Finding Patterns in Three-Dimensional Graphs: Algorithms and Applications to Scientific Data Mining

19 Pages·2002·0.549 MB·English

by  Wang X., Wang J.T.L., Shasha D.

IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002 731
Finding Patterns in
Three-Dimensional Graphs: Algorithms and
Applications to Scientific Data Mining
Xiong Wang, Member, IEEE, Jason T.L. Wang, Member, IEEE,
Dennis Shasha, Bruce A. Shapiro, Isidore Rigoutsos, Member, IEEE, and Kaizhong Zhang
Abstract—Thispaperpresentsamethodforfindingpatternsin3Dgraphs.Eachnodeinagraphisanundecomposableoratomicunit
andhasalabel.Edgesarelinksbetweentheatomicunits.Patternsarerigidsubstructuresthatmayoccurinagraphafterallowingfor
anarbitrarynumberofwhole-structurerotationsandtranslationsaswellasasmallnumber(specifiedbytheuser)ofeditoperationsin
thepatternsorinthegraph.(Whenapatternappearsinagraphonlyafterthegraphhasbeenmodified,wecallthatappearance
“approximateoccurrence.”)Theeditoperationsincluderelabelinganode,deletinganodeandinsertinganode.Theproposedmethod
isbasedonthegeometrichashingtechnique,whichhashesnode-tripletsofthegraphsintoa3Dtableandcompressesthelabel-
tripletsinthetable.Todemonstratetheutilityofouralgorithms,wediscusstwoapplicationsoftheminscientificdatamining.First,we
applythemethodtolocatingfrequentlyoccurringmotifsintwofamiliesofproteinspertainingtoRNA-directedDNAPolymeraseand
ThymidylateSynthaseandusethemotifstoclassifytheproteins.Then,weapplythemethodtoclusteringchemicalcompounds
pertainingtoaromatic,bicyclicalkanes,andphotosynthesis.Experimentalresultsindicatethegoodperformanceofouralgorithmsand
highrecallandprecisionratesforbothclassificationandclustering.
IndexTerms—KDD,classificationandclustering,datamining,geometrichashing,structuralpatterndiscovery,biochemistry,
medicine.
(cid:230)
1 INTRODUCTION
S
TRUCTURALpattern discovery finds many applications in the data mining field, where automated discovery of
natural sciences, computer-aided design, and image patterns, classification and clustering rules is one of the
processing [8], [33]. For instance, detecting repeatedly maintasks.Weestablishaframeworkforstructuralpattern
occurring structures in molecules can help biologists to discovery in the graphs and apply our approach to
understand functions of the molecules. In these domains, classifying proteins and clustering compounds. While the
molecules are often represented by 3D graphs. The tertiary domains chosen here focus on biochemistry, our approach
structures of proteins, for example, are 3D graphs [5], [9], can be generalized to other applications where graph data
[18]. As another example, chemical compounds are also
occur commonly.
3D graphs [21].
In this paper, we study a pattern discovery problem for 1.1 3D Graphs
graph data. Specifically, we propose a geometric hashing Each node of the graphs we are concerned with is an
techniquetofindfrequentlyoccurringsubstructuresinaset undecomposable or atomic unit and has a 3D coordinate.1
Each node has a label, which is not necessarilyunique in a
of3Dgraphs.Ourstudyismotivatedbyrecentadvancesin
graph. Node labels are chosen from a domain-dependent
alphabet (cid:6). In chemical compounds, for example, the
. X. Wang is with the Department of Computer Science, California State alphabet includes the names of all atoms. A node can be
University,Fullerton,CA92834.E-mail:[email protected].
identifiedbyaunique, user-assignednumberinthe graph.
. J.T.L.WangiswiththeDepartmentofComputerandInformationScience,
New Jersey Institute of Technology, University Heights, Newark, NJ Edges in the graph are links between the atomic units. In
07102.E-mail:[email protected]. the paper, we will consider 3D graphs that are connected.
. D. Shasha is with the Courant Institute of Mathematical Sciences, New
For disconnected graphs, we consider their connected
YorkUniversity,251MercerSt.,NewYork,NY10012.
E-mail:[email protected]. components [16].
. B.A. Shapiro is with the Laboratory of Experimental and Computational A graph can be divided into one or more rigid
Biology,DivisionofBasicSciences,NationalCancerInstitutes,Frederick,
substructures. A rigid substructure is a subgraph in which
MD21702.E-mail:[email protected].
. I. Rigoutsos is with the IBM T.J. Watson Research Center, Yorktown the relative positions of the nodes in the substructure are
Heights,NY10598.E-mail:[email protected]. fixed,undersomesetofconditionsofinterest.Notethatthe
. K.ZhangiswiththeDepartmentofComputerScience,TheUniversityof
rigidsubstructureasawholecanberotated(werefertothis
WesternOntario,London,Ontario,CanadaN6A5B7.
E-mail:[email protected]. as a “whole-structure” rotation or simply a rotation when
thecontextisclear).Thus,therelativepositionofanodein
Manuscript received 19 May 1999; revised 18 Oct. 2000; accepted 18 Jan.
2001;postedtoDigitalLibrary7Sept.2001.
For information on obtaining reprints of this article, please send e-mail to: 1.Moreprecisely,the3Dcoordinateindicatesthelocationofthecenterof
[email protected],andreferenceIEEECSLogNumber109849. theatomicunit.
1041-4347/02/$17.00(cid:223)2002IEEE
732 IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002
TABLE 1
Identifiers, Labels andGlobal Coordinates ofthe
Nodes ofthe Graphin Fig.1
Fig.1.AgraphG.
thesubstructureandanodeoutsidethesubstructurecanbe
changed under the rotation. The precise definition of a
“substructure” is application dependent. For example, in
chemical compounds, a ring is often a rigid substructure.
Example 1. To illustrate rigid substructures in a graph,
1.2 Patterns in 3D Graphs
consider the graph G in Fig. 1. Each node is associated
We consider a pattern to be a rigid substructure that may
with a unique number, with its label being enclosed in
occur in a graph after allowing for an arbitrary number of
parentheses. Table 1 shows the 3D coordinates of the
rotations and translations as well as a small number
nodesinthegraphwithrespecttotheGlobalCoordinate
(specified by the user) of edit operations in the pattern or
Frame.Wedividethegraphintotworigidsubstructures:
in the graph. We allow three types of edit operations:
Str andStr .Str consistsofnodesnumbered0,1,2,3,
0 1 0 relabeling a node, deleting a node and inserting a node.
4,and5as,wellas,edgesconnectingthenodes(Fig.2a).
Relabeling a node v means to change the label of v to any
Str consistsofnodesnumbered6,7,8,9,and10as,well
1 valid label that differs from its original label. Deleting a
as, edges connecting them (Fig. 2b). Edges in the rigid
node v from a graph means to remove the corresponding
substructures are represented by boldface links. The
atomic unit from the 3D Euclidean space and make the
edge f5;6g connecting the two rigid substructures is
edges touching v connect with one of its neighbors v0.
represented by a lightface link, meaning that the two
Inserting a node v into a graph means to add the
substructures are rotatable with respect to each other
corresponding atomic unit to the 3D Euclidean space and
around the edge.2 Note that a rigid substructure is not
make a node v0 and a subset of its neighbors become the
necessarilycomplete.Forexample,inFig.2a,thereisno
neighbors of v.3
edge connecting the node numbered 1 and the node
WesaygraphGmatchesgraphG0withnmutationsifby
numbered 3.
applying an arbitrary number of rotations and translations
We attach a local coordinate frame SF0 (SF1, respec- aswellasnnodeinsert,deleteorrelabelingoperations,1)G
tively)tosubstructureStr0(Str1,respectively).Forinstance, andG0 havethesamesize,42)thenodesinGgeometrically
letusfocusonthesubstructureStr inFig.2a.Weattacha matchthoseinG0,i.e.,theyhavecoinciding3Dcoordinates,
0
local coordinate frame to Str whose origin is the node and 3) for each pair of geometrically matching nodes, they
0
numbered 0. This local coordinate frame is represented by have the same label. A substructure P approximately occurs
three basis points P , P , and P , with coordinates in a graph G (or G approximately contains P) within
b1 b2 b3
P (cid:133)x ;y ;z (cid:134), P (cid:133)x (cid:135)1;y ;z (cid:134), and P (cid:133)x ;y (cid:135)1;z (cid:134), re- n mutations if P matches some subgraph of G with
spb1ect0ive0ly.0The ob2rig0in is P0 a0nd the thrbe3e b0asi0s vecto0rs are n mutations or fewer where n is chosen by the user.
V~ , V~ , and V~ (cid:2)V~b1 . Here, V~ represents the Example2.Toillustratepatternsingraphs,considertheset
veb1c;bt2or sbt1a;br3ting at pob1i;nb2t P b1a;bn3d ending ab1t;b2point P . V~ (cid:2) S of three graphs in Fig. 3a. Suppose only exactly
V~ standsforthecrossbp1roductofthetwocorrebs2ponb1d;bi2ng coinciding substructures (without mutations) occurring
b1;b3
vectors. We refer to this coordinate frame as Substructure
Frame 0, or SF . Note that the basis vectors of SF are 3.Here,exactlywhichsubsetoftheneighborsischosenisunimportant,
0 0
astheproposedgeometrichashingtechniquehashesnodesonly,ignoring
orthonormal. That is, the length of each vector is 1 and the
edgesamongthenodes.Noticethatwhenanodevisinsertedordeleted,
angle between any two basis vectors has 90 degrees. Also the nodes surrounding v do not move, i.e., their coordinates remain the
same.Notealsothatwedonotallowmultipleeditoperationstobeapplied
notethat,foranynodenumberediinthesubstructureStr
0 tothesamenode.Thus,forexample,insertinganodewithlabelmfollowed
with global coordinate P(cid:133)x;y;z(cid:134), we can find a local byrelabelingittonisconsideredasinsertinganodewithlabeln.Thethree
i i i i
coordinate of the node i with respect to SF , denoted P0, editoperationsareextensionsoftheeditoperationsonsequences;theyarise
0 i naturallyingraphediting[10]andmoleculeevolution[25].Asshown in
where Section 4, based on these edit operations, our algorithm finds useful
patternsthatcanbeusedtoclassifyandcluster3Dmoleculeseffectively.
P0 (cid:136)V~ (cid:136)(cid:133)x (cid:255)x ;y (cid:255)y ;z (cid:255)z (cid:134): (cid:133)1(cid:134) 4.Thesizeofagraphisdefinedtobethenumberofnodesinthegraph,
i b1;i i 0 i 0 i 0 sinceourgeometrichashingtechniqueconsidersnodesonly.Furthermore,
in our target applications, e.g., chemistry, nodes are atomic units and
2. Such an edge is referred to as a rotatable edge. If two rigid determinethesizeofacompound.Edgesarelinksbetweenthenodesand
substructurescannotberotatedwithrespecttoeachotheraroundtheedge have a different meaning from the atomic units; as a consequence, we
connectingthem,thatedgeisanonrotatableedge. excludetheedgesfromthesizedefinition.
WANGETAL.: FINDINGPATTERNSINTHREE-DIMENSIONALGRAPHS:ALGORITHMSANDAPPLICATIONSTOSCIENTIFICDATAMINING 733
quality of the patterns by using them to classify the
compounds. Here, we extend the work in [35] by
1. considering more general edit operations including
node insert, delete, and relabeling,
2. presentingthetheoreticalfoundationandevaluating
the performance and efficiency of our pattern-
finding algorithm,
3. applying the discovered patterns to classifying
3D proteins, which are much larger and more
complicated in topology than chemical compounds,
and
Fig.2.TherigidsubstructuresofthegraphinFig.1.
4. presentingatechniquetocluster3Dgraphsbasedon
the patterns occurring in them.
inatleasttwographsandhavingsizegreaterthanthree
Specifically, we conducted two experiments. In the first
are considered as “patterns.” Then, S contains one
experiment, we applied the proposed method to locating
pattern shown in Fig. 3b. If substructures having size
frequently occurring motifs (substructures) in two families
greaterthanfourandapproximatelyoccurringinallthe
of proteins pertaining to RNA-directed DNA Polymerase
three graphs within one mutation (i.e., one node delete, and Thymidylate Synthase, and used the motifs to classify
insert, or relabeling is allowed in matching a substruc- theproteins.Experimentalresultsshowedthatourmethod
ture with a graph) are considered as “patterns,” then S achieved a 96.4 percent precision rate. In the second
contains one pattern shown in Fig. 3c. experiment, we applied our pattern-finding algorithm to
discovering frequently occurring patterns in chemical
Ourstrategytofindthepatternsinasetof3Dgraphsis
compounds chosen from the Merck Index pertaining to
todecomposethegraphsintorigidsubstructuresand,then,
aromatic,bicyclicalkanesandphotosynthesis.Wethenused
use geometric hashing [14] to organize the substructures
thepatternstoclusterthecompounds.Experimentalresults
and,then,tofindthefrequentlyoccurringones.In[35],we
showed that our method achieved 99 percent recall and
applied the approach to the discovery of patterns in precision rates.
chemical compounds under a restricted set of edit opera- The rest of the paper is organized as follows: Section 2
tionsincluding nodeinsert andnode delete,andtestedthe presents the theoretical framework of our approach and
Fig.3.(a)ThesetS ofthreegraphs,(b)thepatternexactlyoccurringintwographsinS,and(c)thepatternapproximatelyoccurring,withinone
mutation,inallthethreegraphs.
734 IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002
(cid:18) (cid:19)
4
describes the pattern-finding algorithm in detail. Section 3 5(cid:2) (cid:136)20node-triplets:
3
evaluates the performance and efficiency of the pattern-
finding algorithm. Section 4 describes the applications of There are several alternative ways to decompose
our approach to classifying proteins and clustering com- 3D graphs into rigid substructures, depending on the
pounds. Section 5 discusses related work. Section 6 application at hand and the nature of the graphs. For the
concludes the paper. purposes of exposition, we describe our pattern-finding
algorithm based on a partitioning strategy. Our approach
assumesanotionofatomicunitwhichisthelowestlevelof
2 PATTERN-FINDING ALGORITHM description in the case of interest. Intuitively, atomic units
2.1 Terminology are fundamental building elements, e.g., atoms in a
Let S be a set of 3D graphs. The occurrence number of a molecule. Edges arise as bonds between atomic units. We
break a graph into maximal size rigid substructures (recall
pattern P is the number of graphs in S that approxi-
thatarigidsubstructureisasubgraphinwhichtherelative
mately contain P within the allowed number of muta-
positions of nodes in the substructure are fixed). To
tions. Formally, the occurrence number of a pattern P
accomplish this, we use an approach similar to [16] that
with respect to mutation d and set S, denoted
employs adepth-first search algorithm, referred to as DFB,
occur nodS(cid:133)P(cid:134), is k if there are k graphs in S that contain tofindblocksinagraph.5TheDFBworksbytraversingthe
P within d mutations. For example, consider Fig. 3 again. graphinadepth-firstorderandcollectingnodesbelonging
Let S contain the three graphs in Fig. 3a. Then, toablockduringthetraversal,asillustratedintheexample
occur no0(cid:133)P (cid:134) = 2; occur no1(cid:133)P (cid:134) = 3. below. Each block is a rigid substructure. We merge two
S 1 S 2
Given a set S of 3D graphs, our algorithm finds all the rigidsubstructuresB andB iftheyarenotrotatablewith
1 2
patterns P where P approximately occurs in at least respecttoeachother,thatis,therelativepositionofanode
OccurgraphsinS withintheallowednumberofmutations n1 2B1 and a node n2 2B2 is fixed. The algorithm
Mut andjPj (cid:21) Size,wherejPj representsthe size, i.e., the maintains a stack, denoted STK, which keeps the rigid
substructures being merged. Fig. 4 shows the algorithm,
number of nodes, of the pattern P. (Mut, Occur, and Size
whichoutputsasetofrigidsubstructuresofagraphG.We
are user-specified parameters.) One can use the patterns in
thenthrowawaythesubstructuresP wherejPj<Size.The
several ways. For example, biologists or chemists may
remaining substructures constitute the candidate patterns
evaluate whether the patterns are significant; computer
generated from G. This pattern-generation algorithm runs
scientists may use the patterns to classify or cluster
in time linearly proportional to the number of edges in G.
molecules as demonstrated in Section 4.
Example3.WeusethegraphinFig.5toillustratehowthe
Our algorithm proceeds in two phases to search for the
Find_Rigid_Substructures algorithm in Fig. 4 works.
patterns: 1) find candidate patterns from the graphs in S;
Rotatableedgesinthegrapharerepresentedbylightface
and 2) calculate the occurrence numbers of the candidate
links; nonrotatable edges are represented by boldface
patterns to determine which of them satisfy the user-
links. Initially, the stack STK is empty. We invoke DFB
specified requirements. We describe each phase in turn
to locate the first block (Step 4). DFB begins by visiting
below.
the node numbered 0. Following the depth-first search,
2.2 Phase 1 of the Algorithm DFB then visits the nodes numbered 1, 2, and 5. Next,
DFBmayvisitthenodenumbered6or4.Withoutlossof
In phase1of thealgorithm,we decomposethegraphs into
generality,assumeDFBvisitsthenodenumbered6and,
rigid substructures. Dividing a graph into substructures is
then, the nodes numbered 10, 11, 13, 14, 15, 16, 18, and
necessary for two reasons. First, in dealing with some
17,inthatorder.Then,DFBvisitsthenodenumbered14,
moleculessuchaschemicalcompoundsinwhichtheremay andrealizesthatthisnodehasbeenvisitedbefore.Thus,
exist two substructures that are rotatable with respect to DFBgoesbacktothenodenumbered17,18,etc.,untilit
each other, any graph containing the two substructures is returns to the node numbered 14. At this point, DFB
not rigid. As a result, we decompose the graph into identifies the first block, B , which includes nodes
1
substructureshavingnorotatablecomponentsandconsider numbered 14, 15, 16, 17, and 18. Since the stack STK is
the substructures separately. Second, our algorithm hashes empty now, we push B1 into STK (Step 7).
Initeration2,wecallDFBagaintofindthenextblock
node-triplets within rigid substructures into a 3D table.
(Step 4).DFBreturnsthenonrotatableedgef13;14gasa
When a graph as a whole is too large, as in the case of
block, denoted B .6 The block is pushed into STK
proteins,consideringallcombinationsofthreenodesinthe 2
(Step 7). In iteration 3, DFB locates the block B , which
graph may become prohibitive. Consequently, decompos- 3
includes nodes numbered 10, 11, 12, and 13 (Step 4).
ing the graph into substructures and hashing node-triplets
Since B and the top entry of the stack, B are not
3 2
of the substructures can increase efficiency. For example,
rotatable with respect to each other, we push B into
3
consider a graph of 20 nodes. There are
(cid:18) (cid:19) 5.Ablockisamaximalsubgraphthathasnocut-vertices.Acut-vertexof
20 agraphisonewhoseremovalresultsindividingthegraphintomultiple,
(cid:136)1140node-triplets:
3 disjointedsubgraphs[16].
6.Inpractice,ingraphrepresentationsformolecules,oneusesdifferent
On the other hand, if we decompose the graph into five notation to distinguish nonrotatable edges from rotatable edges. For
example, in chemical compounds, a double bond is nonrotatable. The
substructures, each having four nodes, then there are only differentnotationhelpsthealgorithmtodeterminethetypesofedges.
WANGETAL.: FINDINGPATTERNSINTHREE-DIMENSIONALGRAPHS:ALGORITHMSANDAPPLICATIONSTOSCIENTIFICDATAMINING 735
Fig.4.Algorithmforfindingrigidsubstructuresinagraph.
STK (Step7).Initeration4,wecontinuetocallDFBand isrotatablewithrespecttoB ,wepopoutf5;6gtoform
9
get the single edgef6;10gas the next block,B (Step 4). a rigid substructure (Step 9) and push B into STK
4 9
Sincef6;10gisrotatableandthestackSTKisnonempty, (Step10).Finally,sincethereisnoblockleftinthegraph,
we pop out all nodes in STK, merge them and output we pop out all nodes in B to form a rigid substructure
9
therigidsubstructurecontainingnodesnumbered10,11, and terminate (Step 13).
12, 13, 14, 15, 16, 17, 18 (Step 9). We then push B into
4
STK (Step 10). 2.3 Phase 2 of the Algorithm
Phase 2 of our pattern-finding algorithm consists of two
In iteration 5, DFB visits the node numbered 7 and,
subphases.InsubphaseAofphase2,wehashandstorethe
then,8fromwhichDFBgoesbacktothenodenumbered
candidatepatternsgeneratedfromthegraphsinphase1in
7. It returns the single edge f7;8g as the next block, B
5 a 3D table H. In subphase B, we rehash each candidate
(Step4).Sincef7;8gisconnectedtothecurrenttopentry
pattern into H and calculate its occurrence number. Notice
of STK, f6;10g, via a rotatable edge f6;7g, we pop out
that in the subphase B, one does not need to store the
f6;10g,whichitselfbecomesarigidsubstructure(Step9).
candidate patterns in H again.
We then push f7;8g into STK (Step 10). In iteration 6,
Inprocessingarigidsubstructure(pattern)ofa3Dgraph,
DFB returns the single edge f7;9g as the next block, B
6 wechooseallthree-nodecombinations,referredtoasnode-
(Step4).SinceB andthecurrenttopentryofSTK,B =
6 5 triplets, in the substructure and hash the node-triplets. We
f7;8g, are not rotatable with respect to each other, we
hash three-node combinations, because to fix a rigid
pushB intoSTK (Step7).Initeration7,DFBgoesback
6 substructure in the 3D Euclidean space one needs at least
from the node numbered7to the node numbered6and
three nodes from the substructure and three nodes are
returns the single edge f6;7g as the next block, B
7 sufficient provided they are not collinear. Notice that the
(Step4).Sincef6;7gisrotatable,wepopoutallnodesin
proper order of choosing the nodes i;j;k, in a triplet is
STK, merge them and output the resulting rigid
significant and has an impact on the accuracy of our
substructure containing nodes numbered 7, 8, and 9
approach,aswewillshowlaterinthepaper.Wedetermine
(Step 9). We then push B = f6;7g into STK (Step 10).
7 the order of the three nodes by considering the triangle
In iteration 8, DFB returns the block B = f5;6g formedbythem.Thefirstnodechosenalways opposesthe
8
(Step4).Sincef5;6gandf6;7garebothrotatable,wepop longest edge of the triangle and the third node chosen
out f6;7g to form a rigid substructure (Step 9). We then opposes the shortest edge. For example, in the triangle in
Fig. 6, we choose i;j;k, in that order. Thus, the order is
push B into STK (Step 10). In iteration 9, DFB returns
8
unique if the triangle is not isosceles or equilateral, which
theblockB containingnodesnumbered0,1,2,3,4,and
9
usually holds when the coordinates are floating point
5(Step4).SincethecurrenttopentryofSTK,B =f5;6g,
8
Fig.5.ThegraphusedforillustratinghowtheFind_Rigid_Substructures
algorithmworks. Fig.6.Thetriangleformedbythethreenodesi;j;k.
736 IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002
TABLE 2
The NodeLabels ofthe GraphinFig. 1and
Their Indices inthe Array A
numbers.Ontheotherhand,whenthetriangleisisosceles,
we hash and store the two node-triplets (cid:137)i;j;k(cid:138) and (cid:137)i;k;j(cid:138),
assuming that node i opposes the longest edge and edges
fi;jg, fi;kg have the same length. When the triangle is
equilateral, we hash and store the six node-triplets (cid:137)i;j;k(cid:138),
Fig.7.CalculationofthecoordinatesofthebasispointsP ,P ,P of
(cid:137)i;k;j(cid:138), (cid:137)j;i;k(cid:138), (cid:137)j;k;i(cid:138), (cid:137)k;i;j(cid:138), and (cid:137)k;j;i(cid:138). b1 b2 b3
SubstructureFrame0(SF )withrespecttothelocalcoordinateframe
The labels of the nodes in a triplet form a label-triplet, 0
LF(cid:137)i;j;k(cid:138).
which is encoded as follows: Suppose the three nodes
chosen are v , v , v , in that order. We maintain all node
1 2 3 Let
labels in the alphabet (cid:6) in an array A. The code for the
labels is an unsigned long integer, defined as d (cid:136)(cid:133)l (cid:135)l (cid:134)modPrime modNrow
1 1 2 1
d (cid:136)(cid:133)l (cid:135)l (cid:134)modPrime modNrow:
(cid:133)(cid:133)L (cid:2)Prime(cid:135)L (cid:134)(cid:2)Prime(cid:134)(cid:135)L ; 2 2 3 2
1 2 3
d (cid:136)(cid:133)l (cid:135)l (cid:134)modPrime modNrow
3 3 1 3
where Prime>j(cid:6)j is a prime number, L , L , and L are
1 2 3
theindicesforthenodelabelsofv ,v ,andv ,respectively, Prime , Prime , and Prime are three prime numbers and
1 2 3 1 2 3
in the array A. Thus, the code of a label-triplet is unique. Nrowisthecardinalityofthehashtableineachdimension.
This simple encoding scheme reduces three label compar-
We use three different prime numbers in the hope that the
isons into one integer comparison.
distributionofthehashfunctionvaluesisnotskewedeven
Example 4. To illustrate how we encode label-triplets,
if pairs of l , l , l are correlated. The node-triplet (cid:137)i;j;k(cid:138) is
consider again the graph G in Fig. 1. Suppose the node 1 2 3
hashed to the 3D bin with the address h(cid:137)d (cid:138)(cid:137)d (cid:138)(cid:137)d (cid:138).
labels are stored in the array A, as shown in Table 2. 1 2 3
SupposePrimeis1,009.Then,forexample,forthethree Intuitively, we use the squares of the lengths of the three
nodes numbered 2, 0, and 1 in Fig. 1, the code for the edges connecting the three chosen nodes to determine the
correspondinglabel-tripletis(cid:133)(cid:133)2(cid:2)1;009(cid:135)0(cid:134)(cid:2)1;009(cid:134)(cid:135) hash bin address. Stored in that bin are the graph
1 (cid:136)2;036;163:
identification number, the substructure identification num-
ber, and the label-triplet code. In addition, we store the
2.3.1 Subphase A of Phase 2 coordinates of the basis points P , P , P of Substructure
b1 b2 b3
Inthissubphase,wehashthecandidatepatternsgenerated Frame 0 (SF ) with respect to the three chosen nodes.
0
in phase 1 of the pattern-finding algorithm into a 3D table. Specifically, suppose the chosen nodes i, j, k are not
For the purposes of exposition, consider the example
collinear. We can construct another local coordinate frame,
substructure Str in Fig. 2a, which is assumed to be a
0 denoted LF(cid:137)i;j;k(cid:138), using V~ , V~ and V~ (cid:2)V~ as basis
candidate pattern. We choose any three nodes in Str and i;j i;k i;j i;k
0
vectors. The coordinates of P , P , P with respect to the
calculatetheir3Dhashfunctionvaluesasfollows:Suppose b1 b2 b3
the chosen nodes are numbered i, j, k, and have global localcoordinateframeLF(cid:137)i;j;k(cid:138),denotedSF0(cid:137)i;j;k(cid:138),forma
coordinates P(cid:133)x;y;z(cid:134), P (cid:133)x ;y ;z (cid:134), and P (cid:133)x ;y ;z (cid:134), 3(cid:2)3 matrix, which is calculated as follows (see Fig. 7):
i i i i j j j j k k k k
respectively. Let l , l , l be three integers where 0 1
1 2 3
V~
l1 (cid:136)(cid:133)(cid:133)xi(cid:255)xj(cid:134)2(cid:135)(cid:133)yi(cid:255)yj(cid:134)2(cid:135)(cid:133)zi(cid:255)zj(cid:134)2(cid:134)(cid:2)Scale SF0(cid:137)i;j;k(cid:138)(cid:136)B@V~ii;;bb12CA(cid:2)A(cid:255)1; (cid:133)3(cid:134)
l (cid:136)(cid:133)(cid:133)x (cid:255)x (cid:134)2(cid:135)(cid:133)y (cid:255)y (cid:134)2(cid:135)(cid:133)z (cid:255)z (cid:134)2(cid:134)(cid:2)Scale: (cid:133)2(cid:134) V~
2 i k i k i k i;b3
l3 (cid:136)(cid:133)(cid:133)xk(cid:255)xj(cid:134)2(cid:135)(cid:133)yk(cid:255)yj(cid:134)2(cid:135)(cid:133)zk(cid:255)zj(cid:134)2(cid:134)(cid:2)Scale where
0 1
Here,Scale(cid:136)10pisamultiplier.Intuitively,weroundtothe V~
nearestpthpositionfollowingthedecimalpoint(herepisthe A(cid:136)B@ V~i;j CA: (cid:133)4(cid:134)
i;k
lastaccurateposition)and,then,multiplythenumbersby10p. V~ (cid:2)V~
i;j i;k
The reason for using the multiplier is that we want some
digits following the decimal point to contribute to the Thus,supposethegraphinFig.1hasidentificationnumber
distributionofthehashfunctionvalues.Weignorethedigits 12. The hash bin entry for the three chosen nodes i, j, k is
after the position p because they are inaccurate. (The (cid:133)12;0;Lcode;SF (cid:137)i;j;k(cid:138)(cid:134), where Lcode is the label-triplet
0
appropriate value for the multiplier can be calculated once code. Since there are 6 nodes in the substructure Str , we
0
dataaregiven,aswewillshowinSection3.) have
WANGETAL.: FINDINGPATTERNSINTHREE-DIMENSIONALGRAPHS:ALGORITHMSANDAPPLICATIONSTOSCIENTIFICDATAMINING 737
(cid:18) (cid:19)
6
(cid:136)20node-triplets Recallthatwechoosethethreenodesi,j,kbasedonthe
3
triangle formed by them—the first node chosen always
generatedfromthesubstructureand,therefore,20entriesin opposesthelongestedgeofthetriangleandthethirdnode
the hash table for the substructure.7 chosenopposestheshortestedge.Withoutlossofgenerality,
letus assumethatthe nodesi;j;kare chosenin thatorder.
Example5.Toillustratehowwehashandstorepatternsin
Thus,V~ has theshortest length, V~ isthe second shortest
the hash table, let us consider Table 1 again. The basis i;j i;k
andV~ isthelongest.Weusenodeiastheorigin,V~ astheX-
points of SF of Fig. 2a have the following global j;k i;j
0 axisandV~ astheY-axis.Then,constructthelocalcoordinate
coordinates: i;k
frameLF(cid:137)i;j;k(cid:138)usingV~ ,V~ ,andV~ (cid:2)V~ asbasisvectors.
i;j i;k i;j i;k
P (cid:133)1:0178;1:0048;2:5101(cid:134); Thus,weexclude thelongestvector V~ whenconstructing
b1 j;k
LF(cid:137)i;j;k(cid:138).Hereiswhy.
P (cid:133)2:0178;1:0048;2:5101(cid:134);
b2 Thecoordinates(cid:133)x;y;z(cid:134)ofeachnodeina3Dgraphhave
P (cid:133)1:0178;2:0048;2:5101(cid:134):
b3 anerrorduetorounding.Thus,therealcoordinatesforthe
The local coordinates, with respect to SF , of the node should be (cid:133)x;y;z(cid:134), where x(cid:136)x(cid:135)(cid:15) , y(cid:136)y(cid:135)(cid:15) , z(cid:136)
0 1 2
nodes numbered 0, 1, 2, 3, and 4 in substructure Str of z(cid:135)(cid:15) for three small decimal fractions (cid:15) ;(cid:15) ;(cid:15) . After
0 3 1 2 3
Fig. 2a are as follows: constructing LF(cid:137)i;j;k(cid:138) and when calculating SF (cid:137)i;j;k(cid:138),
0
onemayaddormultiplythecoordinatesofthe3Dvectors.
P0(cid:133)0:0000;0:0000;0:0000(cid:134);
0 WedefinetheaccumulativeerrorinducedbyacalculationC,
P0(cid:133)0:1843;1:0362;(cid:255)0:5081(cid:134);
1 denoted (cid:1)(cid:133)C(cid:134), as
P0(cid:133)0:3782;1:9816;(cid:255)0:5095(cid:134);
2
(cid:1)(cid:133)C(cid:134)(cid:136)jf(cid:255)fj;
P0(cid:133)(cid:255)0:3052;1:0442;0:6820(cid:134);
3
P0(cid:133)(cid:255)0:2568;1:7077;0:5023(cid:134): where f(cid:22) is the result obtained from C with the real
4
coordinates and f is the result obtained from C with
Now, suppose Scale, Prime1, Prime2, Prime3 are 10, rounding errors.
1,009, 1,033, and 1,051, respectively, and Nrow is 31. RecallthatincalculatingSF (cid:137)i;j;k(cid:138),thethreebasisvectors
0
Thus, for example, for the nodes numbered 1, 2, and 3, ofLF(cid:137)i;j;k(cid:138)allappearinthematrixAdefinedin(4).Let
the hash bin address is h(cid:137)25(cid:138)(cid:137)12(cid:138)(cid:137)21(cid:138) and,
0 1 jV~ j(cid:136)jV~ j(cid:135)(cid:14) ;jV~ j(cid:136)jV~ j(cid:135)(cid:14) ;
i;j i;j 1 i;k i;k 2
(cid:255)1:056731 0:357816 0:173981
SF0(cid:137)1;2;3(cid:138)(cid:136)@(cid:255)0:875868 0:071949 0:907227A: (cid:133)5(cid:134) and jV~j;kj(cid:136)jV~j;kj(cid:135)(cid:14)3. Let (cid:14)(cid:136)maxfj(cid:14)1j;j(cid:14)2j;j(cid:14)3jg. Notice
(cid:255)0:035973 0:417517 0:024041
jV~ (cid:2)V~ j(cid:136)jV~ jjV~ jsin(cid:18);
i;j i;k i;j i;k
Asanotherexample,forthenodesnumbered1,4,and2,
the hash bin address is h(cid:137)24(cid:138)(cid:137)0(cid:138)(cid:137)9(cid:138) and where jV~i;jj is the length of V~i;j, and (cid:18) is the angle between
0 1 V~ and V~ . Thus,
0:369457 (cid:255)1:308266 (cid:255)0:242990 i;j i;k
SF0(cid:137)1;4;2(cid:138)(cid:136)@(cid:255)0:043584 (cid:255)0:857105 (cid:255)1:006793A: (cid:133)6(cid:134) (cid:1)(cid:133)jV~i;j(cid:2)V~i;kj(cid:134)
0:455285 (cid:255)0:343703 (cid:255)0:086982
(cid:136)jjV~ jjV~ jsin(cid:18)(cid:255)jV~ jjV~ jsin(cid:18)j
i;j i;k i;j i;k
Similarly, for the substructure Str , we attach a local
1 (cid:136)j(cid:133)jV~ j(cid:135)(cid:14) (cid:134)(cid:133)jV~ j(cid:135)(cid:14) (cid:134)sin(cid:18)(cid:255)jV~ jjV~ jsin(cid:18)j
coordinateframeSF tothenodenumbered6,asshown i;j 1 i;k 2 i;j i;k
1
in Fig. 2b. There are 10 hash table entries for Str , each (cid:136)j(cid:133)jV~ j(cid:14) (cid:135)jV~ j(cid:14) (cid:135)(cid:14) (cid:14) (cid:134)jjsin(cid:18)j :
1 i;j 2 i;k 1 1 2
having the form (cid:133)12;1;Lcode;SF1(cid:137)l;m;n(cid:138)(cid:134) where l;m;n (cid:20)(cid:133)jV~ jj(cid:14) j(cid:135)jV~ jj(cid:14) j(cid:135)j(cid:14) jj(cid:14) j(cid:134)jsin(cid:18)j
i;j 2 i;k 1 1 2
are any three nodes in Str .
1 (cid:20)(cid:133)jV~ j(cid:135)jV~ j(cid:135)(cid:14)(cid:134)(cid:14)
i;j i;k
(cid:136)U
1
7. One interesting question here is regarding the size limitation of
Likewise,
patterns(rigidsubstructures)foundbyouralgorithm.Supposethegraph
identificationnumberandthesubstructureidentificationnumberareboth (cid:1)(cid:133)jV~ (cid:2)V~ j(cid:134)(cid:20)(cid:133)jV~ j(cid:135)jV~ j(cid:135)(cid:14)(cid:134)(cid:14)(cid:136)U
i;j j;k i;j j;k 2
shortintegersof2bytes.Lcodeisalongintegerof4bytes.SF (cid:137)i;j;k(cid:138)isa
0
and
3(cid:2)3matrixoffloatingpointnumbers,eachrequiring4bytes.Thus,each
node-tripletrequires44bytes.ForasetofMpatterns(su(cid:18)bst(cid:19)ructures),each (cid:1)(cid:133)jV~i;k(cid:2)V~j;kj(cid:134)(cid:20)(cid:133)jV~i;kj(cid:135)jV~j;kj(cid:135)(cid:14)(cid:134)(cid:14)(cid:136)U3:
T
havingTnodesonaverage,thehashtablerequiresM(cid:2) (cid:2)44bytesof
3 AmongthethreeupperboundsU ,U ,U ,U isthesmallest.
diskspace.Supposethereare10,000patterns,i.e.,M(cid:136)10;000.IfT(cid:136)20, 1 2 3 1
It’slikelythattheaccumulativeerrorinducedbycalculating
the hash table requires about 502 Megabytes. If T (cid:136)100, the hash table
the length of the cross product of the two corresponding
requiresover71Gigabytes.Thisgivesanestimateofhowlargepatternscan vectors is also the smallest. Therefore, we choose V~ , V~ ,
i;j i;k
beinpractice.Inourcase,thepatternsizesarebetween5and13,sothesize andexcludethelongestvectorV~ inconstructingthelocal
j;k
demands aremodest. Notethat the timeand memory space neededalso coordinate frame LF(cid:137)i;j;k(cid:138), so as to minimize the accumu-
increasedramaticallyaspatternsbecomelarge. lative error induced by calculating SF0(cid:137)i;j;k(cid:138).
738 IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002
2.3.2 Subphase B of Phase 2 nodes u, v, and w geometrically match the nodes i, j, and k,
respectively.
LetHbetheresultinghashtableobtainedinsubphaseAof
phase2ofthepattern-findingalgorithm.InsubphaseB,we Proof. (If) Let A be as defined in (4) and let
calculate the occurrence number of each candidate pattern 0 1
P byrehashingthenode-tripletsofP intoH.Thisway,we B V~u;v C
areabletomatchanode-triplettriofP withanode-triplet B(cid:136)@ V~ A: (cid:133)8(cid:134)
u;w
tri0 ofanothersubstructure(candidatepattern)P0 storedin V~ (cid:2)V~
u;v u;w
subphase A where tri and tri0 have the same hash bin
Note that, if u, v, and w geometrically match i, j, and k,
address.Bycountingthenode-tripletmatches,onecaninfer
whether P matches P0 and, therefore, whether P occurs in respectively, then jBj(cid:136)jAj, where jBj (jAj, respectively)
the graph from which P0 is generated. is the determinant of the matrix B (matrix A, respec-
tively). That is to say, jA(cid:255)1jjBj(cid:136)1.
We associate each substructure with several counters,
From (3) and by the definition of the SF in (7),
which are created and updated as illustrated by the P
we have
following example. Suppose the two substructures (pat-
terns) of graph G with identification number 12 in Fig. 1 0V~ 1 0P 1
havealreadybeenstoredinthehashtableHinsubphaseA. SF (cid:136)B@V~i;b1CA(cid:2)A(cid:255)1(cid:2)B(cid:135)@PuA: (cid:133)9(cid:134)
Supposei;j;k,arethreenodesinthesubstructureStr ofG. P i;b2 u
Thus, for this node-triplet, its entry in the hash ta0ble is V~i;b3 Pu
(cid:133)12;0;Lcode;SF (cid:137)i;j;k(cid:138)(cid:134). Now, in subphase B, consider Thus, the SF basically transforms P , P , and P via
0 P b1 b2 b3
another pattern P; we hash the node-triplets of P using twotranslationsandonerotation,whereP ,P ,andP
b1 b2 b3
thesamehashfunction.Letu;v;w,bethreenodesinP that are the basis points of the Substructure Frame 0 (SF ).
0
have the same hash bin address as i;j;k; that is, the node- Since V~ , V~ , and V~ (cid:2)V~ are orthonormal
b1;b2 b1;b3 b1;b2 b1;b3
triplet (cid:137)u;v;w(cid:138) “matches” the node-triplet (cid:137)i;j;k(cid:138). If the vectors, and translations and rotations do not change
nodes u;v;w, geometrically match the nodes i;j;k respec- this property [27], we know that V~ , V~ , and V~ (cid:2)
tively, i.e., they have coinciding 3D coordinates after V~ are orthonormal vectors. c1;c2 c1;c3 c1;c2
c1;c3
rotations and translations, we call the node-triplet match a (Only if) If u, v, and w do not match i, j and k
truematch;otherwiseitisafalsematch.Foratruematch,let geometrically while having the same hash bin address,
then there would be distortion in the aforementioned
0 1 0 1
SFP (cid:136)SF0(cid:137)i;j;k(cid:138)(cid:2)B@ VV~~uu;;wv CA(cid:135)@PPuuA: (cid:133)7(cid:134) Vt~rca1;nc3sfworomuladtionno.loCnognesreqbueeonrttlhyo,nV~ocr1m;c2a,lV~vce1;cc3t,orasn.d V~c1;c2 (cid:2)tu
V~u;v(cid:2)V~u;w Pu
Theorem2.Iftwotruenode-tripletmatchesyieldthesameSF
This SF contains the coordinates of the three basis points P
P and the codes of the corresponding label-triplets are the same,
oftheSubstructureFrame0(SF )withrespecttotheglobal
0 then the two node-triplet matches are augmentable, i.e., they
coordinate frame in which the pattern P is given. We
can be combined to form a larger substructure match between
compare the SF with those already associated with the
P P and Str .
0
substructure Str (initially none is associated with Str ). If
0 0 Proof.SincethreenodesareenoughtosettheSF atafixed
the SF differs from the existing ones, a new counter is P
P positionanddirection,alltheothernodesinP willhave
created,whosevalueisinitializedto1,andthenewcounter
definitecoordinatesunderthisSF .Whenanothernode-
is assigned to the SF . If the SF is the “same” as an P
P P
existing one with counter value Cnt,8 and the code of the tripletmatchyieldingthesameSFP occurs,itmeansthat
label-triplet of nodes i, j, k; equals the code of the label- geometrically there is at least one more node match
triplet of nodes u, v, w, then Cnt is incremented by one. In between Str0 and P. If the codes of the corresponding
general, a substructure may be associated with several label-tripletsarethesame,itmeansthatthelabelsofthe
different SF s, each having a counter. corresponding nodes are the same. Therefore, the two
P
We now present the theory supporting this algorithm. node-triplet matches are augmentable. tu
Below, Theorem 1 establishes a criterion based on which
Fig. 8 illustrates how two node-triplet matches are
one can detect and eliminate a false match. Below,
augmented. Suppose the node-triplet (cid:137)3;4;2(cid:138) yields the
Theorem 2 justifies the procedure of incrementing the
SF shown in the figure. Further, suppose that the node-
P
counter values.
triplet(cid:137)1;2;3(cid:138)yieldsthesameSF ,asshowninFig.8.Since
P
Theorem1.LetP ,P ,andP bethethreebasispointsforming
the SF definecd1 inc2(7), whecr3e P is the origin. V~ , V~ , thelabelsofthecorrespondingnodesnumbered3and2are
and V~P (cid:2)V~ are orthonormac1l vectors if and oc1n;cl2y ifc1t;ch3e the same, we can augment the two node-triplets to form a
c1;c2 c1;c3 larger match containing nodes numbered 1, 2, 3, 4.
Thus, by incrementing the counter associated with the
8.BysayingSF isthesameasanexistingSF0,wemeanthatforeach
P P
entry e ;1(cid:20)i;j(cid:20)3, at the ith row and the jth column in SF and its SF , we record how many true node-triplet matches are
i;j P P
cpoarrraemspeotenrddinegpeenndtirnygeo0i;nj tinheSdFaP0ta,.jIeni;jt(cid:255)heee0i;jxja(cid:20)mp(cid:15),lewshperreese(cid:15)ntiesdainntahdejupsatapbelre, augmentableunderthisSFP.Noticethatincaseswheretwo
(cid:15)(cid:136)0:01. node-tripletmatchesoccurduetoreflections,thedirections
WANGETAL.: FINDINGPATTERNSINTHREE-DIMENSIONALGRAPHS:ALGORITHMSANDAPPLICATIONSTOSCIENTIFICDATAMINING 739
Fig.8.Illustrationoftwoaugmentablenode-tripletmatches.
ofthecorrespondinglocalcoordinatesystemsaredifferent,
Fig.9.Asubstructure(pattern)P.
so are the SF s. As a result, these node-triplet matches are
P
(cid:18) (cid:19)
not augmentable. 5
(cid:136)10;
Example 6.To illustrate how we updatecounter values for 3
augmentable node-triplet matches, let us consider the
since all matching node-triplets have the same SF as
patternP inFig.9.InP,thenodesnumbered0,1,2,3,4 P
in (10) and the labels of the corresponding nodes are
match, after rotation, the nodesnumbered 5,4, 3,1,2 in
the same.
thesubstructureStr inFig.2a.Thenodenumbered0in
0
Str0 does not appear in P (i.e., it is to be deleted). The Now,consideragaintheSFP definedin(7)andthethree
labels of the corresponding nodes are identical. Thus, basis points P , P , P forming the SF , where P is the
c1 c2 c3 P c1
P matchesStr with1mutation,i.e.,onenodeisdeleted. origin. We note that for any node i in the pattern P with
0
Now, suppose in P, the global coordinates of the globalcoordinateP(cid:133)x;y;z(cid:134),ithasalocalcoordinatewith
i i i i
nodes numbered 1, 2, 3, and 4 are respect to the SF , denoted P0, where
P i
P1(cid:133)(cid:255)0:269000;4:153153;2:911494(cid:134); P0 (cid:136)V~ (cid:2)E(cid:255)1: (cid:133)12(cid:134)
i c1;i
P (cid:133)(cid:255)0:317400;4:749386;3:253592(cid:134);
2 Here, E is the base matrix for the SF , defined as
P (cid:133)0:172100;3:913515;4:100777(cid:134); P
3 0 1
P (cid:133)0:366000;3:244026;3:433268(cid:134): V~
4 B c1;c2 C
E (cid:136)@ V~ A (cid:133)13(cid:134)
RefertoExample5.Forthenodesnumbered3,4,and c1;c3
V~ (cid:2)V~
2 of P, the hashbin address is h(cid:137)25(cid:138)(cid:137)12(cid:138)(cid:137)21(cid:138), which is the c1;c2 c1;c3
same as that of nodes numbered 1, 2, 3 of Str0, and andV~ is the vector starting at P and ending at P.
0 1 c1;i c1 i
(cid:255)0:012200 5:005500 4:474200 Remark. If V~ , V~ , and V~ (cid:2)V~ are orthonormal
SF (cid:136)@ 0:987800 5:005500 4:474200A: (cid:133)10(cid:134) vectors, thecn1;c2jEjc(cid:136)1;c13. Thus, ac1;cp2racticc1;ac3lly useful criterion
P
(cid:255)0:012200 4:298393 3:767093 for detecting false matches is to check whether or not
jEj(cid:136)1. If jEj6(cid:136)1, then V~ , V~ , and V~ (cid:2)V~ are
The three basis vectors forming this SFP are c1;c2 c1;c3 c1;c2 c1;c3
not orthonormal vectors and, therefore, the nodes u, v,
V~ (cid:136)(cid:133)1:000000;0:000000;0:000000(cid:134); and w do not match the nodes i, j and k geometrically
c1;c2
V~ (cid:136)(cid:133)0:000000;(cid:255)0:707107;(cid:255)0:707107(cid:134); (see Theorem 1).
c1;c3 Intuitively, our scheme is to hash node-triplets and
V~ (cid:2)V~ (cid:136)(cid:133)0:000000;0:707107;(cid:255)0:707107(cid:134);
c1;c2 c1;c3 matchthetriplets.Onlyifonetriplettrimatchesanother
tri0doweseehowthesubstructurecontainingtrimatches
which are orthonormal.
thepatterncontainingtri0.UsingTheorem1,wedetectand
Forthenodesnumbered3,1,and4ofP,thehashbin
eliminatefalsenode-tripletmatches.UsingTheorem2,we
address is h(cid:137)24(cid:138)(cid:137)0(cid:138)(cid:137)9(cid:138), which is the same as that of nodes
recordinacounterthenumberofaugmentabletruenode-
numbered 1, 4, 2 of Str , and
0
triplet matches. The following theorem says that the
0 1
(cid:255)0:012200 5:005500 4:474200 countervalueneedstobelarge(i.e.,thereareasufficient
SF (cid:136)@ 0:987800 5:005500 4:474200A: (cid:133)11(cid:134) number of augmentable true node-triplet matches) in
P
(cid:255)0:012200 4:298393 3:767093 order to infer that there is a match between the
corresponding pattern and substructure. The larger the
These two true node-triplet matches have the same
Mut,thefewernode-tripletmatchesareneeded.
SF and,therefore,thecorrespondingcounterassociated
P
with the substructure Str of the graph 12 in Fig. 1 is Theorem3.LetStrbeasubstructureinthehashtableHandlet
0
updated to 2. After hashing all node-triplets of P, the GbethegraphfromwhichStrisgenerated.LetP beapattern
counter value will be where jPj(cid:21)Mut(cid:135)3. After rehashing the node-triplets of P,
740 IEEETRANSACTIONSONKNOWLEDGEANDDATAENGINEERING, VOL.14, NO.4, JULY/AUGUST2002
suppose there is an SF associated with Str whose counter qualified patterns. In Section 3, we will show experimen-
P
value Cnt>(cid:2) , where tallythatthe missedpatterns arefewcomparedwiththose
P
(cid:18) (cid:19) found by exhaustive search.
N(cid:255)1 (cid:133)N(cid:255)1(cid:134)(cid:2)(cid:133)N(cid:255)2(cid:134)(cid:2)(cid:133)N(cid:255)3(cid:134)
(cid:2)P (cid:136) 3 (cid:136) 6 (cid:133)14(cid:134) Theorem4.LetthesetScontainK graphs,eachhavingatmost
N nodes.Thetimecomplexityoftheproposedpattern-finding
and N (cid:136)jPj(cid:255)Mut. Then, P matches Str within Mut algorithm is O(cid:133)KN3(cid:134).
mutations (i.e., P approximately occurs in G, or G
Proof. For each graph in S, phase 1 of the algorithm
approximately contains P, within Mut mutations).
requires O(cid:133)N2(cid:134) time to decompose the graph into
Proof. By Theorem 2, we increase the counter value only substructures. Thus, the time needed for phase 1 is
when there are true node-triplet matches that are O(cid:133)KN2(cid:134). In subphase A of phase 2, we hash each
augmentable under the SF . Thus, the counter value
P candidate pattern P by considering the combinations of
shows currently how many node-triplets from the
any three nodes in P, which requires time
pattern P are found to match with the node-triplets
(cid:18) (cid:19)
fromthesubstructureStrinthehashtable.Notethat(cid:2) jPj
P (cid:136)O(cid:133)jPj3(cid:134):
represents the number of distinct node-triplets that can 3
be generated from a substructure of size N(cid:255)1. If there
Thus, the time needed to hash all candidate patterns is
are N(cid:255)1 node matches between P and Str, then
Cnt(cid:20)(cid:2) .9Therefore,whenCnt>(cid:2) ,thereareatleast O(cid:133)KN3(cid:134). In subphase B of phase 2, we rehash each
P P
candidate pattern, which requires the same time
N node matches betweenStr and P. This completes the
proof. tu O(cid:133)KN3(cid:134). tu
Notice that our algorithm cannot handle patterns with
sizesmallerthan3,becausethebasicprocessingunithereis 3 PERFORMANCE EVALUATION
a node-triplet of size 3. This is reflected in the condition
We carried out a series of experiments to evaluate the
jPj(cid:21)Mut(cid:135)3 stated in Theorem 3. Since Mut can only be
performanceandthespeedofourapproach.Theprograms
zeroorgreaterthanzero,thisconditionimpliesthatthesize
werewrittenintheCprogramminglanguageandrunona
of a pattern must be greater than or equal to 3.
SunSPARC 20 workstation under the Solaris operating
Example7.ToillustratehowTheorem3works,letusrefer
systemversion2.4.Parametersusedintheexperimentscan
to Example 6. Suppose the user-specified mutation
be classified into two categories: those related to data and
number Mut is 1. The candidate pattern P in Fig. 9 has
those related to the pattern-finding algorithm. In the first
sizejPj(cid:136)5.Afterrehashingthenode-tripletsofP,there
category, we considered the size (in number of nodes) of a
isonlyonecounterassociatedwiththesubstructureStr
0 graph and the total number of graphs in a data set. In the
inFig.2a;thiscountercorrespondstotheSF in(10)and
P
second category, we considered all the parameters de-
the value of the counter, Cnt, is 10. Thus, Cnt is greater
scribed in Section 2, which are summarized in Table 3
than (cid:2) (cid:136)(cid:133)5(cid:255)2(cid:134)(cid:133)5(cid:255)3(cid:134)(cid:133)5(cid:255)4(cid:134)=6(cid:136)1. By Theorem 3, P
P
together with the base values used in the experiments.
should match the substructure Str within 1 mutation.
0
Two files were maintained: One recording the hash bin
This means that there are at least four node matches
addressesandtheothercontainingtheentriesstoredinthe
between P and Str .
0
hash bins. To evaluate the performance of the pattern-
Thus,afterrehashingthenode-tripletsofeachcandidate finding algorithm, we applied the algorithm to two sets of
pattern P into the 3D table H, we check the values of the data: 1,000 synthetic graphs and 226 chemical compounds
counters associated with the substructures in H. By obtained from a drug database maintained in the National
Theorem 3, P approximately occurs in a graph G within Cancer Institute. When generating the artificial graphs, we
MutmutationsifGcontainsasubstructureStrandthereis randomly generated the 3D coordinates for eachnode. The
at least one counter associated with Str whose value node labels were drawn randomly from the range A to E.
Cnt>(cid:2) . If there are less than Occur graphs in which P The size of the rigid substructures in an artificial graph
P
approximately occurs within Mut mutations, then we rangedfrom4to10andthesizeofthegraphsrangedfrom
discardP.Theremainingcandidatesarequalifiedpatterns. 10 to 50. The size of the compounds ranged from 5 to 51.
Notice that Theorem 3 provides only the “sufficient” (but Inthissection,wepresentexperimentalresultstoanswer
not the “necessary”) condition for finding the qualified questions concerning the performance of the pattern-
patterns. Due to the accumulative errors arising in the finding algorithm. For example, are all approximate
calculations, some node-triplets may be hashed to a wrong patternsfound,i.e.,istherecallhigh?Areanyuninteresting
bin. As a result, the pattern-finding algorithm may miss patterns found, i.e., is the precision high? In Section 4, we
some node-triplet matches and, therefore, miss some study the applications of the algorithm and intend to
answer questions such as whether graphs having some
9. Notice that we cannot use Cnt(cid:136)(cid:2) here. The reason is that we
P common phenomenological activity (e.g., they are proteins
augment node-triplet matches only when they yield the same SF . In
P
practice,it’slikelythattwonode-tripletmatchescouldbeaugmented,but with the same function) share structural patterns in
yielddifferentSF sduetotheerrorsaccumulatedduringthecalculationof
P common and whether these patterns can characterize the
theSF s.Asaconsequence,ouralgorithmfailstodetectthosenode-triplet
P
matchesandupdatethecountervaluesappropriately. graphs as a whole.