A Distributed Sorting Framework for Ad Hoc Wireless Sensor Networks

Transcription

1 A Distibuted Soting Famewok fo Ad Ho Wieless Senso Netwoks Amitabha Ghosh Honeywell Tehnology Solutions Lab 151/1 Doaisanipalya, Banneghatta Road Bangaloe , India Balamuugan Mohan Honeywell Tehnology Solutions Lab TCE Campus, Thiupaankundam Maduai 65015, India Abstat Wieless senso netwoks (WSN) ae distibuted, self-oganizing, pevasive systems, whih pefom ollaboative omputations to povide useful infomation about the undelying stohasti phenomenon. In this pape, we exploe the anking and soting poblems in distibuted senso netwoks whih povide pespetives in undestanding etain fundamental issues in WSN, as well as motivate many exiting eal life appliations. In the fist pat, we popose a simple, distibuted and paallel soting algoithm in single-hannel, multihop, lattie senso netwoks that has a time omplexity of ( n) and an enegy omplexity of (n n). In the sequel, we onside the soting poblem unde a genealized famewok of geometi andom gaphs and show that finding a time and/o enegy optimal distibuted soting algoithm is NP-had. Finally, we pove the existene of a andomized soting algoithm unde the genei setup. I. INTRDUCTIN Soting is a fundamental poblem in ompute siene that has been analyzed in geat details ove the yeas. We see the value of addessing this poblem in distibuted senso netwoks fom two pespetives. Fistly, distibuted soting being a ollaboative poess, analyzing it fo senso netwoks will allow us to undestand etain fundamental issues. When nodes ollaboate to solve etain poblems in a distibuted manne, unde settings like soting, we will be able to undestand many design issues and limitations, and also find appoahes to solve them effiiently. At the same time, we will be able to exploe the inteation between fomation of netwoks, whih we model as a gaph, synhony and asynhony in media aess potool, boadast natue of ommuniation, outing and flow of data. The seond pespetive to the poblem omes fom etain appliation equiements. Sine in wieless senso netwoks, nodes ollet data samples fom the envionment, many of whih might be eoneous, and hene, they ae often needed to be soted in ode to filte out exteme values. Some of the appliations equie anking of senso nodes based on speifi meties. Fo example, the nodes might be anked aoding to thei esidual battey powe, so that aodingly they an adjust thei tansmission anges to optimize powe onsumption. In some othe appliation senaios, it is not only neessay fo eah senso node to know its own ank, but also to have the knowledge whih of its neighboing nodes ae anked highe o lowe than itself. Anothe example is taget taking with senso netwoks, whee many sensos nea the taget will get ativated in the pesene of a taget and all of them will pefom the detetion. Howeve, sine the data o the measuement in those nodes will be spatiotempoally oelated, it would make sense to quantitatively evaluate suh measuements and assign a value aoding the meit of the data. Suh a meit ould be based on infomation theoeti measues, like mutual infomation ontent between senso values. ne the measuements ae assigned values, the aggeation mehanism may just need to ank and sot the values so that they ould be aggegated meeting the objetive. To ou knowledge, the soting poblem in senso netwoks o simila boadast netwoks, like paket adio netwoks (PRN), so fa has been onsideed unde estited senaios. In [1], Bodim et. al poposed a soting algoithm, whih uses basi opeations of ompaing eithe in ows o olumns and a vaiant of mege sot to sot n data elements plaed in n senso nodes aanged in a two-dimensional gid of size n n in ( n) time slots, fo < n, whee is the tansmission adius. Fo suh shot tansmission anges, this time omplexity mathes with that of the optimal algoithm poposed by Nassimi and Sahni in [], whih is an adaptation of bitoni sot. The authos in [] had shown that on a mesh onneted paallel ompute, n elements an be soted in (n) time. Howeve, fo lage tansmission anges ( n) sine the numbe of nodes that an tansmit simultaneously deeases to avoid intefeene, Bodim s algoithm an take as muh as ( ) time slots. Singh et. al in [3] emulated some of the optimal and distibuted paallel algoithms in two-dimensional gid senso netwoks and showed that an enegy balaned anking algoithm would take ( n/) time with oveall enegy dissipation (n ). They also demonstated that using a lusteing appoah bitoni sot an be implemented on a two dimensional gid of size n n with time omplexity ( n) and with enegy omplexity (n n n), whih onfom to the esults as obtained in [1]. The same authos in [3] pesented an enegy-optimal algoithm fo soting in a single-hop single-hannel senso netwok and showed that thei algoithm an sot n andomly distibuted

2 sensos in (n log n) time and enegy, with no senso being awake fo moe than (log n) time steps. Howeve, all the afoe-mentioned woks have onsideed the anking and soting poblems unde vey estited senaios, eithe by assuming a single-hop senso netwok o by imposing a gid stutue on a senso netwok. It should be noted that a single-hop WSN is of little patial impotane beause in most of the eal wold deployments, thee will be multihop WSN to be onsideed. Similaly, gid stutues imposed on senso netwoks estit the feasibility of the algoithms in patial appliations, beause of the fat that most senso netwoks ae fomed by ad ho andom deployment of nodes athe than following any geometi stutue. In this pape, we onside the soting poblem in a moe genei setup and assume ompaison-only algoithms. We also obseve that the effiieny of the anking and soting algoithms is dependent on the tansmission adii of the senso nodes. A lage tansmission adius edues the netwok diamete but at the same time, esults in highe enegy dissipation and edues the numbe of nodes that an tansmit o eeive onuently beause of intefeene. The est of the pape is oganized as follows. In setion II we fomally intodue the model and the poblem statement fo desibing a simple, distibuted and paallel soting algoithm unde the estited senaio of single-hannel gid senso netwok. Next, in setion III we fomulate the soting/anking poblem unde the genei setup of andom deployment and show that finding a time and/o enegy optimal distibuted soting algoithm is NP-had. We also pove the existene of a shedule of exponential length that allows all the senso nodes to tansmit atleast one with vey high pobability. II. RESTRICTED: GRID-BASED, SINGLE-CHANNEL WSN A. Poblem statement In this setion, we onside the soting poblem on a WSN unde a estited senaio, whee nodes ae aanged in the fom of a two-dimensional gid and thee exists a single hannel fo ommuniation. Let n be the total numbe of nodes plaed on a gid of size, whee is the numbe of ows and is the numbe of olums. Eah node has a data element and a unique identifie stoed in it. Let us also assoiate a vitual identifie ij, with eah node, depending on its position in the gid. Identifie ij uniquely detemines a node plaed on ow i and olumn j, whee 1 i and 1 j. Let us denote the value stoed in node ij by v ij. The goal of the soting algoithm is to edistibute the data among the nodes, suh that at the end of the algoithm the following iteia holds. 1) when i is odd: v i(j1) v ij, fo 1 j < v (i1) v i, fo 1 < i < v (i 1)1 v i1, fo 1 < i ) when i is even: v i(j1) v ij, fo 1 j < v (i1)1 v i1, fo 1 < i < Fig s s' Row-majo snake-like patten, Communiation adius v (i 1) v i, fo 1 < i The above two sets of iteia A and B sots the data in a ow-majo snake-like patten as illustated in Fig. 1. We also make the following assumptions: 1) The distane of sepaation between any two onseutive nodes is the same,. ) All the nodes have unifom ommuniation adius,. 3) A node s an ommuniate only with the nodes that ae loated within a distane fom it, i.e., it annot ommuniate with anothe node s (Fig. 1 ) that is diagonally loated with espet to its own position. B. Algoithm The algoithm is iteative. Eah iteation onsists of fou steps and in eah step, a numbe of nodes goup togethe and pefom loalized soting among themselves. The possible numbe of nodes in a goup an eithe be one, two o fou. In the following, we desibe the fou steps illustating the senaio with 16 nodes aanged in a gid. 1) Step 1: In the fist step, goups of fou nodes ae fomed as shown in Fig.. The top-left nodes (dakened in the figue) in eah goup at as leades, in the sense that all othe nodes within a goup tansmit thei data to this node. The aows shown in the figue signify loal boadasts within a goup. Fo eah goup, it takes five loal boadasts to omplete the following thee steps: all the membe nodes data eah the leade the leade sots the data loally the leade tansmits the soted data bak to the membe nodes. These five loal boadasts will take time slots beause two of the boadasts an happen paallely (aows maked as 1 in the figue). Time equied fo the leade to sot the fou values is 6 time slots (it takes 6 ompaisons in the wost ase to sot values). Thus, the total time taken fo all the thee steps to happen is 10 time slots. Note that, the loal boadasts in all the goups will take plae in paallel. Hene, total time taken fo step 1 to omplete is 10 time slots. ) Step : In this step, the goups will shift one olumn towads ight fom step 1, as illustated in Fig. and the same poess, as desibed in step 1, will ontinue. Hee, goups of two nodes ae fomed in the fist and last olumns of the gid, when is even and >. n the othe hand, when is odd and > 1, goups of two nodes ae fomed only in the fist olumn. Time slots equied to omplete this step is also 10, beause it esentially follows the same thee steps as desibed in step 1.

3 t. The enegy used will be ( ( [5Et 6E v ] [Et E v ] ( [Et E v ]. ) Step : It will have ( 1 goups of nodes fomed, and ( ( 1 goups of nodes fomed. Sine soting in this step will also equie 5 tansmissions and 6 ompaisons, time omplexity of ( step will be t t 6t. The enegy used will be 1 [5Et 6E v ] ( 1 ( [Et E v ] [Et E v ]. () Fig.. Step 1, Step, () Step 3, (d) Step 3) Step 3: In the thid step, the goups will shift one ow towads down fom step 1, as shown in Fig. () and then the same poess follows. Goups of two nodes ae fomed in the fist and last ows, when is even and >, whee as goups of two nodes ae fomed only in the fist ow when is odd and > 1. Time slots equied to omplete this step is again 10. ) Step : This step is simila to step, whee the goups ae shifted one olumn towads ight fom step 3 and the same poess of loalized boadast and soting ontinues. Time equied to omplete this step is 10 slots. The leade node peseves the pevious iteation data of all the membes in its goup. Eah of the membes peseves its one-iteation data duing its assosiation with fou diffeent goups. The membes will tansmit the data to its leade only when thee is a hange fom the pevious value, exept fo the fist time. The leade will undestand that a node s value has not hanged afte the last iteation if the node does not tansmit, and the leade poeeds with ompaison with the pevious data. If thee is no hange in the values fo all the membes they, as well as the leade do not tansmit as the goup is aleady soted. But, if thee is a hange in leade s own data, it ompaes its data with the pevious data of its thee membe nodes and boadast the soted data. The netwok is soted, if all the goups ae settled without any tansmission and eeption. C. Complexity Analysis Consideing ows and olumns we povide the following analysis of the algoithm. We use the following notations: t t is the time equied fo one tansmission, t is the time equied fo one ompaison, E t is the enegy spent duing one tansmission and E v is the enegy spent duing one eeption. 1) Step 1: It will have ( goups of nodes fomed, and ( ( goups of nodes fomed. Sine soting in this step equies time slots fo 5 loal boadasts and 6 time slots by the leade node fo 6 ompaisons, time omplexity of step 1 will be t t (d) 3) Step 3: It will have ( 1 goups of nodes fomed, and ( ( 1 goups of nodes fomed. Sine soting in this step equies 5 tansmissions and 6 ompaisons, time omplexity of this step will be t t 6t. The enegy used will be ( 1 ( [5Et 6E v ] [Et E v ] ( 1 [Et E v ]. ) Step : It will have ( goups of nodes fomed, and ( ( ( goups of nodes fomed. Sine soting in this step equies 5 tansmissions and 6 ompaisons, time omplexity of step will be t t 6t. The enegy used will be ( ( ( [5Et 6E v ] ( ( ) [Et E v ]. D. Total time and enegy omplexity Summing up the total numbe of tansmissions and ompaisons, the total time fo eah iteation will be T = (16t t t ), and the total enegy spent will be E = ( (, as we an obseve that is the lagest tem in total enegy alulation. Now, onside Fig. 3, whee we show the movement of a data element (dakened ile) fom the ight-bottom most position. If the data elements on the left and above this ae all smalle, then ove the fist iteation this data will not move. Howeve, fom the seond iteation and on eah iteation theeafte, it will move atleast by olumns left and ows upwads. This way the wost numbe of iteations will be ( 1 (. Thus, toal time equied fo exeution will be T = 1 ( ( ) ( (16t t t ). Fo the ( ase of squae gid = = n, we will have T = 1 n (16t t t ) = ( n), while en- ) ( ( = egy will be E = ( ( 1 (n n). III. GENERIC: RANDM DEPLYMENT, SINGLE-CHANNEL WSN A. Famewok In this setion, we fomulate the anking and soting poblems unde a genei setup of distibuted senso netwoks. We onside a netwok of n senso nodes distibuted andomly on a two-dimensional plane and assume that eah node has a unique identifie, eah of whih ould be taken fom an index set I. Thus, if S is the set of n senso nodes, then eah node is indexed by an element of the index set, and the set of senso

4 identifies is indiated by S I = {s i i I}. An odeing of these nodes is thus a pemutation on this set indiated as: S π(i) = { s π(i) i I, i = 1,..., n } We assume that all the nodes tansmit at the same powe P, using an omnidietional antenna and has a unifom ommuniation adius, satisfying P = P/ α, whee α 5 depending on envionmental paametes and P is the eeived powe at a distane fom the node. Fo a pai of nodes u, v S, if the Euledian distane between them u v, then thee exists a bi-dietional link between u and v. Thus, we model the senso netwok as an undieted geometi andom gaph, G = (V, E), whee the vetex set V is the set of senso nodes S and the edge set E is geneated by the ommuniation adius, suh that if u, v S, u v then { 1, u v e uv = 0, othewise, and e uv E if and only if e uv = 1. We futhe assume that by a pope seletion of P, the netwok geneated is one-onneted, i.e., between any pai of nodes thee exists a single hop o a multihop ommuniation path. We povide the following definition fo a node s neighbohood. DEFINITIN 3.1: Neighbohood: Fo a node u, we define its one-hop neighbohood as the set of nodes Su, 1 with whih u is able to ommuniate dietly in a single hop, and its twohop neighbohood as the set of nodes Su, with whih u is able to ommuniate eithe in a single hop o in two hops. Thus, Su 1 = {v e uv E, u v} ; Su = { Su 1 v S 1 u Sv 1 } Eah node has knowledge only about its one-hop neighbohood, no othe global pitue is assumed. Evey senso node obseves and measues etain physial quantity of inteest, whih ould be tempeatue, pessue, humidity, hemial ontamination o aousti enegy dessiminated by a moving vehile. Let us denote the measuement of a node u as m u. A anking poblem fo suh a netwok an now be defined as a set odeing π on S I, whih geneates S π(i) suh that if two nodes u, v S have measuements m u and m v espetively, and m u m v, then s π(u) s π(v), whee s π(u), s π(v) S I, and we all this a non-deeasing ode. Similaly, a non-ineasing odeing an be defined as an odeing that ensues s π(u) s π(v), when m u m v. We assume that anking is pefomed on a snapshot of the netwok and measuement does not hange duing the anking poess. Note that, this global odeing is geneated on the identifie set S I, and hene, hold the following popeties. 1) nly an odeing of the identifies is geneated and thus one an quey the netwok with identifie and get the soted values. ) The measuement of a node has not hanged due to anking, only an odeing is eated. Next, we define the notions of onnetivity matix, intefeene gaph, feasability matix and feasable tansmission set fo faming the soting poblem in senso netwoks and poving it to be NP-had. DEFINITIN 3.: Connetivity Matix (C): We epesent the senso netwok gaph G = (V, E) in the fom of a n n symmeti matix, alled the Connetivity Matix C = ( ij ); i, j = 1,..., n, whee n = V and { 1, eij E & i j ij = 0, othewise. The notion of intefeene gaph omes fom the hidden teminal poblem that ous beause of the boadast natue of ommuniation in senso netwoks. A node with a single antenna an eeive data fom only one of its neighbos at a patiula time instant. Hene, hidden teminal poblem aises when two nodes, whih ae not in eah othe s neighbohood, tansmit data to the same node, whih is a ommon neighbo to both of them, at the same time instant. Thus, when a node u is tansmitting to one of its neighbos v, no othe node within the two hop neighbohood (Su) of u should tansmit to v at the same time. DEFINITIN 3.3: Intefeene Gaph (G I ): The Intefeene Gaph is defined as the augmented gaph G I = (V I, E I ) on G = (V, E), suh that the set of veties V I onsists of all the senso nodes and set of edges E I onsists of all the edges in the oiginal gaph G as well as, the new edges between evey node and its two hop neighbos. Thus, V I = V ; & E I = E e uv. (u V I,v Su,v / S1 u ) DEFINITIN 3.: Simultaneous Tansmission Feasability Matix (F ): We epesent a omplimentay n n symmeti matix F = (f ij ); i, j = 1,..., n, with espet to the intefeene gaph G I, suh that n = V I and { 1, eij / E f ij = I & i j 0, othewise. Note that, f ij = f ji fo this matix and indiates that if f ij = f ji = 1, then both i and j an tansmit at the same time instant without ollision. Eah of the olumn vetos of F is alled a feasable tansmission set, denoted by τ = (t i ); i = 1,..., n, and t i = 1 signifies that node i an tansmit without ausing ollision. Eah step of exeution of a anking o soting algoithm will onsist of seveal suh feasable tansmission sets. Thus, we an expess the exeution of a anking o soting algoithm with a n p matix T, eah olumn of whih epesents a feasable tansmission set τ j, j = 1,..., p. We make the following obsevations. 1) The value of p is not neessaily polynomially bounded in the numbe of nodes, n. The optimal anking o soting algoithm will have the minimum value of p. ) Fo a node i to patiipate in the anking o soting algoithm the ondition p j=1 t ij 1, must hold.

5 numbe of iteations(1- steps) B. Complexity of the soting o anking algoithm In this setion, we analyze the omplexity of the anking o soting algoithm that has been fomulated in the pevious setion as an odeing of p feasible tansmission sets. 1) Poblem Instane: SRT-WSN: Given ae the following: A WSN G = (V, E), A funtion m : V Z, An odeing funtion π : V V, suh that i, j V, π(v) satisfies that π(i), π(j) π(v), the anks ae odeed, i.e., m i m j, A positive intege k, A N M matix T, whee eah olumn is a olumn veto t i - a feasible tansmission set. ) Question: Is thee an aangement of the feasible tansmission set vetos t i to onstut T suh that i k and π(v) is ahieved? Theoem 3.5: Soting in WSN unde the genei famewok desibed in setion III-A is NP-had. Poof: It is easy to see that the soting poblem is in NP, beause given a N M shedule it is easy to veify in polynomial time whethe this onstuts a ollision-fee shedule whose length is smalle than k, and whethe this leads to a soted aangemens of the values in the netwok. Next, it is known that the Gaph Coloability poblem (GC) is in NP and also it is NP-Complete. Thus, it follows that the oloability poblem of the intefeene gaph as defined in setion 3.3, an be Tuing eduible to fomation of one ound feasible shedule, suh that in that ound evey node gets a hane to tansmit, i.e., Howeve, M is not polynomially bounded in the numbe of inputs N fo the soting poblem and hene, fo any poblem A in NP, SRT-WSN A. That is, a polynomial time edution (Kap Redution) annot be shown to any poblem in NP fom SRT-WSN. Hene, SRT-WSN is NP-had. C. A Randomized Algoithm fo SRT-WSN Having poved that the poblem SRT-WSN is NP-had, we now desibe a andomized soting algoithm that poves the existene of a shedule of exponential length. We onside G I (n, p) as an intefeene gaph of a WSN gaph G with n nodes, whee eah intefeene edge in G I has been inluded between evey pai of nodes in G with pobability p; i.e., u, v G, P [(u, v) E I ] = p. We define eah feasible tansmission set be geneated ove [0, 1] n, as a sting R. Imagine that someone tosses a fai oin n times to geneate a feasible set and inlude node i in the set to tansmit iff the outome of the i th oin toss is 1, i.e., R i = 1. Theoem 3.6: Fo evey G I (n, p), thee exists a shedule that allows almost all nodes to tansmit one, and is e np long. Poof: We assign a andom vaiable X i with eah i th node s inlusion to a feasible tansmission set, suh that: { 1, node i an tansmit without ollision & Ri = 1 X i = 0, othewise ( n ) numbe of nodes (in 10 Expeimental 3 ) Fig. 3. Simulation esults fo n n lattie netwoks fo n = to n = 10 Wost ase movement. and X = n i=1 X i, indiating how many node feasibly get sheduled in one suh time inteval. Clealy E(X) = E( n i=1 X i) = n i=1 E(X i). Now E(X i ) = P [X i = 1] = j n,j i P [(i, j)e I R j = 1] = (p/) n 1. Thus, E(X) = n(p/) n 1 ne n(1 p/). Now, if we pefom this poess of andom feasible shedule eation fo m times and assign a andom vaiable Y = m i=1 X i, then E(Y ) = m i=1 E(X i) = mne n(1 p/). Sine P [X > E(Y )] > 0, we laim that this poess will eate a shedule whee atleast mne n(1 p/) nodes will get a hane to tansmit atleast one. IV. SIMULATIN AND RESULTS We used MATLAB to simulate the soting algoithm pesented in II. In ou simulation, we onsideed squae gids of sizes, and so on upto 10 10, whih means we simulated a netwok with maximum 0 nodes. In evey un with moe than nodes, we onduted the expeiments 1000 times and plotted the numbe of iteations in the wost ase, equied fo the netwok to get soted. We also plotted the gaph fo n and found out that the wost ase time omplexity is lose to n, negleting the onstant fatos. The gaphs ae shown in Fig. 3. V. CNCLUSINS In this pape, we poposed a simple, paallel and loalized soting algoithm fo WSN that uns in time omplexity ( n). We also fomally pesented the soting/anking poblem unde the genei setup of single-hannel andom deployment of nodes and poved that finding an optimal shedule is NP-had. Lastly, we poved the existene of a andomized soting algoithm unde the genei setup. In ou futue wok, we plan to extend the loalized algoithm fo the genei setup and to povide theoeti bounds of soting/anking poblem in WSN. REFERENCES [1] J. L. Bodim, K. Nakano, and H. Shen, Soting on single-hannel wieless senso netwoks, Intenational Jounal on Foundations of Compute Siene, vol. 1, no. 3, pp , June 003. [] D. Nassimi and S. Sahni, Bitoni sot on a mesh-onneted paallel ompute, IEEE Tansations on Computes, vol. 8, no. 1, pp. 7, Jan [3] M. Singh, V. Pasanna, J. Rolim, and C. Raghavenda, Collaboative and distibuted omputation in mesh-like wieless senso aays, in Pesonal Wieless Communiations (PWC 03), LNCS 775, Venie, Italy, Sept. 003, pp