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.
Module for Grade 9 for Asynchronous/Distance learning
Capacitated Kinetic Clustering in Mobile Networks by Optimal Transportation Theory
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
10. Capacitated Kinetic Clustering Problem
•Assign terminals to corresponding base station
•Energy efficiency
•Capacity constraints
• Terminal data usage
• Base station capacity
•Mobility
10
12. Problem Definition
12
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
13
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
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
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
24
*: 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
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.
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
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
31
• “Move” data from terminal to BS
• Find that meet BS’s capacity
31. Connect OTP to Capacitated Kinect Clustering
32
• “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
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. 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. 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. 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. Iterative Process for Mobile Setting
•Mobile terminals move continuously
•For two contiguous snapshots, is similar
•In step 2, reuse previous for better guess
38
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
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
Not only the BS have capacity constraint, but also each terminal have it’s individual usage.
Free space path loss
Before our solution, we talk about OTP
Ingredient => factory
Solution is a permutation => n!
Ingredient => factory
Solution is a permutation => n!
Ingredient => factory
Solution is a permutation => n!
----- Meeting Notes (8/6/13 03:13) -----
1-1
1- many many - 1
When Cost is Quadratic, solveable by geometric approach
To construct the convex function, the solution we use is Power Voronoi Diagram
construct a plane by h for each factory, then lift
----- Meeting Notes (8/6/13 03:13) -----
we want to find the requied area, what's the relation with u?
After getting u, we define a energy function to find the optimal sol
H0 is convex => energy func is convex, global
Image when h moves, the energy change, and when gradient =0, min
----- Meeting Notes (8/6/13 03:13) -----
because it's gradient map, and convex,
apply newton
To construct the convex function, the solution we use is Power Voronoi Diagram
Is a Partition, given points, partition space into cell
Cell closed to points
and it's dual is Delauney triangulation
Tanget line from point to the ball
if want a bigger cell, assign bigger radius
radius =>weight of points
find a way to move data to some BS
R==H!!!!!!!!!!!!
First consider each user only one msg
Separate user and BS side by side
Complete graph, assign edge weight
N2 to log which user assign to which