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

Mesh network deployment Motivation

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

Mesh network deployment Motivation Candidate location The budget is limited!

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

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

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

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

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

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

Submodular Set Function

Example: Number of covered users

Example: Total Data Rate

Formal Problem Definition

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

Our Algorithm

The Idea The best solution is then the final output

The Solution-Step 1

The Solution-Step 2

User Candidate location

The Solution-Step 2 User Candidate location

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

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

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

[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.

Approximation Ratio

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

Formal Problem Definition

Approximation Ratio

Simulation Results -Use Synthesis Data

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

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

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, Goal = maximum number of covered users Homogeneous costs We compare with Vandin’s algorithm [17]

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

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

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

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

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

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

Simulation Results -Use the Census of Taipei

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

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

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

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

Conclusion

Thank you