Nearly Constant Approximation for Data Aggregation Scheduling in Wireless Sensor Networks

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1 Nearly Contant Approximation for Data Aggregation Scheduling in Wirele Senor Network Scott C.-H. Huang, Peng-Jun Wan, Chinh T. Vu, Yinghu Li and France Yao Computer Science Department, City Univerity of Hong Kong Kowloon, Hong Kong. Department of Computer Science, Illinoi Intitute of Technology Chicago, IL Department of Computer Science, Georgia State Univerity Atlanta, GA Abtract Data aggregation i a fundamental yet timeconuming tak in wirele enor network. We focu on the latency part of data aggregation. Previouly, the data aggregation algorithm of leat latency [] ha a latency bound of ( )R, where i the maximum degree and R i the network radiu. Since both and R could be of the ame order of the network ize, thi algorithm can till have a rather high latency. In thi paper, we deigned an algorithm baed on maximal independent et which ha an latency bound of R + 8. Here contribute to an additive factor intead of a multiplicative one; thu our algorithm i nearly contant approximation and it ha a ignificantly le latency bound than earlier algorithm epecially when i large. I. INTRODUCTION A wirele enor network conit of a elf-configuring network of enor equipped with RF tranceiver and thu connected by wirele link. There i no infratructure in uch network and node are to organize themelve arbitrarily. Thee feature make them uitable for a wide variety of application like emergency medical ituation, battlefield urveillance, traffic monitoring. In thee application, quite often we need to gather data from thoe enor to a fixed ink and proce them. If the data we gathered can be merged uch a taking the maximum or minimum of them, we call thi type of application data aggregation. Otherwie, we call it data collection. In thi paper, we focu on data aggregation and try to reduce it latency by contructing a good chedule. Thi i a challenging iue [] [], ince ingling out the interference iue and doing it at the MAC layer reult in a latency too high to be practical. There i a mut to do cheduling while dealing with it at the ame time. Currently, the tate-of-the-art cheduling algorithm ha a latency bound of ( )R, where i the maximum degree and R i the network radiu. Since both and R could be of the ame order of the network ize, Thi work wa upported in part by the Reearch Grant Council of Hong Kong under Project Number CityU 65/0E and CityU 05 and by the National Science Foundation of the US under Grant Number and CCF thi algorithm can till have a rather high latency. In thi paper, we deigned an algorithm baed on maximal independent et which ha an latency bound of R + 8. Here contribute to an additive factor intead of a multiplicative one; thu our algorithm i nearly contant approximation and it ha a ignificantly le latency bound than earlier algorithm epecially when i large. The ret of thi paper i organized a follow. In II we preent related work. Problem formulation i preented in III. We preent our main data aggregation algorithm and analyze it in IV. Concluion and future work are preented in VI. II. RELATED WORK In wirele enor network, broadcat and data aggregation are the mot fundamental and ueful operation. Broadcat algorithm have been tudied extenively in the literature ince the 80 [],[],[5], [6],[7], [8],[9], [0],[],[], [],[]. Data aggregation, by comparion, i till relatively new but it importance cannot be overemphaized. Data aggregation, ometime called convergecat i about a enor network with a bae tation uch that all (or ome) node collect data and report to the bae tation via wirele communication. Annamalai et al [] deigned a heuritic algorithm for both broadcat and convergecat. The convergecat tree contructed in their algorithm can be ued for broadcat a well. For convergecat alone, Upadhyayula et al [5] deigned another heuritic algorithm aiming at reducing energy and latency. Thee two work mentioned above all ued heuritic approache and ran imulation to verify their reult without doing theoretical analyi. On the theoretical ide, Keelman and Kowalki [6] deigned a randomized, ditributed algorithm that ha latency O(log n). In their model, they aume each node can vary it tranmiion range to reduce link. Chen et al [] deigned a ( )-approximation algorithm for data aggregation, where i the maximum degree of the network graph. They alo proved that the minimum data aggregation time problem i NP-hard.

2 Practical iue of data aggregation, epecially about the MAC layer, have alo been tudied in the literature. Huang and Zhang [7] tudied packet lo and focued on reliability iue in data aggregation. Zhang et al [8] addreed the iue of burty convergecat in real-time application and focued on improving channel utilization and reducing retranmiionincurred channel contention. Krihnamachari et al [9] viewed data aggregation from another apect. They conidered the cae where there i a ubet of node whoe data need to be ent to the bae tation and regard aggregating thee data a a way to ave energy. Intanagonwiwat et al,inahort paper [0], evaluated the impact greedy aggregation to increae the amount of path haring and reduce energy conumption. Yu et al [] alo conidered cheduling packet tranmiion and energy-latency tradeoff. Their goal wa to reduce enor node energy diipation ubject to ome latency contraint. III. PROBLEM FORMULATION We conider a network coniting of n enor node along with one bae tation. Each (enor) node i equipped with an RF tranceiver that can be ued to end or receive data. We conider omni-directional antennae only, and a node tranmiion/reception range i roughly a dik centered at that node. For implicity, we further aume that all node have the ame tranmiion range, and thi type of network can be repreented a a dik graph a follow. Let G be a graph repreenting a network of n node. An arc (or directed edge) exit from u to v if and only if v lie in u tranmiion area (which i a dik). If all node have the ame tranmiion range, then we can normalize their radiu to and ue a unit dik graph to model it. In thi cae, an edge exit between u, v if and only if the ditance between them i le than. For implicity, we only conider unit dik graph. A node can either end or receive data at one time, and it can receive data correctly only if exactly one of it neighbor i tranmitting at that moment. If two or more node are tranmitting imultaneouly and there i a node in their overlapped tranmiion area, then thi node cannot receive the meage clearly ince both tranmiion are interfering with each other. Thi type of ituation i called colliion. The main tak of enor node i to collect data and tranmit back to the bae tation, and data can be aggregated all the way to the bae tation. In other word, if a node ha received one packet from it neighbor before it cheduled tranmiion time, then it can merge thi packet with it own data packet and imply end thi merged packet later. Thi model i particularly ueful in ituation like making a query about the maximum temperature, in which two (or more) piece of data can be merged by taking their maximum. Another ituation where packet cannot be merged doe exit, and it i called data collection. In thi paper we only focu on data aggregation meaning that data can be merged all the way to the bae tation. Given a unit dik graph G = (V,E) along with a bae tation b V. Conider two ubet A, B V with A B. We ay that data are aggregated from A to B by one time lot if all node in A B tranmit in one lot imultaneouly then the data at all node of A B will be received colliion-free by ome node in B. A data aggregation chedule can be thought of a a equence of ender {S,S,...} (in which S i V, i) atifying the data aggregation property. Thi equence repreent that all node in S tranmit in the firt time lot, followed by all node in S tranmitting in the econd time lot and o on o forth. The data aggregation property imply mean that after S tranmit data will be aggregated from V to V S and after S tranmit data will be further aggregated from V S to V S S. If we continue thi proce, finally all data will be aggregated to one ingle node b, which i the bae tation. Without lo of generality we may aume each node only tranmit once, ince multiple tranmiion doe not reult in any advantage for the following reaon. If, in a chedule, a node tranmit more than once, then the data at thi node will be aggregated many time to ome other node. Since a ingle piece of thi information alone i enough a long a it can eventually get to the bae tation colliion-free, we can alway modify the chedule by properly removing multiple tranmiion while preerving the ame latency (number of time lot that all data are aggregated to b). The property of ingle tranmiion i eentially equivalent to all S i being dijoint. The formal definition of the data aggregation property i a follow. A data aggregation chedule i a equence of ender {S,S,...} atifying the following condition. ) S i S j =, i j ) l i= S i = V {b} ) Data are aggregated from k i= S i to k+ i= S i for all k =,,,...,l and finally data are aggregated from S S l to b in time lot l. The number l i called the data aggregation latency. Now,the minimum data aggregation time problem can be formulated a follow. Given a graph G =(V,E) and a bae tation b V, find a data aggregation chedule with minimum latency. Thi problem i proven to be NP-hard even for unit dik graph []. So far, there i no good approximation algorithm for thi problem, ince the bet known algorithm ha ratio (where i the maximum degree and it can be a large a the number of node in the network). In the next ection, we will preent an approximation algorithm for thi problem that ha latency R + 8, where R i the network radiu. Since the farthet node ha to tranmit back to the bae tation, data aggregation latency cannot be le than the network radiu. If R i large, our algorithm ha approximation ratio 7 from an aymptotic point of view. IV. OUR DATA AGGREGATION SCHEDULING ALGORITHM We preent our approximation algorithm in thi ection. Our algorithm ha a data aggregation latency R + 8 where R i the network radiu and i the maximum degree of the network. Thi reult i ignificantly better than the currently

3 bet known algorithm [] whoe latency i ( )R, ince become an additive term intead of multiplicative. If i large, our algorithm achieve a ignificantly horter latency. Scenario with large happen frequently in large-cale, dene network in real life. Our algorithm ha two phae. () Data Aggregation Tree Contruction () Data Aggregation Scheduling. Their detail will be preented in next two ubection. A. Data Aggregation Tree Contruction In thi phae, we firt contruct a breadth firt earch tree for the network and divide all node into layer (where the 0-th layer i the bae tation and the t layer i it neighbor). We then form a maximal independent et BLACK layer by layer (Algorithm ) a follow. Starting from the t layer, we pick up a maximal independent et and mark thee node. We then move on to the nd layer and pick up a maximal independent et and mark thee node again. Note that the node of the nd layer alo need to be independent of thoe of the t layer. We repeat thi proce until all layer have been worked on. Thoe who are not marked are marked white at lat. The peudocode of layered MIS contruction i given in Algorithm. Algorithm Contruct an MIS layer by layer : Divide all node into layer L,L,...,L l : BLACK : for i to l do : Find an MIS BLACK i L i indep. of BLACK 5: BLACK BLACK BLACK i 6: end for 7: return BLACK To contruct the data aggregation tree, we then pick ome of the white node and color them blue to interconnect all node a follow. To connect the t and nd layer, we look at the nd layer node. Each node mut have a parent on the t layer and thi parent node mut be white ince node are independent of each other. We add thi white parent for each node on the nd layer and alo add an edge between them. Thee added white node are colored blue. Moreover, we know that thi blue node mut be dominated by a node either on the t layer or the 0-th layer. We then add an edge between thi blue node and it dominator. We repeat thi proce layer by layer and finally obtain the deired data aggregation tree in thi manner. Note that, in thi tree, each node ha a blue parent at the upper layer and each blue node ha a parent at the ame layer or the layer right above. The peudocode i given in Algorithm. B. Data Aggregation Scheduling The cheduling for data aggregation i quite traightforward. We ue firt-fit cheduling (Algorithm ) throughout thi part. The firt-fit procedure take in an input of a ender et S, a receiver et R, the network graph G, and the data aggregation Algorithm Data aggregation tree contruction : T =(V T,E T ), V T = V, E T : /* Connect node layer by layer */ : for i to l do : for all node v BLACK i+ do 5: Find it parent p(v) in BFS tree 6: Color p(v) blue 7: Find p(v) dominator d p(v) in BLACK i BLACK i 8: Add an edge between p(v),v to E T 9: Add an edge between d p(v),p(v) to E T 0: end for : end for : /* Connect remaining white node */ : for all remaining white node u do : Find u dominator d u 5: Add an edge between u, d u to E T 6: end for 7: return T tree T. The ender are cheduled a follow. Firt we pick a node from S and look at whether it conflict with any node or not. Obviouly ince it i the only node, it doe not conflict with any node. We then add to a temporary et X and remove from S. Now we pick up another node from S and look at whether it conflict with any node in X or not. We look at the parent of all node in X in T. Since there i only one node in X, we only look at the parent of in T, denoted by p T ( ), and ee whether i adjacent to p T ( ) in G. If ye, we do nothing. If not, doe not conflict with X and we add to X and remove from S. We repeat thi proce until we find the larget et X uch that all other node in S will conflict at leat one node in X. ThiX i the maximal poible et that we can chedule them to tranmit for the ame time lot, o we add X to the chedule. Now we initialize X to and repeat thi proce to find the maximal poible et to tranmit imultaneouly for the econd time lot, and o on o forth until all element in S are cheduled thi way. The peudocode for thi i given in Algorithm. Now we have all white node tranmit to node uing the firt-fit ubroutine decribed above. Then, from the lat layer S l we aggregate data layer by layer a follow. The node at layer l, denoted by BLACK l,firttranmitto their parent blue node BLUE l, then BLUE l tranmit to their parent node p T (BLUE l ) in T. Note that p T (BLUE l ) mut be a et of node in BLACK l BLACK l becaue the node were elected layer by layer. Thee tranmiion are all cheduled uing the firt-fit ubroutine to avoid colliion and we repeat thi proce layer by layer. Data will therefore be aggregated layer by layer and going all the way to the bae tation. The peudocode for thi i given in Algorithm.

4 non- 0 5 Fig.. (a) Topology of G (b) Black node election (number repreent hop-ditance from ) non- 0 5 blue white 0 5 Fig.. (a) BFS tree of G (b) Blue node election Algorithm Firt-fit cheduling Input: Sender et S, Receiver et R, network G =(V,E), data aggregation tree T Output: Schedule W, a collection of ubet in V : procedure FIRSTFIT(S, R, G, T ) : X : W { } : while S do 5: for all u S do 6: if x X, ( p T (x),u ) / E then 7: Add u to X and remove u from S 8: end if 9: end for 0: Add X to W : end while : return W : end procedure p T (x) = parent of x in T Algorithm Data aggregation cheduling : Contruct an MIS layer by layer Algorithm : Contruct the data aggregation tree T Algorithm : FIRSTFIT(WHITE,BLACK,G,T) : for i l ( to do 5: FIRSTFIT BLACKi, BLUE i,g,t ) ( 6: FIRSTFIT BLUEi, p T (BLUE i ),G,T ) 7: end for C. An Example Figure (a) how the topology of G, and figure (b) how the election of layered MIS (i.e. node) a decribed in algorithm. In the firt tep, the ource i elected in the MIS and colored. In the econd tep, ince the ource i, all node at layer mut all be white, otherwie it won t be independent of. In the third tep, we will elect an independent et at layer, which mut alo be independent of the node at the previou layer, namely layer, although there i no node at layer and thi retriction doe not have any effect on electing independent et at layer. In figure (b), we how the hop-ditance of each node to. Note that node were elected at layer, node wa elected at layer, and node were elected at layer. We keep doing thi until all layer have been worked on. The node election merely depend on the topology of G and it ha nothing to do with the BFS tree. Not until we elect blue node do we need to conider the BFS tree. Figure (a) how the BFS tree of G a well a the node. Now, we try to add appropriate blue node to interconnect thoe one, a decribed in algorithm. Since the ource doe not have an upper layer and there are no node at layer, we tart from layer directly. For each node v at layer (there are of them), we find it parent p(v) in the BFS tree, color it blue, and connect v to p(v ) a hown in figure (b). Here we ee node at layer are colored blue and connected to their children at layer. Thoe which are not colored blue remain white, and there i uch node. We alo connect thee blue node and the white node to the ource ince they are dominated by. We keep working on layer, find node at thi layer, and color

5 d w blue white w c d Fig.. w6 c c5 d c d6 Data aggregation tree it parent blue. Note that thi node ha to be independent of all node at above layer (i.e. layer 0,,) a well. We then connect the blue node at layer to it dominator, a hown in Figure. We repeat thi proce for every layer until all node are connected and the data aggregation tree i fully built, a hown in figure. To chedule the data aggregation tranmiion, we follow algorithm. Firt we chedule the white node. We arbitrary pick a node, ay w, and chedule it tranmiion for the mot available time lot, which i lot #. We then pick w and find that we cannot chedule it tranmiion in lot # ince w i already cheduled to tranmit in thi lot which will interfere w tranmiion. Then we move to w and find we can chedule it for # a well. However, when we look at w we find we cannot chedule it for # becaue of w. We keep doing thi until we examine all white node and find that w,w,w5,w6 can be cheduled concurrently for time lot #. Now we chedule the remaining white node w and w uing the ame method, and we find that, imilarly, they can be cheduled concurrently for #. Now we chedule the tranmiion from BLACK to BLUE. We find the there are only two node in BLACK, namely d and d5, and we ue the firt-fit algorithm to chedule them. We find that they can both be cheduled for #. Then we chedule BLUE to p T (BLUE ) and chedule c and c for #. Note that p T (BLUE ) could be at layer or. We ue thi method layer by layer until all node have been cheduled thi way. D. Analyi Here we how that it take R + 8 to aggregate data to the bae tation. We look at the abovementioned part eparately. () WHITE to BLACK: Thi part only ha to be done once for the whole chedule and it ha nothing to do with layer. We claim that it take at mot time lot to finih thi part. Thi can be proved in a traightforward manner. Note that each node ha at mot neighbor. Since node are mutually independent, node cannot be neighbor of each other. Thu, thee (or le) neighbor mut be either white or blue. However, each node mut have one blue parent, o there mut be at leat one blue node among thee neighbor. A a reult, a node can have at mot d c w w d5 w5 white neighbor and the firt-fit cheduling i guaranteed to finih within time lot. () BLACK to BLUE: Thi part and the next one have to be done repeatedly. When data are aggregated to the previou layer thi part ha to repeat. We claim that it take at mot time lot to finih the tranmiion. We pick up a node. Thi node ha to tranmit to it blue parent. Now we look at thi blue parent node and oberve that it ha at mot 5 neighbor. Thi follow from the fact that there are at mot 5 independent point in a unit dik. Among thi 5 or le neighbor, one of them mut be thi blue node parent. Therefore, there can be at mot node competing to tranmit with repect to thi blue node and it take at mot time lot to finih the tranmiion. () BLUE to BLACK: Thi part ha to be done layer by layer a decribed above. We claim that it take at mot 9 time lot to finih the tranmiion except on the econd layer tranmiion to the firt, which need 0 time lot. Similarly, picking up a blue node, we need to etimate how many other blue node are competing to tranmit to the ame parent. Fix thi parent node u and conider the unit dic D centered at it, and we count how many blue node lie inide thi dik. Note that each blue node mut have at leat a child becaue blue node are elected to interconnect node. For thi reaon the number of blue node lying inide thi unit dik cannot exceed the number of node lying inide D, the dik of radiu centered at u. According to Lemma. in the appendix, there can be at mot independent point in a dik of radiu. In other word, there can be at mot node in a dik or radiu centered at every node, and each node u can have at mot 0 -hop neighbor (ince we can exclude u itelf). Moreover, we are going to how that we can ubtract one more node except for the econd layer. The reaon i that among thee point in D one of them mut be node u parent parent (which mut alo be and lie inide D ). Therefore, for all other layer, 9 time lot are enough. Now we etimate the data aggregation latency. WHITEto-BLACK need time lot. From layer R to layer, BLACK-to-BLUE and BLUE-to-BLACK together need (9 + ) (R ) time lot. Tranmiion from layer to layer need 0+ time lot and tranmiion from layer to the bae tation need 5 time lot (ince there are at mot 5 independent point in a unit dik). Altogether, it take at mot + (R ) = R + 8 time lot, a deired. V. SIMULATION RESULTS For the imulation part, we randomly deploy enor into a fixed region of ize 00m 00m. All enor have the ame tranmiion range. We compare our algorithm with Chen et al []. All comparion are fairly conducted on the ame graph, and on each graph data are aggregated from the ame et of node to the ame bae tation. Throughout thi imulation, we alway ue number 0 a the root node and all leave of the

6 80 70 Aggregation time when the tramiion range fixed to 0 Our algorithm Chen et al Time when the number of node i 00 Our algorihm Chen et al Time Number of node Time Tranmiion range Fig.. (a) (b) BFS tree rooted at node 0 a the data-toring node, i.e., node that we aggregate data from. Figure (a) and (b) repreent the fluctuation in thoe graph. For each configuration (ame number of node, ame tranmiion range), we do comparion with 00 random graph. In Figure (a), the tranmiion range of each enor i fixed to 0m. We meaure the number of time lot needed to aggregate data from the leave to the root when the number of node varie from 50 to 00. In [], the data aggregation latency (vertical axi: time) i baically proportional to the number of node, while in our algorithm it i not influenced much by the number of node. The pattern of thoe curve matched with our theoretically etimated latency bound. The bigger the number of node, the better the improvement of our algorithm in comparion with Chen. Our algorithm latencie are from.0 to.87 time maller than that in []. In Figure (b), we do it with the graph of 00 node but the tranmiion range varie from 5m to 50m. Similar to Figure (a), the curve repreenting time generated by Chen algorithm i nearly linear, while the one generate by algorithm i o cloe to a horizontal line. The pattern of thoe curve illutrate the theoretical analyi in IV-D. VI. CONCLUSION AND FUTURE WORK In thi paper, we invetigated the data aggregation problem and conidered it latency. We ued the technique of maximal independent et and deigned an algorithm of latency R + 8. For future work, we decribe our hort-term, mediumterm, and long-term goal a follow. Our hort-term goal i applying our technique to directional antennae. We believe that mot technique we developed here can be applied to the cae of directional antennae immediately by imply re-invetigating the geometrical property of it. Alo, we will conider more general model than UDG. For example, ditinguihing the tranmiion range from the interference range and uing two dik or other geometrical object to repreent them. Alo, we ll invetigate the data aggregation cheduling in Signal-to-Noie model. Our medium-term goal i to find a better lower bound for optimal olution. In analyzing the approximation ratio, we actually ued R a the lower bound. Thi i a very looe etimation and according to our tet cae it i ignificantly larger than R, in mot cae at leat the clique number ω. However, we did find a counterexample uch that the optimal latency i O(log ω) and we imply could not find anything better o far. In the future work, we think it i likely to find a tighter lower bound for optimal latency, and the analyi part of thi algorithm may be improved thi way. Our long-term goal i data collection cheduling. We tudied data aggregation firt becaue it ha a imple model. In data collection, ince data cannot be merged, node may need to tranmit multiple time. Thi will eentially make our cheduling impractical. However, we believe that the MIS or ome imilar technique may till be applied. APPENDIX We tate the following theorem from [] and we ll ue it to prove Lemma. later. Theorem. (Wegner Theorem): The area of the convex hull of any n non-overlapping unit-radiu circular dik i at leat (n ) + ( ) n + π. Lemma.: There can be at mot node within any dik of radiu two. (Proof.) FixadikD centered at a point u. LetS denote the et of node in D. Since node are mutually independent. If, for each node in S, we conider a dik of radiu / centered at thi node, then all of thoe dik mut be dijoint. Therefore, the convex hull of S mut be contained in the dik of radiu.5 centered at u. By applying Wegner Theorem with proper caling, we have ( ( S ) + ) S + π<5π.

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