Maximizing Broadcast Tree Lifetime in Wireless Ad Hoc Networks

1. Maximizing Broadcast Tree Lifetime in Wireless Ad Hoc Networks Guofeng Deng, Sandeep Gupta IMPACT Lab, Arizona State University http://impact.asu.edu

2. Broadcast in WANETs • Minimum energy broadcast (minimizing total transmission power) • NP-hard • BIP [Wieselthier Infocom 2000], EWMA [Cagalj Mobicom 2002] • Maximum lifetime broadcast (minimizing maximum transmission power) • Solvable in polynomial time • MST [Camerini IPL 1978][Kang ICC 2003], sub-network solution [Lloyd Mobihoc 2002][Floreen DIALM-POMC 2003] and MDLT [Das Globecom 2003] • Consideration of transmission power alone is insufficient. Receiver power matters. • CORP: constant receiver power model • TREPT: transmitter-receiver power tradeoff model G. Deng & S. Gupta, Globecom'06

3. Outline • Maximizing Broadcast Tree Lifetime (MaxBTL) • Network model • CORP model • TREPT model • Problem statement • MaxBTL under the CORP model • An optimal solution • MaxBTL under the TREPT model • An optimal solution to a special case problem • Conclusion G. Deng & S. Gupta, Globecom'06

4. Maximizing Broadcast Tree Lifetime • Network model • Power consumption is the sum of transmission and receiving power consumption • Transmission power control • Wireless multicast advantage (WMA) • Receiving power will be discussed shortly • Finite battery power capacity and linear battery power model, i.e., the lifetime of a node is the ratio between the amount of battery energy and power consumption. • Problem statement • Broadcast tree lifetime: the period of time for the first node to die • MaxBTL: find a broadcast tree that maximizes the broadcast tree lifetime among all the broadcast trees rooted at the given source node. G. Deng & S. Gupta, Globecom'06

5. Receiver Power Models • CORP • The receiver power, which may vary from node to node, is fixed regardless of the signal strength at the receiver. • E.g., paT = 16mW • If paR = 5mW, then pa = 21mW • TREPT [Cui ICC 2003][Vasudevan et al. Infocom’06] • The receiver power for decoding a signal is a function of the transmission power of the transmitter as well as the distance between them. • E.g., paR = d3/psT and d = 5m. paR = 10.4mW when psT = 12mW; when psT increases to 20mW, paR reduces to 6.25mW. G. Deng & S. Gupta, Globecom'06

6. Roadmap • Maximizing Broadcast Tree Lifetime (MaxBTL) • Network model • CORP model • TREPT model • Problem statement • MaxBTL under the CORP model • An optimal solution • MaxBTL under the TREPT model • An optimal solution to a special case problem • Conclusion G. Deng & S. Gupta, Globecom'06

7. MaxBTL under the CORP model • Define longevity of a transmitter-receiver pair as: • Lemma 1: The lifetime of any broadcast tree T is the minimum longevity of any transmitter-receiver pair in T. • Lemma 2: Given a node v in a link weighted digraph, the spanning tree rooted at v generated using Prim’s algorithm minimizes the maximum link weight among all spanning trees rooted at the same node. • WANET  Inverse longevity graph (ING) • Theorem 1: A rooted spanning tree generated by Prim’s algorithm in an ING is the maximum lifetime broadcast tree. Lemma 2 ING Lemma 1 Prim tree  min max weight  min max inverse longevity  max min longevity  max tree lifetime G. Deng & S. Gupta, Globecom'06

8. Roadmap • Maximizing Broadcast Tree Lifetime (MaxBTL) • Network model • CORP model • TREPT model • Problem statement • MaxBTL under the CORP model • An optimal solution • MaxBTL under the TREPT model • An optimal solution to a special case problem • Conclusion G. Deng & S. Gupta, Globecom'06

9. MaxBTL under the TREPT model • Given a broadcast tree, what is the maximum lifetime? • Power setting of a broadcast tree is a snapshot of transmission and receiving power of each node. • Given a tentative tree lifetime, we can check if there is any valid power setting that satisfies connectivity constraints, i.e., if the given lifetime is feasible. • For example, for a tentative lifetime τ • s: ps = psT = Es / τ, s is OK only if ps ≥ p(s,a). • a: paR = f(psT,d), paT = Ea / τ – paR , a is OK only if paT ≥ max{p(a,c),p(a,b)}. • …… G. Deng & S. Gupta, Globecom'06

10. MaxBTL under the TREPT model • Theorem 2: Under the TREPT model, the binary search algorithm returns the lifetime of any given broadcast tree within ε of the optimal lifetime in O(n log(T/ε)) time, where n is the number of nodes in the WANET and T is an upper bound lifetime. • We suspect that the general problem of finding a maximum lifetime broadcast tree under the TREPT model is NP-hard. G. Deng & S. Gupta, Globecom'06

11. Roadmap • Maximizing Broadcast Tree Lifetime (MaxBTL) • Network model • CORP model • TREPT model • Problem statement • MaxBTL under the CORP model • An optimal solution • MaxBTL under the TREPT model • An optimal solution to a special case problem • Conclusion G. Deng & S. Gupta, Globecom'06

12. Conclusion • Receiver power matters • An optimal solution to MaxBTL under the CORP model • An optimal solution to a special case problem of MaxBTL under the TREPT model • Future directions include a solution to the general MaxBTL problem under the TREPT model or proving it to be NP-hard, and distributed solutions to MaxBTL. G. Deng & S. Gupta, Globecom'06

13. Thank You! G. Deng & S. Gupta, Globecom'06