ELECTROMAGNETIC WAVE PROPAGATION MODELING USING THE ANT COLONY OPTIMIZATION ALGORITHM

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1 Radoengneerng Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm 1 Vol. 11, No. 3, September 2002 P. PECHAČ ELECTROMAGNETIC WAVE PROPAGATION MODELING USING THE ANT COLONY OPTIMIZATION ALGORITHM Pavel PECHAČ Dept. of Electromagnetc Feld Czech Techncal Unversty n Prague Techncká 2, Praha 6 Czech Republc Abstract The Ant Colony Optmzaton algorthm - a multagent approach to combnatoral optmzaton problems - s ntroduced for a smple ray tracng performed on only an ordnary btmap descrbng a two-dmensonal scenaro. Ths btmap can be obtaned as a smple scan where dfferent colors represent dfferent medums or obstacles. It s shown that usng presented algorthm a path mnmzng the wave travelng tme can be found accordng to the Fermat s prncple. An example of practcal applcaton s a smple ray tracng performed on only an ordnary scanned btmap of the cty map. Together wth the Berg s recursve model a non-lne-of-sght path loss could be calculated wthout any need of buldng database. In ths way the coverage predctons for urban mcrocells could become extremely easy and fast to apply. Keywords Wave propagaton modelng, Ant Colony Optmzaton, mcrocell 1. Introducton In the feld of electromagnetc wave propagaton modelng ray tracng technque s often utlzed. Ray tracng or ray launchng algorthms are very powerful provdng excellent outputs. On the other hand the calculaton s usually very complex and tme consumng. One of the man dsadvantages, especally for commercal applcatons, s a necessty of precse geometrcal nputs, e.g. buldng database for urban scenaros. If an nhomogeneous envronment s consdered, both the algorthm and the geometrcal nputs tend to be even more complcated. In ths paper a mult-agent approach to combnatoral optmzaton problems - the Ant Colony Optmzaton - s ntroduced for a smple ray tracng performed on only an ordnary btmap descrbng a two-dmensonal scenaro. Ths btmap can be obtaned as a smple scan where dfferent colors represent dfferent medums or obstacles. Usng the presented algorthm the fastest path for a ray - electromagnetc wave - can be found accordng to the Fermat s prncple. Thanks to the very easy-to-obtan nputs the wave propagaton modelng could become, for selected tasks, smple and fast to apply. 2. Ant Colony Optmzaton The Ant Colony Optmzaton was proposed by Dorgo [1]. Ths mult-agent approach can be used for varous combnatoral optmzaton problems. The algorthms were nspred by the observaton of real ant colones. Ants are socal nsects lvng n colones wth nterestng foragng behavor. In partcular, an ant can fnd shortest paths between food sources and a nest. Whle walkng from food sources to the nest and vce versa, ants depost on ground a substance called pheromone, formng a pheromone tral. Ants can smell pheromone and, when choosng ther way, they tend to choose paths marked by strong pheromone concentratons. It has been shown that ths pheromone tral followng behavor employed by a colony of ants can gve rse to the emergence of the shortest paths. Ths phenomenon s explaned n Fg. 1. In Fg. 1a ants walk between two ponts va unobstructed path. Both the marchng ants (top of the pcture) and correspondng pheromone tral (bottom of the pcture) are dsplayed. When an obstacle breaks the path (Fg. 1b), ants try to get around obstacle randomly choosng ether way. If the two paths encrclng the obstacle have dfferent lengths, more ants pass shorter route on ther contnuous pendulum moton n partcular tme nterval. Whle each ant keeps markng ts way by pheromones the shorter route attracts more pheromone concentratons and consequently more and more ants choose ths route. Ths feedback leads soon to fnal stage n Fg. 1c, where entre ant colony uses the shortest path. There are many varatons of the ant colony algorthm appled on varous classcal optmzaton problems. Many references can be found n [2]. 3. Descrpton of the Algorthm The algorthm s based on the Ant Colony System (ACS) technque [3] orgnally proposed for Travelng Sa-

2 2 Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm Radoengneerng P. PECHAČ Vol. 11, No. 3, September 2002 lesman Problem. It was modfed for needs of the wave propagaton smulaton. The adapted algorthm can be descrbed n fve steps: node medum II. obstacle medum I. arc a b c Fg. 2 Illustraton of an undrected graph generaton. 3.2 Step 2 - A Colony of Ants s Launched In the second step C ants are sequentally launched from N 1. C s a number of ants n the colony. Each ant walks pseudo-randomly from node to node va connectng arcs as far as the N 2 or dead end s reached (Fg. 3). N 1 Fg. 1 Ant Colony Optmzaton: ants searchng the shortest route usng a pheromone tral. 3.1 Step 1 - A Generaton of an Undrected Graph An undrected graph G = (N, A), where N s set of nodes and A the set of arcs connectng the nodes, and two nodes N 1 and N 2 to be connected by the requred fastest ray have to be defned n the frst step. The nodes N are easly generated as a unform grd appled on an nput pxel btmap descrbng the propagaton envronment. The densty of the nodes determnes the precson of a soluton but also memory and computaton tme demands of the algorthm. Then arcs A nterconnect all the nodes of the graph - one wth each other. Usng smple raster graphcs procedures all the nodes and arcs nterferng wth colored elements, whch represent buldngs or other obstacles (e.g. black color), are elmnated. In addton, each arc must traverse through only a sngle medum,.e. only one color n the btmap. Approprate wave travelng tme s assgned to each arc based on ts physcal length and the propagaton medum. The graph generaton s shown n Fg. 2, where a very sparse grd of nodes was used as an llustraton. All arcs A of the graph G are ntalzed wth a small amount of pheromone τ 0, whch s a value correspondng to the drect wave travelng tme between N 1 and N 2 consderng the wave speed n vacuum. N 2 Fg. 3 Nodes and arcs of an undrected graph and ants pseudorandom moton n the graph. When decdng whch arc to go from a specfc node, each -th arc leadng from the node s assgned a probablty: p = β τ η β τ η where τ s the pheromone concentraton on the -th arc, η s an a pror avalable heurstc value for the -th arc - a wave travelng tme to approprate connected node and then contnung drectly to N 2 consderng the wave speed n vacuum, and β s a parameter determnng the relatve nfluence of the heurstc nformaton. Ths parameter plays a key role for the algorthm performance. (1)

3 Radoengneerng Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm 3 Vol. 11, No. 3, September 2002 P. PECHAČ Then a random number q between 0 and 1 s generated. If q < q 0, where q 0 s another parameter of the algorthm, the ant chooses to go va an arc wth the hghest probablty p. Otherwse random selecton of the arc based on the probablty dstrbuton (1) s accomplshed. The prevously vsted arcs are excluded from the selecton. After havng crossed the selected -th arc durng the ant s tour constructon a local update rule s mmedately appled to the pheromone concentraton on the arc: ( 1 ρ ) τ ρτ 0 τ = + (2) where ρ s a parameter 0 ρ 1. The effect of the local updatng rule s to make already chosen arc less desrable for a followng ant. In ths way more route varatons can be explored. 3.3 Step 3 - An Optmzaton of Ants Paths and the Best Ant Selecton After all ants from the colony fnsh ther routs the most successful ant - the ant wth the fastest path from N 1 to N 2 - s selected to update the pheromone trals n the followng step. Before the selecton a determnstc optmzaton s performed on all of the ants paths. Ths very fast and smple procedure tres to elmnate unnecessary nodes on an ant s route. It s based on the followng prncple: f N a, N b and N c are consecutve nodes, the exstence of a drect connecton between N a and N c s tested. If such an arc exsts, the N b s excluded from the path (Fg. 4). Overall performance of the ACS algorthm s sgnfcantly mproved usng ths technque. τ T = T, for the -th arc vsted by the best ant. 0, otherwse. T s the best ant s travelng tme, whch s a sum of wave travelng tmes assgned to all vsted arcs. In homogenous envronments, where the wave speed s the same throughout the map, a value nversely proportonal to the path length of the best soluton can be used for the pheromone update. In ths case the shortest path nstead of the fastest path s searched. It leads to the same results n homogenous medums. There are two dfferent ways to choose the best ant that s allowed to perform the global updatng. In the global-best method only the ant that dd the fastest route snce the very begnnng of the optmzaton process s selected. In the teraton-best method always the best ant from the colony of C ants deposts the pheromones despte of prevous teratons. Whereas the global-best strategy was preferred n [3] for solvng Travelng Salesman Problem, n all presented smulatons teraton-best method was appled. When global-best strategy was used for the ray tracng, a frequent convergence to local mnmums of the optmzaton algorthm was observed. 3.5 Step 5 - A Termnaton of the Algorthm Steps 2 to 4 are repeated for a fxed number of teratons or as long as the desred soluton s reached. After the termnaton of the algorthm a stored soluton of the very best ant ndcates the fastest path between N 1 and N 2. N c N c 4. Evaluaton of the Algorthm N a Fg. 4 Determnstc optmzaton of an ant s path 3.4 Step 4 - A Global Update of Pheromone Trals All arcs of the graph G are updated usng a global update rule: N b ( α ) τ ατ T τ = 1 + (3) where α s a parameter (0 α 1) determnng the evaporaton of pheromone concentratons and N a As shown above there are fve basc parameters controllng the ASC algorthm: number of ants C, q 0 and β for pseudo-random route selecton n (1), ρ for the local update rule n (2), and α for the global update rule n (3). A software tool was developed to test and evaluate the algorthm. Examples of screenshots are shown n Fg. 5 and Fg. 7. As default startng values the parameters from [3] were used: C = 10, q 0 = 0.9, β = 2.0, ρ = 0.1, and α = 0.1. Large set of dfferent parameters was tested on smple scenaros to fnd the optmal values for our knd of problems. Followng values were establshed as the optmum: C = 50, q 0 = 0.5, β = 3.0, ρ = 0.2, and α = 0.2. The heurstc nformaton η together wth the parameter β from (1) was found of crucal mportance for the algorthm. Results of the search process for four dfferent sets of parameters are shown n Fg. 6.

4 4 Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm Radoengneerng P. PECHAČ Vol. 11, No. 3, September 2002 Fg. 5 A screenshot of the software smulaton tool for the ACS algorthm testng - the shortest path search n urban scenaro (n the 2D map pcture the found ray, buldngs, and arcs of the graph are drawn). Path Length β = 0 β = 1 β = 3 β = 5 whch means the heurstc nformaton η n (1) s completely gnored, the global optmum was not found at all. Other parameters were not so fundamental for the optmzaton. For nstance, the number of ants C = 50 s a compromse: the hgher s the value of C the hgher s the certanty that the global soluton wll be found, but, at the same tme, the longer s the computaton tme. 5. Smple Ray Tracng n Inhomogeneous Envronment Iteratons Fg. 6 The shortest path search process (correspondng to the nput scenaro n Fg. 5) for four dfferent sets of parameters: C = 50, q 0 = 0.5, β = 0.0, 1.0, 3.0, and 5.0, ρ = 0.2, α = 0.2. In ths case the nfluence of β s demonstrated. For β = 3 the global soluton was found very quckly. When the value s sgnfcantly ncreased (β = 5) or decreased (β = 1), the same soluton s reached, but the algorthm convergence s much slower. It can be seen that for β = 0, As an example of a practcal applcaton of the algorthm, Fg. 7 shows two smple scenaros wth three dfferent medums wth a dfferent wave propagaton velocty v 1 > v 2 > v 3 (as t s marked n the pctures). The medums are dfferentated by colors n the nput btmap of the optmzaton algorthm. A sold lne represents a result of the smulaton: the fastest path of the wave from lower-left corner to upper-rght corner of the btmap (the path that mnmzes the wave travelng tme). The Fermat s prncple can be ncely demonstrated. The global soluton was usually obtaned wthn 50 teratons for both examples. The number of arcs n the graphs was more than 10,000 and 71,000, respectvely.

5 Radoengneerng Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm 5 Vol. 11, No. 3, September 2002 P. PECHAČ v 1 v 2 v 3 v 3 v 1 v 2 Fg. 7 Examples of a smple ray tracng n nhomogeneous envronments descrbed by a btmap usng the ACS algorthm. The travelng tme of the wave s mnmzed accordng to the Fermat s prncple. 5.1 Path-Loss Calculaton n Mcrocells usng the Berg s Recursve Model One of the practcal applcatons of the algorthm for a propagaton predcton n mcrocells s ntroduced n [4]. Mcrocellular structure s mostly demanded for new wreless personal communcaton systems, such as UMTS. Emprcal propagaton models wdely used n macrocells cannot be utlzed n mcrocells. The nature of wave propagaton n urban mcrocells requres dfferent approach. Mostly the determnstc ray tracng technques were adopted. As t was already mentoned, whle the results of the determnstc modelng are excellent (precson, wde-band outputs) the nput requrements are very extensve. Bul-

6 6 Electromagnetc Wave Propagaton Modelng usng the Ant Colony Optmzaton Algorthm Radoengneerng P. PECHAČ Vol. 11, No. 3, September 2002 dng database ncl. electrcal parameters of used materals has to be defned. That s why there s a challenge to explore alternatve ways of the sgnal propagaton predcton. Sem-determnstc approach for street mcrocells - recursve model - was ntroduced by J-E. Berg [5]. Model does not requre knowledge of buldng materals but only a plan vew buldng database s needed. The shortest path along streets s determned among buldngs between a base staton and a moble antenna. A smple two-dmensonal geometrcal ray tracng technque can be used. Path s break down nto a number of straght segments nterconnected by nodes. The llusory dstance s obtaned recursvely as functon of a real length of the segments and angle of segments crossngs n nodes. It means each tme the path bends llusory dstance s lengthened n comparson to the physcal length. Then a very smple emprcal formula s appled on the llusory dstance to calculate the total path loss. A detaled descrpton of the model can be found n [5]. The model was also ncluded to [6]. Usng the presented ACS algorthm together wth the recursve model, a non-lne-of-sght path loss could be calculated wthout any need of buldng database. Classcal ray tracng would requre some knd of nformaton on buldngs locatons and shapes. If the ACS algorthm s appled to fnd the desred shortest segmented path along streets, only a btmap (scan of the cty map) can be used as an nput. In ths way the coverage predctons for urban mcrocells could become extremely easy and fast to apply. Fg. 5 demonstrates the shortest path search among several obstacles, whch can represent buldngs n mcrocell. 6. Summary For a smple two-dmensonal ray tracng n nhomogeneous medums a new algorthm based on the Ant Colony Optmzaton was ntroduced. A common btmap can be used as an nput descrbng even complcated envronments. Practcal applcaton for coverage predctons n mcrocells, whch s based on two technques orgnated n very dfferent felds of applcaton, was presented. The Ant Colony Optmzaton together wth the Berg s recursve model enables non-lne-of-sght path loss calculatons wthout any need of buldng database. As ndcated above, frst smulatons for smple urban scenaros proved a very promsng effcency and usefulness of the algorthm. The work contnues tunng and evaluatng the method on more complex scenaros. Obvously there are other propagaton predcton applcatons where the presented ACS algorthm can be utlzed nstead of classcal ray tracng. Other well known determnstc algorthms can be used for the ntroduced dea of tracng a ray n an undrected graph as well. As a reference Djkstra s Algorthm [7] was mplemented. It leads always to global best soluton but t s not so effcent for dense and large graphs. More detals about the graph theory and varous relevant algorthms can be found n [7] ncludng many references. Acknowledgements Ths work has been supported by the Grant Agency of the Czech Republc grant No. 102/01/D117 Indoor Propagaton for Wdeband Wreless Systems 2.5 GHz and by the research program No References [1] DORIGO, M., MANIEZZO, V., COLORNI, A. The Ant System: Optmzaton by a Colony of Cooperatng Agents. IEEE Transactons on Systems, Man, and Cybernetcs Part B. 1996, vol. 26, no. 1, pp [2] MIETTINEN, K., MAKELA, M., NEITTAANMAKI, P., PERI- AUX, J., edtors. Evolutonary Algorthms n Engneerng and Computer Scence. Wley [3] DORIGO, M., GAMBARDELLA, L. M. Ant Colony System: A Cooperatve Learnng Approach to the Travelng Salesman Problem. IEEE Trans. on Evolutonary Computaton. 1997, vol. 1, pp [4] PECHAČ, P. Applcaton of Ant Colony Optmsaton Algorthm on the Recursve Propagaton Model for Urban Mcrocells. URSI XXVIIth General Assembly, August 2002, Maastrcht, n press. [5] BERG, J-E. A Recursve Method for Street Mcrocell Path Loss Calculaton. In Proceedngs IEEE Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons PIMRC , vol. 1, pp [6] ETSI techncal report TR , Selecton procedures for the choce of rado transmsson technologes of the UMTS. ETSI, [7] KOLÁŘ, J. Theoretcal Computer Scence. Praha: Vydavatelství ČVUT, About authors... Pavel PECHAČ graduated at the Czech Techncal Unversty n Prague, Faculty of Electrcal Engneerng n He receved hs Ph.D. n Radoelectroncs n 1999 at the Department of Electromagnetc Feld. He s wth the department as an assocate professor. Hs research nterests have been manly n the feld of the electromagnetc waves propagaton predctons for moble wreless systems.