Mon.Not.R.Astron.Soc.000,1–17(2011) Printed4January2012 (MNLaTEXstylefilev2.2) Optimal SKA Dish Configuration using Genetic Algorithms Adam Gauci1, Kristian Zarb Adami2, John Abela3, Babak E. Cohanim4 1Department of Intelligent Computer Systems, Faculty of ICT, University of Malta, Malta. 2Department of Physics, Faculty of Science, University of Malta, Malta. 3Department of Computer Information Systems, Faculty of ICT, University of Malta, Malta. 2 4Mission Design Group Leader, Draper Laboratory, 555 Technology Square, Cambridge MA 02139. 1 0 2 n Released2011XxxxxXX a J 3 ABSTRACT ] M The Square Kilometre Array (SKA) is a radio telescope designed to operate be- tween 70MHz and 10GHz. Due to this large bandwidth, the SKA will be built out I of different collectors, namely antennas and dishes to cover the frequency range ade- . h quately. In order to deal with this bandwidth, innovative feeds and detectors must be p designed and introduced in the initial phases of development. Moreover, the required - o level of resolution may only be achieved through a groundbreaking configuration of r dishes and antennas. Due to the large collecting area and the specifications required t for the SKA to deliver the promised science, the configuration of the dishes and the s a antennas within stations is an important question. This research builds on the work [ done before by Cohanim et al. (2004), Hassan et al. (2005) and Grigorescu et al. (2009) to further investigate the applicability of machine learning techniques to de- 1 v termine the optimum configurations for the collecting elements within the SKA. This 6 work primarily uses genetic algorithms to search a large space of optimum layouts. 2 Every genetic step provides a population with candidate individuals each of which 7 encodes a possible solution. These are randomly generated or created through the 0 combination of previous encodings. In this study, a number of fitness functions that 1. rank individuals within a population of dish configurations are investigated. The UV 0 density,connectingwirelengthandpowerspectraareconsideredtodetermineagood 2 dish layout. 1 Key words: SKA, radio telescope, machine learning, evolutionary programming, : v genetic algorithms i X r a 1 INTRODUCTION desiredspecificationsisofimportancebothintheconstruc- tion phase, but more importantly in the maintenance and TheSKAwillbeaninstrumentthroughwhichmajorscien- runningcostsofthetelescope.Grigorescuetal.(2009)esti- tific discoveries are to be made. Although the construction matethat100MEurowillmostlikelybeallocatedtocabling willfollowaphasedapproach,phase1oftheSKAisalready and trenching that connects the stations together. aformidableinstrumentandwillundoubtedlyshedlighton the evolutionary stages of the universe from the epoch of In this study, the applicability of Genetic Algorithms reionisation as well as improve our understanding of grav- (GA)todeterminethemostoptimumconfigurationsforthe ity through the detection of binary and millisecond pulsars dish array is investigated. Such evolutionary programming (Garrett et al. 2010). approachesarebasedonDarwin’stheoryofnaturalselection Dishes and antenna arrays will, using state of the art inwhichthefittestindividualsfromeachpopulationsurvive receivers,provideunprecedentedsensitivitybetween70MHz and generate offspring chromosomes that encode configura- and 10GHz (Garrett et al. 2010). The required resolving tionswhichareclosertotheoptimumsolution.InSection2, power unavoidably dictates an enormous spatial extent (≈ an introduction to GAs is presented while in Section 3, the 3000km)intheinitialphaseandwillcostaround500MEuro work done for dish array optimisation is discussed. Follow- (Dewdney2010).Apioneeringdesignminimisinginfrastruc- ingdetailsontheimplementedgeneticoperatorsandfitness ture, networking and other costs whilst still achieving the functions, various cases together with the obtained results (cid:13)c 2011RAS 2 A. Gauci et al. are presented in Section 4 and Section 5. Some conclusions are drawn in Section 6. 2 GENETIC ALGORITHMS Genetic Algorithms (GAs) are search heuristics that follow the natural process of evolution to determine the most fit hypothesis from a pool of possible solutions. Unlike other searchtechniquesthatadoptabrute-forceoriterativestrat- egy, GAs combine parts of the best know solutions to try andcreatebetterencodings.Thisevolutionaryprogramming methodologythatusesbothmatingandmutationtocreate betterchromosomeswaspioneeredbyJohnHollandaround the mid 1960’s (Holland 2005). Since then, GAs have been used for a wide range of applications where an optimized solution with a large number of parameters is required. Chromosomes that represent valid hypothesis need to be encoded as streams of data that can be processed by the algorithm. A pool of such encodings is referred to as the population and the GA progresses by updating this set ofsolutions.Ineachgeneration,newindividualsarecreated randomlyorthroughgeneticoperatorssuchascrossoverand mutationthatrecombineormutateparentchromosomesre- spectively. Parent hypothesis from which offsprings are cre- ated, are selected according to a probability function. Figure 1.SKAphase1dishlayout. Ineachgenerationstep,allsolutionsarerankedbyafit- ness function and the population is updated to include the bestindividuals.Theprocessisrepeateduntilthealgorithm stallsandnoimprovementinthefitnessisdetectedwithfur- therprocessing.Inthiswork,eachchromosomerepresented a configuration and stored the dish locations. AsdiscussedbyMitchell(1997),GAssearchthrougha Figure 2.Dishconfigurationchromosomestructure. large space to find the solution that maximises the fitness function. With the adopted approach, the algorithm is less grid. The algorithm was let to evolve for a number of gen- likelytoconvergetowardsalocalminimumsincetheopera- erations until no improvement in the fitness was detected tors can replace parent encodings with completely different and all encodings in the final population were similar. Due offsprings.Asthealgorithmprogresses,onemustmakesure to the required number of dishes and receiver distribution, thatagroupofgoodandsimilarencodingswillnotreplicate the initial population could not be biased with previously and dominate the population. known good encodings. 3.1 Chromosome structure and genetic operators 3 CONFIGURATION Encodings that represented different configurations were The specification document for phase 1 SKA specifies that createdeachonestoringthexandy coordinatesofthedish 250 parabolic dishes each 15m in diameter will be installed locations. As shown in Figure 2, each chromosome stored over a 100km radius region (Dewdney 2010). 125 core sta- the mapping of dishes on the domain grid as a series of 500 tions (50%) will be fixed in the central 500m radius. The integers. An identification number was also associated and inner region and middle region will extend over a radius stored with each encoding. This allowed the properties and of 2,500m and 100,000m and will contain 50 (20%) and 75 status of each chromosome to be monitored and saved. (30%) antennas respectively. Figure 1 shows this layout. The crossover operation was designed to produce off- In this study, a search for the optimum configuration springs whose genes encode combinations of the core, inner that maximises the uniformity of the UV density distribu- andmiddleregion.Fromeverypairofparentchromosomes, tion while keeping the connecting wire length to a mini- afterallgenecombinationshavebeencarriedout,sixnewin- mum was conducted. The goal is to position the dishes in dividualswerecreated.Ifthecore,innerandmiddleregions such a way as to obtain a flat uv distribution with points of the first parent were represented by C1 I1 M1, and the spread uniformly across the uv-plane (Cohanim 2004). A second parent was made from C2 I2 M2, encodings with C1 regular gridded mask representing the domain over which I2 M1,C1 I1 M2,C1 I2 M2,C2 I1 M1,C2 I1 M2andC2 I2 the dishes can be positioned, was initially generated. The M1 were generated. To create more offspring by combining initialencodingswithpossibleconfigurationswerethencre- existing chromosomes, two more encodings were generated ated with antenna locations chosen randomly from such a accordingtoarandomlygeneratedbinaryvector.Inpartic- (cid:13)c 2011RAS,MNRAS000,1–17 Optimal SKA Dish Configuration using Genetic Algorithms 3 Figure 3.Offspringscreatedbythedishconfigurationcrossover operator. Figure 4.UVnominalgridquadrant. ular, dish positions corresponding to one were taken from the first parent while positions corresponding to zero were taken from the second parent. The binary vector was then jectioncanbedeterminedbyequation2.Here,hrepresents bitwise inverted and the same procedure was repeated to the hour angle and δ is the source declination. obtain one more configuration. Since parent chromosomes Inordertobeabletocomparetheresultingoutputs,in may have common dish locations, the last two generated thisworkthedeclinationwasalwayssetto90◦ torepresent offspringswherecheckedbyagenerepairfunctiontoensure aradioobjectatthecelestialnorthpolewhilethehourangle that all 250 locations were distinct. This crossover process was set to range from 0◦ to 345◦ at 15◦ intervals. is graphically shown in Figure 3. Sincethecomputationofthedistancebetweenallbase- The mutation operator was implemented such as to al- lines becomes prohibitively expensive very quickly, we fol- terthepositionsofrandomlyselecteddishlocations.Thisal- lowedtheworkpublishedbyCohanim(2004)andCohanim lowedthealgorithmtokeepsearchingandtoconsiderclosely etal.(2004).Inparticular,thenominalgridpointclosestto related encodings in the multidimensional search space. Al- each UV point was determined and flagged. An analysis of thoughmostpartsofthechromosomesremainedunchanged the non-matched nominal points gave an indication of the after mutation, the shifting of some of the dishes to a new distributionoftheconfigurationandhenceameasureoffit- locationpreventedthealgorithmfromconvergingontoalo- ness. Ideally, the majority of nominal grid points would be cal maximum. In particular, a vector with 250 random val- flagged by at least one UV point. ues between 0 and 1 was populated. Indices of locations to Due to its size and current vision, the SKA will be a which a number less than 0.2 was assigned were identified logbasedstructure.AsshowninFigure4,alogdistribution and new dish locations to the corresponding positions were for the nominal grid was decided to be used. The goal of determined. Chromosomes created by mutation were pro- the GA was then set to minimise the fitness function, i.e. cessedbythegenerepairfunctiontoensurethatnolocation the percentage of non-matched nominal grid points. As in had more than one dish assigned to it. Cohanimetal.(2004),thesewerecalculatedusingequation 3. 3.2 Fitness functions N −N f = total matched (3) 3.2.1 UV density distribution fitness UV N total In order to ensure that the genetic algorithm converged to- Here,N isthetotalnumberofpointsinthenominal total wards a solution that maximised UV coverage, the density grid and N is the total number of matched points. matched mapwascomputedfromeveryuniquepairofdishesbyequa- Thenumeratorequatestothepercentageofgridpointsthat tion 1. were not matched with any UV point. Fitness evaluation of every individual required an effi-     cient calculation of the UV density distribution as well as u x −x i,j 1 i j the mapping onto the nominal grid for a large number of  vi,j = λ yi−yj  (1) chromosomes.Weselectedtouseak-dimensionaltreerepre- w z −z i,j i j sentationofthenominalgridwhichneededtobecomputed Duetothenatureofthedisharray,N(N−1)/2number onlyonceandcouldbethenstoredinmemory.Thenearest ofuniquepoints(whereN isthenumberofdishes)weregen- pointtoeverypositionencodedcouldthenbedeterminedby erated.Incertaintestcases,thefullcoverageofthetelescope traversingtheconstantbinarytreedatastructure.Sincethe aftertakingintoconsiderationtherotationoftheearthwas nominalgridwasdefinedovertwodimensions,eachnon-leaf alsocomputed.AsdiscussedbySegransan(2007),suchpro- node represented a perpendicular hyperplane that divided (cid:13)c 2011RAS,MNRAS000,1–17 4 A. Gauci et al.  ui,j  1 sin(h) cos(h) 0  xi−xj   vi,j =  −sin(δ)cos(h) sin(δ)sin(h) cos(δ)  yi−yj  (2) λ wi,j cos(δ)cos(h) −cos(δ)sin(h) sin(δ) zi−zj Figure 5.Logscalecablelengthfitness(fWireLog). Figure 6.Stepwisecablelengthfitness(fWireStep). the space into two subspaces. The left subtree pointed to 3.2.3 Stepwise wire length fitness other nodes on the left while the right subtree represented Since the majority of chromosomes were found to have a points to the right. cable length of about 1000km, a stepwise function that lin- earlyvariestheoutputbetween0.1and0.8forwirelengths between900kmand1300kmwasimplemented.Morespecif- 3.2.2 Logarithmic wire length fitness ically, the wire length fitness in this case was computed by Various approaches that attempt to compute an accurate equation 5. cost and minimise the required length of cable to connect t(cth2eol0eue0nsd9cti)ostphpreeersnolcvathyoidiogneuegtta,hieansrsef,trwhaosaetfvlrleuaalcbgsteoucerronietn.hpnmIrenecssteCitonhontahetdacon.asiGlmtssoritgetotoarkoaeepls.ctiuin(m2teo0its0ea4ac)la-,. 000..015 iiifff 15000(cid:54)00(cid:54)(cid:54)wiwwreiilrreeenlleegnnthggtt<hh<<1059000;00;; the single linkage algorithm is used. Here, to determine the f = 0.1→0.8 if 900(cid:54)wirelength<1300; WireStep sKmhreounrstkeeatstlalMs.e2iqn0ui0me1nu)cmewatSshpauatsnecndoi.nngneTctrseeal(lMvSerTt)iceaslgtoorgitehtmher(,Ctohre- 001..89 oiiffth1134er00w00i(cid:54)(cid:54)se;wwiirreelleennggtthh<<11450000;; Throughout this work, a cable with unit cost per unit (5) length that connects all dishes in the core, inner and mid- In this way, the algorithm could accurately assign and dle regions, was assumed. Dish locations were connected in rank individuals. Figure 6 depicts this stepwise variation suchawayastocreateanundirectedgraphinwhichedges with wire length more clearly. (connections)betweeneachvertex(dishes)hadnoparticular direction.Theweightofeveryedgewastakentocorrespond totheEuclidiandistancebetweenthetwoconnectingnodes. 3.2.4 Wire length penalty fitness The MST algorithm was then use. Since the UV density fitness corresponds to a percent- Furthertestssuggestedthatawirelengthpenaltyapproach age ranging from 0 (optimum UV distribution) to 1 (worst may be more effective. Individuals encoding dish locations case),anormalizingfunctionthatallowsthecomputedwire that could be connected by a cable length of less than length to be compared and added with the resulting UV 1250km, were not penalised. Chromosomes with a mini- fitness, was required. A log based approach was initially mumwirelengthgreaterthan2250werehighlydiscouraged adoptedandthecablelengthfitnesswascomputedbyequa- throughafitnessassignmentof1.Intermediatecablelengths tion 4. weregivenaweightingwhichvariedlinearlyasdescribedby equation6.Thisvariationofwirelengthfitnessispresented (cid:18) (cid:19) in Figure 7. The threshold values used were determined af- 1 fWireLog =1−log10 wirelength (4) ter noting the results obtained in previous runs. The main advantage of this approach was that it directed the search The wire length is given in kilometers. Figure 5 shows towards solutions with a good UV coverage and penalized the fitness values for cable lengths between 0 and 5000km. encodings that have a wire length above the norm. All en- (cid:13)c 2011RAS,MNRAS000,1–17 Optimal SKA Dish Configuration using Genetic Algorithms 5 Figure 7.Penaltycablelengthfitness(f ). WirePenalty Figure8.Rawangularpowerspectrum(blue)andalogdecaying curveusedasreferenceforfitnesscalculation(red). codingswithacablelengthoflessthan1250kmweretreated equallyandthefitnesswastakentodependsolelyontheUV provementwithsubsequentprocessing.AsdiscussedinSec- distribution. tion 4 below, runs using various combinations of the above mentioned fitness criteria were conducted. 0 if 0(cid:54)wirelength<1250;  f = 0→1 if 1250(cid:54)wirelength<2250; WirePenalty 1 otherwise; 4 RESULTS FOR SKA PHASE 1 (6) Ananalysisofhowtheoptimumconfigurationchangeswith Asdiscussedinsubsequentsections,inordertocompare different fitness functions, population sizes, and criteria for the results obtained in this study with a generic configura- selectingindividualsforsubsequentgenerations,wascarried tion, dishes in the middle region were clustered together. out. In the following subsections the results obtained for This group formation naturally minimised the wire length different cases are presented. and to account for these encodings, another wire penalty fitness function with lower thresholds was defined. This is formally defined by equation 7 below. 4.1 Case 1 - GA with UV and log scaled wire length fitness 0 if 0(cid:54)wirelength<300;  As a first test run, the genetic algorithm was set with an fWirePenaltyLow = 0→1 if 300(cid:54)wirelength<450; initial population of 1024 random chromosomes. For each 1 otherwise; individual, the overall fitness was calculated by equation 8. (7) f =f +f (8) dish1 UV WireLog 3.2.5 Power spectrum fitness Subsequentgenerationswerecreatedafterselectingthe fittest1024individualsfromapoolof4096thatconsistedof Anyimprovementgainedthroughtheintroductionofpower 1024chromosomescreatedbymutation,2048chromosomes spectra calculation as part of the fitness function, was also createdbycrossoverand1024newrandomlygeneratedchro- investigated. Studies such as Parsons et al. (2011) provide mosomes.Theinitialpopulationhad ameanandminimum detailed algorithms of how to compute power spectrum. fitnessof4.788and4.73respectively.After119generations, However, for this study, work done by Green (2007) was the average fitness reduced to 4.421 and the most optimum followedtodeterminetherawangularpowerspectrumfrom individual had a fitness of 4.414. A plot of the resulting the UV-plane. In particular, the number of UV points that dish positions together with the computed wire length is coincided with log spaced annuli of width equal to the re- presented as Figure 9. The corresponding mapping of the stricted zone diameter of the dishes, was determined. The UVdensitydistributionontothenominalplaneisshownin resulting data series was divided by a log decaying curve Figure 10. and a mean value was computed to obtain a measure of fitness proportional to the distance between the two curves (f ).Atypicalrawangularpowerspectrumand PowerSpectrum 4.2 Case 2 - GA with weighted UV and stepwise the considered ideal curve are shown in Figure 8. cable length fitness AstheGAprogressed,thefitnessofindividualsineach population were computed in parallel. The algorithm was In the second case, the UV coverage and wire length were left to evolve until it stalled and there was very limited im- givenaweightingof60%and40%respectivelyasdefinedin (cid:13)c 2011RAS,MNRAS000,1–17 6 A. Gauci et al. Figure 9. Full (top) and zoomed (bottom) dish configuration Figure 10. Mapping of the UV density distribution onto the with shortest wire connecting the middle (blue), inner (green) nominal grid for the full array (top) and core region (bottom) andcore(red)regionsforCase1. showingthematched(blue)andunmatched(red)pointsforCase 1. equation9.Experimentingwithdifferentweightingschemes crossover and the algorithm converged after 111 iterations. allowthestakeholderstohaveabetterunderstandingofthe Figure 11 shows the final dish locations and wire length tradeoffs between performance and cost. whiletheUVdistributionispresentedinFigure12.Thishad aUVdensityfitnessof0.67354andwirelengthof724.74km fdish2 =(0.6×fUV)+(0.4×fWireStep) (9) resulting in f =(0.6×0.67354)+(0.4×0.1)=0.4441. The initial population size was set to 1024. Individuals forsubsequentpopulationswereselectedfromapoolof1024 4.3 Case 3 - GA with UV and cable length chromosomes generated through mutation, 2048 offsprings penalty fitness generatedbycrossoverandanother1024randomencodings. The highest ranking chromosomes were also considered for Inthiscase,theinputtotheGAconsistedofaninitialpop- migration into the next population. ulation with 4096 chromosomes which encoded random po- After the first few iterations, the percentage of ran- sitionsfor250dishesasdefinedinSection3.1.Ateachstep, domlygeneratedchromosomerapidlydecreasedtozero.The parent chromosomes were selected from the population to selectionofindividualsgeneratedthroughmutationalsode- generate4096newoffspringsthroughmutationandanother cayed with time. The strongest genes were created through 8192 new individuals from crossover. The fitness of these (cid:13)c 2011RAS,MNRAS000,1–17 Optimal SKA Dish Configuration using Genetic Algorithms 7 Figure 11. Full (top) and zoomed (bottom) dish configuration Figure 12. UV density distribution for the full array (top) and with shortest wire connecting the middle (blue), inner (green) coreregion(bottom)forCase2. andcore(red)regionsforCase2. new encodings as well as another 4096 randomly generated to have a lower fitness than the new offspring generated individuals were combined with the scores of the previous throughthecombinationofchromosomesalreadyinthepop- populationandrankedtodeterminethefittest4096entries. ulation. Figure 14 shows the typical lifetime for crossover These were selected for the next cycle and the process was chromosomes, mutation chromosomes and randomly cre- restarted. In particular, the fitness was computed by equa- atedindividualsbeforetheygotreplacedbyfittermembers. tion 10. Encodings generated by the implemented genetic operators provedtohavealongerlifetimethanrandomchromosomes. Thisspeededuptheconvergenceofthealgorithmaswellas f =f +f (10) dish3 UV WirePenalty permitted the generation of fitter configurations. Figure13givesanindicationofthepercentageofelite, Figure 15 shows how the fitness improved as the al- crossover, mutation and random chromosomes selected at gorithm progressed. Figure 16 presents a rendering of the each generation. As expected, after the first few iterations, fittest encoding after 102 generations. Dishes in the mid- the genetic operators produced individuals with improved dle,innerandcoreregionsareshowninblue,greenandred fitnessandthealgorithmprogressedbycontinuouslychoos- respectively. The UV density distribution percentage was ing offsprings generated through crossover. Randomly gen- 0.66825 while the minimum wire length computed by the erated individuals became phased out and soon resulted MSTalgorithmwasfoundtobe815.12km.Fullandzoomed (cid:13)c 2011RAS,MNRAS000,1–17 8 A. Gauci et al. Figure13.Percentageofelite(black),crossover(blue),mutation (green)andrandom(red)chromosomesselectedforeachpopula- tionforCase3. Figure14.Lifetimeofcrossover(red),mutation(blue)andran- dom(green)chromosomesforCase3. versions of the UV distribution calculated from all dish po- sitions is presented in Figure 17. 4.4 Case 4 - GA considering randomly oriented grouped outer dishes with UV and cable length penalty fitness Inthiscase,thedishpositioningandgeneticfunctionswere Figure 16. Full (top) and zoomed (bottom) dish configuration modified so that configurations had the elements in the with shortest wire connecting the middle (blue), inner (green) middle region grouped in small random clusters of 3 to 8 andcore(red)regionsforCase3. disheseach.Elementswerepositionedinacircular,triangu- lar or linear fashion and were given a random orientation. ble length was expected to be less and the fitness function Since dishes were not randomly scattered, the required ca- defined by equation 11 was used. f =f +f (11) dish4 UV WirePenaltyLow Thecrossoverfunctionusedinthepreviouscasescould stillbeusedsincethemiddleregionofallchromosomeshad the exact same number of elements. Genes from any two parentscouldbeswappedandstillgeneratevalidoffsprings. However, the mutation operator had to be redefined. If a dish within the middle region was selected for mutation, a new position and shape for the entire group had now to be determined.Sinceeachencodingcouldhaveadifferentnum- ber of groups with different number of dishes, further logic had to be performed before randomising the chromosome. Figure 15. Fitness for the initial individuals (black), random Apart from the chromosome id, an integer with numerals chromosomes (red) and offsprings generated by the mutation that corresponded with the number of dishes in each group (green)andcrossover(blue)operatorsforCase3. wasalsostoredforeachchromosome.Consecutive(x,y)co- (cid:13)c 2011RAS,MNRAS000,1–17 Optimal SKA Dish Configuration using Genetic Algorithms 9 Figure 17. UV density distribution for the full array (top) and coreregion(bottom)forCase3. Figure 18. Full (top) and zoomed (bottom) dish configuration with shortest wire connecting the middle (blue), inner (green) andcore(red)regionsforCase4. ordinates could then be read until the require group of sta- tions was found. Figure 18 and Figure 19 show the resulting configu- snowflake, in a circular pattern and in a reuleaux triangle ration and the connecting wire respectively. Although the orientation,wereinitiallydetermined.Ineachcase,29to30 UV density distribution corresponds to a fitness of 0.77214 elements were used to render well the required shapes.For and a large number of nominal grid points are unmatched, a small number of elements, the configurations and the UV the clustering of dishes allowed for a short cable length of coverage of the circular and reuleaux orientations are very 154.46km. For this run, the algorithm was made to work similarandthealgorithmwassettodistributethedishesin on a population of 1024 chromosomes and evolved for 102 the groups as such. generations before it stalled. Figure21showsoneoftheresultingconfigurationsafter letting the GA run for 102 generations. The corresponding UV pattern and UV mapping onto the nominal grid are 4.5 Case 5 - GA considering grouped outer dishes showninFigure22whileFigure23showstheconstantdecay in a circular orientation with UV and cable in fitness with generations. Here the UV fitness and wire length penalty fitness length resulted to be 0.777321 and 120.94km respectively. Although dishes in the middle region were grouped as de- To determine how the resolution improves with longer scribed for the previous case, clustered elements were now observation times, another run that takes into account the only positioned and oriented in a constant configuration. earth’srotationandwhichconsiderstheUVprojectionover As shown in Figure 20, the corresponding UV distributions 24 hours, was performed. Due to the extra calculations in- for dishes positioned in a straight line, in a triangle, as a volved,thefitnesscomputationofeachchromosomerequired (cid:13)c 2011RAS,MNRAS000,1–17 10 A. Gauci et al. Figure 19. UV density distribution (top) and mapping onto the nominal grid (bottom) showing the matched (blue) and un- matched(red)pointsforCase4. Figure 20.Orientationof29dishes(right)andthecorrespond- ingUVdistribution(left)whenplacedinastraightline(a),tri- angle (b), snowflake (c), circular (d), and reuleaux triangle (e), configurationsforCase5. 4.6 Case 6 - Static SKA CTF and Reuleaux on average 37.26 seconds. To finish processing in reason- triangle configurations able time, a population size of 128 was set. The resulting dish positions and the mapping of the UV points onto the Here, the fitness functions used in the other test cases were nominal grid are presented in Figure 24 and Figure 25 re- computedforstaticconfigurations.Inparticular,thegeneric spectively. As indicated by the shaded tracks, in this case configurationdefinedbytheSKAConfigurationsTaskForce theGAclearlychoseaspiralconfigurationforthedishesin (CTF)aswellasadisharrayspecifiedbyReuleauxtriangles themiddleregion.Moreover,ifthethreearmsaresuperim- were processed in order to be able to evaluate better the posed,thedisheswouldberoughlyequallyspacedalongthe results achieved by GAs. track.Theattaineddistributioniscausingmostofthenom- The generic dish configuration by the CTF is shown in inal grid points to pair after a 360◦ rotation hence giving Figure 26. The provided geographical coordinates were ini- a very good UV fitness. The algorithm converged after 101 tially converted to cartesian points and projected onto the generations when a good compromise between UV density regular spatial grid considered in this work. The UV den- and cable length was found. sity distribution was then computed and mapped onto the (cid:13)c 2011RAS,MNRAS000,1–17