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Capacitated Kinetic Clustering in Mobile Networks by Optimal Transportation Theory

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Presented in INFOCOM 2016
http://www3.cs.stonybrook.edu/~chni/publication/optran/
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We consider the problem of capacitated kinetic clustering in which
n
n
mobile terminals and
k
k
base stations with respective operating capacities are given. The task is to assign the mobile terminals to the base stations such that the total squared distance from each terminal to its assigned base station is minimized and the capacity constraints are satisfied. This paper focuses on the development of distributed and computationally efficient algorithms that adapt to the motion of both terminals and base stations. Suggested by the optimal transportation theory, we exploit the structural property of the optimal solution, which can be represented by a power diagram on the base stations such that the total usage of nodes within each power cell equals the capacity of the corresponding base station. We show by using the kinetic data structure framework the first analytical upper bound on the number of changes in the optimal solution, i.e., its stability. On the algorithm side, using the power diagram formulation we show that the solution can be represented in size proportional to the number of base stations and can be solved by an iterative, local algorithm. In particular, this algorithm can naturally exploit the continuity of motion and has orders of magnitude faster than existing solutions using min-cost matching and linear programming, and thus is able to handle large scale data under mobility.

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Capacitated Kinetic Clustering in Mobile Networks by Optimal Transportation Theory

  1. 1. Capacitated Kinetic Clustering in Mobile Networks by Optimal Transportation Theory Chien-Chun Ni Zhengyu Su, Jie Gao, Xianfeng David Gu Computer Science, Stony Brook University 1
  2. 2. A Simple Case: Assign Terminal to BS 2
  3. 3. + Energy Efficient 3
  4. 4. +BS Station Capacity 4
  5. 5. +BS Station Capacity 5
  6. 6. +BS Station Capacity 6 Drop Signal Congestion …
  7. 7. +BS Station Capacity (reassign) 7
  8. 8. +Terminal Usage 8
  9. 9. +Mobility 9
  10. 10. Capacitated Kinetic Clustering Problem •Assign terminals to corresponding base station •Energy efficiency •Capacity constraints • Terminal data usage • Base station capacity •Mobility 10
  11. 11. Problem Definition 11 k BSs , k << nn Terminals Capacity: Usage: Energy cost
  12. 12. Problem Definition 12 k BSs , k << nn Terminals Capacity: Usage: Energy cost T: X -> Y Min energy cost Fit BS capacity
  13. 13. Our approach •Solve by Optimal Transportation Theory • Optimal result guarantee • Fast and Distributed: • O(n)*O(k log k) • Space efficiency: • O(k) • Mobile ready 13
  14. 14. Optimal Transportation Problem? 14
  15. 15. Optimal Transportation Problem(OTP) • Proposed by Monge, in 1781 • “What is the optimal way to move piles of sand to fill up given holes of the same total volume?” 15
  16. 16. Kantorovich’s Approach(1942) • Transportation plan , solved by LP • 17 y1 y2 … yk x1 x2 … xn
  17. 17. Brenier’s Approach • Geometric approach • Cut domain into different cell w/ required area • Move cell elements to factory 18
  18. 18. Brenier’s Approach • Geometric approach • Cut domain into different cell w/ required area • Move cell elements to factory 19
  19. 19. Capacitated Kinect Clustering: OTP by Geometric Approach 20
  20. 20. Brenier’s Approach •While cost function of OTP is quadratic: • Exist a convex function • Its gradient map gives the solution of OTP • The optimal solution is unique • How to find this convex function? 21
  21. 21. Construct Convex Function • •“Lift” factories to Hyperplanes • Convex function: 22
  22. 22. Geometric Meaning of u • is adjustable, project back get • • equals to power diagram 2323
  23. 23. Energy Function* • Energy function: volume of a cylinder •Gradient: • Since Energy function is convex, global minimum occurred at 24 *: Gu et al. “Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge-Ampere equations ”
  24. 24. Finding • , gives optimal transport map •Hessian of on dual •Iteratively applying Newton’s method 25
  25. 25. In Short: Brenier’s Approach •While cost function of OTP is quadratic: • Exist a convex function • Its gradient map gives the solution of OTP • The optimal solution is unique •Solution: Power Voronoi Diagram [1,2,3] 26 [1]. F. Aurenhammer, F. Hoffmann and B. Aronov, Minkowski-type Theorems and Least-Squares Clustering, vol 20, 61-76, Algorithmica, 1998. [2]. X. Gu, F. Luo, J. Sun and S.-T. Yau, Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations, arXiv:1302.5472, Year 2013. [3]. Bruno L´evy, “A numerical algorithm for L2 semi-discrete optimal transport in 3D”, arXiv:1409.1279, Year 2014.
  26. 26. Voronoi Diagram 27
  27. 27. P Power Diagram •Power diagram (Generalized Voronoi Diagram): • Point weight: circle with radius • Power distance: 28 AB Q r PA PB PD(P,Q) = PA * PB = PQ2 - r2
  28. 28. Power Diagram 29
  29. 29. Revisit OTP by Power Diagram •Finding a radius for each factory st. • Cell area match the requirement 30
  30. 30. Connect OTP to Capacitated Kinect Clustering 31 • “Move” data from terminal to BS • Find that meet BS’s capacity
  31. 31. Connect OTP to Capacitated Kinect Clustering 32 • “Move” data from terminal to BS • Find that meet BS’s capacity
  32. 32. Algorithm 1. Rescale: domain area, user usages 2. Assign initial radius , target BS capacity 3. Compute power diagram, compute 4. Compute dual, form Hessian Matrix H 5. Update 6. If , stop else repeat step 3-5 33
  33. 33. Algorithm 1. Rescale: domain area, user usages 2. Assign initial radius , target BS capacity 3. Compute power diagram, compute 4. Compute dual, form Hessian Matrix H 5. Update 6. If , stop else repeat step 3-5 34
  34. 34. Algorithm 1. Rescale: domain area, user usages 2. Assign initial radius , target BS capacity 3. Compute power diagram, compute 4. Compute dual, form Hessian Matrix H 5. Update 6. If , stop else repeat step 3-5 35
  35. 35. Algorithm 1. Rescale: domain area, user usages 2. Assign initial radius , target BS capacity 3. Compute power diagram, compute 4. Compute dual, form Hessian Matrix H 5. Update 6. If , stop else repeat step 3-5 36
  36. 36. Algorithm 1. Rescale: domain area, user usages 2. Assign initial radius , target BS capacity 3. Compute power diagram, compute 4. Compute dual, form Hessian Matrix H 5. Update 6. If , stop else repeat step 3-5 37
  37. 37. Iterative Process for Mobile Setting •Mobile terminals move continuously •For two contiguous snapshots, is similar •In step 2, reuse previous for better guess 38
  38. 38. Evaluation 39
  39. 39. Evaluation Setting •Compared with LP & perfect matching •Optran by CGAL in C++ •LP by Gurobi, CPLEX •Terminals: ~ 8000 • Different terminal usage: 1~5 •BS: ~ 2000 • Different BS capacity 40
  40. 40. Comparison: •Weighted bipartite perfect matching • Treat BS yj as c(yj) copy • Edge weight: cost(x,y)2 • Hungarian Alg. • Time: O(n3) • Variable: O(n2) 41 C(y1) C(y2) C(y3) x1 x2 x3 x4 x5
  41. 41. Comparison: (conti.) •Linear Programming (LP): • Time: O(n>3.5) • Variable : O(nk) 42 y1 y2 … yk x1 x2 … xn
  42. 42. Optran Result (700 users) 43
  43. 43. Optran Result (4k users) 44
  44. 44. Computation Time 45 ~116 sec 1500x Faster! ~0.1 sec
  45. 45. Time V.S. #Base Stations (w/ 1000 users) 46
  46. 46. Time V.S. #Base Stations 47 Linear Growth
  47. 47. Energy Consumption (dist2 Sum) 48
  48. 48. Mobile Case Scenario (2k users) 49
  49. 49. Mobile Case Time Compare 50
  50. 50. Conclusion •Optimal Transportation Theory & Capacitated Kinect Clustering •Power Diagram •Magnitude faster computation • Less variable: O(k) v.s. O(nk) for LP •Flexible for usage and capacity constraint •Suitable for mobile case 51
  51. 51. Question? Thank you for listening. Contact: Chien-Chun Ni, chni@cs.stonybrook.edu 52

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