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1 1 Supplemetal Materials Xiagmao Chag, Rui Ta, Guoliag Xig, Zhaohui Yua, Cheyag Lu, Yixi Che, Yixia Yag This documet icludes the supplemetal materials for the paper titled Sesor Placemet Algorithms for Fusio-based Surveillace Networks. APPENDIX A HARDNESS OF PROBLEM 2 I this appedix, we discuss the hardess of Problem 2 preseted i Sectio 4.2. Specifically, we discuss the oliearity ad o-covexity of the problem. We ote that o-liear ad o-covex optimizatio problems ofte have expoetial complexities ad the geeral solutio to them does ot exist [1]. I the costrait of Problem 2, the detectio probability P Dj, which is give by (3), has a complex o-liear relatioship with the umber of sesors ad their positios. Therefore, Problem 2 is a o-liear optimizatio problem. We ow discuss the o-covexity of Problem 2. Deote the placemet with sesors as S. Thus, S =. By replacig P Dj i the costrait (i.e., mi 1 j K {P Dj } β) with (3) ad after maipulatio, we have mi 1 j K i=1 W(d i ) σ 2 (χ 1 (1 α) χ 1 (1 β)). where is the umber of sesors i the fusio rage of j spot j. Deote f(s ) = mi i=1 W(d i) σ 2 (χ 1 1 j K j (1 α) χ 1 (1 β)). Suppose S maximizes f(s ), if f(s ), S is a cadidate solutio. By iteratig, we ca fid the optimal solutio to Problem 2, i.e., S = argmi S. If the sub-problem, i.e., maximizig f(s ), is o-covex, Problem 2 is o-covex. We ow prove the o-covexity of f(s ) by exemplificatio. I our example, we radomly choose two surveillace spots ia1 1field,withthe coordiatesof(8.21,.15) ad (.43, 1.69), respectively. We the radomly place 3 X. Chag ad Y. Yag are with Iformatio Security Ceter, State Key Laboratory of Networkig ad Switchig Techology, Beijig Uiversity of Posts ad Telecommuicatios, Beijig 1876, Chia; Y. Yag is also with Research Ceter o Fictitious Ecoomy ad Data Sciece, Chiese Academy of Scieces, Beijig 119, Chia. R. Ta ad G. Xig are with Departmet of Computer Sciece ad Egieerig, Michiga State Uiversity, East Lasig, MI 48824, USA. Z. Yua is with School of Software, Huadog Jiao Tog Uiversity, Nachag, Jiagxi 3113, Chia. C. Lu ad Y. Che are with Departmet of Computer Sciece ad Egieerig, Washigto Uiversity i St. Louis, St. Louis, MO 6313, USA. f(s) = 3 = TheX-coordiate of a sesor Fig. A.1. f(s ) is o-covex. Settigs: W = 4, d = 1, k = 2, σ 2 = 1, α =.1, β =.9. Algorithm 1 The procedure of fidig global optimal solutio Iput: α, β, surveillace field A, a set of surveillace spots T Output: Sesor placemet S where S is miimized 1: N = 1 2: repeat 3: use the CSA solver to fid N sesor locatios i A that maximize mi 1 j K {P Dj } 4: compute η j for each t j T by η j = σ2 X 1 (1 α) j 5: compute P Dj for each t j T by (3) 6: N = N +1 7: util mi 1 j K {P Dj } β 8: retur S or 4 sesors i the field. Whe = 3, the coordiates of sesors are (6.49, 4.51), (7.32, 5.47) ad (6.48, 2.96), respectively. Whe = 4, the coordiates of sesors are (7.45, 3.68), (1.89, 6.26), (6.87, 7.8) ad (1.84,.81), respectively. Fig. A.1 plots f(s ) versus the X-coordiate of the first sesor. We ca see from the figure that f(s ) is a o-covex fuctio with respect to sesor positio. I summary, Problem 2 is a o-liear ad o-covex optimizatio problem. APPENDIX B GLOBAL OPTIMAL PLACEMENT A straightforward optimal solutio for Problem 2 is to icremetally search for the optimal sesor placemets with differet umber of sesors uder the costrait mi 1 j K {P Dj } β. The details are show i Algorithm1.Itbegis with N = 1aditeratesforicremetal N. I each iteratio, the miimum detectio probability amog all surveillace spots, i.e., mi 1 j K {P Dj }, is

2 2 Executio time (secods) e.34n The umber of required sesors N Fig. B.1. The executio time vs. the umber of sesors i the optimal placemet. maximized. Oce the costrait mi 1 j K {P Dj } β is satisfied, the global optimal solutio is foud. The optimizatio step that maximizes the miimum detectio probability (Lie 3 i Algorithm 1) is implemeted by a o-liear programmig solver based o the Costraied Simulated Aealig (CSA) algorithm [2]. CSA exteds covetioal Simulated Aealig to look for the global optimal solutio of a costraied optimizatio problem with discrete variables. CSA allows the objective fuctio ad costrait fuctios to be specified i a procedure istead of i a closedform. Theoretically, CSA is a global optimal algorithm that coverges asymptotically to a costraied global optimum with probability oe (Theorem 1 of [2]). I theory, the complexity of CSA, like other stochastic search algorithms, icreases expoetially with respect to the umber of variables [2]. Therefore, the o-liear programmig solver has a expoetial time complexity with respect to the umber of sesors. Hece, for a large-scale placemet problem, Algorithm 1 becomes prohibitively expesive. Fig. B.1 shows the executio time of Algorithm 1 versus the umber of sesors i the optimal solutio. The dotted curve is the liear regressio of the executio times with differet umbers of sesors. We ca see that the executio time icreases drastically with the umber of sesors. For istace, if 1 sesors are to be placed, the projected executio time of the global optimal algorithm is about e 36 secods, i.e., days. APPENDIX C DIVIDE-AND-CONQUER SENSOR PLACEMENT ALGORITHM The pseudo codes of the divide-ad-coquer sesor placemet algorithm ad the refiemet process proposed i Sectio 5.2 are show i Algorithm 2 ad Algorithm 3, respectively. I Algorithm 2, we treat the surveillace spots oe by oe. Specifically, for spot t j, we place the fewest additioal sesors withi the fusio regio of t j to cover t j ad the eighborig spots i its impact regio. The optimizatio (Lie 5) is implemeted by the CSA solver. Algorithm 2 The divide-ad-coquer sesor placemet algorithm Iput: α, β, impact regio set {A j 1 j K}, ad the set of surveillace spots i each impact regio {T j 1 j K} Output: Local optimal sesor placemet S 1: S = 2: for j = 1 to K do 3: = 4: repeat 5: place additioal sesors i C j (deoted by set ) to maximize mi th T j {P Dh } uder placemet S 6: compute η h for each t h T j uder placemet S 7: compute P Dh for each t h T j uder placemet S 8: = +1 9: util mi th T j {P Dh } β 1: S = S 11: ed for 12: retur S I each roud of Algorithm 3 (from Lie 2 to 2), all surveillace spots are processed oe by oe. The algorithm termiates if the curret roud caot further reduce the umber of sesors i the placemet. Whe the j th spot t j is processed (from Lie 4 to 19), the algorithm first removes all dedicated sesors of t j, ad computes a ew local placemet S usig the CSA solver (from Lie 8 to 15). If S uses fewer sesors tha the origial placemet S, we replace the origial placemet with S (from Lie 17 to 19). Note that we do ot remove ay shared sesors i the placemet computed by Algorithm 2 as otherwise the coverage of eighborig surveillace spots may be affected. We ow discuss the covergece of Algorithm 3. As oly the ew placemet with fewer sesors(from Lie 17 to 19) i each iteratio is acceptable, obviously, the size of the placemet computed i each roud keeps decreasig. Deote N ad N as the sizes of the global optimal solutio (i.e., the output of Algorithm 1) ad the solutio of Algorithm 2, respectively. The upper boud of the umber of rouds of Algorithm 3 is thus N N. APPENDIX D ANALYSIS OF MODEL PARAMETERS I this appedix, we discuss how to set the parameters of the sesig ad data fusio models. D.1 Optimal Upper Boud of Fusio Radius I this sectio, we aalyze the upper boud of fusio radius R, ad give the optimal value regardig sesor desity. Suppose a surveillace spot t is covered by

3 3 Algorithm 3 The sesor placemet refiemet algorithm Iput: α, β, impact regio set {A j 1 j K}, the set of surveillace spots i each impact regio {T j 1 j K}, sesor placemet S computed by Algorithm 2 Output: ew sesor placemet S 1: repeat 2: total sesor umber N = S 3: for j = 1 to K do 4: fid dedicated sesor set D j of t j i S 5: if D j = the 6: skip this iteratio for t j ad cotiue 7: else 8: S = S\D j /* remove dedicated sesors of t j */ 9: = 1 1: repeat 11: place additioal sesors i C j (deoted by ) to maximize mi th T j {P Dh } uder placemet S 12: compute η h ad P Dh for eacht h T j uder placemet S 13: = +1 14: util mi th T j {P Dh } β 15: S = S 16: ed if 17: if S < S the 18: S = S 19: ed if 2: ed for 21: util S = N 22: retur S sesors i the fusio regio of t. Accordig to (3), the detectio probability of t satisfies: ( η i=1 P D = 1 X W(d ) i) σ 2 β. By replacig η with the optimal detectio threshold η = σ 2 X 1 (1 α) obtaied i Sectio 4.2 ad solvig the sum of eergies i=1 W(d i) from the previous iequality, we get: i=1 W(d i) X 1 (1 α) X 1 (1 β). (D.1) σ 2 Iequality (D.1) meas that give sesors, the sum of eergies received by sesors is lower bouded if the correspodig spot is covered. As the sigal eergy atteuates with the distace from the target, i the worst case, every sesor is placed at the edge of the fusio regio. Therefore, the miimum sum of eergies is { } mi W(d i ) = W(R). (D.2) i=1 By replacig the sum of eergies i (D.1) with its miimum value give by (D.2) ad solvig the fusio radius Sesor desity (ρ) σ 2 = 1 σ 2 = 2 σ 2 = Upper boud of fusio radius (R u) Fig. D.1. Sesor desity vs. upper boud of fusio radius. Settigs: W = 4, d = 1, k = 2, α =.1, β =.9. R, we get: ( ( σ R W 1 2 X 1 (1 α) X 1 (1 β) ) ), (D.3) where W 1 ( ) is the iverse fuctio of the sigal atteuatio model W(d). Deote the right had of (D.3) as R u, which is the upper boud of the fusio radius. Iequality (D.3) shows that, give sesors, if the fusio radius does ot exceed R u, the surveillace spot t is always covered o matter how the sesors are placed i the fusio regio of t. If we set R = R u, the sesor desity is ρ = /(πr 2 u), which is the miimum sesor desity that guaratees the coverage of the surveillace spot o matter how the sesors are placed. Because both ρ ad R u deped o, it s hard to derive the margial relatioship betweeρad R u with a aalytical expressio. However, i practice, we ca fid the optimal R u that miimizes ρ by umerical approaches. For istace, Fig. D.1 shows several umerical results uder the sigal atteuatio model defied by (1) uder differet settigs. From the figure, we ca see that the sesor desity ρ is a covex fuctio of R u uder differet settigs ad thus we ca fid the optimal R u, e.g., R u = 7.76 is the optimal value for the settig σ 2 = 1. Furthermore, for a certai fusio radius (e.g., R u = 1), the sesor desity for low SNR settig (e.g., σ 2 = 5) is greater tha the sesor desity for high SNR settig (e.g., σ 2 = 1). This is reasoable as more sesors are eeded to achieve the same sesig quality whe the SNR is low. D.2 Aalysis of Impact Regio Radius I this sectio, we discuss the optmal settig of the impact regio radius. As discussed i Sectio 5.1, the key idea of our divide-ad-coquer approach is to reduce the total umber of sesors by takig advatage of the shared sesors i the overlappig fusio regios of surveillace spots. Moreover, the global optimizatio problem is divided ito the sub-problems of coverig the spots withi the impact regio of each spot. If the distace betwee ay two spots is less tha 2R, the two spots ca have shared sesors. Therefore, i previous sectios, we let the impact regio radius be 2R. I this

4 4 sectio, we discuss the settig of impact regio radius i detail, which provides more isights ito the dividead-coquer approach. We first discuss the impact of placig a additioal sesor for the coverage of a surveillace spot. The ecessary ad sufficiet coditio of the coverage of a surveillace spot is give by (D.1). If a additioal sesor is placed ito the fusio regio of the spot, both sides of iequality (D.1) icrease. We ow explore the sufficiet coditio for maitaiig iequality (D.1). We let f( α,β) = X 1 (1 α) X 1 (1 β) ad g( α,β) = f( + 1 α,β) f( α,β), where 1. Our extesive umerical experimets show that g( α, β) is a mootoically decreasig fuctio if α < β. Note that missio-critical applicatios typically require low false alarm rates (e.g., < 5%) ad high detectio probabilities (e.g., > 9%). Therefore, the maximum value of g( α, β) is g(1 α,β). Whe a additioal sesor is placed i the fusio regio of the spot, the miimum icrease to the left-had side of (D.1) is W(R) σ. Accordigly, if 2 W(R) σ 2 g(1 α,β), iequality (D.1)still holds whea additioal sesor is placed, ad therefore the coverage of the spot will ot be breached. With the above sufficiet coditio for maitaiig coverage whe additioal sesors are placed, we ow discuss the settig of impact regio radius. Our discussio has the followig two cases: Case 1):W(R) < σ 2 g(1 α,β). Suppose the impact regio radius is smaller tha 2R, placig sesors to cover the curretly treated spot ca breach the coverage of the spots out of the impact regio. Therefore, the best choice of impact regio radius is 2R. Case 2): W(R) σ 2 g(1 α,β). I this case, the impact regio radius ca be smaller tha 2R. However, as the sesor placemet is ot kow a priori, it is difficult to derive the optimal impact regio radius that miimizes the etwork desity. I this paper, we evaluate the optimal impact regio radius through umerical experimets, which are preseted i Appedix F.1. D.3 Hadlig Target Locatio Error The problem formulatio i this paper oly esures the detectio performace at the surveillace spots. Whe the target does ot appear at the surveillace spots exactly, the detectio performace of the computed sesor placemet may ot meet the user requiremet. We ow briefly discuss a approach to hadle this issue. Suppose the distace betwee the surveillace spot ad the real positio of the target, referred to as target locatio error, is upper-bouded by ǫ. By replacig W(d) with W(d+ǫ), Eq. (3) computes the lower boud of detectio probability. Therefore, the resulted sesor placemet ca meet the detectio performace requiremet whe the target locatio error does ot exceed ǫ. CDF Detectio probability P D (a) 196 regular spots CDF Detectio probability P D (b) 25 radom spots Fig. E.1. The CDF of the detectio probabilities at regular or radom surveillace spots. APPENDIX E MORE TRACE-DRIVEN SIMULATION RESULTS I the trace-drive simulatios preseted i Sectio 6, we evaluate the performace of sesor placemet for real vehicle trajectories. I order to extesively evaluate the performace of our sesor placemet algorithm, i this appedix, we choose regularly or radomly distributed surveillace spots i a 3 3m 2 area. We evaluate the performace of the cluster-based sesor placemet algorithm i two sets of simulatios. First, 67 sesors are placed to cover 196 surveillace spots regularly distributed at grid poits. Secod, 1 sesors are placed to cover 25 surveillace spots radomly scattered i the field. We the evaluate the effectiveess of the resulted sesor placemets usig the real data traces via the approach discussed i Sectio 6. Fig. 1(a) ad Fig. 1(b) show the CDF of the detectio probability uder the two sesor placemets, respectively. From the figures, we ca see that over 9% surveillace spots are covered i both two placemets, which satisfies the required lower boud of detectio probability. I Sectio 6, we have discussed the reasos for the remaiig 1% surveillace spots that do ot reach the requiremet o detectio probability. APPENDIX F MORE NUMERICAL EVALUATION RESULTS I Sectio 7, we have evaluated the impact of surveillace spot clusterig o our algorithms ad compared the divide-ad-coquer placemet algorithm with the global optimal algorithm as well as a greedy algorithm. I this appedix, we preset more evaluatio results o the impacts of fusio radius, impact regio radius ad decay factor. The evaluatio settigs ca be foud i Sectio 7. F.1 Fusio Radius ad Impact Radius We first evaluate the impact of fusio radius. Total 45 surveillace spots scatter radomly i the field. The cluster-based divide-ad-coquer placemet algorithm is used. We chage the fusio radius R from 3 to 12. Fig. F.1 plots the umber of sesors required i the

5 5 The umber of sesors (N) Fusio rage radius (R) Fig. F.1. The umber of sesors vs. fusio regio radius. The umber of sesors N regular spots radom spots Impact regio radius ( R) Fig. F.2. The umber of sesors vs. impact regio radius. solutio computed by the placemet algorithm versus the fusio radius. From the figure, we ca see that the umber of required sesors drops rapidly from 37 to 7 whe fusio radius R icreases from 3 to 7.6, ad gradually icreases to 17 whe R becomes larger. I Appedix D.1, we discussed the optimal upper boud of fusio radius that miimizes the etwork desity. The simulatio result i this appedix is cosistet with the umerical results show i Fig. D.1 i Appedix D.1, which meas that the aalysis i Appedix D.1 is tight eough to be applicable to computig the optimal fusio radius. We ow evaluate the impact of impact regio radius. As discussed i Appedix D.2, if W(R) σ 2 g(1 α,β), it is difficult to derive the optimal impact regio radius that miimizes the etwork desity. I this appedix, we coduct umerical experimets to explore the optimal impact regio radius. Note that we use the sesor placemet algorithm without surveillace spot clusterig. I the experimets, 255 or 196 surveillace spots are chose regularlyor radomlyithe3 3m 2 area,respectively. Fig. F.2 plots the umber of placed sesors versus the impact regioradius. We casee fromthe figurethat, for both the cases of regular ad radom surveillace spots, the umber of placed sesors has a covex relatioship with the impact regio radius. The ituitio behid this result is as follows. If the impact regio radius is too small (e.g., less tha.2r), the overlappig fusio regio of two adjacet surveillace spots may ot be exploited to place shared sesors. If the impact regio radius is close to 2R, may shared sesors may be placed ito the small area of the shared fusio regio of two surveillace The umber of sesors N clustered uclustered Decayig factor k Fig. F.3. The umber of sesors vs. decayig factor. spots that are early 2R meters apart from each other. Both these two cases ca lead to the iefficiecy of sesor placemet. Moreover, we ca see from Fig. F.2 that fewer sesors are eeded to cover radom spots. Ituitively, if spots are radomly distributed, multiple spots ca be i the same impact regio such that a few sesors ca cover them. I cotrast, the regular spots have eveer spatial distributio tha the radom spots, resultig more sesors placed. F.2 Impact of Decayig Factor I this sectio, we evaluate the impact of the decayig factor k of the sigal decay model o our algorithm. I the experimet, 45 surveillace spots are radomly chose i the 3 3m 2 field. Fig. F.3 plots the umber of placed sesors versus the decayig factor k with ad without the QT clusterig, respectively. We ca see from the figure that the umber of placed sesors icreases with the decayig factor. This result is cosistet with the ituitio that more sesors are required to cover the surveillace spots if the sesors receive weak sigals from the target due to the fast atteuatio of sigal eergy. Moreover, we ca see from the figure that the cluster-based placemet algorithm becomes more effective i reducig sesors whe the decayig factor is larger. F.3 Visualizatio of Sesor Placemets I the first set of experimets, total = 225 surveillace spots are regularly distributed, as show i Fig. 4(a) ad Fig. 4(b). The sesor placemets computed by the cluster-based divide-ad-coquer algorithm ad the greedy algorithm preseted i Sectio 7.2 are also show i the two figures. Total 13 ad 15 sesors are placed by the two algorithms, respectively. I the secod set of experimets, total 196 surveillace spots radomly scatter i the field, as show i Fig. 4(c) ad Fig. 4(d). Total 11 ad 15 sesors are placed, respectively. REFERENCES [1] R. T. Rockafellar, Lagrage multipliers ad optimality, SIAM review, vol. 35, o. 2, [2] B. W. Wah, Y. Che, ad T. Wag, Simulated aealig with asymptotic covergece for oliear discrete costraied global optimizatio, J. Global Optimizatio, vol. 39, 27.

6 6 surveillace spot sesor surveillace spot sesor surveillace spot sesor surveillace spot sesor (a) D&C: K = 225, N = 13 (b) Greedy: K = 225, N = 15 (c) D&C: K = 196, N = 11 (d) Greedy: K = 196, N = 15 Fig. F.4. Sesor placemets usig the cluster-based divide-ad-coquer ad the greedy algorithms. The dotted circles are the impact regios of clusters.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ. 2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For