1. Tung-Wei Kuo, Kate Ching-Ju Lin, and Ming-Jer Tsai Academia Sinica, Taiwan National Tsing Hua University, Taiwan Maximizing Submodular Set Function with Connectivity Constraint: Theory and Application to Networks

2. Mesh network deployment Motivation

3. Mesh network deployment Motivation How should we deploy the network? Candidate location

4. Mesh network deployment Motivation Candidate location The budget is limited!

5. Only one router can access the Internet Mesh networks exploit multi-hop relays Connectivity Candidate location

6. Only one router can access the Internet Mesh networks exploit multi-hop relays Connectivity Candidate location

7. Only one router can access the Internet Mesh networks exploit multi-hop relays Connectivity The network must be connected!

8. Various Performance Metrics A variety of performance metrics – The number of covered users, total throughput, the size of the coverage area, … Given limited resources (routers or budget), deploy a connected mesh that optimizes the performance metric

9. Mesh Deployment Problem This is the optimal solution GOAL: Construct a connected network such that the optimization goal is achieved

10. Design an algorithm for each of the various optimization goals? Many optimization goals can be modeled as submodular set functions Our goal: A universal algorithm for a family of problems whose objective can be modeled as a submodular set function

11. Submodular Set Function

12. Example: Number of covered users

17. Example: Total Data Rate

19. Formal Problem Definition

20. The problem is NP-hard. An approximation algorithm will be given

21. Our Algorithm

22. The Idea The best solution is then the final output

23. The Solution-Step 1

25. The Solution-Step 2

27. User Candidate location

28. The Solution-Step 2 User Candidate location

29. The Solution-Step 3 User Candidate location 3. Use shortest paths to connect routers to the center

30. This is a feasible solution The Solution-Step 3 User Candidate location 3. Use shortest paths to connect routers to the center

31. The Algorithm The best solution is then the final output. How, exactly, should we deploy the routers?

32. [9] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions-I,” Mathematical Programming, vol. 14, pp. 265–294, 1978.

33. Approximation Ratio

34. The Problem with Heterogeneous Deployment Costs Different locations might have different deployment costs

35. Formal Problem Definition

36. Approximation Ratio

37. Simulation Results -Use Synthesis Data

38. Simulation Setting Field size: 1200 m × 1200 m User: – # of users: 200 – Zipf’s law – 802.11b Candidate locations: – Grid network – Grid size: 100 m × 100 m Communication range: 150 m Channel error model: 802.11b PHY Simulink Model

39. Another Common Scenario In some applications, a specific location may need to be included in the solution We modify our algorithm accordingly: How to findthe center? Our algorithm Try all the possible centers and choose the best one Our algorithm w/ specific center Let the specificlocation be the desired center

40. Comparison Schemes [17] F. Vandin, E. Upfal, and B. J. Raphael, “Algorithms for detecting significantly mutated pathways in cancer,” Journal of Computational Biology, vol. 18, pp. 507–522, 2011. Goal = maximum number of covered users Homogeneous costs We compare with Vandin’s algorithm [17]

41. Simulation Scenarios Two types of deployment costs: 1.Homogeneous costs 2.Heterogeneous costs Two performance metrics: 1.Total data rate 2.The number of covered users

42. Maximum Total Data Rate Homogeneous costs Total data rate of covered users (Mb/sec) Number of routers, k Upper bound Arbitrary solution Greedy: max date rate Greedy: max data rate w/ specific center Our algorithm Our algorithm w/ specific center

43. Heterogeneous costs Total data rate of covered users (Mb/sec) Total budget for deployment, B Upper bound Arbitrary solution Greedy: min cost Greedy: min cost w/ specific center Greedy: max data rate Greedy: max data rate w/ specific center Our algorithm Our algorithm w/ specific center Maximum Total Data Rate

44. Maximum Number of Covered Users Upper bound Arbitrary solution Vandin’s algorithm Vandin’s algorithm w/ specific center Our algorithm Our algorithm w/ specific center Homogeneous costs Number of routers, k

45. Heterogeneous costs Total budget for deployment, B Upper bound Arbitrary solution Greedy: min cost Greedy: min cost w/ specific center Greedy: max coverage Greedy: max coverage w/ specific center Our algorithm Our algorithm w/ specific center Maximum Number of Covered Users

46. Summary of the simulation results 1.Our algorithm can be applied to different optimization goals 2.The ratio between the upper bound and our algorithm matches the approximation ratio 3.Our algorithms perform better than the greedy heuristics

47. Simulation Results -Use the Census of Taipei

48. Use the Census of Taipei Use the census to locate the users Heterogeneous deployment costs: – Higher costs are assigned to locations with higher population density Goal: Maximize the number of covered users

49. Input 8 km 12 km Total cost of all locations: 60053 Number of users: 7126

50. Output The output when the available budget = 15000 Number of covered users: 6600 ( ≈93% of the total users ) 8 km 12 km

51. The Results Number of covered users Total budget for deployment, B Upper bound Arbitrary solution Greedy: min cost Greedy: min cost w/ specific center Greedy: max coverage Greedy: max coverage w/ specific center Our algorithm Our algorithm w/ specific center

52. Conclusion

53. Thank you