A Novel Stochastic-Based Algorithm for Terrain Splitting Optimization Problem

Transcription

1 A Nvel Stchastic-Based Algrithm fr Terrain Splitting Optimizatin Prblem Le Hang Sn Nguyen inh Ha Abstract This paper deals with the prblem f displaying large igital Elevatin Mdel data in 3 GIS urrent appraches relate t the splitting algrithms by 2 Plygnal ectr ata such as Particle Swarm Optimizatin (PSO-TSA) and Genetic Algrithm (GA-TSA) We will herein present anther methd based n stchastic ptimizatin fr the cnsidered prblem It als emplys sme ideas f Wife-Selectin scenari and Stick Prcedure The new methd allws us t quickly find the ptimal saving threshld The cmparisn with the state-f-the-art methd will be made t verify the efficiency f the prpsed methd Keywrds igital Elevatin Mdel Gegraphic Infrmatin Systems Stchastic Optimizatin Terrain Splitting I INTROUTION Terrain Splitting Optimizatin (TSO) prblem was presented by the authrs in [4] Its purpse is t reduce the displaying time f large terrain data especially in the frmat f igital Elevatin Mdel () in 3 Gegraphic Infrmatin Systems (GIS) Terrain data are used t represent the three-dimensinal cmpsitins in the relatin with gegraphic factrs epending n the reslutin the size f a terrain is varied t express the infrmatin attached t that terrain Fr example a 3m terrain has a vlume f 28 Megabytes (MB) The smaller the reslutin f terrain is mre details are shwn and its size is increased as a result Indeed it takes lng time t display such a terrain Mathematically this prblem is described belw k () J S i min where S ( i i Si S Si S S i k k i (2) i k ) is the area f a small terrain in a prcessr f a cmputing system S is the area f terrain The parameter is the saving threshld The last parameter is the disparity Nrmally its range falls int ( 5) In equatin (2) the first line states that the memry space in a prcessr is saved by percents The secnd ne cnfirms that the difference between the memry spaces in tw prcessrs is small Several sft cmputing methds were designed fr TSO prblem such as SESA [4] PSO-TSA [6] and GA-TSA [6] They relied n the ideas f using natural evlutin Manuscript received n January 23 Le Hang Sn is with NU University f Science ietnam Natinal University Nguyen inh Ha is with NU Hani Infrmatin Technlgy Institute ietnam Natinal University cmbined with heuristic search methds t specify the slutins These algrithms achieved successful results as shwn in the experiments f equivalent articles The aim f this nte is t investigate anther ptimizatin methd fr this prblem It emplys the new ideas f Wife-Selectin scenari and Stick Prcedure The prpsed methd is named as Stchastic simulatin Test based Terrain Splitting Algrithm (SIT-TSA) and is cmpared with the algrithms abve t verify the efficiency The remainder f this paper is rganized as fllws Sectin 2 presents sme related wrks fr this prblem The prpsed methd is described in Sectin 3 Experimental results and discussins are given in Sectin 4 Finally we will make cnclusins and delineate future wrks in the last sectin II RELATE WORKS The authrs in [4] intrduced a cnditinal parallel partitining methd s-called SESA fr TSO prblem The basic idea f this methd is t traverse all partitins dividing n elements int k blcks Fr each blck SESA calculates its area and checks the cnstraints If a suitable partitin is fund the algrithm will stp and utput the results ertainly t reduce the number f traversed partitins a pre-prcessing step based n gemetric prcessing between plygns is carried ut t arrange sme elements int specific blcks Additinally parallel cmputing is als emplyed t accelerate the running time Hwever the saving threshld fund in SESA is nt ptimal Authrs [6] presented tw algrithms t slve the riginal prblem The first ne based n Genetic Algrithm [2] namely GA-TSA emplys sme ideas f natural evlutin such as inheritance mutatin selectin and crssver fr finding the best saving threshld in a search area In this algrithm an individual is a cllectin f indexes f all plygns that represent fr all blcks in a current slutin Then thrugh a fitness functin all individuals are srted in the ascending rder and half f them are selected t reprduce a new generatin by the mean f rss Over and Mutatin peratins After pre-defined maximal iteratin steps the best generatin is fund and the saving threshld can be extracted frm it The secnd algrithm in that literature started with an idea f Swarm Optimizatin which is cnsidered t be the mst suitable strategy amng all f Heuristic Optimizatin The chsen algrithm t develp is Particle Swarm Optimizatin (PSO) which was invented by Kennedy et al [3] Indeed the algrithm was named PSO-TSA The basic idea f this algrithm lies n Seed Prcedure Basically k seeds are evenly distributed in the space Each seed represents fr a number f plygns If the cnstraints (2) are nt met PSO algrithm is used t generate a new ppulatin until the stpping cnditin is reached In the last iteratin the particle hlding gbest value will be utputted if it satisfies the cnstraints 24

2 A Nvel Stchastic-Based Algrithm fr Terrain Splitting Optimizatin Prblem In the experiments PSO-TSA btains a smaller saving threshld than GA-TSA des Hwever the running time f PSO-TSA is lnger than that ne f GA-TSA ertainly tw algrithms are better than SESA bth by the saving threshld and running time Therefre PSO-TSA is cnsidered as the state-f-the-art algrithm fr TSO prblem III THE PROPOSE METHO A Basic Ideas In this sectin we present anther apprach fr TSO prblem It is a stchastic agent - based apprach named Stchastic simulatin Test based Terrain Splitting Algrithm (SIT-TSA) Keep the prblem in mind and temprarily frget abut the previus ideas f using PSO and GA all we need t knw is a basic principle that cvers all activities f SIT-TSA algrithm In a simple way it can be understd as: It is suppsed t be n satisfied slutin with prbability p after a series f failed stchastic simulatin tests n varius pssibilities derived frm the riginal sample In the ther wrd this principle behaves as a similar way t Mnte arl methd - a class f cmputatinal algrithms that rely n repeated randm sampling t cmpute their results [] Mnte arl methd is widely used in varius fields such as in statistical physics particularly Mnte arl mlecular mdeling as an alternative fr cmputatinal mlecular dynamics as well as t cmpute statistical field theries f simple particle and plymer mdels Therefre a Mnte arl - like apprach is suitable fr ur prblem in case f the number f plygns is large and an intermediate answer is required Nw cnsider the fllwing Wife-Selectin scenari in Fig : Once upn a time there is a great Kingdm with a wise brave king Everybdy admires him a lt Hwever this king has nly a sn and he pays attentin t nthing except hunting r playing with friends The ld man is very sad because he is getting lder and his sn cannt replace him Sharing the wrries sme servants suggest the king t find his sn a fiancée He recgnizes it a gd idea and immediately annunces t all villages in the Kingdm Besides the ld king assigns a duke t fllw with the prince t cme t each village fr searching Nevertheless the prince is very lazy and des nt want t g t all villages The duke wh is a gd mathematician calculates by prbability which village has mre girls satisfying the king s cnditins Hwever these villages are still much and it takes a lt f time t speak with all girls in a specific ne Instead the duke chses randm girls frm each village and tests If he finds a suitable girl then she will be added t his lists The test is repeated in ther selected villages Finally the duke will give the prince his list and let him chse the fiancée If n suitable girl is fund n the duke s list the prince has t wait fr next selectin in sme fllwing years SIT-TSA acts like the scenari abve An agent is randmly initiated at a plygn By using a lcal search methd an rdered list f plygns is established Then by multiple tests with randm sticks dividing this list int k blcks a candidate list f this agent is set up This prcess is repeatedly perfrmed fr ther agents Finally the ptimal slutin with minimal saving threshld parameters will be chsen frm all the candidate lists ertainly we use parallel cmputing as an imprtant aid fr the reductin f ttal cmputing time due t independent wrks between agents B The Algrithm Input: a terrain a plygn shape dataset the ttal number f plygns ( l ) the number f prcessrs ( k ) the disparity and a test range t ] [ t Output: The ptimal saving threshld SIT-TSA: Step : ivide the ttal number f agents fllwing by the number f prcessrs in the system Then the number f agents in each prcessr is calculated as fllws NA l / 2k (3) where l is the ttal number f plygns and k is the number f prcessrs Started ndes fr all agents are ascending chsen frm nde Step 2: Fr each prcessr we find the candidate lists fr all agents in it In specific cnsider a plygn as a nde in a graph Let and are the sets cntaining started ndes f all agents in this prcessr and visited ndes in an agent s life respectively Initially Tw agent parameters a b are als randmly initiated by psitive values a rand() and b a Step 3: Mve the first nde I f (4) t I (5) \ I (6) Step 4: Fr any nde which is nt in calculate the prbability PI a d( I ) b SP (7) I where d ( I ) is the distance between plygns I and Fig Wife Selectin scenari SPI is the area f tw plygns I and 242

3 Step 5: hse nde belw and add it t I max PI P which satisfies the cnditin (9) (8) Step 6: Repeat Step 4 and Step 5 with is the started nde until all ndes are in Then we will have a sequence X X 2 X l extracted frm where X t is a plygn t l Step 7: Select a number f tests fr this agent ( NTest ) by the prbability in equatin (7) NTest MinTest () [( MaxTest MinTest) max Pi ] where [ MinTest MaxTest] is a given range f the number f test cases Indexes i and are tw cnsecutive ndes in Step 8: Fr each test case perfrm Stick Prcedure t find a suitable slutin In essence we select ( k ) randm stick t put int the sequence X X 2 X l 8 Stick ( l k i) () 82 If k then stp; 82 Stick ( i) rand( l k ) ; 83 Stick ( l Stick ( i) k i ) ; 84 End Step 9: Fr each blck in this test case separated by tw cntinuus sticks calculate the area f all plygns in it SP i k ) ( i Step : heck the cnstraints (2) fr the current slutin If they are satisfied we will find the maximum belw and add this test case s slutin int the agent s candidate list (2) SPi max i k Test _ ase S Step : Repeat frm Step 8 t Step with ther test cases We will receive an agent s candidate list with different values Test _ ase agent Find the ptimal slutin fr this agent min (3) Test _ ase Step 2: Repeat frm Step 2 t Step with ther agents in this prcessr Extract the ptimal slutin f the prcessr frm a list f slutins f agents prcessr agent agent min (4) Step 3: Synchrnize all prcessrs in the system and find the ptimal saving threshlds frm if they exist ptimal prcessr prcessr min (5) We then cnclude that fr given the ptimal slutin is ptimal Otherwise n suitable slutin is fund Sme imprtant pints can be drawn frm this algrithm - Firstly tw agent parameters a and b are different t each agent As we can see in equatin (7) the prbability t chse a next nde depends n these parameters Because we have tw criterins SP and d ( I ) I the larger parameter will decide which criterin will be fllwed Mrever the advantage f randm agent parameters can be seen as the way t avid lcal slutins between agents due t different sequences f them - Secndly in Step 8 when selecting sticks t put int a sequence it is pssible that sme test cases are the same Therefre the number f real test will be reduced T increase it we supplement a quantity related t the maximal prbability f tw ndes in the sequence Thus the number f test cases is still in a given range [ MinTest MaxTest] - Thirdly we will try sme cmbinatins f adacency plygns in a sequence t frm slutins In essence ur algrithm belngs t the greedy appraches A sequence in this way is an rdered list f plygns whse cmbinatin between them will create mre slutins than riginal sequence s nes with a specific prbability nsequently it is better t investigate in a gd sequence - Furthly t ensure the time cnditin and avid similar slutins between agents we limit the number f agents t the half f the number f plygns and prcess it by parallel cmputing Then the cmputatinal time and the slutin will be amelirated - Finally SIT-TSA methd cnverges t the glbal slutin instead f the lcal ne due t the best slutin selectin prcess amng all agents Mrever it can answer quickly whether a slutin may exist fr a given parameters r nt An Example Assume that we have a plygn shape dataset belw (Fig2) The number f prcessrs k 3 The number f agents in each prcessr is NA In the first prcessr and Thus a sequence fund by the first agent and sme tests frm it is illustrated thrugh Fig 3 Fig 2 A plygn shape dataset 243

4 A Nvel Stchastic-Based Algrithm fr Terrain Splitting Optimizatin Prblem Fig 3 (a) a sequence f an agent; (b) sme tests frm this sequence Fig 4 The number f cases fllwing by the number f plygns I EXPERIMENTS A Theretical Evaluatin In this sectin we will evaluate the prpsed algrithm bth by time and space cmplexity In SIT-TSA algrithm Step 4 t Step 6 requires l ( l ) / 2 calculatins Step 7 t Step takes NTest l peratins Each prcessr has l / 2k agents Therefre the ttal cmputing time f the algrithm is l l ( l ) NTest l 2k 2 2 O( l k 3 ) (6 ) The memry space in each prcessr t stre prbabilities f all ndes and the sequences f all agents is O (l) Therefre the ttal memry space is k O(l) B Experimental Setup Theretical time cmplexity des nt always indicate clearly the speed f an algrithm Fr this measurements f PU time ften give better infrmatin Therefre in this part we have implemented the prpsed algrithm in additin t PSO-TSA in prgramming language and executed them n a Linux luster 35 with eight cmputing ndes f 52GFlps Each nde cntains tw Intel Xen dual cre 32 GHz 2 GB Ram In SIT-TSA the parameters [ MinTest MaxTest] and are initially set as [ ] and respectively In PSO-TSA the ppulatin and the maximal iteratin are and respectively Terrain data are taken frm Blzan - Blzen prvince [5] Saving Threshld mparisn Firstly we calculate the values f saving threshld fr SIT-TSA and PSO-TSA algrithms fllwing by different number f plygns and number f prcessrs in tw terrain data We cmpare the saving threshlds f tw algrithms fr a specific number f plygns and prcessrs and find the smallest ne Then we increase the number f cases fr the algrithm that has the smallest value f saving threshld by ne The statistics are gruped fllwing by the number f plygns Results are illustrated in Fig 4 Similarly we perfrm anther cmparisn fllwing by the number f prcessrs and summarize the results in Fig 5 Fig 5 The number f cases fllwing by the number f prcessrs Fig 4 clearly shws that the number f cases generated by SIT-TSA is larger than the ne f PSO-TSA Fr example when the number f plygns is 2 SIT-TSA prduces six best cases when PSO-TSA makes three cases nly The maximal difference between tw algrithms is fur cases when the numbers f plygns are 5 and 5 PSO-TSA is nly better than SIT-TSA when the number f plygns is 5 In general SIT-TSA still brings mre cases than PSO-TSA des Fig 5 recnfirms that SIT-TSA generates mre cases than PSO-TSA des The maximal difference between tw algrithms is larger than the result in Fig 4 In fact this number is 7 when the number f prcessrs is 3 PSO-TSA is better than SIT-TSA when the numbers f prcessrs are 8 and 2 Fr the remains SIT-TSA is shwn t btain better results than PSO-TSA Fig 6 Average saving threshlds f algrithms by number f prcessrs 244

5 In Fig 6 we calculate the average numbers f saving threshld fllwing by the number f prcessrs in bth algrithms A cmparisn between them is made in this figure This test clearly pints ut that the values f SIT-TSA are smaller than the nes f PSO-TSA The maximal and minimal differences between tw lines are fund at 96 and 2 when the numbers f prcessrs are 4 and 6 respectively Sme remarks are extracted thrugh this sectin: - Firstly SIT-TSA really brings better results than PSO-TSA - Secndly as illustrated in Fig 5 and Fig 6 we shuld chse the number f prcessrs frm 3 t 8 in rder t btain the best results f SIT-TSA algrithm Running Times mparisn In this sectin we will make a cmparisn f the running times between tw algrithms fllwing by the number f plygns The results are shwn in Fig 7 Obviusly SIT-TSA is faster than PSO-TSA when the number f plygns is smaller than 75 The largest difference between tw algrithms is 595 times when the number f plygns is 2 The difference is getting smaller when the number f plygn increases When the number f plygns is larger than the difference is belw ne and SIT-TSA is slwer than PSO-TSA The reasn fr slw running times f SIT-TSA when the number f plygns increases can be recgnized thrugh the incremental level In PSO-TSA the average increment between tw cnsecutive numbers f plygns is apprximately 25 This number in case f SIT-TSA is 73 As such mre plygns are added the running times f SIT-TSA are lnger Sme remarks can be fund frm this test: - Firstly SIT-TSA is faster than PSO-TSA when the number f plygns is belw 75 - Secndly frm Fig 4 t Fig 7 we get a remark that the number f plygns shuld be smaller than 5 t get the best results f SIT-TSA algrithm in bth the saving threshld and the running times and Stick Prcedure t find the ptimal slutin with the supprts f parallel cmputing It was verified by time and space cmplexity as well as numerical experiments The experimental results shwed that SIT-TSA btains better saving threshlds than PSO-TSA and is suitable fr TSO prblem Future wrks will cncern sme methds t stre terrains in a database as well as perfrm attribute queries in a 3 GIS system I AKNOWLEGMENT This wrk is spnsred by the research prect f NU Hani Infrmatin Technlgy Institute ietnam Natinal University (NU-ITI - Prect N QT27) and the research grant f Natinal Fundatin fr Science and Technlgy evelpment (NAFOSTE - Prect N 2-224) REFERENES [] Andersn H L Metrplis Mnte arl and the MANIA Ls Alams Science vl pp 96-8 [2] Hlland J H Adaptatin in natural and artificial system Ann Arbr: The University f Michigan Press 975 [3] Kennedy J Eberhart R Particle swarm ptimizatin In: Prceedings f IEEE Internatinal nference n Neural Netwrks Piscataway NJ 995 pp [4] Sn L H Thng P H Linh N Ha N ung T Sme Results f 3 Terrain Splitting By 2 Plygnal ectr ata Internatinal Jurnal f Machine Learning and mputing vl n 3 2 pp [5] Sn L H Thng P H Linh N ung T Ha N evelping JSG Framewrk and Applicatins in OMGIS Prect Internatinal Jurnal f mputer Infrmatin Systems and Industrial Management Applicatins vl 3 2 pp 8-8 [6] Thien N Sn L H Lanzi P L Thng P H Heuristic Optimizatin Algrithms Fr Terrain Splitting and Mapping Prblem Internatinal Jurnal f Engineering and Technlgy vl 3 n 4 2 pp Fig 7 The running times fllwing by the number f plygns ONLUSION In this paper we intrduced a nvel stchastic-based ptimizatin algrithm namely SIT-TSA fr TSO prblem This methd emplyed the ideas f Wife-Selectin scenari 245