A Partial Sorting Algorithm in Multi-Hop Wireless Sensor Networks

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1 A Partial Sorting Algorithm in Multi-Hop Wireless Sensor Networks Abouberine Ould Cheikhna Department of Computer Siene University of Piardie Jules Verne Amiens Frane Jean Frédéri Myoupo Department of Computer Siene University of Piardie Jules Verne Amiens Frane Abstrat The partial sorting problem is to sort the k smallest elements of a given set of n integers suh that 1 <k <n in desending order. The algorithms that exist in the literature for the partial sorting in wireless networking solutions are based on single-hop model. In this paper we onsider a multi_hop sensor network. Initially we partition the network into several levels by using the method of Gerla and Tsai. We onsider that eah node has a data item and two linked lists. One the network is partitioned with multiple levels eah luster runs the algorithm MaxCluster seeking the maximum element in that luster. Then the algorithm Partial_Sorting finds separators (separators are identified by the algorithm MaxCluster) and helps to identify the remaining elements of the list to be sorted. We give the upper bound of our algorithm in terms of broadast rounds. We also extend our approah to the ase where eah node has multiple data items instead of a single datum. Finally Experimental results highlight our approah Keywords- Sensor Networks; Cluster head; Clustering; Partial Sort; Wireless ommuniation. I. INTRODUCTION The sorting problem has been extensively studied for deades. They proposed different tehniques to solve this problem. Most of these tehniques fous on how to sort a set of n elements as quikly as possible. Hoare suggested in the early 60's quiksort. Until reently Martinz Shiau and Yang have studied how to sort the k smallest elements of a given set of n elements as 1 <k <n. MARTINZ showed that this problem is known as a partial ordering. Today due to development of personal networks the wireless ommuniation (WN) has beome more interesting and attrative. Single-hop model is a omplete graph and onsists of n stations where eah station listens to all the others. Eah station in this model an ommuniate with eah other via a ommon hannel only. If two or more stations want to send messages simultaneously sending a onflit ours. In this model we assume that eah station has the ability to detet the onflit. When a onflit is deteted the pattern of resolution being put in plae. A. Related work Quiksort is one of the most influening algorithms in sorting. It has been studied and hosen to appear among the best ten algorithms in the 20 th entury [7]. The average number of omparisons for the quik sorting is O(nlogn+ O(n))[7]. One of the variants of quik sorting alled median-of-three runs in (12/7nlogn+ O(n)) [ ]. The amelioration of 2n to 12/7n is proved by Chern and Hwang [3] who disussed the generalized quiksort. Martinez [9] has reently introdued partial sorting. The problem is not the sorting of n elements but rather seleting the smallest k elements among n elements 1 < k < n. Martinez [9]. Now the question is how to resolve the partial sorting problem when 1 < k <n. Martinez [9] has proposed an algorithm alled partial quiksort to solve the partial sorting. Shiau and Yang [11] also proposed another algorithm generalization of partial sorting for resolving the same problem. Both two algorithms were based on Quiksort and some of their results are almost the same. The main differene between the two papers is the original funtion and the methods of analysis they use. The method in [7] onsists of seleting the j -th element of a file ontaining n elements. Prondinger [11] derives an algorithm to selet the k elements j 1 -th j 2 - th j k -th. It may also resolve the partial sorting problem if we let k seleted elements be the k smaller elements. Martinez [9] mentioned that his algorithm an resolve the partial sorting problem but with two onstraints. The first onstraint is that k must be large enough. The author does not mention what value of k is large enough. The seond onstraint is that the n elements must be treated offline. Furthermore Sedgewik and Flajolet [13] show that the standard deviation the

2 number of omparisons used by Quiksort is 0.6n. This means that the auray of performane predition is not very good and need to be further disussed. The first onstraint means that some algorithm better results than the algorithm of Martinez [9] when k is not big enough. The extreme ase is when k = 1. Of Clearly resolution of this ase the smallest element seleted by the algorithm is the best and it will take (n- 1) omparisons. The algorithm of Martinez [9] needs 2H n 2n omparisons where H n is harmoni number. The two algorithms have less seletion good results of the algorithm when k = 1. How quikly an we sort the two smaller elements of set n elements? A simple method is by exeuting the algorithm that allows the smallest elements fall twie. The seond iteration will hoose the seond smaller element starting from (n-1) of remaining elements. The time of exeution of the method of seletion is n to selet the smallest n elements in a set of n elements. Generally this method for researhing the k smaller elements among n elements needs 1 [( n 1) + ( n k )] k 2 omparisons. It proves that partial sorting loses its effiieny with respet to Partial Quiksort when k>=3. One an think that the algorithm of Martinez [7] and the algorithm of Shiau and Yang [11] may take leadership position for k>=3 on the traditional model. It is not the ase: Shyau-Horng Shiau has proposed an algorithm for resolving the partial sorting problem online. The algorithm outperforms both algorithms and assumes the position of leader k =1 with 3 k = n omparisons. The algorithm is an 5 adaptation of sorting insertion and an be treated online. Some sorting algorithms for single hop wireless networks appeared in the literature [ ]. In [10] a wireless network on p station is onsidered to sort n data items using k hannels k<p. Their algorithm runs in n p O ( n ) broadast rounds provided that k k p k. 2 Shiau et al [14] proposed an algorithm for partial sorting in single up network with the number of slot time bounded by 23 6 where k is the number of hannels. k + 5 ln( n ( k 1 )) B. Main ontribution This paper proposes an algorithm for solving the partial sorting problem (sorting k the smallest elements in a set of n elements) in wireless multi-hop sensor networks. Initially we onsider that eah node in the network has one and only one data. One we sueed in solving the 7 6 problem we proposed a generalization of our approah to solve the partial sorting problem in the ase where a node an have one or more data items. Our approah does not need to have omplex nodes: eah node is apable of turning a sorting algorithm in order to loally selet the largest value in the list it holds. More preisely we show the number of broadast rounds for a partial sorting problem annot exeed O( n) + S * Pr * max * max log( max min where n is the number of sensors in the network S is the st of separators Pr is the number of layers obtained after the lustering proess C max is the luster of maximum size among all lusters of the different layers. C min the luster of minimum size among all lusters of the different layers The rest of paper is organized as follows: the following Setion presents our approah in the ase where eah node has a single data. Setion 3 is a detailed desription for the generalization for our approah for the ase when one node has multiple data s. Setion 4 presents a simulation for our approahes and finally a onlusion to our paper. II. PARTIAL SORITNG ALGORITHM ON THE MODEL MULTI-HOP After the network lustering the method onsists for eah station to send the value it ontains to the lusterhead whih is its leader. One the lusterhead reeives all messages it seeks the maximum among all the reeived elements. We will use the notation < x 0 x 1 xt > to denote a linked list where x t is the head of the list. Let us suppose L 1 = < x 0 x 1 xt > and L 2 = < y 0 y 1 y t > the notation < L 1 L 2 > represents onatenation of L 1 and L 2 whih is < x 0...x t y 0 y t >. In the desription of our algorithm we also use the term data elements that represents the data given by the station. A. Clustering algorithm We use the algorithm Gerla &Tsai [5] for the hierarhial lustering of our network. This algorithm onsists in finding a set of interonneted lusters. More preisely the topology of the system is separated into sattered partitions. One the network is partitioned we repeat the proess until we find a single luster that is named the Super Clusterhead. The deision is based on data held by the nodes: we onsider that the node holds

3 the smallest value and prioritize the most adapted for this task to be lusterhead. An interesting point in this algorithm is that the luster head nodes and the ordinary nodes all do the same task when working in the onstrution of luster. Thus they spend as muh energy as eah other. The omplexity of this algorithm is O( V whih V all nodes of the graph represent the network. B. Assumptions Let M be a set of data items assoiated with a network whih onsist of a group of sensors that an ommuniate with eah other via a ommuniation hannel. M is also assoiated with a graph where verties are the sensors. In this direted graph there is an ar from u to v if u an send a message toward v. We assume also that the n nodes are deployed; eah nod has a data and two linked lists. We onsider the following notations: - M: set of data assoiated with a sensor network; - UC ij : The tree partitioning of the network M using the algorithm of Gerla and Tsai; - P r : The depth of tree lustering; - Node v: Eah node in progressive L' g and L g with L' g ontains the values hold by their neighbors in their luster L g list that ontains the most great value in the luster - C ij : The luster j in layer i; - C ij : The size of the luster j in layer i (number of nodes); - x ij k : Evaluate the hold by the node k in luster j layer i; - x ij : Evaluate the hold by the lusterhead of the luster j in layer i; - k : Represents respetively lusterheads and ordinary nodes; - L : The hained list that detains the largest values of the lusterhead of node luster j in layer i; - L g = k: Chained list named list of separators whih ontains the largest values of eah luster that is part of the tree portioning. - L m : Chained list named list of separators whih ontains the largest values of eah luster that is part of the tree partitioning. III. DESCRIPTION OF OUR ALGORITHM A. Algorithm MaxCluster i.- Step 1: Multi-level lustering We use Gerla & Tsai lustering algorithm to yield a multi-layer luster organization: UC ij with C ij in the luster j in layer i. As mentioned above it is obvious that eah luster has a leader named lusterhead. ii.- Step 2: Finding the greatest data item of eah luster. The algorithm MaxCluster is a method of finding the maximum: it onsists for an ordinary node in a luster to send its value to its luster head. One it reeives all data items finds the greatest data item among all those it reeived. The luster heads work in parallel in this step. Step 3: Conatenation method of the data in the hained list of luster head of the luster using the algorithm MaxCluster that is presented in algorithm 1 below. Algorithm 1: MaxCluster 1. Input: C ij where C ij is luster j in layer i. 2. Output: Chained list L g the list of separators 3. L k ij hained list of node k of luster j in layer i x max =NULL L g = L p =NULL P r the depth of the tree 4. For i=1 Until P r { 6. For j=1 Until q i where q i is the number of luster in layer i. 7. { 8. For l=1 Until C ij 9. { 10. eah sensor in C ij sends its value to its luster head l 11. L =< L x ij > 12. If x l ij > xmax 13. l x max : = x ij 14. } 15. L : = Quiksort( L ) 16. ' ' L g : =< ( L g xmax > 17. } 18. } 19. ' L g : = Quiksort( L g ) 20. Return L ' g

4 B. Partial soritng algorithm For distint n elements the partial sort problem is to find the first k k 1 the most k greatest elements in the non-inreasing order. Our partial sorting algorithm may be represented as a funtion of reursive researh: ( L g )= Sort-partial (M k) where L g is the hained list that stores the sorting sequene. is the number of elements in L g and M represents a set of data items (sensors network). In our algorithm eah sensor has exatly one data item and maintains two hained lists as mention above. Finally we will get a hained sorting list L g ontaining the first greater k elements. For any set of data items there exists at least one sensor network that is its representation For our partial sorting algorithm we use the algorithm MaxCluster that is a method of researhing the maximum. A set of elements are onsidered as separators to bring the data items remaining in the layers properly. Then the lists of separators will be deomposed in the form of intervals. The reursive funtion is based on these intervals in order to identify the remaining elements whih will be in the sorted list. The algorithm an be represented as a reursive funtion of researh framed through interval. Our algorithm of partial sort is depited in the algorithm 2 Algorithm 2: Partial Sort (L g )=Partial-Sort(M K) 1: Input: M the set of items to sort (sensor network) 2: Output: hained list L g that ontains the k greatest Values 3: Step 1: Hierarhial lustering of the network 4: Step 2: Algorithm MaxCluster 5: Step 5: L m =MaxCluster L m =<x 1 x 2 x t-1 xt > with x i-1 x i 1 i t. 6: Set L g =NULL 7: For i=0 Until L g1( 1) 8: { 9: L g =< xt Lg > 10: L g 11: Return 12: Else 13: { 14: If ( x < L [ i] < x ) pr t 1 g1( p 1) t r 15: L =< L [ i] L > m g1 ( pr 1) 16: } 17: } 18: Return ( L g ) m Theorem: The number of broadast rounds required by our partial sorting algorithm in a multi hop Wireless sensors networks annot exeed O( n) + S * Pr * max * max log( max min Where n is the number of sensors in the network S is the set of separators Pr is the number of layers obtained after the lustering proess. C max is the luster of maximum size among all lusters of the different layers. C min is the luster of minimum size among all lusters of the different layers. Proof: It is well known that the omplexity Gerla and Tsai [5] lustering algorithm in terms of broadast rounds is O (V ) V is the set of sensors of the original network. Sine we set n as the number of sensors in the network we have learly O ( V ) = n (1) In our partial sorting algorithm we have the following symbols: Pr :The number of layer in the lustering proess ; C : The size of luster i in layer j ; ij Also note that our partial sorting algorithm uses the quik sort tehnique in the algorithm MAxCluster to identify the greatest element in eah luster. The lusters in all layers run MaxCluster in parallel. Thus the omplexity of this algorithm is a funtion of luster size and is given by the following formula: ij log( ) max ij max for the lusters in layer j. Note C max the luster of maximum size among all lusters of the different layers. Thus the omplexity of MaxCluster is dominated by max log( max We use the following strategy to derive the number of broadast rounds of our partial sorting algorithm : ij max : The size of the luster that ontains the largest number of elements from all the lusters in layer j ; : The size of the luster that ontains the ij min smallest number of elements from all the lusters in layer j; : The largest number of lusters in layer j ; ij min One an easily dedue that the largest number of rounds in the layer j is: ij max * ij max log( ij max ) ij min

5 Note C min the luster of minimum size among all lusters of the different layers. Thus the largest number of broadast rounds in any layer is dominated by max * max log( max min Sine P r is the number of layers after lustering we an easily dedue that the total number of broadast rounds after lustering annot exeed P r max * max log( max (2) min Our partial sorting approah uses a list of separator for the onstrution of intervals of searh: let S be the set of separators and S the number elements in this set. Aording to (1) and (2) we an onlude that the number of rounds of our sorting partial algorithm annot exeed: O( v) + S * Pr * max * max log( max min C. An example The input file ontains 20 elements whih are { }. Let us suppose we want to sort the largest first 9 elements. Let us also onsider the example of topology shown in Figure 1 for a good understanding of the onstrution method. Subsequent to partitioning to several level we obtain Figure 2 where we find a tree omposed of a set of lustering that are: {1 0 2} { } { } { } { } { } { } { } {1 2 5} the index represents the layer in the tree partition. From here it is easy to set up that any node at the end of the algorithm finally got to take but single luster. Indeed the identifier of the luster to whih the node is reattahed is either holds the value of the node itself or the greatest value holds by its neighbors. It is also important to note that with this algorithm even in a luster two nodes are at most at a distane of 2 hops from eah other. For this it is enough to onsider several nodes of the same luster. Eah node must be able to diretly reah the Clusterhead of his luster. Thus two nodes of the same luster must be at distane of at most 2 from one of the other. Fig. 1. Network that represent the data set. As already mentioned above our algorithm proeeds through the hierarhial partitioning of the network whih orresponds to the whole data elements. The following figure shows the results of the hierarhial lustering applied to the network whih represents the whole data s. Fig. 2. The hierarhial tree orresponds to the data set. For example the input file ontains 20 elements whih are { }. Let us suppose we want to sort the first 9 greatest elements. The whole proess is shown in Fig. 3. After the exeution of the algorithm MaxCluster we find the greater elements whih are 20 and the list separator omposed is as follows: (( ). One have the separator list we an easily onstrut intervals for the rest of elements to be sorted. These intervals are: [17 20] [13 17] [11 13] [9 11] and [2 9]. The algorithm of partial sorting is alled in a reursive manner with eah of these preedent intervals. For the first interval we find the two elements that will be injeted into the set ontaining the sorted sequene. The partial sorting algorithm stops one the size of the whole stuff L g = k.

6 Fig. 3. An example of partial sorting algorithm with 20 elements. IV. GENERALIZATION OF OUR PARTIAL SORTING ALGORITHM ON THE MODEL MULTI-HOP In this setion we genralize our algorithm the general soritng problem on the multi-hop. For this we onsider that the sensors the network an ontain multiple data items. These data items must be sorted in a linked list. So we end up with a system whih is omposed of p sensors in whih the a entire sequene of n elements suh that p < n is stored. For this generalization we model our network sensors suh that eah node ontains three-linked list of data items. These lists are L L' g and L g. L is the list of data items ontained in their lusters and L g is the list that ontains the greatest data items in the lusters. V. SIMULATION RESULTS To evaluate the auray of our algorithms ompared to the results of the theoretial analysis that exists in the literature for sort partial we perform simulations of our approah. Our algorithm has been exeuted on the platform OMNET with a network of sensors. The sensors are distributed in a uniform manner. Fig. 4 show the simulation of our approah to sort partial a set ontaining 1000 data eah sensor has one and only data item. For the generalized approah for sort partial we have a variable p that represents the number of data items held by a sensor. Fig 5 show the results for p=10 p=20 and p=40. The algorithm of generalization ( L gg ) is based on our algorithm for the lassial sorting problem part. It assumes that eah node in the network ontains one and only one data item. The algorithm for generalized sorting problem uses the partial searh method developed earlier in MaxCluster. We an desribe the stages of our algorithm as follows: STEP 1: Hierarhial Clustering the networks by using algorithm Gerla & Tsai; STEP 2: The partial sorting algorithm ( L gg ) = ( L g )= Partial-sorting (M k). Fig. 4. Simulation performane of our sort partial approah with 1000 elements.

7 RE FE R E NC E S Fig. 5. Simulation performane of our generalized sort partial approah with 1000 elements and p=10 p=20 and p=40. The simulation results show that our algorithms math the theoretial bound. The simulations also show that when we need to sort the first k largest elements not all elements of the set our approahes an be good hoies. VI. CONCLUSION The main motivation of this work is to give a solution of the partial sorting problem in multi-hop sensor networks. Our algorithms are the based on multi-layer lustering of the sensor network based on the algorithm of Gerla & Tsai. The first approah resolves the problem of partial sorting with a number of broadast rounds not exeeding O( n) + S * Pr * max * max log( max min where n is the number of sensors in the network S is the st of separators Pr is the number of layers obtained after the lustering proess C max is the luster of maximum size among all lusters of the different layers. C min the luster of minimum size among all lusters of the different layers The seond algorithm provides a generalization of first approah to the lassial sorting problem. This generalization for the partial sorting problem onsists in assuming that eah node that is part of the network may ontain several data items. [1] J. L. Bordim Koji Nakano and Hong Shen Sorting on Single-Channel Wireless Sensor Networks. Proeedings of the International Symposium on Parallel Arhitetures Algorithms and Networks (ISPAN.02). P [2] H. H. Chern and H. K. Hwang Phase hanges in random m-ary searh trees and generalized quiksortin Phase hanges in random m-ary searh trees and generalized quiksort Random Strutures and Algorithms Vol. 19 pp [3] H. H. Chern and H. K. Hwang Transitional behaviors of the average ost of quiksort with median-of-(2t+1) In Algorithmia Vol. 29 pp [4] R. Dehter and L. Kleinrok. Broaast ommuniation and distributed algorithms IEEE Transations on Computers C [5] M. Gerla and J T.C. Tsai. A Multiluster Mobile Multimedia Radio Network Wireless Networks vol. 1 no.3 p [6] C. A.. Hoare Find (algorithm 65). In Communiations of the ACM Vol. 4 pp [7] J. JaJa A perspetive on quiksortin Computing in Siene and Engineering2000. [8] J. M. Marberg and E. Gafni. Sorting and seletion in multi-hannel broadast neworks ICPP pp [9] C. Martinez Partial quiksortin Pro. of the 6th ACM- SIAM Workshop on Algorithm Engineering and Experiments and the 1st ACM-SIAM Workshop on Analyti Algorithmis and Combinatoris pp [10] K. Nakano S. Olariu and J. L. Shwing Broadast- Effiient protools for Mobile Radio Networks. IEEE T.P.DS. vol.10 pp [11] H. Prondinger Multiple quikselet - hoares nd algorithm for several elements In Information Proessing Letters Vol. 56 No. 3 pp [12] R. Sedgewik Algorithms In C. USA : Addison-Wesley Publishing Company 3 ed [13] R. Sedgewik and P. Flajolet An Introdution to the Analysis of Algorithms. In USA : Addison-Wesley Publishing Company1996. [14] Shyue-Horng Shiau and Chang-Biau Yang. Generalization of Sorting in Single HopWireless Networks. IEICE Trans. Inf. annd Syst. Vol. E89 D no.4 p [15] C. B. Yang R. C. T. Lee and W..-T. Chen. Conflit-free sorting algorithms under single and multi-hannel broadast ommuniation models. ICCI LNCS 497 P