Joint Routing and Scheduling in Multi-hop Wireless Networks with Directional Antennas

1. Joint Routing and Scheduling in Multi-hop Wireless Networks with Directional Antennas* Partha Dutta, IBM Research India Vivek Mhatre, Motorola Inc Debmalya Panigrahi, MIT Rajeev Rastogi, Yahoo Research India Infocom 2010 * Part of this work was done at Bell Labs India 1

2. Network Model Multi-hop Wireless Network Directional Antennas Long-distance vs Local access links Low-cost networking in developing countries Wireless backhaul for 3G cellular networks 2

3. Interference Constraint Raman Chebrolu �05 Dutta Jaiswal P Rastogi �08 3

4. Schedules and Link Utilization Sequence of bi-partitions of nodes into transmitters and receivers (Directed cuts) Utilization of link = Fraction of time for which link is active 4

5. Demand Matrix 5

6. Routing Demands 6

7. Joint Routing-Scheduling Problem Input Bi-directional capacitated graph G Demand matrix D Output Schedulable flow f satisfying D Optimization version Maximize ? = 1 such that ?D is satisfied 7

8. Related Work [Kodialam Nandagopal �05a, �05b] joint routing-scheduling for omni-directional antennas with single and multi-radio nodes high-level approach is similar significant difference in problem structure [Narlikar Wilfong Zhang �06] joint routing-scheduling for directional antennas no provable guarantees of proposed heuristics 8

9. Schedulable flows A flow f is schedulable if there exists a schedule S such that for each edge e, utilization of e in S = fe/ce A flow f is tenable if for each vertex v, and every pair of incoming and outgoing edges e, e� fe/ce + fe�/ce� = 1 9

10. Schedulable flows Lemma: Every schedulable flow is tenable (but not vice-versa) 10

11. Relaxed Problem Input Bi-directional capacitated graph G Demand matrix D Output Schedulable flows f satisfying demands ?D, for max ? = 1 Is a solvable linear program! 11

12. Roadmap Solve the linear program to obtain tenable flow f satisfying ?0D demands Design scheduling algorithm to schedule af flow (a = 1) ?0 = max ? for schedulable flows ? a-approx algorithm! 12

13. Goal Scheduling algorithm for flow f such that for each vertex v, and every pair of incoming and outgoing edges e, e� fe/ce + fe�/ce� = a (a-tenable flows) 13

14. Schedule Split 1 second into t scheduling intervals Schedule specifies directed cut for each interval Utilization of edge e in schedule must be = fe/ce Edge e must be in t(fe/ce) cuts 14

15. Notation Multiplicity of node = max mult. of incoming edge + max mult. of outgoing edge Tight node: one having max mult. (?) Almost simple graph: Simple graph + extra copy of each edge incident on a tight node with in/out-degree = 0 a-tenable flows: ? = at 15

16. The Scheduling Problem Input: Multi-graph with ? = at Output: Partition edges into t sets s.t. No two parallel edges in the same set Every set of edges forms a directed cut 16

17. The Scheduling Problem Input: Multi-graph Output: Partition edges into t sets s.t. No two parallel edges in the same set Every set of edges forms a directed cut 17

18. Scheduling Algorithm 18

19. Analysis ? decreases by 2 in each iteration Thm: Multigraph can be partitioned into ? ?(K)/2 subsets of edges satisfying the two constraints Recall: we wanted to partition into t subsets So, if t = ? ?(K)/2 we can schedule the flow i.e. ? = (2/?(K))t ? Approx factor a = 2/?(K) 19

20. ?(K) In most practical networks, our algorithm provably achieves at least 40-50% of the best achievable throughput 20

21. Open Problems Better approx factor? Hardness NP-hard? Probably yes, and probably as hard as MAXCUT 21