Lower Bounds on the Maximum Energy Benefit of Network Coding for Wireless Multiple Unicast

1. Lower Bounds on the Maximum Energy Benefit of Network Coding for Wireless Multiple Unicast (To be published, 2010 EURASIP J. Wirel. Commun. Netw., Special Issue on Wireless Network Coding.) • Introduction • Model and Problem Statement • Results • Theorem 1 • Theorem 2 • An Efficient Code on the Hexagonal Lattice • Configuration • Network Code • Validity of the Network Code • Energy Consumption • Discussion & Future Work

2. 1. Introduction Limited resources : Power, Frequency, Time, # of the sensors Minimizing energy consumption: Minimum cost routing[1]-[3], Power control algorithm[4]-[6], Cross-layer protocol design[7], Use of network coding[8]-[14] If so, how much amount of the minimization can we reduce the energy consumption by using the network coding? (Motivation)  It is to know what the benefits of network coding are compared to routing. That is the ratio of the minimum energy solution in a routing solution compared to the minimum energy network coding solution. How can we approach to verify the ratio of the minimum energy between them?  Multiple unicast traffic, not multicast traffic and broadcast. How can we know the minimum-cost routing and network coding in the multiple unicast?  In routing, the shortest path is one for each session individually.  In network coding, the hexagoanl and rectangular lattice are considered. This work extends the lower bound approach in the multiple unicast.

3. 2. Model and Problem Statement To establish lower bounds

4. 4. An Efficient Code on the Hexagonal Lattice(configuration) Position Neighborhoods of a node in the interior

5. 4. An Efficient Code on the Hexagonal Lattice(Network Code) Left border Right border Bottom border In the interior

6. 4. An Efficient Code on the Hexagonal Lattice(Validity)

7. 4. An Efficient Code on the Hexagonal Lattice(Energy Consumption)

8. 3. Results(Theorem 1) The ratio of the minimum energy consumption of routing solutions and the minimum energy consumption of network coding solutions, maximized over all nodes location, multiple unicast sessions and transmission ranges, with the transmission range equal for the routing and network coding solutions, is at least like that:

9. # of nodes in the interior # of hops with the shortest path # of sessions

10. 3. Results(Theorem 2) For 2-dimentional wireless networks, the ratio of the minimum energy consumption of routing solutions and the minimum energy consumption of network coding solutions, maximized over all node locations, and multiple unicast sessions, with the transmission range optimized individually for the routing and network coding solution, is at lease 3, Lower bound [21] : 2.4 (2dim.) Upper bound using decode-and-recombine[26] : 3 (3dim.) This research show lower bound 3 (2dim.)  Other coding strategies can be obtained the large energy benefits.

11. 5. Discussion and Future work • Network coding have the potential of reducing energy consumption in wireless networks. • It remains to be shown how this potential can be exploited using practical codes. • What if other topologies(random networks), not lattice, be considered the energy-benefit? • Another open problem is to find upper bounds on the benefit of network coding for wireless multiple unicast.

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