1. An Efficient Clustering-based Heuristic for Data Gathering and Aggregation in Sensor Networks Wireless Communications and Networking (WCNC 2003). IEEE, Volume: 3, 16-20 March 2003. Koustuv Dasgupta, Konstantinos Kalpakis, Parag Namjoshi

2. Outline System Model The Data Gathering Problem MLDA Finding a near-optimal admissible flow network Constructing a schedule CMLDA Experiment

3. System Model sensor numbers <- n t <- the only one base station locations are fixed and known apriori round <- each time unit packet generating rate <- one data packet per round all data packet size <- k bits transmission ability of each sensor <- to any other sensor through the network or directly to the base station

4. Energy Model A sensor consumes to run the transmitter or receiver circuitry for the transmitter amplifier Thus,

5. The Data Gathering Problem We define lifetime T of the system to be the number of rounds until the first sensor is drained of its energy Data gathering schedule <- a collection of T directed trees, each rooted at the base station and spanning all the sensors Objective: Find a schedule that maximizes the system lifetime T

6. MLDA: Maximum Lifetime Data gathering with Aggregation Assumption: that an intermediate sensor can aggregate multiple incoming packets into a single outgoing packet f i,j <- total number of packets i transmits to j Energy Constraints:

7. MLDA Flow network G = (V,E) <- a directed graph where V <- all the nodes, E 0 Theorem 1: Let S be a schedule with lifetime T and G be the flow network induced by S then (->) for each sensor s, the maximum flow from s to he base station t in G is >= T Prove : Each packet from a sensor must reach the base station Thus, a necessary condition for a schedule to have lifetime T is that each node in the induced flow network can push flow T to the base station t

8. Solution of MLDA admissible flow network with lifetime T 1.allow each sensor to push flow T to base 2.respecting the energy constraints in (3) optimal admissible flow network A admissible flow network with maximum lifetime First we find a near-optimal admissible flow network G Then, we construct a schedule from G

9. Finding a near-optimal admissible flow network <- the flow k send to t over the edge (i,j) //Energy constraint i kk k +T //The flow k send out is T and will all arrive at t Integer Program NP complete Linear Relaxation Linear Relaxation <- Polynomial time Allow fractional values We find G with maximum T

10. Schedule Fig. 1. An admissible flow network G with lifetime 100 rounds, and two aggregation trees A1 and A2 with lifetimes 60 and 40 rounds respectively.

11. Constructing a schedule Discuss how to get a schedule from an admissible flow network f <- the life time of the aggregation tree Def 1: Given an admissible flow network G with lifetime f,we define the (A,f)-reduction G ’ of G to be the flow network that result from G after reducing by f, the capacities of all of its edges that are also in A. We call G ’ the (A,f)-reduced G. Def 2: An (A,f)-reduction G ’ of G is feasible if the maximum flow from v to the base station t in G ’ is >= T – f for each vertex v in G ’.

12. Constructing a schedule If A is an aggregation tree, with lifetime f, for an admissible flow network G with lifetime T, and the (A,f)-reduction of G is feasible Then (->) the (A,f)-reduced flow network G ’ of G is an admissible flow network with lifetime T-f Therefore we can devise a simple iterative algorithm

13. Fig. 2. Constructing aggregation tree A with lifetime f from an admissible flow network G with lifetime T. //Aggregation Tree Find a (i,j) that makes Gr feasible //The running time of this algorithm is polynomial of n We can prove that it is always possible to find a collection of aggregation trees based on a powerful theorem in graph theory j i G

14. CMLDA Objective: The MLDA algorithm involves solving a linear program with O(n^3) variables and constraints. For large values of n, this can be computationally expensive.

15. CMLDA – Clustering-based MLDA heuristic m <- numbers of clusters Øi <- ith cluster |Øi|<= c for i = 1,2,…,m super-sensor <- cluster ε Øi <- energy of cluster i <- total energy in cluster i Distance between Øi and Øj <- the maximum distance between any two nodes in each cluster Base station defined as Ø m+1 (with single node)

16. CMLDA We can use previous method to find a schedule consists of T1,T2, …,Tk, each rooted at Ø m+1 AS-tree <- such aggregate tree (Aggregation super-tree) <- residual energy at sensor I Initially = for all sensor

17. CMLDA We use BUILD-TREE procedure to construct an aggregation tree A from AS-tree Objective: construct aggregation trees such that minimum residual energy among the n sensors is maximized (thereby maximizing the lifetime)

18. Pre-order Traversal

19. BUILD-TREE procedure Include all nodes in Ø to the required Aggregation Tree A Def: residual energy of a pair (i,j) <- Distance and Residual energy update //pre-order The running time of the procedure is O(n^3) There could be more than one AS-tree We choose the AS-tree in decreasing order of their lifetimes

21. Experiments R <- CMLDA lifetime / LRS lifetime Depth of a sensor v <- its average depth in each of the aggregation trees D <- depth of the schedule <- Give an estimate of the average delay that is incurred in sending data packets to the base station

22. Experiments Initial Energy 1J, Packet size 1000 bits Tradeoffs between delays and system lifetime fractional

23. Experiments We cannot see the improvement in CMLDA compared to MLDA with the increasing network size

24. Future Work Investigate modifications to the MLDA algorithm that would allow sensor to be added to (or removed from) the network, without having to re-compute the entire schedule Study the data gathering problem with depth (delay) constraints for individual sensors, in order to attain desired tradeoffs between the delay experienced by the sensors and the lifetime achieved by the system

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