=Paper= {{Paper |id=Vol-1513/paper-04 |storemode=property |title=Partition Around Medoids Clustering on the Intel Xeon Phi Many-core Coprocessor |pdfUrl=https://ceur-ws.org/Vol-1513/paper-04.pdf |volume=Vol-1513 |authors=Timofey V. Rechkalov }} ==Partition Around Medoids Clustering on the Intel Xeon Phi Many-core Coprocessor== https://ceur-ws.org/Vol-1513/paper-04.pdf

Partition Around Medoids Clustering on the
Intel Xeon Phi Many-Core Coprocessor

Timofey V. Rechkalov

South Ural State University, Chelyabinsk, Russia
trechkalov@yandex.ru

Abstract. The paper touches upon the problem of implementation Par-
tition Around Medoids (PAM) clustering algorithm for the Intel Many
Integrated Core architecture. PAM is a form of well-known k-Medoids
clustering algorithm and is applied in various subject domains, e.g. bioin-
formatics, text analysis, intelligent transportation systems, etc. An op-
timized version of PAM for the Intel Xeon Phi coprocessor is introduced
where OpenMP parallelizing technology, loop vectorization, tiling tech-
nique and efficient distance matrix computation for Euclidean metric are
used. Experimental results for different data sets confirm the efficiency
of the proposed algorithm.

Keywords: data mining · clustering · k-Medoids · Partition Around
Medoids · Intel Many Integrated Core architecture · Intel Xeon Phi co-
processor · parallel computing · tiling · vectorization · OpenMP

1 Introduction
Clustering is one of the basic problems of data mining aimed to organizing a set
of data objects into subsets (clusters) such that objects in a cluster are similar to
one another, yet dissimilar to objects in other clusters. Similarity is commonly
defined in terms of how close the objects are and is based on a specified distance
metric.
The most fundamental method of clustering is partitioning, which organizes
the objects of a set into several exclusive groups. More formally, given a set of
n objects, a partitioning algorithm constructs k partitions of the data, where
each partition represents a cluster and k ≤ n. The algorithm divides the data
objects into k clusters. An object is assigned to a closest cluster based on the
distance measure between the object and the cluster center. Then algorithm
iteratively improves the within-cluster variation by computing the new cluster
center using the objects assigned to the cluster in the previous iteration. After
This work was financially supported by the Ministry of education and science
of the Russian Federation (“Research and development on priority directions of
scientific-technological complex of Russia for 2014–2020” Federal Program, contract
No. 14.574.21.0035).
30 Timofey V. Rechkalov

this cluster centers are updated and all the objects are then reassigned using the
new cluster centers. The iterations continue until the assignment is stable, that
is, the clusters formed in the current round are the same as those formed in the
previous round.
Partitioning clustering algorithms differ in a way of calculation cluster cen-
ters, e.g. k-Means [11] and k-Modes [5] algorithms uses mean and mode values
of clustered objects respectively, whereas k-Medoids algorithm uses an object of
clustered data set (called medoid ).
The Partition Around Medoids (PAM) [18] is a variation of k-Means, which
is used in a wide spectrum of applications, e.g. text analysis, bioinformatics,
intelligent transport systems, etc. The complexity of each iteration in the PAM
algorithm is O(k(n − k)2 ). For large values of n and k computations are very
costly. That is why there are approaches to speed up k-Means and PAM al-
gorithms by means of GPU [3, 10]. At the same time there none for modern
accelerators based on the Intel Many Integrated Core (MIC) [8] architecture. In
spite of many recent developments for manycore platforms in data mining [13,
15, 23] and databases [1, 7] there are few ones for the Intel MIC architecture.
In this paper we present a parallel version of PAM for MIC accelerators. The
remaining part of the paper is organized as follows. Section 2 gives an overview
of serial PAM algorithm and discusses related work. In section 3 we describe
parallelization of PAM adapted for the Intel MIC architecture. The results of
the experiments evaluating the algorithm are presented in section 4. Section 5
contains summary and directions for future research.

2 Background of the Research

2.1 Serial PAM Algorithm

To provide formal description of the PAM [9] algorithm we will use the following
notation. Let O = {o1 , o2 , . . . , on } is a set of objects to be clustered where each
object is a tuple consisting of p real-valued attributes. Let k is the number
of clusters, k n, and C = {c1 , c2 , . . . , ck } is a set of medoids, C⊂O, and
ρ : O × C → R is a distance metric.
The algorithm takes the form of a steepest ascent hill climber, using a simple
swap neighbourhood operation. In each iteration medoid object ci and non-
medoid object oj are selected that produce the best clustering when their roles
are switched. The objective function used is the sum of the distances from each
object to the closest medoid:
n
X
E= min ρ(ci , oj ). (1)
1≤i≤k
j=1

Algorithm 1 depicts PAM pseudocode. PAM consists of two phases, namely
BUILD and SWAP. In the first phase an initial clustering is obtained by the
successive selection of representative objects until k objects have been found.
Partition Around Medoids Clustering on the Intel Xeon Phi 31

Input : Set of objects O, number of clusters k
Output: Set of k clusters
1 Init C ; /* BUILD phase */
2 repeat /* SWAP phase */
3 Calculate Tmin ;
4 Swap cmin omin ;
5 until Tmin < 0;

Fig. 1. PAM

The first object c1 is the one for which the sum of the distances to all other
objects is as small as possible:
n
X
c1 = arg min ρ(oh , oj ). (2)
1≤h≤n j=1

Object c1 is the most centrally located in O set. Subsequently, at each step
another object is selected, which decreases the objective function as much as
possible. This object is the one for which the minimal distance to all selected
medoids and distance to this object is as small as possible:

n
X
c2 = arg min min(ρ(c1 , oj ), ρ(oh , oj )), (3)
1≤h≤n j=1

Xn
c3 = arg min min( min (ρ(cl , oj )), ρ(oh , oj )), (4)
1≤h≤n j=1 1≤l≤2
...
n
X
ck = arg min min( min (ρ(cl , oj )), ρ(oh , oj )). (5)
1≤h≤n j=1 1≤l≤k−1

This process is continued until k objects have been found.
In the second phase of the algorithm, it is attempted to improve C (i.e.
set of medoids) and therefore also to improve the clustering yielded by this
set. Algorithm searches for a pair of objects (cmin , omin ), which minimizes the
objective function. This is done by considering all pairs of objects (ci , oh ) where
ci is a medoid and oh is not a medoid. It is determined what effect is obtained
on the objective function when a swap is carried out, i.e., when object ci is no
longer selected as a medoid but object oh is. Let denote this effect as Tih , then
minimum value of Tmin is achieved with (cmin , omin ) pair. If Tmin > 0 then C
set can not be improved so the algorithm stops.
Let us consider calculation of the Tih effect using the following notation. Let
D = {d1 , d2 , . . . , dn } is a set of distances from each object to the closest medoid.
Let S = {s1 , s2 , . . . , sn } is a set of distances from each object to second closest
medoid. Let Cjih is a contribution of non selected object oj to the effect Tih of
32 Timofey V. Rechkalov

a swap between ci and oh on the objective function. In this case Tih is the sum
of the contributions Cjih :
n
X
Tih = Cjih . (6)
j=1

Algorithm 2 [9] depicts pseudocode of calculating Cjih .

Input : oj , ci , oh , dj , sj
Output: Cjih
1 if ρ(oj , ci ) > dj and ρ(oj , oh ) > dj then
2 Cjih ← 0
3 else if ρ(oj , ci ) = dj then
4 if ρ(oj , oh ) < sj then
5 Cjih ← ρ(oj , oh ) − dj
6 else
7 Cjih ← sj − dj
8 end
9 else if ρ(oj , oh ) < dj then
10 Cjih ← ρ(oj , oh ) − dj
11 end

Fig. 2. Calculating Cjih

2.2 Related Work

A significant amount of work has been done in the area of cluster analysis. The
classical k-Means and k-Medoids algorithms was suggested in [5, 11]. The original
PAM algorithm was proposed in [9].
The research devoted to accelerating clustering algorithms using parallel
hardware includes the following. In [6] FPGA and GPU implementations of
k-Means are compared. Authors of [20] describe improvements of k-Means re-
ducing data transfers between CPU and GPU. In [21] a technique improving
data distribution among GPU threads in k-Means is suggested. k-Means imple-
mentation for Hadoop framework with GPUs is described in [22]. In [3] several
clustering methods on GPU including k-Medoids are implemented. A GPU-based
framework for clustering genetic data using k-Medoids described in [10].
In our opinion currently the potential of the Intel MIC accelerators for clus-
ter analysis is underestimated. Paper [16] proposes modification of the DBSCAN
density-based clustering algorithm for the Intel MIC architecture. In [19] a ver-
sion of k-Means for CPU and Intel MIC heterogeneous architecture is presented,
where authors used vectorization and sophisticated layout scheme to improve
data locality. The contribution of this paper is technique of acceleration of
Partition Around Medoids Clustering on the Intel Xeon Phi 33

the Partitioning Around Medoids clustering algorithm with the Intel Xeon Phi
many-core coprocessor.

3 Parallel PAM Algorithm for MIC Accelerators
In this section we describe an approach to implementation of PAM algorithm for
the Intel Xeon Phi coprocessor [17]. The Intel Xeon Phi coprocessor is an x86-
based SMP-on-a-chip with over fifty cores. It supports 4× hardware threads per
core and contains 512-bit wide vector processor unit (VPU). Each core has two
levels of cache memory: a 32 Kb L1 data cache, a 32 Kb L1 instruction cache,
and a core-private 512 Kb unified L2 cache. The Intel Xeon Phi coprocessor
is connected to other devices via the PCIe bus. Intel Xeon Phi coprocessor is
based on Intel x86 architecture and it supports the same programming tools and
models as a regular Intel Xeon processor. Our approach is based on the following
principles.
Data parallelism and vectorization. Using OpenMP technology we perform si-
multaneous execution on multiple cores of the same function across the elements
of a dataset. Most loops of the original PAM algorithm with arithmetic oper-
ations were implemented to provide conversion of such operations from scalar
form to vector form to be effectively computed by the coprocessor’s VPUs.
Our implementation strives to provide data locality as much as possible, i.e.
the program uses data close to recently accessed locations. Since the coprocessor
loads a chunk of memory around an accessed location into the cache, locations
close to recently accessed locations are also likely to be in the cache so finally it
increases algorithm’s performance.
Algorithm 3 depicts PAM pseudocode adapted for use on the Intel Xeon Phi
many-core coprocessor.

Input : Set of objects O, number of clusters k
Output: Set of C clusters
1 Offload O, k from CPU to coprocessor;
2 M ← P repareDistanceM atrix(O);
3 C ← BuildM edoids(M ) ; /* BUILD phase */
4 repeat /* SWAP phase */
5 Tmin ← F indBestSwap(M, C) ;
6 Swap cmin and omin ;
7 until Tmin < 0;
8 Offload C from coprocessor to CPU;

Fig. 3. Parallel PAM for Intel Xeon Phi coprocessor

The summary of parallel PAM subalgorithms is presented in Tab. 1.
To improve performance we use precomputing technique by means of calcu-
lating distances between all objects of O set in advance. There is no need for
34 Timofey V. Rechkalov

Table 1. Summary of parallel PAM subalgorithms

Name Complexity Parallelizing technique(s)
PrepareDistanceMatrix O(pn2 ) OpenMP, vectorization
BuildMedoids O(kn2 ) OpenMP, vectorization
FindBestSwap O(k(n − k)2 ) OpenMP

repeated calculation of distances at each iteration, since distances simply can be
looked up in M matrix.
The PAM algorithm deals with a lot of data arrays which are not fit into Intel
Xeon Phi L2 memory cache. We process data by chunks of L bytes to satisfy data
locality requirement. It is recommended [8] to set L to 16 and try multiplying
or dividing by 2 and use n divisible by L. In our work we use L = 32.
The PrepareDistanceMatrix subalgorithm initializes distance matrix (see Al-
gorithm 4). Unlike in [9] we store matrix in full form (not in upper triangular
form) to provide better data locality for the rest of subalgorithms. To achieve
better performance of this subalgorithm we use tiling technique [8].

Input : Set of objects O
Output: Distance matrix M
1 parallel for oi such that 1 ≤ i ≤ n do
2 for j = 1 to n step L do
3 for k = 1 to p do
4 for l such that j ≤ l ≤ j + L do /* vectorized */
5 mil ← mil + (oi [k] − ol [k])2 ; /* access to ol is tiled */
6 end
7 end
8 for l such that j ≤ l ≤ j + L do /* vectorized */
√
9 mil ← mil ;
10 end
11 end
12 endfor

Fig. 4. Prepare Distance Matrix

Tiling is a technique for improving data reuse in cache architectures. Cache
architectures generally employ least recently used (LRU) methods to determine
which data is evicted from the cache as new data is requested. Therefore, the
longer data remains unused, the more likely it will be evicted from the cache
and no longer available immediately when needed. Tiling the access pattern can
exploit data that remains in the cache from recent, previous iterations.
The BuildMedoids subalgorithm implements BUILD phase (see Alg. 5) ac-
cording to formulas (2)–(5). The FindBestSwap subalgorithm implements SWAP
phase (see Alg. 6). It checks all pairs of (ci , oh ) objects where ci is a medoid and
Partition Around Medoids Clustering on the Intel Xeon Phi 35

oh is not a medoid, calculates the effect for each Tih swapping and returns the
minimal one.

Input : Distance matrix M
Output: Set of medoids C
1 parallel for i = 1 to n do
Pn
2 if mij is minimal then /* sum is vectorized */
j=1
3 c1 ← oi ;
4 end
5 endfor
6 Init D distances to nearest medoid;
7 for l = 2 to k do
8 parallel for i = 1 to n do
Pn
9 if min(dj , mij ) is minimal then /* sum is vectorized */
j=1
10 c l ← oi ;
11 end
12 endfor
13 Update D;
14 end

Fig. 5. BUILD phase

Input : Distance matrix M , set of medoids C
Output: Tmin
1 Init T array of swap effects;
2 parallel for oh such that 1 ≤ h ≤ n and oh is not a medoid do
3 for l = 1 n L do
4 for i = 1 k do
l+L
P
5 Tih ← Tih + Cjih ;
j=l
6 end
7 end
8 endfor
9 Tmin ← min Tih ;
1≤h≤n,1≤i≤k

Fig. 6. SWAP phase

SWAP phase executes many logical operations in (see Alg. 2). By this reason
two versions of the PAM algorithm were implemented. PAM-1 executes more
logical operations with lesser temporary data. PAM-2 executes lesser logical op-
36 Timofey V. Rechkalov

erations but stores more temporary data. Preparing matrix function and BUILD
phase are the same in both versions.

4 Experimental Evaluation
To evaluate the developed algorithm we performed experiments on the hardware
specified in Tab. 2. Experiments were performed on single precision data, the
coprocessor was used in offload mode. We measured PAM runtime while varying
number of clustered objects and investigated the influence of dataset properties
on runtime of PAM subalgorithms.

Table 2. Specifications of hardware

Specifications Processor Coprocessor
Model Xeon X5680 Xeon Phi SE10X
Cores 6 61
Frequency, GHz 3.33 1.1
Threads per core 2 4
Peak performance, TFLOPS 0.371 1.076

Datasets used in experiments are summarized in Tab. 3.

Table 3. Datasets Summary

n, ×210
Dataset p k Max data size, Mb Time to transfer to
min max coprocessor, sec
FCS Human [2] 423 10 2 18 29.74 0.005
Corel Image Histogram [14] 32 15 5 35 4.38 0.001
MixSim [12] 5 10 5 35 0.68 0.001
Letter Recognition [4] 16 26 2 18 1.13 0.001

Experimental results for FCS Human dataset are introduced in Fig. 7(a). FCS
Human dataset has large dimension so the most time is taken by calculation of
distance matrix. Calculation of distance matrix on the Intel Xeon Phi is two
times faster then on the Intel Xeon. There is no significant difference between
PAM-1 and PAM-2 for this dataset.
Experimental results for Corel Image Histogram dataset are introduced in
Fig. 7(b). Data dimension is small so preparing distance matrix does not require
much time. PAM-1 shows similar performance on both CPU and Intel Xeon Phi.
The PAM-2 algorithm is two times slower on the Intel Xeon than on the Intel
Xeon Phi.
Experimental results for MixSim dataset are introduced in Fig. 7(c). Again
PAM-1 shows similar performance on both CPU and the Intel Xeon Phi. PAM-2
on the Intel Xeon Phi shows best result on this dataset.
Partition Around Medoids Clustering on the Intel Xeon Phi 37

(a) FCS Human dataset (b) Corel Image Histogram dataset

Fig. 7. Performance of the PAM algorithm

Experimental results for Letter Recognition dataset are introduced in
Fig. 7(d). PAM-1 shows the best result on the Intel Xeon. Both PAM-1 and
PAM-2 shows similar results on the Intel Xeon Phi.
Intuitively PAM-2 is a better implementation for the Intel Xeon Phi. This
suggestion is confirmed by experiments. In all tests PAM-2 is twice better on the
Intel Xeon Phi than the Intel Xeon. In the same time PAM-1 is the best with
the Intel Xeon only once. In other tests there is no significant difference.
To investigate this fact deeper we made more experiments to see contribution
of every PAM subalgorithm in Fig. 8. Figures 8(a) and 8(c) show time of matrix
calculation and BUILD phase. The Intel Xeon Phi outperforms the Intel Xeon
in both subalgorithms. Figures 8(b) and 8(d) show average time of one iteration
in SWAP phase. In these figures we can see that Intel Xeon Phi performance
degraded faster then Intel Xeon. PAM-2 implementation looses to PAM-1 in
38 Timofey V. Rechkalov

SWAP phase for big datasets so we need to continue PAM-2 improvements for
Intel Xeon Phi.

(a) MixSim: BUILD phase and prepare (b) MixSim: PAM-1 and PAM-2 itera-
distance matrix timings tion timings

(c) Letter Recognition: BUILD phase (d) Letter Recognition: PAM-1 and
and prepare distance matrix timings PAM-2 iteration timings

Fig. 8. Deep comparison of the PAM algorithm implementations

Experiments show that PAM performance depends on clustered data nature.
The most complex thing for large dimension data is calculation of distance ma-
trix. In case of small dimension data the rest of the PAM subalgorithms take
significantly larger part of runtime than distance matrix calculation. BUILD
phase is more effective on the Intel Xeon Phi. SWAP phase perform better on
the Intel Xeon. PAM execution on Letter Recognition dataset requires more it-
erations than MixSim experiment. By this reason PAM-1 shows best result with
Letter Recognition and PAM-2 shows best result with MixSim.
Partition Around Medoids Clustering on the Intel Xeon Phi 39

5 Conclusion

The paper has described a parallel version of Partitioning Around Medoids clus-
tering algorithm for the Intel Xeon Phi many-core coprocessor. An optimized
version of PAM for the Intel Xeon Phi coprocessor is introduced where OpenMP
parallelizing technology, loop vectorization, tiling technique and efficient dis-
tance matrix computation for Euclidean metric are used. Algorithm stores data
in continuous arrays and process data by chunks to achieve data locality for
better performance.
Experimental results show effectiveness of suggested approach. Experiments
show that PAM performance depends on clustered data nature. The most com-
plex thing for large dimension data is calculation of distance matrix. In case of
small dimension data the rest of the PAM subalgorithms take significantly larger
part of runtime than distance matrix calculation. BUILD phase is more effective
on the Intel Xeon Phi. SWAP phase perform better on the Intel Xeon. PAM-1
shows best result with Letter Recognition dataset and PAM-2 shows best result
with MixSim.
As future work we plan to extend our research in the following directions:
implement our algorithm for the cases of several coprocessors and cluster system
based on nodes equipped with the Intel Xeon Phi coprocessor(s).

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