Presented in INFOCOM 2016 http://www3.cs.stonybrook.edu/~chni/publication/optran/ -- 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.

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
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2. A Simple Case: Assign Terminal to BS
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3. + Energy Efficient
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4. +BS Station Capacity
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5. +BS Station Capacity
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6. +BS Station Capacity
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Drop Signal
Congestion
…

7. +BS Station Capacity (reassign)
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8. +Terminal Usage
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9. +Mobility
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10. Capacitated Kinetic Clustering Problem
•Assign terminals to corresponding base station
•Energy efficiency
•Capacity constraints
• Terminal data usage
• Base station capacity
•Mobility
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11. Problem Definition
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k BSs , k << nn Terminals
Capacity: Usage:
Energy cost

12. Problem Definition
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k BSs , k << nn Terminals
Capacity: Usage:
Energy cost
T: X -> Y
Min energy cost
Fit BS capacity

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
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14. Optimal Transportation Problem?
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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?”
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16. Kantorovich’s Approach(1942)
• Transportation plan , solved by LP
•
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y1 y2 … yk
x1
x2
…
xn

17. Brenier’s Approach
• Geometric approach
• Cut domain into different cell w/ required area
• Move cell elements to factory
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18. Brenier’s Approach
• Geometric approach
• Cut domain into different cell w/ required area
• Move cell elements to factory
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19. Capacitated Kinect Clustering:
OTP by Geometric Approach
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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?
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21. Construct Convex Function
•
•“Lift” factories to Hyperplanes
• Convex function:
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22. Geometric Meaning of u
• is adjustable, project back get
•
• equals to power diagram
2323

23. Energy Function*
• Energy function: volume of a cylinder
•Gradient:
• Since Energy function is convex, global minimum
occurred at
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*: Gu et al. “Variational principles for
Minkowski type problems, discrete optimal
transport, and discrete Monge-Ampere
equations ”

24. Finding
• , gives optimal transport map
•Hessian of on dual
•Iteratively applying Newton’s method
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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]
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[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. Voronoi Diagram
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27. P
Power Diagram
•Power diagram (Generalized Voronoi Diagram):
• Point weight: circle with radius
• Power distance:
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AB Q
r
PA
PB
PD(P,Q) = PA * PB = PQ2 - r2

28. Power Diagram
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29. Revisit OTP by Power Diagram
•Finding a radius for each factory st.
• Cell area match the requirement
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30. Connect OTP to Capacitated Kinect Clustering
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• “Move” data from terminal to BS
• Find that meet BS’s capacity

31. Connect OTP to Capacitated Kinect Clustering
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• “Move” data from terminal to BS
• Find that meet BS’s capacity

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
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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
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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
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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
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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
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37. Iterative Process for Mobile Setting
•Mobile terminals move continuously
•For two contiguous snapshots, is similar
•In step 2, reuse previous for better guess
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38. Evaluation
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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
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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)
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C(y1)
C(y2)
C(y3)
x1
x2
x3
x4
x5

41. Comparison: (conti.)
•Linear Programming (LP):
• Time: O(n>3.5)
• Variable : O(nk)
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y1 y2 … yk
x1
x2
…
xn

42. Optran Result (700 users)
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43. Optran Result (4k users)
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44. Computation Time
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~116 sec
1500x Faster!
~0.1 sec

45. Time V.S. #Base Stations (w/ 1000 users)
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46. Time V.S. #Base Stations
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Linear Growth

47. Energy Consumption (dist2 Sum)
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48. Mobile Case Scenario (2k users)
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49. Mobile Case Time Compare
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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
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51. Question?
Thank you for listening.
Contact:
Chien-Chun Ni, chni@cs.stonybrook.edu
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