The document discusses the parallel complexity of the Minimum Sum of Diameters Clustering Problem (MSDCP). It begins with introducing the MSDCP and basic concepts in parallel complexity theory. Then, it outlines the authors' contributions which include showing that for partitioning into two clusters, the MSDCP is solvable in polylogarithmic parallel time using a polynomial number of processors, meaning it is in class NC. The authors also present a more practical parallel algorithm for the problem that runs in logarithmic cube time using a sub-polynomial number of processors. In conclusion, the work analyzes the MSDCP through the lens of parallel computation to better understand its theoretical and practical complexity.

1. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
.
......
On the Parallel Complexity of Minimum Sum
of Diameters Clustering
Nopadon Juneam 1,2 and Sanpawat Kantabutra 2
1Department of Computer Science, Chiang Mai University, Chiang Mai, Thailand
2The Theory of Computation Group, Department of Computer Engineering
Chiang Mai University, Chiang Mai, Thailand
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 1 / 19

2. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Outline
1 Introduction
Minimum Sum of Diameters Clustering Problem
(MSDCP)
2 Basic Parallel Complexity Theory
Basic Concepts
3 Our Contributions
Parallel Complexity
More Practical Parallel Algorithm
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 2 / 19

3. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Outline
1 Introduction
Minimum Sum of Diameters Clustering Problem
(MSDCP)
2 Basic Parallel Complexity Theory
Basic Concepts
3 Our Contributions
Parallel Complexity
More Practical Parallel Algorithm
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 3 / 19

4. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Background
Conceptually, a cluster is a collection of similar entities, where
entities within the same cluster are more similar to each other than
to those in different clusters.
A basic problem of clustering is to partition a given finite set of
entities into clusters.
Applications of clustering are known in a variety of areas such as
data mining, machine learning, pattern recognition, bioinformatics
and social networking.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 4 / 19

5. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Basic Problem of Clustering
Given a set of points on the plane.
(e.g. climate data such as GPS locations)
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 5 / 19

6. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Basic Problem of Clustering (cont.)
The set of points partitioned into three clusters.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 6 / 19

7. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Minimum Sum of Diameters Clustering
Problem (MSDCP)
Objective: Partition a given set of entities into clusters such that the sum
of clusters’ diameters is minimized.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 7 / 19

8. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Minimum Sum of Diameters Clustering
Problem (MSDCP) (cont.)
“In this work, we studied the complexity of the MSDCP
on a parallel computer.”
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 8 / 19

9. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Outline
1 Introduction
Minimum Sum of Diameters Clustering Problem
(MSDCP)
2 Basic Parallel Complexity Theory
Basic Concepts
3 Our Contributions
Parallel Complexity
More Practical Parallel Algorithm
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 9 / 19

10. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Key Concepts
Researches in computational complexity theory are centered on the
question how difficult it is to solve a computational problem in the
term of some computational resources.
On a parallel computer, important computational resources are the
computational time and the number of processors.
Problems can obtain some speed improvement while being solved
on a parallel computer. However, a significant speed up may not
possible for every problem.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 10 / 19

11. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Time Order
source: https://apelbaum.wordpress.com/2011/05/05/big-o/
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 11 / 19

12. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Theory of P-Completeness [2]
The parallel complexity class NC contains all the problems that
have a polylogarithmic-time parallel algorithm using a polynomial
number of processors.
Such a parallel algorithm is considered as a highly parallel solution,
or efficient parallel algorithm for the problem.
In contrast, the parallel complexity class P-complete contains all
the problems for which no highly parallel solution is known; up to
some technicality.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 12 / 19

13. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Research Question
“Whether the MSDCP is in NC or P-complete? If it is in NC,
then how good its highly parallel solution can be?
in the use of parallel algorithms in practice.”
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 13 / 19

14. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Outline
1 Introduction
Minimum Sum of Diameters Clustering Problem
(MSDCP)
2 Basic Parallel Complexity Theory
Basic Concepts
3 Our Contributions
Parallel Complexity
More Practical Parallel Algorithm
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 14 / 19

15. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Previous Studies
For the case of a partition into three or more clusters
In 1978, Brucker showed that the MSDCP is NP-hard [1]
The problem is extremely difficult.
An exact solution to the problem requires at least exponential time.
For the case of a partition into two clusters
In 1987, Hansen and Jaumard gave an algorithm in O(n3
log n)
time [3].
The problem becomes easy.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 15 / 19

16. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Our Contributions
For the case of a partition into two clusters, we show that the
MSDCP is in class NC. That is, the problem has highly parallel
solutions.
1 The problem can be fasteastly solved in O(log n) parallel time
and n6
processors on the Common CRCW PRAM model
of parallel computer.
2 A more practical NC algorithm can be implemented in
O(log3
n) parallel time using n3.376
processors on the EREW
PRAM model of parallel computer.
In addition, these results will be published in [4].
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 16 / 19

17. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
References
P. Brucker.
On the complexity of clustering problems.
In R. Henn, B. Korte, and W. Oettli, editors, Optimization and
Operations Research, volume 157 of Lecture Notes in Economics and
Mathematical Systems, pages 45–54. Springer Berlin Heidelberg, 1978.
R. Greenlaw, H. J. Hoover, and W. L. Ruzzo.
Limits to Parallel Computation: P-completeness Theory.
Oxford University Press, Inc., New York, NY, USA, 1995.
P. Hansen and B. Jaumard.
Minimum sum of diameters clustering.
Journal of Classification, 4(2):215–226, 1987.
N. Juneam and S. Kantabutra.
On the parallel complexity of minimum sum of diameters clustering.
Journal of Internet Technology, 2017 (in press).
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 17 / 19

18. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Acknowledgment
Financial support from the Thailand Research Fund through
the Royal Golden Jubilee Ph.D. Program
(Grant No. PHD/0267/2552) is acknowledged.
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 18 / 19

19. .....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
....
.
....
.
.....
.
....
.
.....
.
....
.
....
.
Thanks You
juneam.n@gmail.com
Nopadon Juneam (Chiang Mai University) 114th RGJ Seminar Series April 29, 2016 19 / 19