Ant Colony Optimal Algorithm: Fast Ants on the Optical Pipelined R-Mesh

Transcription

1 At Coloy Optimal Algorithm: Fast Ats o the Optical Pipelied R-Mesh Ke D. Nguye ad Au G. Bourgeois Departmet of Computer Sciece Georgia State Uiversity P.O. Box 3994 Atlata, Georgia {kguye, abourgeois}@cs.gsu.edu Abstract I this paper, we demostrate how to implemet ad improve two At Coloy Optimiatio (ACO) algorithms o the Optical Pipelied Recofigurable Mesh (PR-Mesh): the geeric ACO ad the Fast At Coloy Optimiatio (FACO) algorithm. The ru-time complexity of our improved geeric ACO algorithm, with x geeratios each geeratio havig m ats, o a x PR-Mesh is O((x m + )log), which outperforms the curretly best kow electrical model implemeted i [3] with ru-time complexity of O(x (m + )log). Our FACO algorithm o PR-Mesh yields O((( log 2 )+ log ) loglog) ru-time complexity for 2 jobs while the existig FACO algorithm o the electrical model yields a ru-time complexity of O(( + )log ) but ca oly hadle log 2 jobs, where is the total umber of ats from all geeratios. I additio, we propose a theoretical FACO algorithm o a three dimesioal PR-Mesh solvig 2 jobs i O(x (m + ) log) time. I. Itroductio The At Coloy Optimiatio (ACO) algorithm is a greedy approach. It has bee used to solve may combiatorial NP problems such as the Travelig Salesma Problems (TSP) [1], the Flow-Shop problems [2], the Sigle Machie Total Tardiess problem (SMTTP) [3], the Resource Costraied Project Schedulig problem [4], the Quadratic Assigmet problem (QAP) [5] [6], graph colorig [7], ad routig i commuicatio etworks [8] [9], etc. The ACO algorithm was itroduced by Marco Dorigo [10] i his dissertatio i 1992 ad is described by the behavior of ats fidig food. I geeral, ats from a coloy will dispatch searchig for food i ay radom path. Each at will leave some pheromoe o its path to, ad from, the food source. The amout of pheromoe ca be differet depedig o the goodess of the food foud. A portio of pheromoe o the path will be evaporated after a period of time. All followig ats leavig the coloy will follow the path(s) that has the strogest pheromoe. A geeratio of ats is defied by the order i which they are leavig the coloy. All ats i a geeratio are workig idepedetly. Sice the ats keep followig the strogest pheromoe path(s), oly the best foud path(s) will remai. Because of the ature of the ACO algorithm, it is best to implemet the algorithm o a parallel computer model, where each at ca be simulated by a processig elemet (PE) ad the pheromoe values o the path(s) are represeted by a two dimesioal matrix. There are may approaches to implemet the ACO. For example, Delisle P. et al. [17] implemeted the ACO o OpeMP machies with a ru-time complexity of O( 2 ), where is the umber of jobs, Merkle ad Middedorf [3] implemeted the ACO algorithm o the electrical Recofigurable Mesh (R- Mesh) with a ru-time complexity of O(x (m+) log ), where x is the umber of geeratios ad each geeratio has m ats. Merkle ad Middedorf s implemetatio is curretly the fastest amog all implemetatios due to the characteristics of the recofigurable mesh. However, all ACO algorithm implemetatios o electrical models are costraied by the commuicatio delays betwee processig elemets. I this paper, we preset several techiques to implemet the ACO algorithm ad its fast versio FACO o the Optical Pipelied Recofigurable Mesh (PR-Mesh) model efficietly ad effectively. With our ew approach o the PR-Mesh model, the ru time complexity of FACO algorithm is improved sigificatly. The FACO algorithm

2 ca be scaled-up, scaled-dow or extedig to a three dimesioed PR-Mesh depedig o the umber of jobs ad available resources. The R-Mesh model Merkle ad Middedorf used ad the PR-Mesh model utilied i this work belog to the same architecture class called Recofigurable Architectures. Traditioally, a recofigurable architecture is oe that ca dyamically alter the structure coectig its compoets at every step of a computatio leadig to a chage i its fuctioalities ad capabilities. This paper is orgaied as follows: Sectio II describes the Recofigurable Mesh models ad their operatios, Sectio III describes the existig ACO ad FACO algorithms ad their implemetatios o a R-Mesh, Sectios IV ad V describe our implemetatios of the ACO ad FACO algorithms o the PR-Mesh model, Sectios VI ad VII describe ew techiques to improve the performace ad exted the capability of the FACO algorithm, ad the last sectio summaries our work. II. Recofigurable Mesh Models The commo characteristic of all recofigurable models is the capability to recofigure themselves at every step of computatio. All compoets coected to a bus ca receive data i costat time. The feature that leads to differet recofigurable models is the types of the buses coectig the compoets i each model. The model usig electrical buses ca cofigure cycles while the oe usig optical buses caot. O the other had, the model usig optical buses ca trasmit multiple messages to multiple destiatios o a sigle bus simultaeously, while the other caot. The followig sectios describe more about these two models. A. Optical Pipelied Recofigurable Mesh A optical PR-Mesh [12] is a two dimesioal grid of processors, i which each processor has four ports coected by optical buses. Eight buses coect the four ports of a processor to its eighbors as i Figure 1. Each processor cotrols a set of local switches that allows the processor to segmet or fuse the buses at each itercoectio. The local fusig cofigures a liear bus coectig eighborig processors. Each such liear bus exactly resembles a Liear Array with a Recofigurable Bus Systems (LARPBS) [14]. A LARPBS is a array of N processors p 0,p 1,...,p 1 coected by a optical bus. The optical bus is costructed from three idetical waveguides: the message, select ad referece waveguides. The waveguides are divided ito two segmets: receivig segmet ad trasmittig segmet. The message Directioal couplers West T R F C C T F R Left T Trasmittig segmet R Receivig segmet North Top Bottom South Right R C Coditioal delay loop F Fixed delay loop F T C C F R T East Fusig coectio Fig. 1. PR-Mesh processor coectios waveguide is used for sedig data, the select ad referece waveguides are used for sedig addressig iformatio. Each processor p i is coected to the optical bus with two directioal couplers. Oe coupler is used for writig data o the trasmittig segmet (upper segmet) of the bus ad the other oe is used for readig the data from the receivig segmet (lower segmet) of the bus. Thus, the LARPBS uses six couplers for each processor p i to coect to the three waveguides. Two cosecutive processors are coected by a optical fiber segmet of oe uit pulse-legth,, as i Figure 2,(for simplicity, the figure omits the data waveguide, which resembles the referece waveguide). The optical sigals are propagated uidirectioal from left to right o the trasmittig segmet ad from right to left o the receivig segmet. The LARPBS has a set of coditioal-delay switches o the trasmittig segmet of the select bus ad a set of fixed-delay switches o the receivig segmet of the referece bus. The LARPBS uses the coicidet pulse techique [15] to route messages by maipulatig the select ad referece sigals, through the switches, o separate bus so that they will coicide at the desired destiatio processor(s). Each processor has a select frame ad a referece frame of N-bit each, which ca be used to sed a message by ijectig, up to N slots, its pulse sigals ito the frames. Whe processor p i detect a coicidece of the select ad referece pulses o the receivig segmet, it reads the data frame. Ay subsequece coicideces at processor p i i the same bus cycle will be igored. I additio, the LARPBS cotais a set of segmet switches o the trasmittig ad receivig segmet of the bus. Whe all the segmet switches are set to straight, the bus system operates exactly like a regular pipelied bus system. Settig the switches at p i to cross, the bus is segmeted ito two separate bus systems, oe cosists of p 0,p 1...,p i ad the other cosists of p i+1,p i+2,...,p 1.

3 Trasmittig segmet P0 P1 P2 P3 P4 Receivig segmet Fig. 2. A five-processor LARPBS model {NS,E,W} {EW, N, S} {NW, S, E} {NE, S, W} {SW, N, E} {SE, N, W} {NW, SE} {NE, SW} {NS, EW} {N,S, E,W} it goes through the R-Mesh. The iformatio about which items have ot bee selected is kept i a selectable set S i the at s memory. The ext item is always chose from S accordig to the Pseudo-Radom-Proportioal Actio Choice Rule [3], usig probability q 0, where 0 q 0 1 is a parameter for the algorithm. A at chooses item j that has ot bee selected so far which maximies τ α ij η β ij (1) where α d β are costats that determie the relative ifluece of the pheromoe value o the at s decisio i selectig the ext item. The probability of item to be selected is 1 q 0. This probability is computed as q ij {NSE, W} {NSW, E} {NEW, S} {SEW, N} {NSEW} Fig. 3. local possible coectios of ports withi a PE(right), R-Mesh (left) B. Electrical Recofigurable Mesh Descriptio A electrical Recofigurable Mesh (R-Mesh) [11] is a two dimesioal grid of PEs coected by electrical buses. Each processor ca locally cofigure ay or all of its ports to the coectig buses, as i Figure 3, withi oe time step. Sigals propagatig o a bus are odirectioal. Thus, the R-Mesh ca form cycles. Due to the ature of the bus, to avoid collisio, the COMMON write rule is used for cocurret write operatios. III. ACO ad FACO o R-Mesh Geerally, permutatio problems require oe to fid the optimal permutatio from a set of give items. These items represet the distaces betwee cities as i the Travellig Salesma problems (TSP) [1] or the jobs as i Sigle Machie Total (Weighted) Tardiess problems (SMTWTP) [3]. A geeric at coloy optimiatio algorithm uses a x pheromoe matrix to fid the permutatios of the items. I this sectio we describe Middedorf s implemetatio of a geeric at algorithm (ACO) ad its fast versio (FACO) o the R-Mesh model. A. ACO algorithm o R-Mesh I [3], Middedorf embedded the x pheromoe matrix M ito a x R-Mesh. Each processor p ij cotais a pheromoe value τ ij ad a heuristic value η ij, (i, j [1,]). The ats are pipelied through the R-Mesh from the first row. Each at selects a uselected item from row i as τij α q ij = ηβ ij h S τ ij α ηβ ij (Most ofte, heuristic values η ij are ot used). A at stops its jourey whe it reaches row, at which poit its solutio is foud. O a system of m ats, the last at reaches row i after +(m 1) steps. Whe all ats have foud their solutios, their solutios will be compared to existig solutios i order to determie which ats ca update the solutios. The selected ats update their solutios ad the pheromoe o their paths. This completes a at geeratio. The algorithm will ru for x geeratios, x 0. I geeral, a at selects a item i row i as follows: whe the at is i row i 1, p i 1,j kows whether item j has bee selected i oe of the rows 1...(i 1) or ot, p i 1,j seds this iformatio to p i,j as the at moves to row i. This step ca be doe i time O(1) by cofigurig p i 1,j {NS, E, W}. Next, the at selects a item i row i by computig the prefix-sums of τ ij η ij, where τ ij η ij = τ ij1 η ij1 + τ ij2 η ij2 + + τ ijk η ijk,k [1, (i 1)], which is the prefix-sums of τ ij η ij of all elemets i S, S = {j 1,j 2,...,j (i 1) },j 1 <j 2 < <j (i 1), across row i. The first PE i row i the radomly chooses a value, [0, j S τ i,j α ηβ i,j ), ad broadcasts to all PEs i the row. p i,j is selected whe [ l<j,l S τ ij α ηβ il, l j,l S τ ij α ηβ il, ). The prefixsums ca be doe i time O(log) usig biary tree reductio method [11](page 5), ad the selected PE ca be determied by comparig the prefix-sums of p i 1,j ad p i,j i time O(1). Whe all the ats i a geeratio have foud their solutios, which at is allowed to update the pheromoe is determied. The selected ats store their solutios ad the pheromoe value is evaporated from every PE before the ext geeratio of ats starts. The amout of pheromoe evaporated is defied follow: (2) τ ij =(1 ρ) τ ij (3)

4 where ρ is ay predefied factioal parameter for pheromoe evaporatig fuctio. To force ats from followig geeratios to follow the ear best foud solutios, a elitist at is desigated to update the pheromoe accordig to the best foud solutio i each geeratio. Sice each step of the algorithm ca be carried out i costat time, except the prefix-sums computatio, the algorithm ru time is O(x (m + ) log), where x is the umber of geeratios. B. FACO algorithm o the R-Mesh The fast versio of ACO algorithm is the result of abadoig the geeratioal pheromoe update ad simplifyig the probability distributio. First, istead of updatig the pheromoe at the ed of each geeratio, a at is allowed to update the pheromoe whe some criteria is met [18]. I particular, a at waits util the followig (m 1)/2 ats have foud their solutios. The at is allowed to update the pheromoe oly if its solutio is better tha the m 1,m m, best solutios that have bee foud by the precedig (m 1)/2 ats ad the followig (m 1)/2 ats. If multiple ats with equal solutios, oly the (m/m )th at is allowed to update. Evaporatio is doe whe a at updates the pheromoe. Therefore, the ruig time of the updated ACO algorithm is O((x m + ) log) istead of O(x (m + ) log). Secodly, each item per row is assiged two bit-values h ad l, where h>l>0, for low ad high probabilities of beig selected ext. These values are determied by a threshold fuctio t. For each item j S, defie { g(τij α η β h if τ ij )= α ij η β ij >t (4) l otherwise I p ij, for each j S, a bit l ij is set to 1 if g(τij α ηβ ij )=l, otherwise, a bit h ij is set to 1. Next, the umber of PEs with bit l set to oe, l, ad the umber of PEs with bit h set to oe, h, i each row are determied. A radom umber, [0, l l + h h), is selected as i the geeric ACO algorithm. If [(p 1) l, p l), the p th PE i the row with a l bit set to 1 is selected. If [ l l +(p 1) h, l l + p.h) the the p th PE i the row with a h bit set to 1 is selected. The t, h, ad l values could be chaged durig the ru of the algorithm. Similarly, the Pseudo-Radom-Proportioal Actio Choice Rule is modified as follows: let t τ > 0 be a pheromoe threshold ad h τ > l τ > 0. For each item j S defie { hτ if τij g(τ ij )= α >t τ (5) l τ otherwise a at will select the item that maximies the value of g(τ ij ) η β ij. Fig. 4. Embeddig a row of the pheromoe matrix i a submesh of the R-mesh C. Implemetatio of FACO algorithm o the R-Mesh ad its Ru Time Middedorf proposed [3] to use a log 2 x (/log 2 ) R-mesh. The a row of pheromoe matrix forms a log 2 x(/log 2 ) submesh as i Figure 4. The sums l ad h ca be computed i O(log ) as stated i [16] ad [3]. After selectig a radom umber, the p th PE with bit l set to oe is selected. By [3] ad [16], the sum of all l bits i the first i log 2 colums, i [1, (/log 4 )], of all sub-meshes ca be foud i time O(log ). The p th PE with l-bit is set is determied usig the same techique. Therefore, the total time to select a item is O(log ). At every step, there will be at most + m/2 3 2 solutios eed to be stored. I order for a at to update its solutio, it must kow the rak of its solutio. Therefore, the stored solutios must be raked by their goodess. The rak of the ew solutio ca be computed i O(log ) by comparig the solutio to the stored solutios i parallel, ad a bit is set whe it is worse. Additio of the resultig bits gives the rak of the ew solutio. The pheromoe ca be updated i O(1) as the at updatig the solutio usig the Formula 3. Therefore, this FACO algorithm ru-time complexity is O(( + ) log ), where is the umber of ats o a log 2 x /log 2 R-mesh. This ru time ca oly be achieved if the umber of differet weights is at most log 2 [3]. I the ext sectio, we describe techiques to implemet the ACO o PR-Mesh model. IV. ACO algorithm o PR-Mesh (ACO-PR) As stated i Sectio II the optical model caot cofigure ay cycles. To implemet the ACO algorithm o a PR-Mesh, we have to esure that oe of the steps i the algorithm cofigures a cycle i the PR-Mesh. Whe a at moves from row i 1 to row i, p i 1,j passes this iformatio to p i,j usig oly colum j ad forms o cycles. p i,j broadcasts its ew iformatio to all PEs i row i, which does ot form ay cycle either. Next, the prefixsums of row i is computed usig biary tree reductio o

5 row i, which also uses o cycle. The first PE i row i chooses the value ad broadcasts it to all PEs i row i, ad p i,j determies whether it is selected or ot usig oly row i for commuicatio. The last step, whe all ats have foud their solutios, the ew foud solutios must be routed to the PEs that holdig the previously foud solutios to determie whether ay at has foud a better solutio. Let us assume the m best solutios foud so far are positioed i the last row of the PR-Mesh. If we keep each at i a colum, i.e. at k, 1 k m moves from row1torow usig oly colum k, the at k will store its solutio to p,k whe it reaches row. Obviously, the ACO algorithm ca be implemeted o a x PR-Mesh with ru time O(x (m + ) log). I the ext sectio, we show how to determie the goodess of a solutio ad update the pheromoe i costat time. Usig that techique, each at ca update its solutio ad the pheromoe immediately if its solutio is better tha previously foud solutios. The ru time will be reduced to O((x m + ) log). V. FACO algorithm o PR-Mesh (FACO-PR) The FACO algorithm ca be simulated i the PR-Mesh as follows: istead of fidig the sums of of l bits l ad h bits h l of each row, we compute the prefix-sums of these bits. The prefix-sums ca be computed i time O(1) [11](page 374). After computig the prefix-sums, the p th PE with l bit or h bit set to 1 is the oe has the prefixsums equals to p. This iformatio is obviously ca be determied i time O(1). Updatig the solutios i PR-Mesh is as follows: first, we use the last two rows of the PR-Mesh to store 3 2 possible solutios. These two rows are coected i a sake-like array, where the last PE of the mesh, p,,is the head of the bus ad the last PE of row (-1), p ( 1),, is the tail of the bus, actually, we oly use the lower half, p 1,0,...,p 1, /2, of row (-1). We call this bus the solutio bus. Wheever a at fids a solutio, this solutio is broadcasted to all the PEs i the solutio bus. Each PE i the first 3 2 of the bus, startig from the bus head, holdig a solutio will set a bit to 1 if its solutio is better tha or equal to the ew solutio, otherwise set its bit to 0. The sum of these bits is the rak of the ew solutio. Each PE that set a bit to 0 (holdig worse solutio) i the previous step icreases its rak by 1. The all the solutios are routed to their locatios based o the rak. The best solutio will have the lowest rak. The sum of these bit ca be added i O(1) time [11](page 374). The routig ca be doe i O(1) time [11](page 369). Evaporatig the pheromoe is doe similar to the origial algorithm. The updatig at seds a sigal to all PEs i its path to evaporate the pheromoe. Sice the goodess of each solutio ca be determied i costat time, we ca allow the solutio ad the pheromoe to be updated as soo as the solutio emerges to make it available for m 1 ats i the pipelie. I additio, we oly eed store at most m solutios, which ca be stored i the last row of the PR-Mesh. Sice each step of the FACO algorithm o PR-Mesh rus i O(1) time, the algorithm yields ( 1+) steps or O( + ), for is the total umber of ats leavig the coloy, which is better tha the oe ruig o the R-Mesh by log factor. O( + ) is the lower boud for ACO algorithm o the pipelied recofigurable mesh, ad this is ot hard to verify. By the ature of the ACO algorithm, the decisio of choosig the ext move of a at depeds o the locatios it has visited. It ca ot visit a locatio twice or it ca visit locatio i before visitig locatio i 1. Therefore, the at takes exactly steps to fid its solutio. I additio, a states pipelie with jobs take at least +( 1) time slots to fiish. Thus, the ru time of the FACO algorithm o the PR-Mesh is optimal. VI. FACO Algorithm o sub-pr-meshes (FACO-PR-Sub) Sice the ACO algorithm practices the greedy approach, where each at chooses the best possible path adjacet to its curret locatio, the over all solutio may ot be optimal. I this sectio we preset a divide ad coquer approach, i which the solutio will be the optimal of subproblems ad each sub-problem is solved by applyig the FACO-PR algorithm. A. Selectio o PR sub-mesh I the ACO algorithm, a at chooses the ext item by selectig a item i the ext row that maximies the pheromoe ad heuristic values τij α ηβ ij (similarly for the ats i the FACO algorithm). If all the items are equally likely to be selected, the probability of this selectio behavior ca be represeted as: P = 1 1 j=0 j = 1!.To speedup the algorithm, we propose the followig: First, we divide the x PR-Mesh ito blocks (sub-mesh) of sie log x log, givig log2 such blocks. The solutio for each block is carried out the same as described i the FACO-PR algorithm. Next, the solutio for all the blocks is determied usig oe of the followig two methods. (i) sum the sub-solutios of all blocks i the same blockrow, the fid the maximum of all the block-rows (Max Row method). (ii) fid the maximum sub-solutios of each block-colum, the add all the maximum values to obtai a solutio (Max Colum method). The two methods share almost the same behavior. The selectio probability for all

6 Selectio Methods b 1,1 b 2,1 b 1,2 b 1, b 2, Solutio Matrix sie ACO Max Row Max Colum Fig. 5. Simulatio of the origial ACO selectio, Max Row selectio ad Max Colum selectio Methods - The Max Colum selectio method resembles the origial ACO selectio method the disjoit blocks is P b = log 1 j=0 1 log j =(log )!. I additio, the probability of pickig the best sub-solutio 1 block out of a block-colum, or block-row, is log, ad P c = ( 1 log 1 log... 1 log ) = ( 1 log ) 1 log = 1 (log) log for log block-colums. Therefore, the ew selectio log! probability is P = P b P c = (log) log!. Sice the solutio of the problem closely depedig o the distributio of the items, it is ot clear which selectio method, the origial oe or the proposig oes, yields better solutio. Based o our simulatio (see Figure 5) of the three selectio methods, where the pheromoe values are radomly geerated ad distributed, the Max Colum method resembles the origial selectio method. Next, we will describe the Max Colum method i details, (the Max Row Method is similar). B. Applyig Max-Colum Selectio Method o PR-Mesh b,1 b,2 b, Fig. 6. A x PR-Mesh partitioed ito logxlog blocks block-colums is determied. I order to fid the best subsolutio of a colum, all sub-solutios s 1,s 2,...,s log of block-colums are routed to the last row of block b log,j of each block-colum j, where p,i is the destiatio of s i. Next, i each block b log,j the solutios are broadcasted to all other PEs through the colums of the block. p i,i of the block broadcasts its solutio alog row i, aype holdig a worse solutio sets its bit to 1. The bitwise-or over every colum bit of the blocks is computed. A result ero of a colum sigifies the solutio i that colum is the best solutio. Each oe of these steps take O(1) time. Next, the solutio to the problem is determied by addig all the best sub-solutios. This solutio is the routed to the solutio bus for rakig ad storig as described i the previous sectio. The ru time of every step ivolved i this method is either costat or similar to the FACO- PR, except the summig of partial solutios. The sum of these log solutios ca be computed i O(loglog) time usig biary-tree reductio. It takes O( log ) time to fill the pipelie. Therefore, the ru time of the FACO-PR-Sub is O((( log 2 )+ log ) loglog). VII. FACO Algorithm o 3D PR-Mesh Ats used i the ACO models are cotrol logics ad ofte represeted as iteratios of ruig a algorithm; therefore, requirig more ats may ot impose ay physical restrictio o the algorithm. Iitially, we partitio a x PR-Mesh ito log 2 sub-meshes, or blocks, of sie. The x PR-Mesh M ca be viewed as a log x log log x log sub- logxlog matrix, where each elemet is a mesh as i Figure 6. Assumig we have at least ats, log 2. We desigate the last row of the x PR-Mesh as the solutio bus as i Sectio V. First we distribute /log 2 ats to each block. The ats are pipelied through the sub-mesh i the same way as described i Sectio V. After a at fids its solutio i its block b i,j, where i, j [1,log], the best of all the solutios foud by these Combiatorial problems that have their weights chagig dyamically durig the executio of a algorithm ad require iteratig over the solutios may times for the solutios to coverge to a optimal solutio ted to use a large umber of ats to solve. I this sectio, we describe a techique to exted the FACO-PR-Sub to multidimesioal PR-Mesh to solve such problem effectively. A. Jobs FACO Algorithm o 3D PR- Mesh (FACO-PR 3D) I geeral, if the umber of ats used i the ACO algorithms is relatively large, the FACO ca be exteded to ru o a k-dimesioal PR-Mesh. I particular, we will

7 slice j across all the slices Slice k Slice 1 j i row 1 row j s s s s s Fig. 7. A xx PR-Mesh Soluio mesh Head of the bus Fig. 8. A x PR-Mesh cofigured it a sake-like liear bus describe a techique to exted the FACO to ru o a xx PR-Mesh i logarithmic time. A 3D PR-Mesh ca be viewed as of x PR-Meshes ( slices) coected by the optical buses i the third dimesio k, where all the last rows of these slices form a x solutio mesh (see Figure 7). Let row t h of each slice holds the best solutios foud so far - we call these rows as solutio buses. The FACO algorithm ca be implemeted as follows: First, let the ats pipeliig through the slices as i the 2D PR-Mesh. After steps, there will be ats arrivig to the last row with their solutios, oe i every slice. For the first arrivig solutios, we must sort them by their raks. This ca be achieved by usig Leighto s colum sort algorithm o a x mesh, as described i [11] (page 377), i O(1) time. After sortig, these solutios are broadcasted to all other slices. O every subsequece step, the raks of ew solutios are determied ad routed to their positios i the solutio buses. To extract the best solutios we perform the followig stages. First, the solutios i p,j,1 are swapped with the solutios i p,j, if j is eve. Secod, p,j,1,p,j,2,...,p,j, are coected to form a bus goig across all slices. p,j,k is the head of the bus if j is odd, otherwise, p,j, is the head of the bus. All PEs holdig ew solutios ad head PEs are set to active. All active PEs have their solutios raked ad routed to their positios. At this momet, all the ew solutios are packed to the head of the buses. Next, the solutio mesh is cofigured ito a sake-like bus,(see Figure 8), where p,1,1 is the head of the bus. The best solutios are routed to the head of the bus by computig the prefix-sum of the iactive PEs ad lettig all active PEs sed their solutios to the PEs of distaces equal to their prefix-sums toward the head of the bus. The fial step is to propagate the best solutios to each solutio bus i the solutio mesh. This is doe i two steps:p,1,k seds its solutio to p,k,k ad p,k,k broadcasts its value across slices p,k,1,p,k,2,...,p,k,. All of these steps ru i O(1) time. The pheromoe is updated i each slice similar to the earlier described algorithm. p,k,k updates slice k if the its solutio is ew. Next, 2 buses coectig PEs i the first slice to the last slice are formed to broadcast the ew pheromoe values to all the slices. As a result, the ru time for this algorithm is O(( )+ ), where is the total umber of ats. Usig the same method described i Sectio VI, the ru time of this algorithm ca be lowered to O((( log 2 )+ log ) loglog). B. x Jobs At Coloy Algorithm o 3D PR-Mesh (FACO-PR 3Dx) To solve a problem with 2 jobs usig the PR-Mesh as described i Sectio V we will eed a 2 x 2 PR- Mesh to store the pheromoe values. I this sectio, we will use the result of the previous sectio to solve this problem efficietly. Let the 2 jobs be distributed alog the j ad k directios of the 3D-PR-Mesh such that each slice cotaiig jobs. The solutios are computed i both directios i ad j as i Figure 7. This algorithm requires at least +1 ats. Each at steps dow from the first row to the last row of its slice ad selects a item similar to the the geeral ACO described i Sectio III-A simultaeously. We call the at from the first slice the leader at. Whe the leader at is o row i ad selectig item j, the weight of that item w j is set to the crossed slice j (cotaiig p 1,j,1,p 2,j,1,...,p,j, ). This is possible sice each PE has may separate buses, (see Figure 1). For clarity, we ca visualie the first slice as a additioal x mesh attachig to the xx cube. The solutio of slice j is the sum of its solutio ad the weight w j of the leader at. After steps, the leader at tailors its solutio by summig the solutios of all crossed slices i O(log) time. The pheromoe is updated o each crossed slice as described i previous sectios. At the ed of each geeratio, the maximum values τ ij ad η ij, (i, j [1,]) must be routed to the first slice for the leader at to work properly. This step is doe as follows: First, each p i,j o slice 1 becomes the head of a bus coectig all other p i,j from all other slices. Next, the max values are computed ad routed to the heads of

8 TABLE I. Algorithms Ru times Algorithm Ru Time ACO O(x (m + ) log) ACO-PR O(( + ) log) FACO O(( + ) log ) FACO-PR O( + ) FACO-PR-Sub O(( log 2 + log ) loglog) FACO-PR 3D O(( log 2 + log ) loglog) FACO-PR 3Dx O(x (m + ) log) TABLE II. Algorithms Characteristics Algorithms Number of ats Jobs Max. Wts. Mesh Sie ACO x m 2 x ACO-PR x m 2 x FACO x m log 2 log 2 x log 2 FACO-PR x m 2 x FACO-PR-Sub log2 x 2 x FACO-PR 3D log2 x 2 xx FACO-PR 3Dx xx the buses. Each bus works idepedetly. These steps ca be doe i O(1) time [11] (page 376). The total ru time to compute the first solutio is O(+ log). The ru time of the 3D PR-Mesh is O(x (m + ) log), where x is the umber of geeratios each havig m ats. This ru time ca be further reduced by applyig the partitioig techique i Sectio VI to the algorithm. VIII. Coclusio I this paper, we show that the At Coloy Optimiatio (ACO) ad its simplified versio Fast At Coloy Optimiatio (FACO) algorithms with ats ca be simulated o the Optical Pipelied Recofigurable Mesh (PR-Mesh) with better time complexity tha the electrical recofigurable mesh. Ulike the electrical model, which restricts the umber of differet weights to be at most log 2, the optical model imposes o restrictio o the weights. I particular, we preset techiques to implemet the FACO algorithm o the PR-Mesh with ru time O(( log 2 + log log 2 + log ) loglog) ad O(( ) loglog) o 2D PR-Mesh ad 3D-PR-Mesh (FACO-PR-Sub ad FACO-PR 3D), ad a versio of the FACO algorithm with 2 jobs o a 3D-PR- Mesh (FACO-PR 3Dx) i O(x (m + ) log). The rutime complexity compariso of the two models is give i Table I, ad the characteristics of the algorithms o these models are give i Table II, (ote: x is the umber of geeratios, m is the umber of ats per geeratio, = x m, ad italicied algorithms are o the electrical recofigurable mesh). Refereces [1] M. Dorigo ad L. M. Gambardella, At Coloy System: A Cooperative Learig Approach to the Travelig Salesma Problem, IEEE tras. o Evolutioary Comp. Vol. 1: 53-66, [2] T. Stütle, A at Approach for the Flow Shop Problem, Proc. 6th Europea Cogress o Itelleget Techiques ad Soft Computig, Verlag Mai: Aache, Vol. 3: , [3] D. Merkle ad M. Middedorf, Fast At Coloy Optimiatio o Rutime Recofigurable Processor Arrays,. Joural of Geetic Programmig ad Evolvable Machies, Vol. 3: , [4] D. Merkle, M. Middedorf, ad H. Schmeck, At Coloy Optimiatio for Resource-Costraied Project Schedulig, IEEE tras. o Evolutioary Comp. Vol. 6: , [5] L. M. Gambardella, E. Taillard, ad M. Dorigo, At Coloies for the Quadratic Assigmet Problem, Joural of the Operatioal Research Society, Vol. 50: , [6] V. Maieo, A. Colori, ad M. Dorigo, The At System Applied to the Quaratic Assigmet Problem, IEEE Tras. Kowledge ad Data Egieerig, Vol. 11: , [7] D. Costa ad A. Hert, Ats ca colour graphs, Joural of the Operatioal Research Society, Vol. 48: , [8] G. Di Caro ad M. Dorigo, At coloies for adaptive routig i packet-switchied commuicatios etworks, Proceedig of PPSN- V, Fifth Iteratioal Coferece o Parallel Problem Solvig from Nature, Vol. 1498: , [9] R. Schooderwoerd, O. Hollad, J. Brute, ad L. Rothkrat, At-based load balacig i telecommuicatio etworks, Adaptive behavior Vol. 5: , [10] M. Dorigo, Optimiatio, learig ad atural algorithm, Ph.D Thesis, DEI, Politecico di Milaio, Italy, [11] R. Vaidyaatha ad J. Traha, Dyamic Recofiguratio - Architectures ad Algorithms, Kluwer Academic/Pleum Publishers, ISBN , [12] J L. Traha, A. G. Bourgeois, ad R. Vaidyaatha, Tighter ad Broader Complexity Results for Recofigurable Models, Parallel Processig Letters, Vol. 8: , [13] M. Middedorf, Bit-summatio o the Recofigurable Mesh, Parallel ad Distributed Computig, Proc. of the 11 IPPS/SPDP 99 Workshops, 6th Recofigurable Architectures Workshop RAW-99, J. Rolim et al. (eds.), Spriger: Berli, LNCS, Vol. 1586: , [14] Y. Pa ad K. Li, Liear Array with a Recofigurable Pipelied Bus Systems: Cocepts ad Applicatios, Iformatio Sciece 106: , (1998). [15] C. Qiao ad R. Melhem, Time-Divisio Optical Commuicatio i Multiprocessor Arrays, IEEE Tras. Comput. 42: , [16] K. Nakao. Prefix-sums Algorithms o Recofigurable Mesh, Par. Process. Letter, 5:23-25,1995. [17] Delisle P., Krajecki M., Gravel M., Gag C., Parallel implemetatio of a at coloy optimiatio metaheuristic with OpeMP, Iteratioal Coferece o Parallel Architectures ad Compilatio Techiques, Proceedigs of the 3rd Europea Workshop o OpeMP (EWOMP01), September 2001, Barceloe, Espage. [18] V. Maieo, Exact ad approximate oditermiistic tree-search procedures for the quaratic assigmet problem, INFORMS Joural o Computig, Vol. 11: , [19] T. Stütle, H.H. Hoos, The MAX-MIN at system ad local search for the travelig salesma problem, Proceedigs of the IEEE Iteratioal Coferece o Evolutioary Computatio(ICEC 97): , 1997.