Minimum-Energy Broadcast in All-Wireless Networks: NP-Completeness and Distribution Issues

1. Minimum-Energy Broadcast in All-Wireless Networks: NP-Completeness and Distribution Issues

2. Wireless Sensor Networks • Numerous sensor devicesequipped with • Modest wireless communication, processing, and memory capabilities • Form ad hoc network (self-organized) • Low mobility (static) • Distributed system • Broadcasting - important communication primitive

3. pi pk pj Design goals • Find a minimum-energy broadcasttree l i o j k p n m • Minimize power consumption per packet

4. Peculiarities of wireless media • Wireless Multicast Advantage (WMA) [WieselthierNE00] • Nodes equipped with omnidirectional antennas • i transmits at and reaches both j and k • Energy expenditure k pik i pij j • Wireless media is a node-basedenvironment

5. s.t. are said to be covered by node • Captures the WMA property Node-based network model • Graph i o j k p n l m

6. Formal problem definition • Minimum Broadcast Cover (MBC) problem • Given , , edge costs , the source and an assignment Find a power assignment vector s.t. it induces graph , , in which there is a path from r to any node of V, and . is minimized • Special case: Geometric MBC (GMBC) • MBC in two-dimensional Euclidean metric space • Edge costs are given by

7. Complexity issues • THEOREM 1.Minimum Broadcast Cover (MBC) is NP-complete. • Set Cover (SC) is NP-complete [M. Garey and D. Johnson] • MBCNP and SC  MBC ( - polynomial transformation) • COROLLARY 1.MBC cannot be approximate better than O(log ) if P  NP. • SC to MBC transformation preserves approximation ratio achievable for SC • No polynomial-time algorithm approximates SC better than O(log ) if P  NP [D.S. Hochbaum] • THEOREM 2.Geometric MBC (GMBC) is NP-complete. • Planar 3-SAT (P3SAT) is NP-complete [M. Garey and D. Johnson] • GMBCNP and P3SAT GMBC

8. 8 4 pb=8 1 pa=2 5 2 pe=4 5 pd=4 [WanCLF01] 5 pc=5 5 4 Heuristicsbased approach • Broadcast Incremental Power (BIP)[WieselthierNE00] h f b a d e c i j g • Embedded Wireless Multicast Advantage (EWMA) • Begin with an initial feasible solution (minimum spanning tree) • Improve the initial solution by embedding the WMA property while preserving the feasibility of the solution

9. Calculate gains 5 5 • Calculate new transmission power EWMA by example (1/2) h f 8 4 pb=8 b 1 a pa=2 5 2 d pe=4 pd=4 e 4 c pc=5 4 i j g

10. h h f f 13 13 8 4 b b 1 a a pa=13 pb=8 1 pa=2 d d 9 pe=4 10 5 e 2 e pe=4 5 13 pd=4 c c 5 9 4 i i pc=5 5 4 j j g g [WanCLF01] EWMA by example (2/2)

11. Phase 2: Local EWMA • Node x waits before it potentially becomes a forwarder Distributed EWMA (1/2) • Phase 1: Distributed-MST[GallagerHS83] i o j k p n l m

12. Distributed EWMA (2/2) • Lack of global information • Propagate information about forwarding nodes along the transmission chain • Phase 2 organized in rounds of duration • Probation , correction and active periods i o j k p n l m

13. Performance evaluation • Simulation setup • 100 instances x 10,30,50,100-nodes networks • Spatial Poisson distribution of nodes • Cost of links • The performance metric is a normalized power • For k-nodes network

14. Performance evaluation (1/2) Avg. Normalized Power vs. Network Size • = 3 • = 4

15. Performance evaluation (2/2)Avg. Normalized Power vs. Network Size • = 2 • = 2

16. Conclusion • Achievements • Proved that building the minimum-energy broadcast tree in wireless networks is NP-complete • Devised a new algorithm:Embedded Wireless Multicast Advantage (EWMA) • Shown that it can be distributed • Future work • Extend the network model (i.e. mobility, interference) • Take into account the battery lifetime • Consider the minimum-energy multicast problem http://www.terminodes.org

17. Synchronization of DEWMA • Necessary conditions • Duration of the second phase is bounded by