SCHOOL OF INFORMATION TECHNOLOGY & ELECTRICAL ENGINEERING THE UNIVERSITY OF QUEENSLAND Brisbane Queensland 4072 Australia. Technical Report No.

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1 SCHOOL OF INFORMATION TECHNOLOGY & ELECTRICAL ENGINEERING THE UNIVERSITY OF QUEENSLAND Brisbne Queenslnd 407 Austrli Phone: Fx: Emil: trici@itee.uq.edu.u Technicl Report No. 465 On optiml route computtion of mobile sink in wireless sensor network S. Nesmony, M. K. Virmuthu, M. E. Orlowsk nd S. W. Sdiq //006

2 On optiml route computtion of mobile sink in wireless sensor network S. Nesmony, M. K. Virmuthu, M. E. Orlowsk nd S. W. Sdiq School of ITEE, The University of Queenslnd, Austrli ABSTRACT There is evidence of rnge of sensor networks pplictions where mobile sink entity (node) is utilised for dt collection from stticlly positioned sensor nodes in sensor field. The mobile sink is typiclly required to cover the sensor field by physicl motion in order to obtin the vlues from the sensor nodes in periodic fshion. This chrcteristic leds to very interesting problem of determining the optiml route of the mobile sink, in terms of distnce trvelled, to ccomplish the dt collection from ll the sensor nodes. This minimum distnce problem tht is spnned from the design nture of the network hs very intriguing nd motivting connections with set of clssic computtionl problems. These cohesions nd similrities re explored in this pper, nd the computtionl complexity is nlysed. The pplicbility of numericl solutions to the current problem is discussed nd numericl heuristic is provided to rrive t n pproximte nswer tht is close to the ctul solution. An evlution of the proposed pproch is lso provided through experimentl results. Keywords: Sensor Networks, Sensor Network Design, Optimistion problem, Trvelling Slesperson Problem with Neighbourhoods, TSPN, Numericl Method Heuristic. INTRODUCTION Wireless sensor networks re becoming indispensble on mny fronts including consumer pplictions, civil pplictions, wrehousing pplictions nd militry pplictions becuse of severl distinct dvntges over trditionl methods. The

3 perfect juxtposition of utomtion, computtion nd sustennce long with the decresing cost of deployment mkes sensor networks perfect ingredient in mny dt collection, monitoring nd distribution pplictions. In this pper, we re concerned with specific potentil sensor network ppliction tht is bsiclly chrcterised s follows: A collection of n sensor nodes re positioned sttionry in the sensor field. There is bse sttion or sink node which collects dt from ll the positioned sensors. The ction tht the sink cn tke cn be mere propgtion of dt to further systems or computtion nd initition of some events bsed on the collected dt or computed result. Consider for exmple, sensor field which is lrge with respect to the trnsmission rnge of the individul sensors nd contins hevy popultion of the sensors. An pproch, to collect the dt from these sensors which my hve non-overlpping rnges, is proposed to be to use the sink s mobile gent tht physiclly moves round the sensor field nd collecting the dt from the sensors by visiting their trnsmission rnges. The communiction rchitecture of the network could be designed bsed on the requirements nd limittions of the ppliction. For exmple, the sensor nodes cn be designed to trnsmit dt if nd only if requested by the sink node thereby reducing the energy consumption or wstge. Figure. Sensor nodes with rnges nd the mobile sink s pth through the field

4 For collecting the dt, the sink hs to move round the field nd position itself within the trnsmission rnge of every sensor node, query nd collect the dt thereby covering ll the sensor nodes present in the field. The sought objective function in this scenrio is to find such route for the mobile sink to trvel in the sensor field subject to the constrint tht the distnce trced by tht route is minimum. A more forml representtion of the problem is presented in the following sections. The pper is orgnised s follows. Initilly, the foundtion concepts of the problem scenrio nd the domin it fits into re exmined in section. The ssocited computtion complexity is explored s well in the sme section. Following the discussion bout the complexity of the problem, is the section devoted to comprehensive coverge of the relted work tht hs been done towrds the combintoril problem to which our problem mps to. There is brief discussion in section 4, bout the pproch tken towrds solving the problem. It lso contins the description of the sub problems tht form the building blocks of the solution to the min problem. Section 5 is dedicted to the detiled discussion of the solution to the min problem. It is followed by the discussion of ordering of nodes to visit which plys crucil role in the effectiveness nd the efficiency of the finl solution, in section 6. The retrospection nd the nlysis performed over the experimentl results nd the resultnt observtions re presented in the penultimte section 9. The summry of the overll pper long with the res of further extension is provided in the concluding section.. FOUNDATION CONCEPTS Evidently, the problems of shortest pth or minimum distnce hve been considered from centuries go in vrious forms nd pplictions. The generic form of these problems is to determine pth or route with minimum cost where the cost is modelled in different metric prmeters depending on the ppliction. In our cse the metric considered for optimlity is the distnce trvelled by the sink node.

5 Finding the minimum distnce route tken by the mobile sink in order to visit the trnsmission rnge of ll the sensor nodes, is to find loop tht touches set of circles (ssuming tht the trnsmission re of ll the sensor nodes re circulr in shpe) nd given point (the strting point of the sink node) whose length is minimum. Thus there is perfect geometric inclintion from the given rel world problem. We operte on two dimensionl Eucliden geometric which is explined further in this section. With circles involved in estimtion of the minimum distnce loop, we consider the representtion of circles to be set of discrete points rther thn vlues from the intervls of the rel line. This discretistion is the first pproximtion step towrds the sought best possible solution. The discretistion of the representtion of circles will result in smll error fctor of the solution reched depending on the frequency of discretistion of the circulr segments s well s the rounding off errors. The problem t hnd is combintoril optimistion problem which cn be formulted in terms of discrete optimistion where the vribles present in the objective function re llowed to ssume discrete vlues, nmely the x, y coordintes identifying the point on the circumference of circle in the two dimensionl spce. The input to geometric minimum distnce lgorithms is set of points in d - dimensionl rel spce d R given by their coordintes. For p, the distnce between two points ( x,..., x d ), ( y,..., y d ) d R in the p norm is defined s d p p xi y []. i i= When p =, the norm will be the Eucliden norm which is used in this pper. The distnce metric for our problem is mesured in Eucliden geometric in two dimensions. Conceptully the optimistion problem to be ddressed is detiled s follows: Given set of points P R in the Eucliden plne, nd n+ subsets { S0, S, S,..., S n } of P where S 0 represents fixed point nd S through Sn contins the points tht re on the circumference of the circles thereby representing connected geometric regions in the Eucliden plne.

6 S represents the fixed strting position of the mobile sink nd { S } n i i = represent the 0 points tht enclose the trnsmission region of the sensor nodes. The set S 0 is singleton contining only one point nd other sets contin points tht lie on the circumference of the circles. Thus, S = { }, S = {,,..., },..., S = {,,..., } nd in generl, S { } Si i = ij j=. 0 0 S n n n n Sn The problem t hnd is to construct route on set P ' P such tht P ' contins t lest one point from ech subset S i. The objective is to minimize the length of such route. Given set of points P R in the convex Eucliden region, nd n+ subsets { S, S, S,..., S } of P, the objective is to find 0 where n = n min d πϕ.. () d πϕ d π ( i) ϕ( π ( i)) π (( i+ ) mod ( n+ )) ϕ ( π (( i+ ) mod ( n+ ))) i= 0... () such tht d being the Eucliden distnce between the points ( x, y ) nd ( x, y ) ij kl ij ij kl kl given by ij nd kl d = ( x x ) + ( y y ). (3) ij kl ij kl ij kl π is the permuttion over n π :{0,..., n} {0,..., n}. (4) π ( i) denotes the point which is t i th loction in the tour ϕ is the permuttion over π ( i) ϕ :{,,... S } {,,... S }. (5) π ( i) π ( i) In the objective function () which is to be minimised, there is permuttion over nother permuttion thus contributing to the explosively lrge solution spce where

7 the explortion of the solution occurs. Without ny loss of generlity, if we cn ssume tht ll the sets S through Sn re of sme crdinlity of size M, tht is, ll the circles re chrcterised by M points tht lie on the circumferences. Then, the brute force serch of ll the minimum length tours covering ll the sets will be of the order n of n! M (hlved becuse of the cyclic permuttion of ordered tours) which is exorbitntly lrge. The computtionl complexity of this combintoril optimistion problem cn lso be determined by considering it generlistion of the well-known Trvelling Slesperson Problem (TSP). When the rdii of ll the circles (trnsmission rnges of ll the sensors in the field), becomes zero or very close to zero, then this problem reduces to the TSP for n + points on n Eucliden plne which is NP-hrd [][3]. Hence the problem t hnd is t lest NP-Hrd, becuse ech circle cn be point t the simplest possible cse. When the circles re of non-zero rdii, this complexity will only esclte owing to ll the points on the circumference of the circles. There is n exct isomorph of our problem which is more generlised vrint of TSP which is termed s TSPN (Trvelling Slesperson Problem with Neighbourhood). A discussion bout TSP nd TSPN is presented in the following section. 3. RELATED WORK Although the origins of the TSP cn be trced bck to 9 th century, it is the pper [4] which cme in 954 tht brought it into limelight. Since then this problem s populrity hs sored high nd it hs been clssic problem in the domin of combintoril optimistion. Generl heuristic mechnisms tht re devised for combintoril optimistion problems re used to pproch generl TSP problem. Also, given tht it is very hrd problem to solve, severl pproximtion lgorithms hve been proposed nd the common line of interest is to find the pproximtion lgorithm tht finds TSP tour which is t most c times the optiml one, where c is positive constnt. The populr Christofides lgorithm [5] gve n pproximtion lgorithm which gives tour not

8 more thn.5 times the optiml tour. The TSP for this lgorithm considers the distnce function to be symmetric nd constrined to stisfy the tringle inequlity. In fct this lgorithm ws one of the first pproximtion lgorithms nd cemented the position of pproximtion lgorithms s prcticl pproch to intrctble problems. The pproximtion lgorithms generlly provide provble solution qulity nd provble run time bounds s ginst heuristics which provide resonble good solutions in resonble time. Considering the PTAS (Polynomil Time Approximtion Scheme) for the TSP, it is proved tht it is NP-Hrd to find TSP tour tht is t most more thn the optiml tour [6]. With bounded metrics, it is lso 9 proved tht there is no polynomil lgorithm tht finds tour tht is not more thn 389 times the optiml tour [7] unless P = NP. 388 Considering the extensive ttention pid to the TSP nd relted problems, it is nturl to observe tht most of the different forms, flvours nd vritions of the clssic TSP problem hve been identified nd ttempted for deriving solutions. There re vrious forms of the problem like Mx TSP, Min Are TSP, Mx Are TSP, TSP with Neighbourhoods, Lwn Mowing Problem, Milling Problem etc. A brief discussion of these problems nd other relted network optimistion problems is presented in [8]. These problems tht revolve round the ctul core TSP hve been studied exhustively. So is our problem, which is in fct specific version of the problem tht is referred to s Trvelling Slesperson Problem with Neighbourhood (TSPN). Although TSP hs been under significnt ttention from the reserch community for more thn hlf century, the geometric TSPN hs ctully been formlly ddressed nd treted only since 994 in the pper [9] where the uthors introduce it s n extended geometric version of the covering slesmn problem [0]. The TSPN problem is defined over the scenrio where the slesperson wnts to meet set of potentil buyers ech of who specifies compct set in the plne, clled his/her neighbourhood, within which he/she is willing to meet. The slesperson wnts to compute tour of shortest length tht intersects ll of the buyers neighbourhoods nd finlly returns to his initil deprture point. In the sme pper [9], there re heuristic procedures for neighbourhood types such s prllel unit segments, trnslte of convex regions e.g. unit circle or rectngle, etc. The neighbourhoods re represented

9 by single points over which pproximtion lgorithms re pplied. The uthors lso present Combintion Lemm which pproximtes the problem with regions of different types, by combining pproximtions of ech type. So, [9] gives constnt pproximtion rtio lgorithm where the neighbourhoods re well behved in specific forms. There is n O( n ) time neighbourhoods re stright lines in the plne given by []. pproximtion lgorithm for the cse where the For the generl cse of connected rbitrry polygonl neighbourhoods, [] gve n O( log k ) pproximtion lgorithm tht is bsed on guillotine rectngulr subdivisions [3] for TSPN with time complexity Ω n 5 ( ) in the worst cse, where k is the number of neighbourhoods (polygon regions) nd n is the totl complexity of the input. [4] cme up with significntly improved logrithmic pproximtion lgorithm for the generl cse with running time O( n log n ). To be specific, the pper offers severl results. The uthors provide n lgorithm tht genertes tour with logrithmic pproximtion fctor when the strt point is known. If there is no strt point given, then it is shown how to compute good strt point in O( n log n ). In ddition, the uthors provide n lgorithm tht performs t lest one of the tsks tht re listed below: (i) It outputs TSPN tour of length O( log k ) times the optimum tour in O( n log n ) time (ii) It outputs TSPN tour of length tht is t most ( + ε ) times the optimum tour in time 3 O( n ) if ε 3 or O( n log n ) otherwise, for ny fixed ε > 0 rbitrry rel constnt s n optionl prmeter. It is not known in dvnce which of the bove tsks will be ccomplished s it depends on the instnce of the neighbourhoods. However, no polynomil time method gurnteeing constnt fctor pproximtion is known for generl connected regions s neighbourhoods. In generl, if the neighbourhoods hve similr size nd shpe, then one cn usully find constnt fctor pproximtion lgorithm. If the neighbourhoods re of rbitrry size, then there could be logrithmic pproximtion lgorithm. Nevertheless, in [5], the uthors give n O () pproximtion lgorithm, where the neighbourhoods re connected, disjoint, convex nd ft. This ws the first lgorithm tht did not require the neighbourhoods to hve roughly the sme size. There is generl grph version of the

10 TSPN problem which is clled One-of--Set TSP [6], where the neighbourhoods my be disconnected. The direct use of the TSPN problem hs been seen in pplictions like communiction network design [7], VLSI routing [8] etc. It is very interesting to note the notoriety of the TSPN problem nd the opque resistnce it exhibits to the ttempt t solutions. It is very hrd to pproch nd solve the problem in generl. The works tht hve been discussed so fr concentrte on chrcterising the input i.e. identify the different styles of instnces of the TSPN problem, nd provide smrt lgorithms hving theoreticl bounds. Vrious chrcteristic setups of the problem instnce re considered nd the behviour of execution of finding the TSPN tour is nlysed in ech cse. It ws shown tht TSPN problem is in fct APX-hrd when the input neighbourhoods re very long, skinny nd overlpping [5][9] nd cnnot be pproximted within fctor of 39 unless 390 P=NP. i.e. it is NP-Hrd to find 39 pproximtion to TSPN. The APX hrdness 390 hevily relies on different sizes nd overlp. If n optimistion problem belongs to the clss of APX-hrd, it is very hrd problem to pproch s it denies the existence of PTAS nd very difficult even to pproximte. This APX-hrdness proof ws derived from the reduction of the known APX-hrd problem, Min Vertex Cover problem. With this pprecition of the deep complexities of the TSPN problem, let us proceed to nlyse the problem tht ws identified in specific design requirement of sensor network nd try to propose numericl solution to it by reduction of serch spce nd improved itertive pproximtion. 4. BASIC BLOCKS OF MINIMUM DISTANCE PROBLEMS In this pper, we ttempt to describe prcticl problem, which emerges during the design phse of specific wireless sensor network ppliction scenrio. In this network setting, the sensor nodes deployed in the field re positioned t fixed loctions nd the sink node tht collects dt from the sensors is mobile. The problem is to find the optiml route of the sink node in terms of the distnce trvelled such tht the sink returns to its strting position nd must hve collected dt from ll the

11 sensor nodes in the field by positioning itself within the trnsmission rnge of every node. This rel problem in the sensor network domin is identified to be n instnce of the TSPN problem. In the previous section, the ppers tht ddress the hrdness of the TSPN problem nd give pproximtion lgorithms for specific instnces hve been discussed. This pper provides geometry bsed heuristic tht uses itertive procedures over the numericl method pproch for the problem t hnd. The evlution of the performnces is supported by experimenttions. The heuristic is developed bsed on few components or sub-procedures. They provide conceptul resoning nd understnding of the orienttion of the pproch. Also they ly the foundtion for the finl solution s well. These individul elements re described in detil in the following section. Some of these sub problems re hierrchiclly linked so s to offer the finl solution which is ought to be very resonble both in terms of the running time s well s ccurcy. Thus, in order to chieve the finl objective of computing the miniml distnce route in the given setup, we consider set of minimum distnce sub problems of incresing conceptul nd computtionl complexity. They re individully discussed in this section nd solutions re provided which re either direct or heuristic tht ims t the best possible solution. These problems will provide the bsic blocks of resoning nd understnding tht will contribute to the finl solution of our problem. We follow bottom up pproch of defining nd solving individul problems nd then using them s building blocks or components to compose the eventul solution. These sub problems re ll minimum distnce problems contining points, lines nd circles. There re three entities in every problem which re either points nd lines or points nd circles. There re four sub problems discussed viz. (i) Two Points nd One Line (ii) Two Points nd One Circle (iii) Two Lines nd One Point nd (iv) Two Circles nd One Point. In line with these nmes our problem in fct cn be termed s n-circles nd One Point which is discussed in the next section.

12 4. Two Points nd One Line The fundmentl bsic block of the whole problem of the minimum pth distnce is expressed in this sub problem. Let us consider stright line nd two points in two dimensionl Eucliden spce. With respect to the nottions mentioned erlier, we hve S = { S0, S, S} where S 0 nd S re the single element sets contining point ech nd S is the set of points tht denote stright line. The objective is to find point on the line S such tht the sum of distnces between the points to the point on the line is minimum. It is note here tht since we re considering whole line rther thn just segment, the set S contins infinite elements, but this not of concern s the solution point in S will be in between the (perpendiculr) projections of S 0 nd S on S. The solution is trivil if the points were on the opposite side of the line, it being the intersection point between the line S 0 nd S, nd the given line S. If the points S0 nd S re on the sme side of S, then there is direct method to solve this problem in rel spce, given by Heron of Alexndri [0] in his work Ctoprtic. The procedure given by Heron is very simple nd direct tht gives the exct solution. In ccordnce with our erlier nottions, the elements of the sets S0 nd S re given S by, 0 0 = { } nd S = { }. Considering S s mirror, plot nother point ' on the other side of the line S tht will be mirror imge of the point. Thus, S will be perpendiculr bisector of the line segment tht connects the points nd '. Now, the point of intersection of the line connecting ' nd 0 nd S, which we my cll x is the required point on S. This proof for this procedure is very simple nd is bsed on the tringle inequlity. The solution given by Heron lso follows the Principle of lest ction in Optics [], nd consequence is tht, the given two points form lines to the optimum point on the line, which sweep equl ngles to the norml to the given line drwn t the optimum point.

13 Figure. Equl ngles to the norml t the optimum point Although this lest distnce problem hs no direct relevnce with our problem, it is presented here s n cknowledgement to the simplest version of minimum distnce loop problem. Also, this problem helps in supporting the future rguments in lter sub sections with the concept behind equl ngle sweeping. It lso demonstrtes the underlying semntics of minimum distnce loop computtion bsed on the concept of stright line being the shortest distnce between ny two points in two dimensionl spce. 4. Two Points nd One Circle In this sub section we consider nother version of minimum distnce problem which is quite delt in detil becuse it is the crucil element in building the finl solution. Here we hve the set S contining three elements S = { S0, S, S}. S0 = { 0} nd S = { } re two points nd S is set of points tht forms the circumference of circle. This is specific cse of problem tht is referred to s milkmid problem []. Loosely stted, the milkmid problem is the problem of finding point on the river to which the milkmid, whose position is given, should visit for rinsing her bucket nd then go to the cow t nother specific loction. The cow nd the milkmid re on the sme side of the river, which tkes the shpe of ny curvture nd tht the route tken by the milkmid is shortest. There re different methods in clculus tht is directed towrds pproching these problems like Lgrnge s multipliers nd Newton s method. We hve our current sub problem Two-Points-nd-One-Circle (TPOC) if the curve in the milkmid problem is perfect circle.

14 Thus we re fter the minimum collective distnce d 0 x + d x ( d s given in (3)) where x is ny element of the set S. Here the set S is the set with the elements following cyclic order s they re the points on the circumference of the circle being represented which re rrnged in cycle. There is no exct solution to this problem s we hd in the erlier cse of two points nd one line. Geometriclly, the non trivil cse is only when ny one point is outside the tngentil spce formed by the tngents tht enclose the circle by the other point nd lso when the two points re not in line with the centre of the circle. Otherwise, the trivil solution is the stright line formed tht joins the two points. The optimum point on the circle will be ny one point of intersection between the circle nd the line joining the given two points. In the non trivil cse, ny one point hs to be outside the tngentil spce formed by the tngents to the circle from the other point. If one of the two points is positioned outside the tngents of the other point enclosing the circle, then the vice vers is lso true. Figure 3. Trivil cse For the non trivil cse, finding the best point on the set of points on the circumference cn be n exhustive serch. However, rther thn hving every element of the set S to be considered for potentil cndidte, the serch spce cn be very much reduced. It cn be reduced to the set of points on n rc segment rther thn the points on the entire circumference of the circle trced by the elements (points) of the set S.

15 Let S be the point on S tht is closest to the point 0 nd let S, be the k closest point to. Geometriclly, k is the point of intersection of the circle trced by the elements of S nd the line joining 0 nd the centre of the circle. Now, the serch spce is confined to the points tht re within the smller rc formed between the points k nd l on S. If the set S contins set of points tht represents points on the circumference of the circle in sequence, then the serch spce is within the closed intervl [, ] or [, ] k l l k of the cyclic ordered l S. The smller one is considered s the reduced serch spce for the optimum solution. The points k nd l re on the circumference of circle, nd they induce two rc segments between them nd the short rc lone represents the reduced serch spce. Figure 4. Reduced serch spce Figure 5. Optimum solution criteri It cn be proved tht the smll rc between the points optimum point bsed on the equl ngles towrds the norml pproch. k nd l contins the Any point on the rc segment could be cndidte for the optimum point nd we consider α being the ngle between the norml t the cndidte point nd the line segment joining 0 nd the cndidte point. Similrly, let β be the ngle between the norml t the cndidte point nd the line segment joining nd the cndidte point. As we sw in the previous cse, the cndidte point x will be the optimum point if nd only if ngles α = β. This is illustrted in the bove figure. Consider the following figure. At k, α = 0 nd β is some non-zero entity. At the point l, β = 0 nd α is non-zero. Thus, trcing every point from the ordered

16 sequence x l x= k, α increses from zero to finite vlue nd β reduces from some finite vlue to zero. Thus, for α to be equl to β, it hs to hppen somewhere in between k nd l nd thus proved tht the reduction of serch spce still retins the optimum point without ny loss. Figure 6. Proof of correctness of reduced serch spce It cn lso be noted tht the reduced serch spce is lwys less thn t lest hlf the size of the originl serch spce nd tries to rech hlf the size s the distnce between the points become fr. Now, hving reduced the serch spce, there cn be numerous methods to find the best point for the optimum solution in the serch spce. We cn use binry serch style method or brnch nd bound style methods in order to rrive t the optimum point. This Two-points-nd-One-Circle procedure for finding the minimum length loop touching ll the entities is the key procedure in the instrumenttion of the pproximtion of the finl solution for our problem. 4.3 Two Lines nd One Point This sub problem Two-Lines-nd-One-Point (TLOP) is covered primrily for the comprehensive coverge of the minimum loop problem of three entities with two being the unknown. This problem lso demonstrtes the key ide of the heuristic which is the regressive itertion nd provides bsic understnding of the execution of the procedure. This problem module involves two lines nd point. Given point nd two lines, the objective is to find point on ech of the two lines such tht the perimeter of the tringle formed by these points nd the given point is miniml. Agin S is the set of three elements S = { S0, S, S}, where S0 = { 0}, S nd S denote the

17 points representing the two lines. The objective is to find x nd y on S nd S respectively, such tht the perimeter of the tringle formed by 0, x nd minimum. y is This cse is quite different from the previous cses becuse we hve to find two points on two sets. It is more complex cse thn the previous problems nd there is no direct exct solution for the non-trivil cse. Also, since S nd S represent the discretised points on the lines, these sets contin infinite elements, s discussed erlier in the TPOL problem. However, there is no impct of this issue becuse the solution points on S nd S will not be very fr from the projections of the point 0 on the lines S nd S. There re two trivil cses for this problem: () One trivil cse is tht the lines S nd S re prllel to ech other nd the point 0is positioned in between them. The solution will contin the points x on S which is closest to 0 nd y on S tht is closest to 0. Thus, x will be the point on S, where the norml drwn from 0 to S intersects S nd similrly with y on S. (b) The other trivil cse will be tht the point 0 is not positioned between the lines S nd S. In this cse, the solution will be the points of intersection on S nd S with the norml drwn from 0 to the frthest line. This cn be proved gin using the tringle inequlity. For the non trivil cses, this problem is hrder thn the previous ones. We try to chieve the gol of finding the tringle with minimum perimeter with one fixed point nd two vrible points (on ech of the lines) by following n pproximtion procedure through regressive itertions. The concept is so very simple but yet mzingly powerful. First, rndom point is tken on S nd is clled k. With 0 nd k s the two fixed points nd the line S, the best point on S clled l is found such tht the perimeter of the tringle formed by the points 0, k nd l is minimum. This cn be effected by using the

18 heron s procedure for Two-Points-nd-One-Line (TPOL). Now, with the newly found point l on S nd 0 s the two fixed points nd S s the line the sme procedure is pplied gin to find the best point in the line S to compute the minimum perimeter. This procedure is repeted gin with 0 nd the newly found best point on S to find the best point on S. This procedure is itertively repeted nd it cn be observed tht the totl length of the perimeter of the tringle formed in every step reches to convergence very quickly. It is lso observed in empiricl results tht the vlue of the converged loop length is very close nd lmost the bsolute minimum vlue. Figure 7. Evlution of optimum points for Two Lines nd One Point Although we re deling with circles nd points in our originl problem, the hndling of lines nd points will provide cler understnding on solid bse nd will stnd for the soundness of the pproch which will be lter employed in the cse of circles nd points Heuristic for computing the optiml route for Two Lines nd One Point Inputs: { } S 0 0 i i i i S { } = S { } = procedure TWO-LINES-ONE-POINT ( S 0, S, S ) /* initilistion */ oldperimeter 0 newperimeter 0

19 temppoint null temppoint null temppoint null tempbestpoint null /* pick rndom point temppoint from S */ /* perimeter computes the perimeter of the tringle formed by given points */ temppoint TPOL( 0, temppoint, S ) newperimeter perimeter( 0, temppoint, temppoint) templine S tempbestpoint temppoint while newperimeter!= oldperimeter do end do oldperimeter newperimeter temppoint tempbestpoint tempbestpoint TPOL( 0, temppoint, templine) newperimeter perimeter( 0, temppoint, tempbestpoint) if templine = S then templine S else if templine = S then templine S /* output method outputting the best points */ output(temppoint, tempbestpoint) Two Circles nd One Point This problem of Two-Circles-nd-One-Point (TCOP) is similr to the bove problem of TLOP except tht the lines re replced by circles. We now hve S = { S0, S, S}, where S0 = { 0} nd S nd S represent the points on the circumference on the circles. The point is ssumed to be outside of both the circles. The objective is to find point ech on the sets S nd S, such tht the tringle formed by those two points long with 0 hs the minimum perimeter. The solution for this problem in the non-trivil cse is not esy to compute s in the previous sub problem. The trivil cse will be the cse where the point 0 is exctly on the line connecting the centre points of the circles S nd S. The solution would

20 then be points of intersection on the circumferences of the circles nd the line joining their centre points. The other trivil cse is depicted in the following figure. If the point is nywhere on the grey re, then the best points on the circles will lie on the line connecting 0 nd the centre of the frthest circle. Figure 8. Trivil cse for Two Circles nd One Point For the non trivil cse, the sme procedure employed in the bove sub problem cn be used in this cse s well. A rndom point k from S is tken. With the points 0 nd k nd the circle S, the procedure TPOC( 0, S, k ) is used to find the best point in S, which we shll cll l, such tht the perimeter of the tringle formed by the points 0, k nd l is minimum. Now, we follow the sme methodology of successive nd itertive regression tht hs been used in the erlier problem of TLOP. We compute the new best point on S using the procedure TPOC( 0, S, l ), tht is gin coupled with 0 to find the next best vlue on S. This procedure is repeted until stbilistion is reched with respect to the loop distnce. It ws seen tht the convergence of the loop distnce ws rther rpid giving the best points bsolutely close to the optiml points on S nd S. The discrepncies cn be ttributed to the errors due to discretistion nd rounding off, s the experiments were performed on integer bsed pixel spce. The pseudo code of this procedure is congruent to the procedure of Two-Lines-One-Point except in using TPOC insted of TPOL nd supplying rguments ccordingly.

21 Figure 9. Computtion of optimum points for Two Circles nd One Point 5. OPTIMAL ROUTE COMPUTATION FOR N CIRCLES Hving built sound understnding of constructing resonble solutions for minimum distnce loop problems tht were discussed, let us proceed to build the solution of the min problem. The problem of computing the optiml route for the mobile sink in the sensor field s described in the initil sections is now trimmed down to be problem similr to the TCOP problem where there re n circles nd one point. We hve n circles nd single point in dimensionl spce nd the objective is to find point on ech of the circle (on its circumference) such tht the cumultive length of the loop strting nd ending t the fixed point nd pssing through the points found is minimum. Given the set S of finite element sets S0, S, S,..., S n we hve to find sequence of ordered elements strting nd ending with the sme element such tht there is one element chosen from every set S, i = 0,,..., n nd tht opertion performed on the i sequence gives n optimum result, the opertion being the cumultive distnce between the points represented by the elements of the sets S i. As discussed erlier, n there will be n exhustive totl of n! M loops to exmine for the shortest loop, where n is the totl number of sets with crdinlity greter thn unity (sets representing circles excluding the one representing the point) nd M is the mximum crdinlity of the sets { S } n i i =. This explosive number is due to the combintoril explosion of the points in the serch spce. It is very unlikely to even pproximte the

22 optiml solution to this problem unless dditionl constrints re dded to reduce the dimensionlity in the complexity of the problem. One plusible constrint tht cn be imposed to reduce the hrdness is tht the order of visiting the sets is known priori. We proceed to present the method tht reches the pproximte solution to the problem, given tht the order to visit the sets is known. The decision regrding the order gretly ffects the nture of the finl solution in terms of its optimlity. Thus there re two phses in finding the best possible solution. The first one is the determintion of the order of visiting the sets nd the second one being computing the optiml points on the sets for the given order. We will focus on the ltter prt in this section nd will reserve the discussions regrding the ordering to the further section where the ordering of sets is nlysed in detil nd the experimentl results re used to compre the reltively better orderings. Let us ssume tht the order of visiting the sets is given the by the indices on the sets. Thus, S is the first set to visit, followed by S nd then the lst set will be S n. We consider the principle of elstics conceptully to derive the best point in every set. Considering the geometric scenrio with n circles nd one point, we re given the order of visiting the circles. The best point tht will be prt of the optimum loop on every circle will be the best point with respect to the best points of the djcent circles i.e. for the circle Si,< i < n, the best point should be found with respect to the best points of the circles Si nd S i +. When i =, for the circle S, the best point on it is found with respect to the strting point given by the sole element of the set S 0 nd the best point on the circle S. Similrly, the best point of the circle S n is evluted with respect to the best point on circle Sn nd the strting point 0 S0. This is like putting stretched piece of elstic over the circles in order nd then llowing it to shrink such tht it leves none of the circles. The shrinking t every circle hppens with respect to the djcent circles. Hence, we employ this pproch in order to rrive t resonble solution becuse of the sensibility of the pproch. Now, the tsk is to find the best points t ech of the circles. We extensively use the solution of the sub problem which dels with two points nd one circle (TPOC).

23 Initilly, we tke one rndom point from every set S i which will be { }, where the domin of the vlues of i nd j re i = {0,,..., n} nd j = {,,..., S }. i i ij i Thus the initil set of best points will be {,,,..., }. Given the procedure 0 j j nj n for two points nd one circle s TPOC(point,circle, point ), first we execute, TPOC ( 0, S, j ) nd this will return the best point on circle S with respect to the points 0 nd j. This will give new vlue for j tht will override the previous vlue. Now the procedure TPOC(, S, j j 3 j 3 is found. Finlly the execution of the procedure TPOC( gives the new best vlue of ) is run nd new vlue for n jn, n S, 0} tht nj n will mrk the completion of the first itertion. At the end of this itertion we will hve new set of best points {,,,..., } 0 j j nj n different from the erlier set nd tht gives better solution. By better solution, we men tht the current loop distnce with these points is smller thn the loop distnce with the previous set of best points. Now, this itertive procedure is repeted gin with the new set of best points tht will result in better set of best points. The itertion is repeted until the loop length stbilises. From the experiments, it cn be observed tht the stbilistion of the loop length hppens firly quickly within few itertions nd thus it cn be sfely concluded tht this procedure does give very resonble better pproximte solution considering the smller liner time it tkes for execution for mssive problem Heuristic for computing the optiml route for n-circles nd One Point Inputs: { } S 0 0 S { } = S i i S { } =.. S i i S n Sn { ni} i = S S S S ) procedure N-CIRCLES-ONE-POINT ( 0,,,..., n /* initilistion */

24 oldloopdistnce 0 newloopdistnce 0 /* Pick rndom points from S to S n & cll them j, j,..., nj n */ /* FindLoopDistnce clcultes the length of the loop formed by n + ordered points */ newloopdistnce FindLoopDistnce(,,,..., while newloopdistnce!= oldloopdistnce do end do oldloopdistnce newloopdistnce for i = to n do i j i = TPOC( i ji ) 0 j j nj n, S i, i+ mod n+ ji + mod n + ) end do newloopdistnce = FindLoopDistnce(,,,..., ) /* output method outputting the best points */ output(,,..., ) j j nj n 0 j j nj n ORDERING OF CIRCLES It hs been observed tht the bove procedure gives n pproximte solution firly quickly for given order of visiting the circles (sets). Hence, given n order, we hve method tht computes very resonble nswer. However, this is not sufficient for the broder problem tht we hve defined erlier. The current tsk is to find the best order for which the finl solution becomes better. The order is evluted with respect to the ccurcy tht it injects to the finl solution s well s the overhed time it tkes for determining the order. Ascertining the ordering of circles is not trivil tsk either. There re n! orderings possible for n circles becuse of their cyclic symmetry. This is huge number nd choosing t lest better ordering rther thn the best ordering could be certinly rewrding. The notion of better is subjective to two principl prmeters: precision of the result nd the running time for execution. In generl, they both re conflicting prmeters nd typiclly trde-off is estblished between them. In this section we exmine few orderings of the circles nd nlyse their performnce with respect to both these contrsting benchmrks experimentlly nd conceptully.

25 The most intuitive nd direct wy to order the circles is by ordering them bsed on the TSP order of their centre points. This ordering is conceptully similr to pproximting the circles to their corresponding centre points. The min issue in this cse is the running time for the execution of TSP procedure over the n points. Also, if the circles re of lrge nd vrying rdii, then the centre points TSP ordering my not provide the best order to visit the circles. This is primrily becuse of the fct tht the inccurcy of representing circle with its centre point grows with the circle s size. Figure 0. Centre point TSP ordering not being the best ordering Another defult ordering to consider will be rndom ordering of the circles. This ordering (or rther no ordering) will lift the intensity of the rigorous TSP computtion from the procedure but will very likely to provide results tht re fr from being close to optimum. Nevertheless, it is method which mesures up the comprison of the efficiency of other methods with respect to optimum nture of the solution nd running time. Similrly there is nother method which ws used for experimenttion purposes tht used the closest points on the circles with respect to the strting point 0, to be fed to the TSP procedure. This is just nother vrint of uniformly representing ll the circles with points on them. This ordering is not of theoretic interest. It cn be intuitively observed tht proper ordering of the circles requires the TSP ordering of points on the circles tht represent them. Points re chosen from every circle which closely represent the corresponding circles nd re then fed into the TSP procedure. Obviously, this mens more computtion becuse of the TSP procedure, but the precision of the solution becomes much better.

26 The choice of points tht represent the circles cn be done in multiple wys. In the centre point TSP ordering, we generlised ll the circles in common wy to their centre points. Hving the centre points represent the circles my not yield the best generlistion when the circles hve lrger rdii. Also, insted of hving common representtion method for ll the considered circles (in terms of identifying points on the circles tht represent the very ones) in the plne, we cn pply different types of representtions for different circles. This crries hevy importnce becuse when we choose point tht represents circle, we intuitively consider the position of the circle in the entire field, its neighbouring circles, the concentrtion of the circles in its re, the distribution of circles nd lso the orienttion of the presence of circles in the considered re. These prmeters vry for every circle, nd thus hving uniform representtion of points for ll the circles becomes is not pproprite. Hving greed on the pposite generlistion of the circles (to single points) which my not be uniform, we present nother method to choose the points to represent circles tht is conceptully clen nd better thn choosing the centre points of the circles or the closest points on the circles to the strting point. Initilly convex hull is formed for ll the circles in the plne. The circles tht re present on the vertices of the convex polygon thus formed will be the circles tht re on the outmost side. Now, with ll the circles, it is required to find centre point for the whole region, which in generl cn be the centre of grvity of ll the circles in the region. Now, the points tht represent the outwrd circles (present on the vertices of the convex hull formed), will be the closest point on the circle towrds the new centre point of the region which is computed. The rest of the circles cn hve the initil representtion which will be their centre points. Thus, the TSP procedure will tke the centre points of the circles which do not lie on the convex hull nd the closest points (towrds the centre of grvity) on the remining circles. On creful inspection, it becomes obvious tht this method of choosing the points for TSP procedure is better nd never worse thn the method of hving the centre points of ll the circles s the input for TSP. When we don t know where exctly the best point will lie on circle in the finl loop, it is best to pproximte or represent the circle using the centre point. But, if we know the rcul segment of circle which will contin the resulting best point for the minimum distnce loop, then rther thn

27 representing tht circle with its centre point, it is very resonble to represent the circle with point long tht rcul segment for computing the TSP ordering. Thus this more precise representtion of the circles will led to better TSP ordering of the circles s the procedure s inputs re more relevnt. This cn be seen in the next figure where the highlighted portions of the outmost circles re the potentil segments where the best points for minimum loop computtion will be present. Hence this segment is considered for finding point tht represents the circle nd the closest point towrds the centre of grvity which will definitely lie within the highlighted segment. This method of ordering is currently not implemented nd is not considered for experimentl nlysis. It is discussed here becuse of its conceptul ppel nd the performnce nlysis of this method long with explortions of more resonble orders re reserved for further study in this line of reserch. Figure. Highlighted rcul segments contining best points of representtion We lso propose nother ordering of the circles for observing their performnce in the experiments. It is more of heuristic which does not involve hevy computtions of the TSP thereby reducing the time tken for processing. However, it pys in the re of optimlity s the finl result of the minimum distnce loop computtion is lrger thn the TSP bsed ordering of circles. This method orders the circles in cyclic fshion strting from the top-left qudrnt nd then sweeping the other qudrnts in circulr mnner. This method is termed NEWS sort s the circles re ordered bsed on the direction. The ctul outputs of the experimenttion re nlysed in detil in the following section. As in ny sensor network ppliction, the pplictions hving the discussed scenrio of mobile sink nd fixed sensor nodes my hve certin chrcteristics tht could

28 impct the design nd functioning of the network. It could be requirement tht the order of visiting the nodes is known nd fixed. This knowledge cn be influenced by the ppliction itself. For instnce, the circles re ordered in terms of the scending order of their rdii s the ppliction would require the sink to collect dt first from nodes tht hve smller trnsmission rnge s they could be dying. These externl constrints reduce the dditionl dimension of the problem specifiction nd re not uncommon in the sensor networks scenrio. 7. ANALYSIS OF RESULTS For the purpose of nlysing the performnce of the heuristic nd the vrious orderings tht were described in the erlier sections, testing pltform ws developed. This testing pltform is prt of the Simulted Environment for Networked Sensor Experiments (SENSE) [3]. In ddition to the pplets provided, individul progrms were built to test the performnce efficiency of the methods over rndom smples of problem instnces. There were rndomly generted circles with rdii flling within preset rnge in smple pixel spce of 000 X 000 grid. The testing environment contins the following: objects such s points nd circles; methods for generting rndom circles; methods tht run different lgorithms to compute optimum route over the generted circles; nd methods tht nlyse the performnce every method nd report the results. The following results nd grphs re outcomes of the numericl nlysis performed over rndomly generted dt on the testing pltform. 7. Test Dt The experiments re simulted nd re primrily to estblish the credibility nd the correctness of the pproch tht we dhered to. Vrious instnces of smple dt were creted over which the procedures re executed nd re observed for their performnce. The smple test dt is generted rndom i.e. the circles were rndomly generted. Every circle creted for smple hs centre point determined rndomly (within the 000 X 000 pixel spce) nd hs the vlue of the rdius rndomly chosen between 30 nd 60 pixels. The strting point (of the sink node) is lso rndomly selected. In every instnce of the problem, there were fixed number of circles nd

29 there were mny smples for every instnce. The vrious instnces considered were 3, 4, 0, 5 nd 50 number of circles. In order to hve n unbised nd generlised view regrding the running time, 00 smples were considered for every instnce. For eg. there were 00 smples of 0 rndomly generted circles nd the strting point, over which the procedures re run nd the results re studied. The verge of the vlues of the results is then considered to be the result tht closely represents the procedure implementtion for the instnce contining 0 circles nd strting point. Obviously, the more smples being used, more closer the representtion is. For the 3 nd 4 circle instnces only 5 smples were used becuse of the intensity of the brute force computtion. 7. Tested Methods As discussed erlier, the performnce of our procedures re nlysed bsed on two contrsting prmeters, running time nd ccurcy. Hving discussed in depth bout the computtionl complexity of our problem, the ide is to find resonbly precise solution within resonble time. The simulted experiments record the minimum distnce clculted for every ordering discussed nd the time tken by the clcultion for every smple. Ech smple is fed to four of the mentioned trversl lgorithms s input viz. Rndom ordering, Centre Point TSP, Close Point TSP nd NEWS sort. A performnce monitor object cptures the informtion regrding the durtion of ech method over given smple nd the clculted minimum distnce covered by the trversl method. The fundmentl objective of the experiments is to derive knowledge bout the performnce of the vrious orderings discussed in the previous section. It hs been verified tht for the TCOP method, the regressive itertion produces nswers tht re very close to the optimum. We need to check tht for n-circles-nd-one-point which involves the regressive pproch long with the elsticity principle, our procedure finds close-to-optimum solution. However, given tht the brute force check is fr too intensive to compute even for resonbly sized problems, the experiments compute the bsolute optimum (minimum distnce of the route) only for given smple of 3 circles nd one point which is compred with our method for vrious orderings. It is

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