# Decentralized Utility-based Sensor Network Design

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3 4.1 Game Description We consider an iterative game. Here, we define an iteration as a round in which all children decide on the parent they want to attach. This game is played on the edges e E of the bipartite graph mentioned in section 3. The players in this game are the children i such that i N. For each player i, let Sk i M i be the strategy space at iteration k. Let u i k (p) be the utility of sensor i to choose sensor p as its parent in iteration k. Further, u i k(p) = C p k (1) where C p k is the bandwidth guaranteed by parent p at iteration k. i.e. the parent p is committed to provide a bandwidth of at least u i k (p) to child i for all iterations after k so long as i is a child of p. Let u i k = Max p S i k {ui k(p)} = Max p S i k {Cp k } (2) This utility function in Eqn. 2 dictates that at each iteration, a sensor prefers to connect to a parent that gives it the maximum bandwidth guarantee. i.e. each sensor is selfish about the bandwidth guarantee it receives. At every iteration, each parent p gives equal bandwidth guarantees to its children. In section 4.2, we describe the algorithm that constructs a load-balanced tree in the presence of selfish sensors. 4.2 Algorithm for Load Balanced Tree Construction In this section, we describe an iterative distributed algorithm for the load balanced tree construction. This algorithm assumes that each sensor s decision making is governed by the utility function described in Eqn. 2 in section 4.1. Before describing the algorithms of the parent and child, we introduce a few notations and definitions. Let 1. deg(j) be the degree of a node j in the bipartite graph described in section N p k be the set of children currently attached to parent p at the end of iteration k. 3. n p k be the number of new children that wish to attach to parent p at iteration k. 4. a p k be the number of new children that actually attach to parent p at iteration k1. 5. y p k be the number of additional children that parent p can take at iteration k, giving a bandwidth guarantee of C p k. Initially yp 0 = deg(j) (as Cp 0 = 1 deg(p) ) 6. zp k be the number of children that leave parent p at iteration k. 7. Pk i be the parent of node i at iteration k. 8. B p k be the total bandwidth guaranteed by parent p at iteration k. i.e. B p k = N p k Cp k if N p k > 0 (3) = C p k otherwise (4) Definition 1. A parent p is considered to be saturated at iteration k iff B p k = 1. If N p k > 0, saturation implies that p cannot take any more children at the guaranteed bandwidth of C p k. If N p k = 0, Bp k = Cp k = 1 implies that each of p s children have a bandwidth guarantee of 1 from some other parent p. p: p M, y p k = 0 iff p is saturated at iteration k. We now describe the algorithms of the Child and the Parent. Child: At an iteration k, the strategy space of child i, Sk i = {p p M i, B p k < 1}. i.e. at each iteration k, each child i will try to connect to an unsaturated parent p that provides the best possible bandwidth guarantee. If there are multiple such parents, the child will indicate a willingness to connect to all of them 2. After it has been chosen by one parent, the child node will inform the other parents that it does not wish to connect to them. 1 A child might request to be attached to several parents at iteration k, but will ultimately choose exactly one parent that is willing to take it at iteration k 2 The child has to respond to all such parents because a previously unsaturated parent might have to reject some children if it gets saturated during the current iteration.

4 Parent: Essentially, until a parent p saturates, it successively increases its bandwidth guarantees from 1 deg(p), 1 deg(p) 1, 1 deg(p) 2, so on to 1. Initially, a parent p sets Cp k = 1. While p is not saturated deg(p) (may also happen if a child leaves a saturated parent making B p k < 1), it executes the following iterative algorithm (here, k is a global variable that keeps track of iterations previously executed by this parent): Parent(p) { While (B p k < 1) { C p k+1 Cp k Announce C p k+1 to all its children /* Find the upper bound on the number of children that can be taken */ y p k+1 1 N p C p k. k+1 x p k+1 min{np k+1, yp k+1 } N p k+1 N p k + ap k+1 zp k+1. If (B p k+1 = 1){ exit } /* If some child left or a child joined don t increase the bandwidth guarantee */ If (z p k+1 > 0) or (np k+1 > 0) { C p k+1 Cp k }else{ C p k+1 1 } k k + 1 }. 1 C p 1 k Termination: The algorithm terminates at the smallest iteration k at which all parents are saturated i.e. M p=1 Bp k = M. 4.3 Analysis In this section, we analyze the running time of the algorithm. We show that at termination, a Nash Equilibrium exists. We conjecture that the Nash Equilibrium coupled with the termination condition mentioned at the end of section 4.2 results in an optimally load balanced tree (or max-min allocation of bandwidths to the children). Theorem 1. The total number of iterations taken by the distributed algorithm is O(Nγ), where γ is the maximum degree of a parent in the graph G. At termination, a Nash Equilibrium exists. Proof: In every iteration, at least one parent increases its bandwidth guarantee. Moreover, each parent can increase it bandwidth guarantee at most γ times. On termination at iteration k, all parents are saturated i.e. p, B p k = 1. Thus, the strategy space of every child i and iteration k, Sk i = φ. Hence, by definition, a Nash Equilibrium exists at termination. We are also interested in proving the correctness of the algorithm. We are in the process of working out the details of the proof. However, we outline the preliminary progress made in this direction. As mentioned in section 2, a load balanced tree corresponds to an optimal semi-matching. Harvey et al. characterize an optimal semi-matching using the notion of cost reducing paths (defined below) [3].

5 Figure 1. An illustration of the execution of the algorithm. The matched edges are marked as solid lines. The set of parents M = {1, 2, 3}, while the set of children N = {4, 5, 6}. The bandwidth guarantee offered by each parent is shown on top of the parent. Definition 2. Given a semi-matching S in G = (V, E), a cost reducing path R is a sequence of alternating matched and unmatched edges given by R = {(v 1, u 1 ), (u 1, v 2 ),...(u z 1, v z )} where v i M, u i N and (v i, u i ) S for all i, such that deg S (v 1 ) deg S (v z ) > 1. deg S (v i ) is the number of children matched to parent v i in the matching S. Moreover, the authors of [3] also prove the following: Theorem 2. A semi-matching is optimal iff no cost-reducing path exists. We are interested in proving the following: Conjecture 1. On termination at iteration k, the algorithm produces a load balanced tree (or an optimal semi-matching) i.e. the allocation of bandwidths x k = (x 1 k, x2 k,...xn k ) is max-min fair. For this purpose, we define a simple cost reducing path as follows: Definition 3. A simple cost reducing path is a cost reducing path that does not contain any other cost reducing path within it. Theorem 3. If at each iteration, no simple cost reducing path terminates at a saturated parent, the algorithm on termination, results in a load-balanced tree (optimal semi-matching). Proof: On termination, all parents are saturated, implying no simple cost reducing paths can exist in the resulting matching and hence by theorem 2, the semi-matching produced is optimal. Moreover, simple cost reducing paths terminating at a saturated parent have the following definitive characteristic: Theorem 4. Given a semi-matching S in G = (V, E), a simple cost reducing path R = {(v 1, u 1 ), (u 1, v 2 ),...(u z 1, v z )} where v z is saturated should satisfy the following conditions: 1. k > 0 i.e. the path should have at least one intermediate parent between v 1 and v z. 2. i : i 1, z : deg S (v i ) = deg S (v z ) + 1

6 3. deg S (v 1 ) = deg S (v z ) + 2. Proof: 1. If v z is saturated with x children (deg S (v z ) = x), v z 1 can have at most x + 1 children (deg S (v z 1 ) x + 1). This can be proved by contradiction. Let deg S (v z 1 ) > x + 1. Consider the child u z 1 such that (v z 1, u z 1 ) E and (u z 1, v z ) E. Let this child be attached to parent v z 1 in the current matching S, implying that u z 1 never got an opportunity to attach to a parent that offered it a bandwidth greater than or equal to 1 x+1. At some iteration k, v z would not have been saturated and hence increased its bandwidth guarantee from 1 x+1 to 1 x. i.e. Cvz k 1 = 1 x+1. Thus, at iteration k 1, u z 1 had at least one unsaturated parent 1 v z offering a bandwidth guarantee equal to x+1. This contradicts the claim that u z 1 never got an opportunity to attach to a parent that offered it a bandwidth greater than or equal to 1 x+1. Thus, deg S (v z 1 ) deg S (v z ) 1. Thus, v z 1 cannot be the starting vertex of the cost reducing path R, because the deg S (v 1 ) deg S (v z ) > Consider v i such that i 1, z. Now, deg S (v 1 ) deg S (v z ) > 2. Hence, deg S (v i ) deg S (v z )+1. Otherwise if deg S (v i ) deg S (v z ), then deg S (v 1 ) deg S (v i ) 2. Thus, R contains a simpler cost reducing path that terminates at v i. Also, deg S (v i ) deg S (v z )+1. Otherwise, if deg S (v i ) deg S (v z ) 2, R contains a simpler cost reducing path from v i to v z. Hence i : i 1, z, deg S (v i ) = deg S (v z ) For R to be a cost reducing path deg S (v 1 ) deg S (v z ) 2. If R is a simple cost reducing path terminating at a saturated parent v z, deg S (v 1 ) deg S (v z ) = 2. Otherwise, if deg S (v 1 ) deg S (v z ) > 2, implies deg S (v 1 ) > (1+deg S (v z ))+1. Thus, deg S (v 1 ) > deg S (v i )+1. Hence, R contains a simpler cost reducing path that terminates at one of the intermediate parents v i, i z because, deg S (v i ) = deg S (v z ) + 1. We are currently working on proving that such simple cost reducing paths do not arise during the execution of our algorithm. Our confidence in conjecture 1 is increased by the simulations that we describe next. 5 Simulations In this section, we compare the tree resulting from an implementation of our distributed algorithm and the centralized algorithm for producing an optimally load balanced tree described in [3]. In our simulation scenarios, sensors were randomly distributed in a 500 x 500 m area. There were a total of 7 parents and 30 children. In the first set of simulations (using 8 different random seeds), the transmission range R = 250m. In the second set simulations (using 8 different random seeds), R = 200m. In all these simulations, we observed that the tree produced by the distributed algorithm and the centralized algorithm were similar. i.e. the number of parents of degree x were same in both cases. 6 Conclusions & Future Work In this study, we advocate a utility function based approach for designing sensor networks. Unlike previous related studies, we illustrate that treating the sensors as selfish (using appropriate utility functions) enables the design of distributed algorithms for optimizing the network performance as a whole. This is done for the case study of constructing a load-balanced data gathering tree in a sensor network. As part of future work, we would like to prove the correctness of our algorithm. It would also be interesting to extend our mechanism of load balancing for more general scenarios that account for heterogeneity in the energy of the sensors, quality of the links, etc. We also plan to investigate other desired global objectives of data collection trees for which decentralized mechanisms using local utility functions can be designed.

7 References 1. N. Nisan and A. Ronen. Algorithmic mechanism design. In Proc. 31śt ACM Symposium of Theory of Computing, , R. Kannan, S. Sarangi, S. Sitharama Iyengar and L. Ray. Sensor Centric Quality of Routing in Sensor Networks, IEEE INFOCOM N. J. Harvey, R. E. Ladner, L. Lovasz and T. Tamir. Semi-Matchings for Bipartite Graphs and Load Balancing. Workshop of Algorithms and Data Structures (WADS) J. Aspnes, Y. Azar, A. Fiat, S. Plotkin and O. Waarts. On-line Machine Scheduling with Applications to Load Balancing and Virtual Circuit Routing. In Proc. ACM Symposium of Theory of Computing (STOC) D. Grosu and A. Chronopoulos. A Game-Theoretic Model and Algorithms for Load Balancing in Distributed Systems. International Parallel and Distributed Processing Symposium (IPDPS) workshop C. Zhou and B. Krishnamachari, Localized Topology Generation Mechanisms for Self-Configuring Sensor Networks, IEEE Globecom, San Francisco, December J. Byers and G. Nasser. Utility-Based Decision-Making in Wireless Sensor Networks. IEEE MOBIHOC P. Berenbrink, F. Heide and K. Schroder. Allocating Weighted Jobs in Parallel. ACM Symposium on Parallel Algorithms and Architectures. p , 1997.

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