A Serial Multilevel Hypergraph Partitioning Algorithm

0.65 MB·

by  Foad Lotfifar

A Serial Multilevel Hypergraph Partitioning Algorithm
Foad Lotfifar · Matthew Johnson
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Abstract The graph partitioning problem has many edges (or hyperedges) can contain any number of ver-
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applications in scientific computing such as computer tices rather than just two. Thus hypergraphs are able

n aideddesign,datamining,imagecompressionandother to represent more complex relationships and model less
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applications with sparse-matrix vector multiplications structured data [7,23,44] and thus can, for some ap-
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as a kernel operation. In many cases it is advantageous plications in High Performance Computing, provide an
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to use hypergraphs as they, compared to graphs, have improved model. For this reason, hypergraph modelling

] a more general structure and can be used to model and hypergraph partitioning have become widely used
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more complex relationships between groups of objects. in the analysis of scientific applications [2,7,9,22,24,32,
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This motivates our focus on the less-studied hypergraph 42,47].
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s partitioning problem. We propose a Feature Extraction Hypergraph Parti-
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[ In this paper, we propose a serial multi-level bi- tioning (FEHG) algorithm for hypergraph partitioning
partitioning algorithm. One important step in current which makes novel use of the technique of rough set
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heuristics for hypergraph partitioning is clustering dur- clustering. We evaluate it in comparison with several
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6 ing which similar vertices must be recognized. This state-of-the-art algorithms.
3 can be particularly difficult in irregular hypergraphs
3 with high variation of vertex degree and hyperedge size;
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0 heuristics that rely on local vertex clustering decisions 1.1 The Hypergraph Partitioning Problem
. often give poor partitioning quality. A novel feature of
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theproposedalgorithmistousethetechniquesofrough We first define the problem formally. A hypergraph
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6 setclusteringtoaddressthisproblem.Weshowthatour H = (V,E) is a finite set of vertices V and a finite
1 proposedalgorithmgivesonaveragebetween18.8%and set E ⊆ 2V of hyperedges. Each hyperedge e ∈ E can
v: 71.1% better quality on these irregular hypergraphs by contain any number of vertices (constrast with a graph
i comparingittostate-of-the-arthypergraphpartitioning where every edge contains two vertices). Let e∈E and
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algorithms on benchmarks taken from real applications. v ∈V be a hyperedge and a vertex of H, respectively.
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a Keywords HypergraphPartitioning·LoadBalancing· Then hyperedge e is said to be incident to v or to
contain v, if v ∈e. This is denoted e(cid:46)v. The pair (cid:104)e,v(cid:105)
Multi-level Partitioning · Rough Set Clustering ·
is further called a pin of H. The degree of v is the
Recursive Bipartitioning
number of hyperedges incident to v and is denoted d(v).
The size or cardinality of a hyperedge e is the number
1 Introduction of vertices it contains and is denoted |e|. An example
of a hypergraph with 16 vertices and 16 hyperedges is
This paper is concerned with hypergraph partitioning. given in Fig. 1.
A hypergraph is a generalization of a graph in which Let ω:V (cid:55)→ N and γ:E (cid:55)→ N be functions defined
on the vertices and hyperedges of H. For v ∈ V and
SchoolofEngineeringandComputingSciences,DurhamUni-
e∈E,wecallω(v)andγ(e)theweights ofvande.Fora
versity,SouthRd,Durham,CountyDurhamDH13LE,United
Kingdom non-negativeintegerk,ak-waypartitioning ofH isacol-
E-mail:[email protected],[email protected] lection of non-empty disjoint sets Π={π ,π ,···,π }
1 2 k
2 Foad Lotfifar, MatthewJohnson
v13 now briefly give an overview of the methods that have
been developed.
e13
As graph partitioning is a well-studied problem and
e5 v10 e15 e4 e12
good heuristic algorithms are known [16] one might ex-
pecttosolvethehypergraphpartitioningproblemusing
e10 v1 e1 v2 v15 v4 v8 v12 graph partitioning. This requires a transformation of
thehypergraphintoagraphthatpreservesitsstructure.
v5 e7 e2 e3 e8 e16 There is no known algorithm for this purpose [25].
Move-based or flat algorithms start with an initial-
v7 e6 v9 v3 v6 v16 isation step in which vertices are assigned at random
to the parts. Then the algorithm builds a new prob-
lem solution based on the neighbourhood structure of
v14 e14
e9 the hypergraph [15,36]. The main weakness of these
algorithms is that while the solution found is locally
v11 e11 optimal,whetherornotitisalsogloballyoptimalseems
to depend on the density of the hypergraph and the
Fig. 1: A sample hypergraph H with 16 vertices and 16
initialrandomdistribution[21].SaabandRao[35]show
hyperedges.Verticesandhyperedgesarerepresentedassquare
that the performance of one such algorithm (the KL
and circular nodes, respectively. The weights of the vertices
andhyperedgesareassumed tobe1. algorithm) improves as the graph density increases.
Recursive bipartitioning algorithms generate a bi-
partitioning of the original hypergraph and are then
such that (cid:83)k π =V. If k =2, we call Π a bipartition.
i=1 i recursively applied independently to both parts until a
We say that vertex v ∈ V is assigned to a part π if
k-way partitioning is obtained [15,6,11]. Direct k-way
v ∈π. Furthermore, the weight of a part π ∈Π is the
algorithms calculate a k-way partitioning by working
sum of the weights of the vertices assigned to that part.
directly on the hypergraph [36]. There is no consensus
A hyperedge e∈E is said to be connected to (or spans
on which paradigm gives better partitioning. Karypis
on) the part π if e∩π (cid:54)=∅. The connectivity degree of a
reports some advantages of direct k-way partitioning
hyperedge e is the number of parts connected to e and
over recursive algorithms [20]. Cong et al. report that
is denoted λ (H,Π). A hyperedge is said to be cut if its
e direct algorithms are more likely to get stuck in local
connectivity degree is more than 1.
minima [8]. Aykanat et al. report that direct algorithms
We define the partitioning cost of a k-way partition-
give better results for large k [4]. In fact, recursive al-
ing Π of a hypergraph H:
gorithms are widely used in practice and are seen to
achieve good performance and quality.
(cid:88)
cost(H,Π)= (γ(e)·(λ (H,Π)−1)) (1) We mention in passing that other algorithms con-
e
e∈E sider the hypergraph partitioning problem with further
The imbalance tolerance is a real number (cid:15)∈(0,1). restrictions. Multi-constraint algorithms, are used when
Thenak-waypartitioningΠofahypergraphH satisfies more than one partitioning objective is defined by as-
the balance constraint if signingaweightvectortothevertices[4].Inthecontext
ofVLSIcircuitpartitioning,thereareterminals,suchas
I/O chips, that are fixed and cannot be moved. There-
W ·(1−(cid:15))(cid:54)ω(π)(cid:54)W ·(1+(cid:15)), ∀π ∈Π (2)
fore when partitioning these circuits some vertices are
(cid:80) fixedandmustbeassignedtoparticularparts.Problems
where W = ω(v)/k is the average weight of a
v∈V of this type are easier to solve and faster running times
part.
can be achieved [1,6,4].
The Hypergraph Partitioning Problem is to find a
Inthispaper,weintroduceamulti-level algorithmso
k-way partitioning with minimum cost subject to the
let us describe this general approach. These algorithms
satisfaction of of the balance constraint.
have three distinct phases: coarsening, initial partition-
ing and uncoarsening. During coarsening vertices are
1.2 Heuristics for Hypergraph Partitioning mergedtoobtainhypergraphswithprogressivelysmaller
vertex sets. After the coarsening stage, the partitioning
The Hypergraph Partitioning Problem is known to be problem is solved on the resulting smaller hypergraph.
NP-Hard [18], but many good heuristic algorithms have Then in the uncoarsening stage, the coarsening stage
been proposed to solvethe problem [6,11,15,28] andwe is reversed and the solution obtained on the small hy-
ASerialMultilevelHypergraphPartitioningAlgorithm 3
pergraph is used to provide a solution on the input ing phase and make better vertex clustering decisions
hypergraph. The coarsening phase is sometimes also and achieve better partitioning on the hypergraph.
known as the refinement phase. The FEHG algorithm proposed in this paper makes
novel use of the technique of rough set clustering to
During coarsening the focus is on finding clusters
categorise the vertices of a hypergraph in the coarsen-
of vertices to merge to form vertices in the coarser
ing phase. FEHG treats hyperedges as features of the
hypergraph. One wishes to merge vertices that are, in
hypergraph and tries to discard unimportant features
some sense, alike so a metric of similarity is required;
to make better clustering decisions. It also focuses on
examplesareEuclideandistance,HyperedgeCoarsening,
the trade-off to be made between local vertex matching
First Choice (FC), or the similarity metrics proposed
decisions (which have low cost in terms of the space
by Alpert et al. [3], and Catalyurek and Aykanat [7].
required and time taken) and global decisions (which
A problem encountered with high dimensional data
can be of better quality but have greater costs). The
sets is that the similarity between objects becomes very
emphasis of our algorithm is on the coarsening phase:
non-uniform and defining a similarity measure and find-
good vertex clustering decisions here can lead to high
ing similar vertices to put into one cluster is very dif-
quality partitioning [20].
ficult. This situation is more probable when the mean
and standard deviation of the vertex degrees increases
(see,forexample,theanalysisofEuclideandistanceasa 2 Preliminaries
similaritymeasuresbyErto¨zetal.[14]).Althoughother
measures, such as cosine measure and jaccard distance, 2.1 Rough Set Clustering
address the issue and resolve the problem to some ex-
tent, they also have limitations. For example, Steinbach Rough set clustering is a mathematical approach to
etal.[40]haveevaluatedthesemeasuresandfoundthat uncertainty and vagueness in data analysis. The idea
they fail to capture similarity between text documents was first introduced by Pawlak in 1991 [33]. The ap-
indocumentclusteringtechniques(usedinareassuchas proach is different from statistical approaches, where
text mining and information retrieval). The problem is the probability distribution of the data is needed, and
that cosine and jaccard distances emphasise the impor- fuzzy logic, where a degree of membership is required
tance of shared attributes for measuring similarity and for an object to be a member of a set or cluster. The
they ignore attributes that are not common to pairs of approach is based on the idea that every object in a
vertices. Consequently, other algorithms use other clus- universe is tied with some knowledge or attributes such
tering techniques to resolve the problem such as shared that objects which are tied to the same attributes are
nearest neighbour (SNN) methods[13]andglobalvertex indiscernible and can be put together in one category
clustering information. [34].
The data to be classified are called objects and they
Decisions for vertex clustering are made locally and
are described in an information system:
global decisions are avoided due to their high cost and
complexity though they give better results [43]. All pro-
Definition 1 (InformationSystem)Aninformation
posed heuristics reduce the search domain and try to
system is a system represented as I = (U,A,V,F)
findverticestobematchedusingsomedegreeofrandom-
where:
ness [11]. This degrades the quality of the partitioning
• U is a non-empty finite set of objects or the uni-
by increasing the possibility of getting stuck in a local
verse.
minimum and they are highly dependent on the order
theverticesareselectedformatching.Abettertrade-off • A is a non-empty finite set of attributes with size
is needed between the low cost of local decisions and |A|=t.
the high quality of global ones. • V=(V ,V ,...,V ) where, for each a∈A, V
1 2 t a
Furthermore,thereisadegreeofredundancyinmod- is the set of values for a.
elling scientific applications with hypergraphs. Remov- • F :U×A(cid:55)→V is a mapping function such that,
ingthisredundancycanhelpinsomeoptimisationssuch for u∈U,a∈A, F(u,a)∈V .
a
asclusteringdecisions,storageoverhead,andprocessing
requirements. An example is proposed by Heinz and For any subset of attributes B = {b ,b ,···,b } ⊆
1 2 j
Chandra [22] in which the hypergraph is transformed A, the B-Indiscernibility relation is defined as follow:
into a Hierarchy DAG (HDAG) representation and re-
ducingthememoryrequirementforstoringhypergraphs.
Similarly, one can identify redundancies in the coarsen- IND(B)=(cid:8)(u,v)∈U2 |∀b∈B, F(u,b)=F(v,b)(cid:9)
4 Foad Lotfifar, MatthewJohnson
(3) similarity measure that reduces the number of edges in
the HCG. Different similarity measures, such as Jaccard
When (u,v) ∈ IND(B), it is said that u and v are
Index or Cosine Measure, can be used for measuring
indiscernible under B. The equivalence class of u with
the similarity. As the hyperedges of the hypergraph are
respect to B is represented as [u] and includes all
B weighted, similarity between two hyperedges is scaled
objects which are indiscernible with u. The equivalence
according to the weight of hyperedges: for e ,e ∈ E
relation provides a partitioning of the universe U in i j
the scaling factor is γ(ei)+γ(ej) . One of the charac-
whicheachobjectbelongsonlyonepartanditisdenoted 2×maxe∈E(γ(e))
teristics of the HCG is that it assigns hyperedges to
U/IND(B) or simply U/B.
non-overlapping clusters (that is, the connected compo-
The set of attributes can contain some redundancy.
nents of the HCG).
Removing this redundancy could lead us to a better
clustering decisions and data categorisation while still
preserving the indiscernibility relation amongst the ob-
3 The Serial Partitioning Algorithm
jects[46,48].Theremainedattributesafterremovingthe
redundancy is called the reduct set [41]. More precisely,
TheproposedFeatureExtractionHypergraphPartition-
ifB⊆A,thenBisareductofAifIND(B)=IND(A)
ing(FEHG)algorithmisamulti-levelrecursiveserialbi-
and B is minimal (that is, no attribute can be removed
partitioningalgorithmcomposedofthreedistinctphases:
from B without changing the indiscernibility relation).
coarsening, initial partitioning, and uncoarsening. The
Butthereductisnotuniqueanditisknownthatfinding
emphasisofFEHG isonthecoarseningphaseasitisthe
a reduct of an information system is an NP-hard prob-
most important phase of the multi-level paradigm [20].
lem[39].Thisisoneofthecomputationalbottlenecksof
We provide a general outline of the algorithm before
rough set clustering. A number of heuristic algorithms
going into further detail. In the coarsening phase, we
have been proposed for problems where the number of
transfer the hypergraph into an information system and
attributes is small. Examples have been given by Wrob-
we use rough set clustering decisions to match pairs of
lewski using genetic algorithms [45,46], and by Ziarko
vertices. This is done in several steps. First, the reduct
and Shan who use decision tables based on Boolean
set is found to reduce the size of the system and remove
algebra[48].Thesemethodsarenotapplicabletohyper-
superfluous information. Second, the vertices of the
graphs which usually represent applications with high
hypergraph are categorised using their indispensability
dimensionality and where the operations have to be
relations. These categories are denoted as core and non-
repeated several time during partitioning. We propose
core where, in some sense, vertices that belong to the
a relaxed feature reduction method for hypergraphs by
same core are good candidates to be matched together
defining the Hyperedge Connectivity Graph (HCG).
andthenon-coreverticesarewhatisleftover.Thecores
aretraversedbeforethenon-coresforfindingpair-vertex
2.2 The Hyperedge Connectivity Graph matches. The whole coarsening procedure is depicted in
Fig. 2.
The Hyperedge Connectivity Graph (HCG) of a hyper-
graph is the main tool used used in our algorithm for
removingsuperfluousandredundantinformationduring 3.1 The Coarsening
vertex clustering in the coarsening phase. The similarity
betweentwohyperedgese ande isdenotedsim(e ·e ) The first step of the coarsening stage is to transfer the
i j i j
(we will discuss later possible similarity measures). hypergraph H = (V,E) into an information system.
The information system representing the hypergraph is
Definition 2 (HyperedgeConnectivityGraph)For I =(V,E,V,F);thatis,theobjectsaretheverticesV
H
a given similarity threshold s ∈ (0,1), the Hyperedge and the attributes are the hyperedges E. We define the
ConnectivityGraph(HCG) ofahypergraph H =(V,E) set of values as each being in [0,1] and the mapping
isagraphGs =(V,E)whereV =E andtwohyperedges function is defined as:
e ,e ∈V are adjacent in Gs if sim(e ,e )(cid:62)s.
i j i j
Thedefinitionissimilartothatofintersectiongraphs f(e)
F(v,e)= , (4)
[12] (graphs that represent the intersections of a family (cid:80) γ(e(cid:48))
∀e(cid:48)(cid:46)v
of sets) or line graphs of hypergraphs (graphs whose
vertexsetisthesetofthehyperedgesofthehypergraph where f(e)=γ(e) if e(cid:46)v and is otherwise 0.
and two hyperedges are adjacent when their intersec- ThetransformationofthehypergraphgiveninFig.1
tion is non-empty). The difference is the presence of the into an information system is given in Table 1.
ASerialMultilevelHypergraphPartitioningAlgorithm 5
Transferthehypergraphintoaninformationsystem
Findredundantattributesandremovethem
Categoriseverticesascoreandnon-corevertices
Yes No
Cores?
Traverseeachcorelistandfindbestpairmatches Traversethenon-corelistandfindbestpairmatches
Fig. 2: The coarsening phase at a glance. The dashed arrow means that non-core vertex list is processed after all
cores have been processed.
Table 1: The transformation of the hypergraph H depicted in Fig. 1 into an information system. The values are
rounded to 2 decimal places.
e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16
v1 0.33 0 0 0 0.33 0 0 0 0 0.33 0 0 0 0 0 0
v2 0.25 0.25 0 0 0 0 0.25 0 0 0 0 0 0 0 0.25 0
v3 0 0 0.33 0 0 0 0.33 0 0.33 0 0 0 0 0 0 0
v4 0 0.16 0.16 0.16 0 0 0 0.16 0 0 0 0.16 0 0 0 0.16
v5 0.33 0 0 0 0 0 0.33 0 0 0.33 0 0 0 0 0 0
v6 0 0 0.33 0 0 0 0 0 0.33 0 0.33 0 0 0 0 0
v7 0 0 0 0 0 0.33 0.33 0 0 0 0 0 0 0.33 0 0
v8 0 0.2 0 0.2 0 0 0 0.2 0 0 0 0.2 0 0 0 0.2
v9 0 0 0 0 0 0.25 0 0 0.25 0 0.25 0 0 0.25 0 0
v10 0 0 0 0 0.33 0 0 0 0 0.33 0 0 0.33 0 0 0
v11 0 0 0 0 0 0 0 0 0 0 0.5 0 0 0.5 0 0
v12 0 0 0 0.33 0 0 0 0 0 0 0 0.33 0 0 0 0.33
v13 0 0 0 0 0.5 0 0 0 0 0 0 0 0.5 0 0 0
v14 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0 0 0
v15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0 0
v16 0 0 0 0.25 0 0 0 0.25 0 0 0 0.25 0 0 0 0.25
Calculating the HCG: Initially, hyperedges are not same edge partition are removed from the information
assigned to any cluster. Then, until all hyperedges are system I and replaced by their representative. After
H
assigned,ahyperedgeeisselected,anewclusternumber this reduction, the new information system is built and
isassignedtoitandtheclusterisdevelopedaroundeby represented as IR =(cid:0)V,ER,VR,FR(cid:1) in which the at-
H
processingitsadjacenthyperedgesusingDefinition2.At tribute set is replaced by ER. In the new information
the end of HCG calculation, each hyperedge is assigned system, the set of values is VR ⊆N. Furthermore, the
eR
to exactly one cluster. We refer to the cluster set as mapping function for ∀e ∈ER is redefined as follows:
R
edge partitions and denote it ER. The size and weight
ofeache ∈ER isthenumberofhyperedgesitcontains
R
and the sum of their weights, respectively. An example
of the HCG for the sample hypergraph in Fig. 1 and FR(v,e )=|{e(cid:46)v ∧ e∈e |e∈E }|. (5)
R R
similarity threshold s=0.5 is depicted in Fig. 3.
Hyperedgesbelongingtothesameedgepartitionare We can further reduce the set of attributes by picking
considered to be similar. For each edge partition, we the most important ones. For this purpose, we define a
choose a representative and all other hyperedges in the clustering threshold c∈[0,1] and the mapping function
6 Foad Lotfifar, MatthewJohnson
Table 2: The reduced information system that is built
u2
based on the HCG in Fig. 3.
u5 u13 u15 u12 u8
C C C C C C
C1 C2 C3 v1 31 02 03 04 05 06
u10 u1 u4 u16 v2 1 1 1 1 0 0
v 0 0 1 1 0 2
3
v 1 0 3 0 0 0
4
u6 u7 u9 v5 2 0 0 1 0 0
v 0 0 0 0 0 3
C5 C4 u11 C6 v67 0 0 0 1 2 0
u14 u3 v8 0 0 3 0 0 0
v 0 0 0 0 2 2
9
Fig. 3: AnexampleofHyperedgeConnectivityGraph(HCG) v 3 0 0 0 0 0
10
of the hypergraph H depicted in Fig. 1 and the similarity v 0 0 0 0 1 1
11
thresholds=0.5. v 0 0 3 0 0 0
12
v 2 0 0 0 0 0
13
v 0 0 0 0 1 0
14
is changed according to Eq.(6) below to construct the
v 0 1 0 0 0 0
15
final information system If.
v 0 0 1 0 0 0
16
(cid:40)1, if FR(v,eR) (cid:62)c Table 3: The final information system for clustering
Ff(v,e )= |{e(cid:46)v,∀e∈E}| (6) threshold c=0.5.
R
0, otherwise.
C C C C C C
1 2 3 4 5 6

An example of the reduced information system and vv1 11 00 00 00 00 00 
the final table using the clustering threshold c=0.5 for 5 Core 1
v 1 0 0 0 0 0
thehypergraphofFig.1isdepictedinTable3.Thefinal v10 1 0 0 0 0 0 
13
table is very sparse compared to the original table. At v 0 0 0 0 0 0
2
(cid:27)
this point, we use rough set clustering. For every vertex, v3 0 0 0 0 0 1 Core 2
v 0 0 0 0 0 1
we calculate its equivalence class. Then, a partitioning 6 
v 0 0 1 0 0 0
U/IND(ER) on the vertex set is obtained using the v4 0 0 1 0 0 0 
equivalence relations. We refer to parts in U/IND(ER) v8 0 0 1 0 0 0 Core 3
as cores such that each vertex belongs to a unique v1126 0 0 1 0 0 0 
(cid:27)
v 0 0 0 0 1 1
core. For some of the vertices in the hypergraph, the 9 Core 4
v 0 0 0 0 1 1
mapping function gives zero output for all attributes 11 (cid:27)
v 0 0 0 0 1 0
that is Ff(v,eR) = 0, ∀eR ∈ ER. These vertices are v174 0 0 0 0 1 0 Core 5
(cid:9)
assigned to a list denoted as the non-core vertex list. v15 0 1 0 0 0 0 Core 6
We observe that cores are built using global cluster-
inginformation.Thefinaloperationistomatchpairsof
better than Euclidean based similarity measures. Ver-
vertices.Coresarevisitedsequentiallyoneatatimeand
tices that do not find any pair matches during core
they are searched locally to find pair matches. Inside
searcharetransferredintothenon-corevertexlist(such
eachcore,avertexuisselectedatrandomandmatched
as v in Table 2).
with its “most similar” neighbour. Then the process is 15
One possible problem is that only a small fraction
repeated with the remaining vertices. These decisions
of the hypergraph’s vertices might belong to cores. This
are made with a local measure of similarity. We use the
can happen, for example, the average vertex degree in
Weighted Jaccard Index that is defined as follows:
the hypergraph is high so there is a large denominator
(cid:80) in Eq.(6).
γ(e)
{e(cid:46)v∧e(cid:46)u}
J(u,v)= (cid:80) , v,u∈V, and∀e∈E. As proposed by Karypis [20], we define the com-
γ(e)
{e(cid:46)v∨e(cid:46)u} pression ratio between two levels of coarsening in the
(7) multi-level paradigm with c levels as follows:
This is similar to non-weighted jaccard index in PaToH
(where it is called Scaled Heavy Connectivity Matching). |V |
r = i ,∀0(cid:54)i<c (8)
This captures similarity in high dimensional datasets |V |
i+1
ASerialMultilevelHypergraphPartitioningAlgorithm 7
An advantage of the multi-level approach is that Then partitioning algorithms refine the partitioning
it provides a trade-off between quality and speed-up. as the hypergraph is projected back in the uncoarsen-
The more coarsening levels, the better the partitioning ing phase. Due to the success of the FM algorithm in
quality, but the longer the running time and the greater practice, we use a variation of FM algorithm known
the memory consumption. On the other hand, with few as Early-Exit FM (FM-EE) [20] and Boundary FM
coarsening levels, we end up with a large hypergraph (BFM) [6].
in which it is not possible to find a good partitioning.
Having many coarsening levels will give a very small
4 Evaluations
coarsest hypergraph with perhaps few feasible solutions.
This may result in a poor partitioning quality. Karypis
Inthissectionweprovidetheevaluationofouralgorithm
[20]providesatradeoffbetweenqualityandlevelsofpar-
compared to state-of-the-art partitioning algorithms in-
titioning by limiting the compression ratio r to between
cluding PHG (the Zoltan hypergraph partitioner) [11],
1.5 and 1.8. In order to satisfy a certain compression
PaToH [6] and hMetis [26]. PaToH is a serial hyper-
ratio, we will visit the non-core vertex list and select
graph partitioner developed by C¸atalyu¨rek and it is
vertices randomly. For every selected vertex, the algo-
claimedtobethefastestmultilevelrecursivebipartition-
rithm finds a pair match among its unmatched adjacent
ing based tool. The earliest and the most popular tool
vertices in the non-core list.
for serial hypergraph partitioning is hMetis developed
When pair matches are found, the hypergraph is
byKarypisandKumar;itisspeciallydesignedforVLSI
contracted to build a coarser hypergraph for the next
circuitpartitioningandthealgorithmsarebasedonmul-
coarsening level. This is done by merging matched ver-
tilevel partitioning schemes and support recursive bisec-
tices. The weight of the coarser vertex is the sum of
tioning(shmetis andhmetis)anddirectk-waypartition-
the weight of two merged vertices and its set of incident
ing (kmetis). Finally, Zoltan data management services
hyperedges is the union of the hyperedges incident on
for parallel dynamic applications is a toolkit developed
each of the merged vertices. After building the coarser
at Sandia National Laboratories. The library includes
hypergraph, we perform two final operations on the hy-
a wide range of tools for problems such as Dynamic
peredge list. First, hyperedges of unit size are removed
Load Balancing, Graph/Hypergraph Colouring, Matrix
as they do not have any impact on the partitioning
Operations, Data Migration, Unstructured Communica-
cut. Second, identical hyperedges (that is, those hav-
tions, Distributed Directories and Graph/Hypergraph
ing the same vertex set) are detected and each set of
Partitioning.Itfollowsadistributedmemorymodeland
identical hyperedges is replaced by a single hyperedge
uses MPI for interprocessor communications. It is avail-
whose weight is the sum of the weight of hyperedges in
able within Trilinos, an open source software project
the set. To find identical hyperedges, each hyperedge is
for scientific applications, since version 9.0 [37].
hashed to an integer number based on its vertex list. To
Allofthesealgorithmsaremulti-levelrecursivebipar-
avoid hash conflicts, the content of the hyperedges are
titioning algorithms and PHG is a parallel hypergraph
compared if they hash to the same number.
partitioner while the other two are serial partitioning
tools.
3.2 Initial Partitioning and Refinement Fortheevaluation,wehaveselectedanumberoftest
hypergraphsfromavarietyofscientificapplicationswith
The coarsening phase ends when the number of the ver- different specifications. The hypergraphs are obtained
tices in the coarsest hypergraph is less than a threshold from the University of Florida Sparse Matrix Collection
(that is 100 in our algorithm). The size of the coarsest [10]. It is a large database of sparse matrices from real
hypergraph is very small compared to the original hy- applications. Each sparse matrix from the database is
pergraph and its partitioning can be calculated quickly. treated as the hypergraph incident matrix with the
We use a series of algorithms for this purpose. The best vertices and hyperedges as rows and columns of the
partitioning that preserves the balancing constraint is matrix, respectively. This is similar to the column-net
selected and it is projected back to the original hyper- model proposed in [7]. The weight of each vertex and
graph.Thecategoryofalgorithmsusedforthisstageare each hyperedge is assumed to be 1. The list of test data
random (randomly assigns vertices to the parts), linear used for our evaluation is in Table 4.
(linearly assigns vertices to the parts starting from a The simulations are done on a computer with In-
randompart),andFM Based (selectsavertexrandomly tel(R) Xeon(R) CPU E5-2650 2.00GHz processor, 8GB
and assigns it to part 1 and all other vertices to part 0; of RAM and 40GB of disk space and the operating
then the FM algorithm is run and the bipartitioning is system running on the system is 32-Bit Ubuntu 12.04
developed). LTS. Furthermore, we set the imbalance tolerance to
8 Foad Lotfifar, MatthewJohnson
Table 4: Tested hypergraphs forsequentialalgorithm simulationandtheir specifications
Hypergraph Description Rows Columns Non-Zeros NSCC1
CNR-2000 SmallwebcrawlofItalianCNRdomain 325,557 325,557 3,216,152 100,977
AS-22JULY06 Internetrouters 22,963 22,963 96,872 1
CELEGANSNEURAL NeuralNetworkofNematodeC.Elegans 297 297 2,345 57
NETSCIENCE Co-authorshipofscientistsinNetworkTheory 1,589 1,589 5,484 396
PGPGIANTCOMPO LargestconnectedcomponentingraphofPGPusers 10,680 10,680 48,632 1
GUPTA1 LinearProgrammingmatrix(A×AT) 31,802 31,802 2,164,210 1
MARK3JAC120 JacobianfromMULTIMODMark3 54,929 54,929 322,483 1,921
NOTREDAMEWWW Barabasi’swebpagenetworkofnd.edu 325,729 325,729 929,849 231,666
PATENTSMAIN Pajeknetwork:mainNBERUSPatentCitations 240,547 240,547 560,943 240,547
STD1JAC3 Chemicalprocesssimulation 21,982 21,982 1,455,374 1
COND-MAT-2005 Collaborationnetwork,www.arxiv.org 40,421 40,421 351,382 1,798
1NSCCstandsfortheNumberofStronglyConnectedComponents.
2% and the number of parts (the value of k) is each In graph theory, a set of vertices are said, informally,
of {2,4,8,16,32}. The final imbalances achieved by the tobeclustered iftheyinducemoreedges(orhyperedges)
algorithms are not reported because the balancing re- than one would expect if the edges had been placed
quirement was always met. in the graph at random. There are various ways in
which one can formally define a clustering coefficient
(CC) of either a vertex or a graph, and for hypergraphs
4.1 Algorithm Parameters various definitions have been proposed [19,29,30]. We
need something a little different as we are interested in
Each of the evaluated tools has different input parame- theclusteringofhyperedgesandwewanttotakeaccount
ters that can be set by the user. We use default param- of their weights. Given a hypergraph H = (V,E), we
eters for each tool. All algorithms use a variation of the define CC for a hyperedge e∈E as follows:
FM algorithm (FM-EE and BFM) in their refinement
CC(e) (10)
phase.PHG usesanagglomerative coarseningalgorithm
that uses the inner product as the measure of similarity (cid:80){e(cid:48)∩e(cid:54)=∅}(cid:18)(cid:18)||ee∩|−e(cid:48)1|(cid:19)·γ(e(cid:48))(cid:19), ∀e(cid:48),e(cid:48)(cid:48) ∈E\e, if |e|>1
between the vertices [11]. This is also used in hMetis. = (cid:80){v∈e}(cid:80){e(cid:48)(cid:48)(cid:46)v}γ(e(cid:48)(cid:48))
The default partitioning tool for hMetis is shmetis and 0, otherwise,
thedefaultcoarseningschemeistheHybrid First Choice
(HFC)schemewhichisacombinationoftheFirstChoice andtheCCofH iscalculatedastheaverageofCCover
(FC) [20] and Greedy First Choice scheme (which is a all its hyperedges:
variation of the FC algorithm in which vertices are
grouped and the grouping is biased in favor of faster
(cid:88) CC(e)
reductioninthenumberofthehyperedgesinthecoarser CCH = |E| . (11)
hypergraph).PaToH isinitialisedbysettingtheSBProb- e∈E
Type parameter to PATOH SUGPARAM DEFAULT. It
Duringcoarsening,eachtimeweproceedtothenext
uses Absorption Clustering using Pins as the default
coarseninglevel,thestructureofthehypergraphchanges
coarsening algorithm (this is an agglomerative vertex
and so the value of CC in the coarser hypergraph is dif-
clustering scheme). The similarity metric, known as ab-
ferent. To recalculate the CC value in each coarsening
sorption metric, of two vertices u and v is calculated
level could be costly so we are interested in finding a
as follows:
way of updating the CC value without a complete recal-
(cid:88) 1 culation (possibly sacrificing accuracy in the process).
. (9)
|e|−1 Foudalis et al. [17] studied the structure of social net-
{∀e∈E|u∈eandv∈e} work graphs and looked at several characteristic metrics
including the clustering coefficient. They found that
The algorithm finds the absorption metric for every
social networks have high CC compared to random net-
pin that connects u and the cluster that vertex v is
works, and that the CC is negatively correlated to the
already assigned to. See the manuals of the tools for the
degree of vertices. Two vertices with low vertex degree
full description of the parameters [6,27,38].
aremorelikelytoclustertoeachotherthantwovertices
The FEHG algorithm has two parameters that need
with higher degree. In addition, Bloznelis [5] has theo-
to be set: the similarity threshold in Definition 2 for
retically investigated random intersection graphs1 and
buildingtheHCG,andtheclusteringthresholdinEq.(6).
In the rest of this subsection, we look further at these 1 Random intersection graphs can be obtained from ran-
two parameters. domlygeneratedbipartitegraphswhichhavebipartitionV∪W
ASerialMultilevelHypergraphPartitioningAlgorithm 9
edge partitions C and C can be removed from the
2 4
reduced information system in Table 3 without causing
any changes in the cores. Using this strategy, the parti-
tioning cut for CNR-2000 is 29.9% better than the best
bipartitioningcutreportedinFig.4andthisisachieved
for c=0.6.
4.2 Simulations
In the first evaluation, we assume unit weight for both
vertices and hyperedges of the hypergraph. In this situ-
Fig. 4: The variation of bipartitioning cut based on the ation, a partitioning algorithms performs well if it can
clustering threshold. Values are normalised with the best cut
capture strongly connected components of the hyper-
foreachhypergraph.
graph. A strongly connected component is a group of
the vertices that are tightly coupled together. Assuming
shown that the CC is inversely related to the average that the hypergraph represents a graph (by limiting
vertex degree in the graph. Based on these results, we the hyperedge cardinality to two), a strongly connected
update the value of CC from one coarsening level to the component is a clique. Because the weight of all hy-
next based on the inverse of the average vertex degree. peredges is 1, the aim of the partitioning algorithm is
Finally, the similarity threshold is set as the CC of the to take these cliques out of the cut as they are the
hypergraph at the beginning of the coarsening phase. In major cause of increasing the cut; therefore, the ver-
thesimulationsection,weinvestigatehowouralgorithm tex connectivity is more important for identifying those
performs,intermsofpartitionqualityandrunningtime, cliques. An algorithm that identifies those cliques and
when the similarity threshold is updated compared to merged their vertices to build a coarser vertex is the
when it is recalculated in each coarsening level. one that gives better partitioning quality and the clus-
In order to investigate the impact of the clustering tering algorithm that captures those strongly connected
threshold on the quality of the partitioning, we have components in the first few levels of coarsening would
checked the dependency of the algorithm on the cluster- obtain competitive partitioning qualities.
ing threshold. Figure 4 depicts the quality of the 2-way Eachalgorithmisrun20timesanditsaveragecutis
partitioning of the hypergraphs for variable clustering stated in Table 5 as well as the best cut among all runs.
threshold values. The cut is normalised based on the The results in the table are normalised with respect
best partitioning cut for each hypergraph. The average to the minimum value among algorithms. For example,
correlation between the cut and the clustering thresh- FEHG finds the minimum average cut of 79 for a bi-
old over all hypergraphs is calculated to be 0.2442 and partitioning on the CELEGANSNEURAL hypergraph. PHG,
excluding CNR-2000 is 0.2096, a very weak correlation. hMetis and PaToH find, respectively, average biparti-
The standard deviation of the changes with respect to tioning cuts that are worse by factors of 1.07, 1.17, and
the average cut values is also less than 4.2% over all hy- 1.01. The results show that FEHG performs very well
pergraphs. Therefore, changing the clustering threshold compared to PHG and hMetis and it is competitive
has a very small effect on the partitioning quality. (An with PaToH; considering the best cut, this is found in
exceptionoccursfortheCNR-2000hypergraphsuchthat 30 (of the 55 cases) by FEHG and in 27 of the cases by
the variation of change is very high.) Hyperedges with PaToH (this includes some ties). As can be seen from
high CC values are those that are more likely clustered the results, all algorithms give similar partitioning cut
with other hyperedges and those with low CC values when the hypergraph has only a few strongly connected
do not form any cluster and form edge partitions in components. In this situation, even the local clustering
HCG of size 1. Consequently, we can remove every edge algorithms can capture these strongly connected compo-
partition in HCG of unit size and set the value of the nents and merge their vertices; therefore the differences
clustering threshold to zero in Eq.(6). As an example, inpartitioningcutfordifferentalgorithmsthatareusing
different clustering methods (either global or local) is
ifeachvertexv inV ={v ,v ,···v }selectsthesetD ⊂W very small.
i 1 2 n i
ofitsneighboursinthebipartitegraphrandomlyandindepen- As the number of strongly connected components
dently such that the elements of W have equal probability to
increases, it is harder to identify those components es-
beselected.Thiscanbeconsideredtodefineahypergraphby
pecially when the boundaries between them are not
assumingthatV andW arethesetsofverticesandhyperedges,
respectively. clear and they have overlaps. This situation happens
10 Foad Lotfifar, MatthewJohnson
Table 5: Qualitycomparison ofthealgorithmsfor differentpartsizes and2%imbalance tolerance.Thevaluesarenormalised
accordingtotheminimumvalueforeachhypergraph;therefore,thealgorithmthatgives1.0cutvalueisconsideredtobethe
best.Unitweightsareassumedfor bothverticesandhyperedges.
NumberofParts
2 4 8 16 32
AVE BEST AVE BEST AVE BEST AVE BEST AVE BEST
FEHG 1.11 1.00 1.02 1.00 1.04 1.01 1.01 1.00 1.01 1.03
AS-22JULY06 PHG 2.90 2.46 1.77 1.56 1.64 1.36 1.43 1.34 1.37 1.32
hMetis 1.34 1.95 1.19 1.30 1.16 1.18 1.04 1.06 1.09 1.04
PaToH 1.00 1.43 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00
MinValue 136 93 355 319 629 599 1051 995 1591 1529
FEHG 1.00 1.00 1.09 1.00 1.10 1.06 1.11 1.08 1.07 1.03
CELEGANSNEURAL PHG 1.07 1.00 1.04 1.03 1.02 1.00 1.06 1.00 1.00 1.00
hMetis 1.17 1.21 1.00 1.05 1.00 1.04 1.00 1.02 1.00 1.00
PaToH 1.01 1.04 1.00 1.06 1.03 1.07 1.03 1.06 1.05 1.05
MinValue 79 77 195 184 354 342 548 536 773 769
FEHG 1.37 1.00 1.71 1.07 1.59 1.41 1.53 1.45 1.63 1.51
CNR–2000 PHG 35.88 45.62 12.48 9.17 5.73 4.84 3.54 2.98 2.42 2.02
hMetis 12.19 18.82 8.24 8.43 5.08 4.71 3.46 3.29 2.66 2.50
PaToH 1.00 1.71 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
MinValue 81 45 244 202 569 509 1014 911 1927 1830
FEHG 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.02 1.01 1.00
COND–MAT–2005 PHG 1.17 1.17 1.11 1.10 1.05 1.05 1.03 1.03 1.02 1.01
hMetis 1.05 1.07 1.11 1.12 1.11 1.12 1.11 1.10 1.01 1.01
PaToH 1.02 1.02 1.03 1.03 1.00 1.00 1.00 1.10 1.00 1.00
MinValue 2134 2087 5057 4951 8609 8485 12370 12150 16270 16150
FEHG 0.0 0.0 0.0 0.0 2.00 1.50 1.50 1.00 2.08 1.81
NETSCIENCE∗ PHG 0.0 0.0 0.0 0.0 1.50 1.00 1.40 1.00 1.87 1.5
hMetis 2.0 2.0 5.0 5.0 4.22 3.50 1.75 1.75 1.99 1.87
PaToH 0.0 0.0 0.0 0.0 1.00 1.00 1.00 1.00 1.00 1.00
MinValue 0 0 0 0 2 2 8 8 16 16
FEHG 2.12 1.27 1.00 1.00 1.04 1.00 1.00 1.08 1.00 1.00
PGPGIANTCOMPO PHG 13.23 1.83 1.44 1.04 1.25 1.04 1.02 1.00 1.08 1.00
hMetis 9.7 9.61 1.46 1.71 1.04 1.40 1.31 1.40 1.26 1.27
PaToH 1.00 1.00 1.04 1.27 1.00 1.04 1.02 1.15 1.08 1.06
MinValue 18 18 242 200 419 400 695 617 956 930
FEHG 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
GUPTA1 PHG 1.58 1.45 1.31 1.24 1.15 1.04 1.07 1.04 1.09 1.05
hMetis 1.73 1.82 1.61 1.69 1.58 1.64 1.60 1.57 1.51 1.48
PaToH 1.22 1.17 1.08 1.09 1.04 1.05 1.05 1.07 1.08 1.09
MinValue 486 462 1466 1384 3077 2893 5342 5134 8965 8519
FEHG 1.01 1.01 1.02 1.01 1.01 1.00 1.00 1.00 1.06 1.07
MARK3JAC120 PHG 1.00 1.01 1.02 1.02 1.02 1.00 1.00 1.00 1.72 1.78
hMetis 1.00 1.00 1.00 1.02 1.00 1.00 1.30 1.00 4.20 1.78
PaToH 1.00 1.02 1.00 1.00 1.00 1.00 1.26 1.20 1.00 1.00
MinValue 408 400 1229 1202 2856 2835 6317 6245 3142 2944
FEHG 0 0 1.00 1.00 1.12 1.12 1.09 1.03 1.06 1.07
NOTREDAME∗ PHG 4326 4326 158.56 288.69 13.82 16.78 2.09 3.06 1.72 1.78
hMetis 880 707 67.92 129.92 10.98 12.65 3.36 3.37 2.23 2.30
Patoh 24 22 1.90 3.31 1.00 1.00 1.00 1.00 1.00 1.00
MinValue 0 0 27 13 316 259 1577 1484 3142 2944
FEHG 1.20 1.00 1.03 1.01 1.05 1.03 1.00 1.00 1.00 1.00
PATENTS–MAIN PHG 12.49 13.19 2.52 2.30 1.79 1.65 1.42 1.38 1.23 1.18
hMetis 2.38 2.77 1.16 1.24 1.26 1.43 1.26 1.31 1.21 1.22
PaToH 1.00 1.02 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.00
MinValue 643 528 3490 3198 6451 6096 11322 10640 16927 16460
FEHG 1.01 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00
STD1–JAC3 PHG 1.15 1.08 1.16 1.10 1.18 1.13 1.28 1.35 1.33 1.29
hMetis 1.05 1.00 1.52 1.03 1.54 1.23 1.70 1.53 1.71 1.51
Patoh 1.00 1.00 1.08 1.00 1.16 1.14 1.00 1.26 1.30 1.29
MinValue 1490 1371 3735 3333 7616 6167 13254 11710 22242 21200
∗Whentheminimumcutfortheaverageorbestcasesarezero,thevaluesshownareactualcutvaluesratherthannormalisedvalues.