Evolution of Non-Deterministic Incremental Algorithms. Hugues Juille. Volen Center for Complex Systems. Brandeis University. Waltham, MA

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1 Evolution of Non-Determinitic Incremental Algorithm a a New Approach for Search in State Space Hugue Juille Computer Science Department Volen Center for Complex Sytem Brandei Univerity Waltham, MA Abtract Let u call a non-determinitic incremental algorithm one that i able to contruct any olution to a combinatorial problem by electing incrementally an ordered equence of choice that dene thi olution, each choice being made non-determinitically. In that cae, the tate pace can be repreented a a tree, and a olution i a path from the root of that tree to a leaf. Thi paper decribe how the imulated evolution of a population of uch non-determinitic incremental algorithm oer a new approach for the exploration of a tate pace, compared to other technique like Genetic Algorithm (GA), Evolutionary Strategie (ES) or Hill Climbing. In particular, the eciency of thi method, implemented a the Evolving Non-Determinim (END) model, i preented for the orting network problem, a reference problem that ha challenged computer cience. Then, we hall how that the END model remedie ome drawback of thee optimization technique and even outperform them for thi problem. Indeed, ome 16-input orting network a good a the bet known have been built from cratch, and even a 25- year-old reult for the 13-input problem ha been improved by one comparator. 1 INTRODUCTION In the eld of optimization and machine learning technique, ome very ecient and promiing tool like Genetic Algorithm (GA), Evolutionary Strategie (ES), Hill Climbing or Simulated Annealing (SA) have been deigned. Each of thee technique ue a et of operator to generate new tate (opring) from a current et of tate (parent). Thee operator are croover and mutation for GA, recombination and mutation for ES ([Back et al., 1991]), and \jump" to a local neighboring tate for claical Hill Climbing or SA. Thee approache appear to be very ecient for many NP-complete problem like the Traveling Saleman Problem (TSP), the deign of VLSI circuit or the Job Shop Scheduling problem (JSP). However, when the ubpace of valid olution i dened by a et of complex contraint, the deign of a good repreentation for a tate, and therefore the deign of ueful operator, can be very dicult. For example, ecient GA or ES implementation for TSP or the Number Partitioning problem ([Ruml et al., 1995]) involve non-trivial operator in order to introduce ome problem-pecic knowledge. Thi paper preent another well-known problem for which there i only little information about the topology of the ubpace of valid olution: the orting network problem. Hilli ([Hilli, 1992]) and Drecher ([Drecher, 1994]) ued GA to tackle thi problem. However, becaue no operator i known that retrict the earch in the ubpace of valid orting network, it i only at the cot of a very large population ize that ignicant reult have been achieved. On the contrary, the earching technique preented in thi paper manage to retrict the exploration only to valid olution. Indeed, the tate pace that i explored i decribed by a tree in which leave correpond to valid olution and a path from the root of that tree to a leaf repreent the equence of choice neceary to generate the correponding olution. So, thi earching technique require the deign of an incremental algorithm which i able to generate any valid olution repreented in the tree. Such an algorithm begin at the root of the tree and, at each tep of the incremental contruction, elect a node among the children of the current nodeuntil a leaf i reached. Then, a population-baed model we called Evolving Non-Determinim (END) imulate the evolution of a population of uch incremental algorithm for which the election of the ucceor to the current nodei performed uniformly randomly. The law that drive the evolution of thi population of non-determinitic incremental algorithm are uch that individual whoe rt choice eem to be more promiing to generate a good olution reproduce more than other.

2 Thi paper preent the END model and compare it eciency to GA for the orting network problem. Thi i the follow-up of an etablihed problem for which everal approache have been ued to try to improve ome 25 year old reult ([Belew & Kammayer, 1993, Drecher, 1994, Hilli, 1992, Levy, 1992, Parberry, 1991]). Actually, thi problem wa alo the origin of an early paper [Tuft & Juille, 1994] in which GAwere ued to try to replicate Hilli' experiment ([Hilli, 1992]) for the 16-input problem and in which ome idea of the END model were preented. Encouraging reult decribed in thi paper how how the END model both outperform GA for thi problem and even let u expect that a broader eld of application can be tackled by thi model. Thi paper i organized a follow: Principle and parameter of the END model are preented in detail in Section 2. Section 3 decribe the orting network problem and Section 4 analyze GA and the END model approache for thi problem. Section 5 preent a ummary and poible future reearch. 2 EVOLVING NON-DETERMINISM 2.1 PRINCIPLES In thi paper, it i aumed that the earch tate pace can be repreented by a tree. Leave of thi tree repreent valid olution and internal node repreent partial olution (or tep neceary to reach valid olution). There i a well-known AI algorithm for earch in directed graph and tree called Beam earch ([Winton, 1984]). Beam earch examine in parallel anumber of nearly optimal alternative (the beam). Thi earch algorithm progree level by level in the tree of tate and it move downward only from the bet w node at each level. Conequently, the number of node explored remain manageable: if the branching factor in the tree i atmot b then there will be atmot wb node under conideration at any depth. The END model i imilar to thi algorithm in the ene that each individual of the population can be een a one alternative in the beam. Moreover, a tne (or core) mut be aigned to internal node of the tree in order to determine which node will be explored further and which node will be ignored by the earch algorithm. Here, beam earch ue heuritic to core the dierent alternative and to elect alternative that are mot promiing, i.e., the node with larget core. However, thi approach aume the exitence of a core operator ued to evaluate the node. The new idea propoed by the END model i to etimate the core of a given internal node by doing a random ampling from thi node. That i, a path i contructed incrementally and randomly until a leaf (or valid olution) i reached. Then, the core of thi olution i directly computed according to the problem objective function. If the problem i to nd a hortet path, the core can be the length of the path neceary to reach the correponding leaf in the tree where the dierent edge are eventually weighted according to a cot function. If the problem i to nd a trategy to play a game, the core can be the evaluation of the game conguration dened by the leaf. Finally, the core of thi non-determinitically and incrementally contructed olution i aigned to the initial node. In fact, the END model perform the earch by driving the evolution of the population of incremental algorithm according to the following procedure: 1. Initially, the earch i retricted to the rt level of the tree and each individual of the population randomly elect one of the rt-level node. 2. Each individual core it aociated node (or alternative) by doing a random ampling a dicued above. 3. A reproduction round i operated in the population. The purpoe of thi operation of reproduction i to imulate the Darwinian mechanim of natural election and urvival of the ttet. 4. A tet i performed and the reult determine whether the level of earch i increaed by one or not. In the armative, each individual elect uniformly randomly one of the children of it aociated node and thi node become the new alternative aigned to the individual. 5. The earch top if no new node can be explored; otherwie it continue with tep 2. So, the imulated evolution i a equence of competitive round. In the END model, the level of earch in the tree i called the commitment degree ince it correpond to a commitment to the rt choice of the incremental contruction of an optimal olution. The next two ection decribe the reproduction operation and the management of the commitment degree. 2.2 REPRODUCTION The mechanim ued for the reproduction operation have been inpired by the parallel architecture ued for the implementation of the END model. Indeed, the current implementation of the END model ue a Mapar MP-2 parallel computer. Thi ytem i a SIMD 2D wrap-around meh (i.e., a toru) architecture with a number of proceor element (PE) ranging from 1K to 16K (our conguration i compoed of 4K PE). So, our population ha been modeled a a 2D wraparound meh where an individual i aigned to each point of interection of the meh and, therefore, ha four neighbor. Then, the reproduction operation i performed in the following way:

3 Each individual compare it core with the core of the individual in it neighborhood. Thi neighborhood i compoed of the individual whoe Hamming ditance from the rt individual i leer than a given value of the parameter: radiu. If one individual in the neighborhood ha a better core than all the other then it i copied intead of the current individual. If everal individual in the neighborhood have an identical core which i better than all the other then one i elected uniformly randomly and copied intead of the current individual. If no individual in the neighborhood i better than the current individual, then thi individual remain unchanged. So, if a given node of the current level of earch in the tree i more likely to lead to a more promiing olution then the reproduction operation allow individual aociated to thi node to reproduce more than other individual (therefore focuing the beam). 2.3 MANAGEMENT OF THE COMMITMENT DEGREE Becaue of the reproduction operation, alternative that eem to be more promiing are repreented by a larger and larger number of individual. Ultimately, if one wait until all the individual in the population agree on the ame alternative before continuing the earch on the next level, thi mean that the beam i reduced to the exploration of a ingle alternative. On the contrary, if the level of earch i increaed too frequently, the beam can become very wide. Indeed, in that cae, many dierent alternative are repreented in the population and each alternative i repreented by a mall number of individual. So, the diappearance of a promiing alternative during the reproduction operation becaue of a non-favourable random ampling core evaluation i more likely. Clearly, for the earch to be ecient, thee two extreme mut be avoided. Thi goal i reached by uing a trategy that manage the level of earch. In fact, the purpoe of thi trategy i to drive the earch by deciding when the commitment degree i incremented. In the current implementation, two trategie have been deigned. The rt one i the impler ince the commitment degree i increaed every n round, where n i xed. n ha to be choen atutely o that the numberofindividual that correpond to better alternative can reach a ignicant ize. The problem with thi trategy i that the value of n i dicult to etimate a priori. The econd trategy ue a meaure of the tate of the model called diorder meaure. The diorder meaure evaluate the width of the beam. To compute thi meaure, each individual of the population count the number of individual among it four nearet neighbor that correpond to a dierent alternative than itelf. Then, thi number i ummed over all the member of the population and the reult i the diorder meaure. Thi meaure reect the degree of convergence of the population. If a large part of the population correpond to a few alternative then thi meaure i mall becaue mot of the individual are identical. On the other hand, if many alternative are explored, and each alternative i repreented by a few individual, then the diorder meaure i large. A individual of the population focu on mot promiing alternative becaue of the reproduction operation, the diorder meaure decreae. When thi meaure reache an arbitrarily xed threhold (which i a parameter of the model), one conider that the width of the beam i mall enough and the earch can continue on the next level of the tree. The drawback of thi trategy i that it can take a long time for the diorder meaure to reach the given threhold if everal alternative lead to equivalent optimal olution. Thi problem doen't appear with the rt trategy. Mot of the time, a combination of thee two trategie oer a good compromie. 2.4 PARAMETERS OF THE MODEL The decription of the END model in the previou ection how that it i characterized by the following parameter: Population ize : If the number of individual in the population increae then the width of the beam can alo be larger without decreaing the eciency of the earch. Therefore, the ize of the explored tate pace i directly related to the ize of the population. Neighborhood ued for reproduction : The ize of the neighborhood for which the tne of individual i compared during a reproduction round drive the dynamic of evolution of the population. Indeed, if the radiu of thi neighborhood i large then the earch focue quickly on the bet individual and dicard apparently le promiing alternative. On the contrary, a mall radiu allow the earch to converge lowly. Therefore, thi parameter manage the tradeo between exploration and exploitation. Management of the commitment degree :Ait i decribed in the previou ection, thi parameter alo play an important r^ole in managing the balance between exploration and exploitation. The above decription of the inuence of thee parameter on the earch ha been conrmed experimentally in [Juille, 1994].

4 2.5 IDEA OF THE END APPROACH Another approach to decribe how the END model work i to make the following analogy: Children of the root of the tree of olution can be een a a partition of the pace of tate, each child correponding to a particular ubet of thi partition. Then, the reproduction proce allow alternative for which the core (or tne) evaluation by random ampling i better on average to be repreented by a larger number of individual than other alternative. Such alternative correpond to the domain of the pace of tate for which the mean value for the tne i larger. Therefore, at thi tage, detail and gradient of the landcape of the pace of tate are not conidered. Then, a the commitment degree increae, each domain i partitionned into maller ub-domain and, therefore, detail of the landcape take more and more importance. Schraudolph and Belew ([Schraudolph & Belew, 1992]) implemented a imilar idea for GA by tracking the convergence of the population to retrict ubequent earch uing a zoom operator. Of coure, it i eay to dene a landcape uch that thi trategy doen't work. For example, take a tne function uch that the optimal correpond to a peak located in a region with a very low tne and for which another region, far from thi optimal peak, ha a high average value. Thi trategy will be inclined to nd out a local optimum in the region of high average tne. A it i hown in [Juille, 1995], the landcape of the pace of tate for the orting network problem i of thi kind. However, the ability of the END model to maintain diverity by managing the balance between exploration and exploitation allow it to be an ecient earch algorithm. 3 THE SORTING NETWORK PROBLEM An obliviou comparion-baed algorithm i uch that the equence of comparion performed i the ame for all input of any given ize. Thi kind of algorithm ha received much attention ince it admit an implementation a circuit: comparion-wap can be hard-wired. Such an obliviou comparion-baed algorithm for orting n value i called an n-input orting network (a urvey of orting network reearch iin [Knuth, 1973]). There i a convenient graphical repreentation of orting network a hown in gure 1, which i a 10-input orting network (from [Knuth, 1973]). Each horizontal line repreent an input of the orting network and each connection between two line repreent a comparator which compare the two element and exchange them if the one on the upper line i larger than the one on the lower line. The input of the orting net Figure 1: A 10-input orting network uing 29 comparator and 9 parallel tep. Table 1: Current upper and lower bound on the depth of n-input orting network. Input Upper Lower Input Upper Lower work i on the left of the repreentation. Element at the output are orted and the larget element migrate to the bottom line. Performance of a orting network can be meaured in two dierent way: 1. It depth which i dened a the number of parallel tep that it take to ort any input, given that in one tep dijoint comparion-wap operation can take place imultaneouly. Current upper and lower bound are provided in [Parberry, 1991]. Table 1 preent thee current bound on depth for n It length, that i the number of comparion-wap ued. Optimal orting network for n 8 are known exactly and are preented in [Knuth, 1973] along with the mot ecient orting network to date for 9 n 16. Table 2 preent thee reult. The 16-input orting network ha been the mot challenging one. Knuth [Knuth, 1973] recount it hitory a follow. Firt, in 1962, Boe and Nelon dicovered a method for contructing orting network that ued 65 comparion and conjectured that it wa bet poible. Two year later, R. W. Floyd and D. E. Knuth, and independently K. E. Batcher, found a new method and deigned a orting network uing 63 comparion. Then, a 62-comparator orting network wa found by G. Shapiro in 1969, oon to be followed by M.W. Green' 60 comparator network (ee [Green, 1969] and [Knuth, 1973]).

5 Table 2: Bet upper bound currently known for length of orting network. Previouly, the bet known upper bound for the 13-input problem wa 46. Input Comparator Input Comparator A the length of network increae, the ratio of valid orting network increae ince adding a comparator cannot tranform a valid orting network into an incorrect one. However, thi ratio become very mall when one come cloe to the optimal length. In that cae, the probability that croover or mutation improve individual i very low and, a i hown in the next ection, the population ize ha to be large to counterbalance thi undeirable property. 4 Comparion of GA and END approache 4.1 GA approach Decription A in [Drecher, 1994] and [Hilli, 1992], the intuitive repreentation of orting network i that each genome encode a orting network a a equence of pair, each pair repreenting a comparator. Then, croover i performed by exchanging group of comparator between two mating individual. A mutation conit in modifying one of the two indice dening a comparator. The tne of individual i then cored uing the following criteria: Uing the zero-one principle ([Knuth, 1973]), it i ucient to tet the 2 n poible binary input vector (where n i the number of input of the orting network) to determine if a orting network i correct. Hilli ([Hilli, 1992]) and Drecher ([Drecher, 1994]) ued the ratio of correctly orted binary vector to core the tne of individual in the population. Some comparator in the repreentation can be non-ignicant becaue they don't reduce the ize of the unorted vector et. Thi criterion i ued by Drecher to allow convergence toward hort network. The technique ued by Hilli i a little dierent becaue of hi repreentation of orting network a pair of chromoome. Shorter network are created when identical comparator occur at the ame poition in the two chromoome of a pair. Uing thi repreentation for network, one can ee that it i highly probable that croover or mutation create opring outide the pace of correct orting network. For example, let u tudy the pace of tate in the very imple cae of a 4-input orting network. Table 3 preent, for network of a given length, the proportion of valid orting network among all poible network (retricted to network for which no two conecutive comparator are identical) that contitute the tate pace explored by GA Reult Hilli and Drecher both tackled the 16-input orting network problem. However, the ize of the earch pace i coniderably reduced ince their population i initialized with the rt 32 comparator of Green' network and thi \prex" i protected from changing. Indeed, ince there are no regular pattern for the lat 28 comparator of Green' contruction, one can think that a better olution exit with the ame initial 32 comparator. Moreover, ince there are only 151 remaining unorted vector after thi initial contruction, the tne can be computed within reaonable time. Detail of GA implementation are not relevant here and are decribed in [Drecher, 1994] and [Hilli, 1992]. Reult along with population ize and number of generation for Hilli and Drecher' experiment are preented in table 4. Drecher' GA evolved orting network a compact a the bet known. However, in all experiment, a very large population ize i ued to reach thee reult. The next ection preent experiment for the END model approach and how that thi model i able to tackle even much more complex intance of the orting network problem. 4.2 END approach Decription A non-determinitic incremental algorithm (ee gure 2) i run by each individual of the population to generate incrementally valid orting network. A run of thi algorithm correpond to the incremental contruction of a path in the tree repreenting all valid and fair orting network; that i, valid orting network with no uele comparator. Valid orting network are built uing the zero-one principle. So, only the 2 n poible binary input vector need to be conidered (intead of the n! permutation of n ditinct number). The tne of a orting network i dened a it length. However, for the reproduction operation, tie are broken uing the depth of orting network. In that way, ecient orting network are generated regarding the number of parallel tep.

6 Begin with an empty or a partial network N DO BEGIN Compute the et S of ignificant comparator IF (S i not empty) THEN Pick randomly a comparator from S and append it to N END_IF UNTIL (S i empty) /* Now N i a valid and fair orting network */ Figure 2: Non-determinitic incremental algorithm run by each individual Reult Experiment were performed on a Mapar MP-2 parallel computer. The conguration of our ytem i compoed of 4K proceor element (PE). The peak performance of thi ytem for 32-bit integer computation i 17; 000Mip. In the maximal conguration a MP-2 ytem ha 16K PE and a peak performance of 68; 000Mip. Each PEimulate one individual if one want to tudy a 4K population, but it can alo imulate everal individual to evolve a larger population. Reult for the 16-input problem initialized with the rt 32 comparator of Green' orting network will not be preented. The lat verion of the model i able to evolve a orting network a good a the bet known with a 4K population ize and a ucce rate of almot 100% within 5 to 10 minute. Thi time performance i comparable to Drecher' reult ([Drecher, 1994]) preented in ection Actually, the intereting comparion between hi GA approach and the END model i that: GA evolved a population of 2 19 (= 524; 288) orting network (compared to a 4,096 population ize for END), 29 to 100 generation are enough for GA to nd the optimum but 150 to 200 generation are often required for END. To how the eciency of the END model for the orting network problem, the contruction of network from cratch, i.e., without any initialization, ha been tudied. So, there i no retriction on the earch in the pace of tate. Then, the END model ha been able to nd all bet known upper bound for the length of orting network (for the number of input in the range of 9 to 16) and even improved the upper bound for the 13-input problem. Table 5 preent parameter and reult of experiment for the 13-input and the 16-input problem 1. 1 A new algorithm that doen't ue the zero-one principle but that maintain the et of unorted vector uing a lit of mak ha been recently deigned and implemented. Figure 3: Two 13-input 45-comparator orting network uing 10 parallel tep. For the 13-input problem, the END model dicovered two orting network uing only 45 comparator (preented in gure 3), one comparator le than the bet current known. Moreover, thee two orting network ue 10 parallel tep which ivery good ince to get maller delay time one often ha to add one or two extra comparator module ([Knuth, 1973]) and the bet known delay for 13-input orting network i 9. For the 16-input orting network problem, two 60 comparator orting network have been evolved from cratch, each of them uing 10 parallel tep. Thi i a good a the bet human-built orting network deigned by M. W. Green. Figure 4 preent one of thee two 16-input orting network. 5 CONCLUSION AND FUTURE RESEARCH Thi paper preented a new and very promiing earch algorithm. By uing a population-baed approach for the earch in the tate pace and by contructing o- Thi algorithm improve the execution time by a factor of about even for the 16-input problem. Now, reliable reult can be obtained within an execution time of 12 hour for thi problem. Uing the maximal conguration for the Mapar, a run would take about 3 hour.

7 lution incrementally, thi model outperform GA in the cae of problem for which the contraint that dene the ub-pace of valid tate are complex and reult in a topology for which no ueful operator i known to explore thi ub-pace eciently. We were intereted in tudying the orting network problem becaue it i a very challenging problem and reult of experiment could be compared to the GA approach that have been ued by Hilli and Drecher. Moreover, analyi of thi problem alo revealed that the topology of the ub-pace of valid orting network make the ue of croover and mutation harmful. In another eld, board game can alo be conidered a a ubet of thi cla of \topologically" complex problem ince rule of uch game dene valid conguration of the board and thee valid conguration often repreent only a very mall ubet of the whole et of poible conguration. In [Juille, 1995], an example of a board game i preented to how how a trategy i evolved by the END model to play thi game. Moreover, the END model i intrinically highly parallelizable and calable. Uing a 2-D meh architecture where each proceor imulate one or everal individual, it i poible to evolve a very large population. The END model could alo be enhanced by adding ome feature like: Allowing the ue of ome heuritic for olution generating in order to reduce the number of potential extenion at each node of the tree of olution. Managing a local memory for each individual that would memorize it \pat" and would allow learning. Each individual could look for a local optimum before reproduction round. When poible, thi technique allow a fater convergence. Finaly, we are currently working to replace the global trategy for the commitment degree management by a local trategy that would be managed by the individual of the population themelve. Acknowledgement I am pleaed to acknowledge the upport of the W. M. KECK Foundation for the Volen National Center for Complex Sytem. I would like to thank Jordan Pollack, Patrick Tuft and Martin Cohn for their valuable advice and dicuion. Thank alo to the NSF whoe grant allowed the Brandei Computer Science Department to get a Mapar computer. Finally, Iwant to thank my wife, Anne, for the moral upport he provided me while I wa working on thi project and for her contant curioity. Reference [Back et al., 1991] Back, T., Homeiter, F., & Schwefel, H.-P. (1991). A urvey of evolution trategie. In Belew, R. K. & Booker, L. B. (Ed.), Proceeding of the Fourth International Conference on Genetic Algorithm, pp. 2{9, San Mateo, California. Morgan Kaufmann. [Belew & Kammayer, 1993] Belew, R. K. & Kammayer, T. (1993). Evolving thetic orting network uing developmental grammar. In Forret, S. (Ed.), Proceeding of the Fifth International Conference on Genetic Algorithm, San Mateo, California. Morgan Kaumann. [Drecher, 1994] Drecher, G. L. (1994). Evolution of 16-number orting network reviited. Peronal communication. [Green, 1969] Green, M. W. (c.1969). Some improvement in nonadaptive orting algorithm. Technical report, Stanford Reearch Intitute, Menlo Park, California. [Hilli, 1992] Hilli, W. D. (1992). Co-evolving paraite improve imulated evolution a an optimization procedure. In Langton, C. et al. (Ed.), Articial Life II. Addion Weley. [Juille, 1994] Juille, H. (1994). Evolving nondeterminim: An inventive and ecient tool for optimization and dicovery of trategie. Draft paper. [Juille, 1995] Juille, H. (1995). Incremental coevolution of organim: A new approach for optimization and dicovery of trategie. To appear in the proceeding of the Third European Conference on Articial Life. [Knuth, 1973] Knuth, D. E. (1973). The Art of Computer Programming, volume 3: Sorting and Searching. Addion Weley. [Levy, 1992] Levy, S. (1992). Articial Life: the Quet for a New Creation. Pantheon Nook. [Parberry, 1991] Parberry, I. (1991). A computeraited optimal depth lower bound for nine-input orting network. Mathematical Sytem Theory, 24:101{116. [Ruml et al., 1995] Ruml, W., Ngo, J. T., Mark, J., & Shieber, S. (1995). Eaily earched encoding for number partitioning. To appear in the Journal of Optimization Theory and Application. [Schraudolph & Belew, 1992] Schraudolph, N. N. & Belew, R. K. (1992). Dynamic parameter encoding for genetic algorithm. Machine Learning, 9:9{21. [Tuft & Juille, 1994] Tuft, P. & Juille, H. (1994). Evolving non-determinitic algorithm for ecient orting network. Poter, Articial Life IV Conference. Firt 60-comparator reult. [Winton, 1984] Winton, P. H. (1984). Articial Intelligence. Addion-Weley. Second edition.

8 Table 3: Ratio of valid orting network in the pace of all poible network Network length Size of tate pace 750 3,750 18,750 93, ,750 2,343,750 11,718,750 Valid orting network , , ,412 5,097,960 Ratio Table 4: Reult for Hilli and Drecher' GA experiment W. David Hilli Gary L. Drecher Population ize 65, ,288 Number of generation up to 5, to 100 Reult 61 comparator uing 60 comparator, co-evolution of paraite 100% ucce for 10 conecutive run, 6 contruction ue 10 parallel tep (like Green' orting network) Parallel computer ued 64K proceor CM-1 64-node CM-5 Execution time 100 to 1,000 generation 5to18minute per minute Table 5: Reult for the END model experiment for the 13-input and the 16-input problem 13-input problem 16-input problem Population ize 65,536 65,536 each PEimulate 16 individual each PEimulate 16 individual Number of generation 160 to to 500 Neighborhood radiu 3or4 5 for reproduction Reult Number of run: 6 Number of run: 3 For 2 run: 45 comparator For 2 run: 60 comparator Execution time about 8 hour for each run about 48 to 72 hour for each run Figure 4: A 16-input 60 comparator orting network uing 10 parallel tep.