1. Maximizing Broadcast Tree Lifetime in Wireless Ad Hoc Networks
Guofeng Deng, Sandeep Gupta
IMPACT Lab, Arizona State University

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