Research Part B, Vol.41, No.2, 27, pp [6] H. Goto, Efficiet Calculatio of the Trasitio Matrix i a Max-Plus Liear State-Space Represetatio, IEICE Trasactios o Fudametals, Vol.E91-A, No.5, 28, pp [7] H. Goto ad H. Taahashi, Fast Computatio Methods for the Kleee Star i Max-Plus Liear Systems with a DAG Structure, IEICE Trasactios o Fudametals, Vol.E92-A, No.11, 29, pp [8] P. Cocshott ad K. Refrew, SIMD Programmig Maual for Liux ad Widows, Spriger, Heidelberg, 24. [9] T. Corme ad C. Leiserso, Itroductio to Algorithms, MIT Press, Massachusetts, 21. [1] H. Goto ad A. Kawamiami, A Efficiet Solver for Schedulig Problems o a Class of Discrete Evet Systems Usig CELL/B.E. Processor, Proceedigs of the 9th Iteratioal Coferece of Itelliget Systems Desig ad Applicatios, 29, pp ISSN: ISBN: