Li (Erran) Li Joint work with R. Bhatia, A. Kashyap (U. of Maryland) Special thanks to Alon Efrat

1. The “power” of 2 Power AssignmentsThe power balancing problem in energy constrained sensor networks Li (Erran) Li Joint work with R. Bhatia, A. Kashyap (U. of Maryland) Special thanks to Alon Efrat

2. Given a network G=(V,E): For each edge e=(u,v), P(e) is the minimal power needed for u to reach v A power assignment {Pu} induces a subgraph G’=(V,E’) An edge e=(u,v) exists In G’ iff Pu >P(e) Topology control seeks to find a power assignment {Pu} such that G’=(V,E’) is connected Maxu Pu is minimized or the total power is minimizedΣu Pu Topology control in sensor networks

3. Optimal one power assignment: Max power: 1 Optimal two power assignments: Max average power: (1+ ε)/2 1 ε ε 1 Example Network 1 1 ε ε 1 1 ε ε Optimal one power assignments The case for more than 1 power assignments

4. Multiple Power assignments • For ease of implementation, we consider 2 power assignments • Two network interface cards • Routing in two parallel networks • Given a network G=(V,E), we seek to find 2 power assignments and f1 such that Maxu f1P1u + (1-f1)P2u

5. Hardness Results • For arbitrary number of power assignments [G. Calinescu, S. Kapoor, A. Olshevsky and A. Zelikovsky, “Network lifetime and power assignment in ad hoc networks, ESA 2003]: • NP-hard • (1+ ε)-approximation algorithm • For 2 power assignments without unit disk graph assumption: • The problem is NP-hard via reduction from 3SAT • Hard to approximate with a factor better than 2 • MST gives a trivial 2-approximation algorithm

6. Hardness Results (Cont’d) • Open problem • How hard is the 2 power assignment problem if G is a unit disk graph?

7. Centralized algorithm: CAA • Given power model Pr = cPt/dα • If α =2, a 1.76-approximation algo • If α=3, a 1.966-approximation algo

8. Localized algorithm: one power assignment • Each node knows only its visibility graph (the subgraph induced by the set of nodes a node can reach using max power) • For one power assignment, LMST [N. Li, J. Hou, and L. Sha, Design and analysis of an MST-based topology control algorithm for wireless multi-hop networks, INFOCOM’03] • Each node u constructs a local MSTu using its visibility graph • An edge e=(u,v) exists in the final graph G’ iff e is in LMSTu and LMSTv • G’ is connected if G is connected • The max power is the minimal among all algorithms using only visibility graph

9. Localized algorithm: two power assignments • A naive algorithm LTD: duplicate LMST • A 2-approx with respect to optimal localized 2 power algorithm • Algorithm LA • Run LMST using edge weight w(e)=max(Pminu, Pminv)+P(e) where e=(u,v) and Pminu is the minimal power to reach any node; Let the resulting topology be G1 • Let the power of node u in G1 be P1u • Set edge weight w’(e)= max(P1u, P1v)+P(e) • Prune edges with weight w’(e)>2 times the max node power in visibility graph of u and v • Run LMST using w’(e), let the resulting topology be G2 • LA is a 2-approx with respect to optimal localized 2 power algorithm

10. 2 1 1 2 (a) Input graph 2 2 3 2 2 1 1 3 (c) G1 (b) Phase I edge weights 1 1 4 3 3 2 2 3 (e) G2 (d) Phase II edge weights Localized algorithm LA Example:

11. Performance Evaluation • Nodes deployed in unit square • Max transmission range 0.5 • Compare with • CMSTD (centralized MST, one power assignment) • LTD (localized MST, one power assignment) • CAA (centralized algorithm, two power assignments) • LA (localized algorithm, two power assignments) • Average over 20 random deployments

12. Performance evaluation (cont’d) • Improves over LMST by as much as 30% (a) α =4 (a) α =2

13. Conclusion and future work • Power balancing is important in energy constrained sensor networks • Details in the paper that will appear in Infocom’07 • Open problems • Multi-criterion optimization problems: consider average power in conjunction with • total power of all nodes • throughput between any pair of nodes • broadcast latency • total interference • Consider application traffic pattern

14. NP-hard proof • Gadget

15. Performance evaluation (cont’d) • CMSTD same as LTD (a) Double transmission range (α =2) • N=40, varying transmission range (α =2)