HQuad: Statistics of Hamiltonian Cycles in Wireless Rechargeable Sensor Networks

Transcription

1 HQuad: Statistics of Hamiltonian Cycles in Wieless Rechageable Senso Netwoks Yanmao Man Dept. of ECE Univesity of Aizona Tucson, AZ, U.S.A. Jing Deng Dept. of CS UNC at Geensboo Geensboo, NC, U.S.A. Geoge T. Amaiucai Dept. of CS Kansas State Univesity Manhattan, KS, U.S.A. Shuangqing Wei School of EECS Louisiana State Univesity Baton Rouge, LA, U.S.A. Abstact The ise of wieless echageable senso netwoks calls fo an analytical study of planned chaging tips of wieless chaging vehicles (WCVs). Often times, the WCV eceives a numbe of chaging equests and fom a Hamiltonian cycle and visit these nodes one-by-one. Theefoe, it is impotant to lean the statistics of such cycles. In this wok, we use a heuistic algoithm, which we tem HQuad, that takes O(N) to geneate a Hamiltonian cycle in a 2-D netwok plane befoe we analyze its statistics. HQuad is based on a ecusive appoximation of dividing the egion into fou quadants and the non-empty quadants will be visited one-by-one. Ou analysis is based on Poisson point distibution of nodes and models such Hamiltonian cycles supisingly well in both expected values and the distibution functions of lengths as a function of diffeent netwok paametes. Numeical esults of ou analysis model ae compaed with simulations and demonstated to be accuate. Index Tems wieless echageable; Hamiltonian Cycle; wieless senso netwoks I. INTRODUCTION Wieless devices usually un on enegy stoed on batteies. Due to thei physical constaints, these batteies will un out of enegy soone o late. Wieless powe tansfe technology is expected to pave the way fo the so-called wieless echageable netwoks, in which wieless devices will have thei batteies eplenished fom time to time, vitually extending thei lifetimes foeve [1] [3]. Most often, thee is one o multiple WCVs in such netwoks. WCV eceives chaging equests fom wieless devices and chooses a list of them to visit/chage one afte anothe [], [5]. In this wok, we ty to answe the following fundamental question: based on a cetain node density and andom distibution, how to compute o estimate the lengths of the chaging cycle without knowing the exact location of these nodes on the plane? Any of such chaging cycles will be consideed a Hamiltonian cycle, but we ae inteested in the shotest one that will save time and tavel distance fo WCV. Answes to the above question obviously have many pactical applications such as netwok planning and scheduling. Fo instance, the expected length of such shotest Hamiltonian cycles in a known netwok setting (of cetain node density and netwok egion) can ensue that the battey capacity of WCV is Y. Man s wok was done while with UNCG, suppoted in pat by NSF gant CCF able to sustain such a chaging tip befoe etuning fo its own chage o battey eplacement, o maybe an additional WCV is equied fo the nomal opeation of the entie netwok. The poblem is closely elated to Taveling Salesman Poblem (TSP) but diffeent. TSP aims to give shotest Hamiltonian cycles and sometimes appoximates with shot unning times [6], while this pape tagets the estimation of such cycles by giving a pobabilistic model [7]. In this wok, we popose HQuad, Quadtee-based heuistic algoithm, that takes O(N) to geneate a Hamiltonian cycle given a netwok within a squae plane. HQuad is an appoximation method to come up with a Hamiltonian cycle that is vey close to the actual shotest Hamiltonian cycle. With the help of HQuad, we can deive chaging path statistics such as aveage length and pobability density function (PDF). The main objective of this wok is to povide an efficient method to compute the statistics of Hamiltonian paths so that netwoks can be planned with sufficient esouces, e.g., chaging capacity, numbe of chaging vehicles, and/o numbe of devices to be chaged. II. RELATED WORK The well-known Taveling Salesman Poblem (TSP) is cetainly elated to what we ae tying to achieve. Many woks have been done on TSP [8]. Aoa [9] poposed an appoximation algoithm fo Euclidean TSP poblem, whee coodinates of nodes ae at fist petubed, then a shifted Quadtee is constucted. Finally, dynamic pogamming is pefomed to find the optimal Hamiltonian cycle. Howeve, we focus on path statistics hee. Wieless Senso Rechaging poblem has been investigated on in ecent yeas. The poblem is to schedule a oute fo one o moe WCVs so that they could echage those devices in low battey enegy. A hexagon-based scheme was poposed in [10]. Matello pesented an algoithm to identify Hamiltonian cicuits in diected gaphs [11]. The algoithm used banching decisions and backtacking to filte and ecod potential Hamiltonian cicuits. Kivelevich et al. investigated Hamiltonian cycles in andom gaphs, but did not focus on its length distibution [12].

2 S 1 S S 1 S n 3 n 9 n 7 n 8 S 2 S 3 S 2 S 3 n 10 (a) (b) n 2 n 1 n 6 S 1 S n 5 n S 2 S 3 Fig. 2: An example of blocks and echaging oute. (c) Fig. 1: HQuad Algoithm illustations. Vig and Paleka [13] investigated the pobability distibution of estimated TSP tou lengths using heuistics and fits fo the fist fou moments wee deived. Diffeent fom these pio ats, the main taget of this pape is not to povide a echaging dispatch algoithm. Instead, we aim to give a pobabilistic distibution of Hamiltonian cycle given the density and the size of squae plane whee the devices ae located. III. HQUAD: HEURISTIC QUADTREE-BASED ALGORITHM A. Poblem Statement FOR HAMILTONIAN CYCLES We focus on a netwok with a size of and N nodes, each of which needs to be chaged. Note that the investigation of an abitay ectangle aea of w is essentially the same with each quadant in simila shape as the oiginal one and some of the esults extended fom squaes. Theefoe, we focus on squae aeas in this wok: Hamiltonian Cycles (HAM): What is the distibution of the shotest Hamiltonian cycle in a egion of with N nodes distibuted? The exact deivation of such a distibution is impossible because Hamiltonian cycle/path poblem is NP-complete. Yet, such a distibution can be immensely helpful in netwok planning as well as algoithm design in Wieless Rechageable Senso Netwoks (WRSNs). Theefoe, we use a heuistic appoach, temed HQuad [9], to identify the statistics of the shotest Hamiltonian cycles. B. Pimay HQuad is based on Divide-and-Conque stategy that takes the locations of N nodes in a squae block as input and geneates the Hamiltonian Cycle fo WCV to follow. As shown in Figue 1.a, N nodes ae placed in an squae block S 0, which is divided into fou /2 /2 subblocks that ae maked as S i, i = 1,2,3,. WCV visits each sub-block fom one to anothe befoe etuning to the stating point. Each sub-block is futhe divided quaduply into smalle ones, whee WCV follows simila oute to visit the nodes and then goes to the next bigge block as illustated in Figue 1.b and 1.c when we zoom in. Nomally, some blocks may be empty and some do not need to be divided any futhe with only one node in each of them. Moe ealistic example of division is shown in Figue 2, whee we can obseve that all blocks contain at most one node and WCV only visits the non-empty blocks athe than all (sub)blocks. C. Algoithm Details Because the blocks ae divided into equal sub-blocks when necessay, we use Quadtee T as the data stuctue whee evey paent owns exactly childen, simila to [1]. Each vetex in T epesents a block and its childen epesent the sub-blocks. In this subsection, the tems block and vetex ae intechangeable, so ae the tems sub-block and child. The biggest block is epesented by vetex oot, which is the oot of T. Fo each vetex paent, it contains two elements, seq and nodes. seq decides the visiting sequence of its sub-blocks. Fo example, if seq = (S 3,S,S 1,S 2 ), then the oute is shown in Figue 1.b. nodes is the set of nodes that locate within this block. Take S in Figue 2 fo an example, nodes = {n 7,n 8,n 9 }. The Quadtee T is built up in Algoithm 1 fom Line 1 to Line 20 in the way of Beadth-fist Seach (BFS) with a queue Q. Fo evey vetex paent in Q, we fist test if its nodes set contains only one node o nothing at all. If so, then we continue to the next iteation because we do not have to futhe divide this block. Othewise, we have to at fist deive the sequences belonging its childen seq i, i = 1,2,3,. We need to detemine the seq of evey paent. Suppose sequence of thepaent isseq = (S a,s b,s c,s d ), the sequence of its fist child S a is seq a = (S a,s d,s c,s b ). Note that the fist child might not be S 1. Fo example, in Figue 1.b,

3 {n 3 } {n 1, n 2,..., n 10 } {n 2,n,n 5,n 6,n 10 } {n 1 } {n 7, n 8, n 9 } {n 10 } {n 5,n 6 } {n } {n 2 } {n 7,n 8 } {} {} {n 9 } {} {n 5 } {} {n 6 } {} {} {} {n 7,n 8 } {} {} {n 7 } {n 8 } Fig. 3: An example of Quadtee T geneation. the fist child S a is S 3. The second and thid childen shae the same sequence with seq. The fouth one depends on whethe the paent is the oot of T o not. If so, then seq d = (S c,s d,s a,s b ). Else, then seq d = (S c,s b,s a,s d ). The eason of such a diffeence is that if thepaent is the oot, which epesents the biggest block, then when WCV finishes the fouth sub-block, it goes back to the stating point (see S 2 in Figue 1.b). If paent is not the oot, WCV goes to the in Figue1.c). Having the seq i and nodes i of the sub-blocks, we attach them topaent as its childen in the ode of howseq indicates and put them into Q. When Q becomes empty and the while loop beaks, we un Depth-Fist-Seach (DFS) on T and output those nodes having exactly one node in them. Given the locations of M = 10 nodes in Figue 2, an example of Quadtee geneated by Algoithm 1 is shown in Figue 3. In Algoithm 1, the while loop takes O(N) and the DFS takes O(N). Finally, the time complexity of poposed algoithm HQuad is O(N). next bigge block (see S IV. ANALYSIS We focus on an aea of and mak fou sub-block as illustated in Figue 1.a. Each of these blocks has a size of /2 /2. We name these blocks S i, i = 1,2,3,, counteclockwise stating fom top left. We intoduce a binay tuple A = (b 1,b 2,b 3,b ) to epesent whethe each sub-block contains at least one node. Fo example, A = (1,1,0,1) means each of the sub-blocks s 1,s 2,s contains at least one node and the sub-block s 3 does not contain any node. Obviously thee ae 15 diffeent settings fo A, which we define as A, anging fom (0, 0, 0, 1) to (1,1,1,1). Define f(,ρ,n) as PDF of numbe of nodes, n, within a block, given density ρ. We futhe assume Poisson point Algoithm 1 HQuad(nodes 0 ) 1: seq (S 1,S 2,S 3,S ) 2: oot (seq, nodes) 3: Add oot into a queue Q : while Q do 5: paent Q.poll() 6: (seq, nodes) paent 7: if Size of nodes 0 is 1 then 8: Continue 9: 10: (S a,s b,s c,s d ) seq 11: seq 1 (S a,s d,s c,s b ) 12: seq 2 seq 3 seq 13: if paent is oot then seq (S c,s d,s a,s b ) 1: else seq (S c,s b,s a,s d ) 15: 16: fo i = 1 to do 17: nodes i nodes within the sub-block seq[i] 18: child i (seq i,nodes i ) 19: Add child i into Q 20: Attach child i as No.i child of paent 21: Run Depth-Fist Seach (DFS) fom oot, meanwhile output non-empty set nodes of evey leaf. distibution fo ease of analysis (although we do not expect to see any diffeence in the opeation of HQuad in othe node distibutions), we have f(,ρ,n) = (ρ2 ) n e ρ2. (1) n! Define g(a,,ρ) as the pobability that, fo an block with node density ρ, the nodes ae distibuted as A indicates. g(a,,ρ) = ( 1 f( ) A ( 2,ρ,0) f( ) A 2,ρ,0), (2) whee A is the numbe of 1 s in tuple A. Assume that each node is at the cente of its block. Define h(a,) as the Euclidean distance needed to visit all the nodes located at the cente of sub-blocks and the path should ente the top-left sub-block and exit the top-ight sub-block. Fo example, when A = (1,1,1,1), then h((1,1,1,1),) = 2 as illustated in Figue a. When A = (1, 1, 0, 1), then h((1,1,0,1),) = ( ) as illustated in Figue b. Thee ae 16 mappings as listed in Table I. Define D(,ρ) as the mean of length of Hamiltonian cycle needed fo an block with node density ρ. It can be calculated ecusively. D(,ρ) = A A [ h(a,)(1 1 A ) + A D( 2,ρ) ] g(a,,ρ), (3)

4 TABLE I: Function h(a, ) A h(a, 1) A h(a, 1) Least/Geatest hop distance in Hamiltonian cycle, d L / (d G /) Least hop distance Geatest hop distance (a) A = (1,1,1,1) (b) A = (1,1,0,1) Fig. : Illustations of h(a, ) with diffeent A. in which the tem1 1 A educes the ovelap of length between diffeent ecusive levels. The A D( 2,ρ) epesents the length sum of sub-blocks. Define Θ L (,ρ,l) as CDF of the length, l, of Hamiltonian cycle fo an block with node density ρ. It can be calculated ecusively as well. Denote I(A) the set of index of tems as 1 in A, e.g., fo A = (1,1,0,1), I(A) = {1,2,}. Θ L (,ρ,l) = P(L l,ρ) = A A{ µ l i=0,i I(A) [ P i I(A) i I(A) l i +h(a,)(1 1 A ) l ] Θ L ( 2,ρ,l i)dl i } g(a,,ρ) Recusion continues untilf(,ρ,2) < ǫ (an accuacy contol paamete), in which case Θ L (,ρ,l) = l v=0 () λ(,v), (5) wheeλ(,v) is defined as the PDF of the distance, v, between two andom points in an squae [15]. Let s = v 2, define π s 2 + s 3, 0 < s 2 Λ(,s) = sin 1 ( 2 s ) + 3 s 2 π s 2, 2 < s 2 2. Then λ(,v) = Λ(,s) ds dv = 2vΛ(,v2 ). In (), the integal cap µ is supposed to be. In pactice, µ = 2D( 2,ρ) Numbe of Nodes, N Fig. 5: Least and Geatest hop distance in each Hamiltonian cycle. V. PERFORMANCE EVALUATION We pesent pefomance evaluation in this section. We fist show esults fom simulations, in which N nodes ae andomly placed on an egion and we used a Genetic Algoithm [7] to identify the nea optimum Hamiltonian cycles. Then the esults of ou HQuad algoithm and computation ae shown and compaed to the simulation esults. Finally, we use an example of netwok planning to futhe demonstate how to use these PDF esults. In Figue 5, we show the least and the geatest hop distance in each Hamiltonian cycle as a function of N. These epesent the shotest and longest hop in the entie Hamiltonian cycle when N nodes ae andomly deployed in the egion. The 95% confidence intevals ae also shown in the same figue. In geneal, a deceasing tend can be obseved in both geatest and least hop distance as N inceases afte, caused by the inceasing numbe of nodes in the netwok. The initial incease of geatest hop distance when N = 2,3 is because of the small numbe of hops in the entie Hamiltonian cycle. Fo example, thee ae only two hops between N = 2 nodes a egion and thee hops between N = 3 nodes. In Figue 6, we compae simulation and numeical esults on Hamiltonian cycle lengths as a function ofn fo diffeent (the egion size). The 95% confidence intevals ae also shown fo the simulation esults. It can be seen that these two sets of esults match quite well with each othe. Unde lage n and lage, ou numeical esults ae lowe than simulation esults with lage gaps (due to ecusive appoximations). We demonstate how to use these PDF esults in a hypothetical netwok planning. Suppose N nodes will be deployed to a 1,000 1,000 egion and we have only one WCV to chage these devices tying to keep all of these active. The WCV has a battey capacity of 120 KJ, consumes 50 Joules pe mete on taveling and 1 J/s in geneal, chages sensos at the ate of 2 J/s, and moves at a speed of 1 m/s [5]. We will use ou numeical esults to compute the pobability of successful chaging suppot of N nodes with the above

5 TABLE II: Battey Capacity Requiement N E WCV (80%) E WCV (95%) Fig. 6: Compaison between simulation and numeical esults (3) on Hamiltonian cycle lengths, D. Pobability of successful chaging, p =0.5 KJ =2 KJ = KJ =8 KJ Numbe of nodes, N Fig. 7: Pobability of successful chaging with diffeent senso battey capacities as a function of N. netwok paametes fo diffeent senso battey capacities, = 0.5,2,,8 KJ, espectively. Such pobabilities ae shown in Figue 7. When is elatively small (assuming that enegy consumption ate of the sensos is also small enough to wait fo next ound of chaging), oughly 25 to 35 sensos can be suppoted. As inceases, fewe and fewe sensos can be suppoted. This is because of the heavie load to chage sensos and the longe Hamiltonian cycles fo the WCV to tavese befoe it can etun and eplace/chage its own battey. Fo example, when = 8 KJ, the netwok can suppot only 7 sensos. Suppose ou goal is to ensue that the WCV should have a battey that is lage enough to povide successful chaging to all sensos in the netwok with 80% and 95% pobability, espectively. We can use the numeical esults to compute the equied capacity of the WCV battey, based on a fixed value, e.g., KJ. We show the esults on Table II. The incease in equied WCV battey capacity is due to both of the additional enegy needed to chage the sensos as N inceases. VI. CONCLUSION In this pape, we have investigated the chaging tip poblem of the WCV in wieless echaging netwoks. We have intoduced a fast heuistic algoithm, HQuad, that geneates Hamiltonian cycle in O(N), based on but not esticted to which we have analyzed and modeled the length of Hamiltonian cycle by giving a cumulative distibution function (CDF) Θ L (,ρ,l), as well as a mean function D(,ρ). Simulation esults and Calculation esults not only show that HQuad woks well, but also demonstates that the pobabilistic models ae pecise. REFERENCES [1] L. Xie, Y. Shi, Y. T. Hou, and H. D. Sheali, Making senso netwoks immotal: An enegy-enewal appoach with wieless powe tansfe, IEEE/ACM Tans. Netw., vol. 20, no. 6, pp , Dec [2] P. Cheng, S. He, F. Jiang, Y. Gu, and J. Chen, Optimal scheduling fo quality of monitoing in wieless echageable senso netwoks, IEEE Tansactions on Wieless Communications, vol. 12, no. 6, pp , June [3] C. Wang, J. Li, F. Ye, and Y. Yang, A mobile data gatheing famewok fo wieless echageable senso netwoks with vehicle movement costs and capacity constaints, IEEE Tansactions on Computes, vol. 65, no. 8, pp , Aug [] S. Guo, C. Wang, and Y. Yang, Joint mobile data gatheing and enegy povisioning in wieless echageable senso netwoks, IEEE Tansactions on Mobile Computing, vol. 13, no. 12, pp , Dec 201. [5] G. Jiang, S. K. Lam, Y. Sun, L. Tu, and J. Wu, Joint chaging tou planning and depot positioning fo wieless senso netwoks using mobile chages, IEEE/ACM Tansactions on Netwoking, vol. PP, no. 99, pp. 1 17, [6] G. Reinelt, Fast heuistics fo lage geometic taveling salesman poblems, ORSA Jounal on Computing, vol., no. 2, pp , [7] A. M. Fieze and J. E. Yukich, Pobabilistic Analysis of the TSP. Boston, MA: Spinge US, 2007, pp [8] D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook, The taveling salesman poblem: a computational study. Pinceton univesity pess, [9] S. Aoa, Polynomial time appoximation schemes fo euclidean taveling salesman and othe geometic poblems, Jounal of the ACM (JACM), vol. 5, no. 5, pp , [10] L. Xie, Y. Shi, Y. T. Hou, W. Lou, H. D. Sheali, and S. F. Midkiff, Multi-node wieless enegy chaging in senso netwoks, IEEE/ACM Tansactions on Netwoking (ToN), vol. 23, no. 2, pp , [11] S. Matello, Algoithm 595: An enumeative algoithm fo finding hamiltonian cicuits in a diected gaph, ACM Tansactions on Mathematical Softwae (TOMS), vol. 9, no. 1, pp , [12] M. Kivelevich, C. Lee, and B. Sudakov, Compatible hamilton cycles in andom gaphs, Random Stuctues & Algoithms, vol. 9, no. 3, pp , [13] V. Vig and U. S. Paleka, On estimating the distibution of optimal taveling salesman tou lengths using heuistics, Euopean Jounal of Opeational Reseach, vol. 186, no. 1, pp , [1] H. Samet, The quadtee and elated hieachical data stuctues, ACM Computing Suveys (CSUR), vol. 16, no. 2, pp , 198. [15] J. Philip, The pobability distibution of the distance between two andom points in a box, se. TRITA / MAT / MA: TRITA. KTH mathematics, Royal Institute of Technology, 2007.