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Theories of Hypergraph-Graph (HG(2)) DataStructure

Shiladitya MunshiMeghnad Saha Institute of Technology

Kolkata, India

Web Intelligence and

Distributed Computing research Lab

Golfgreen, Kolkata: 700095, India

Email:[email protected]

Ayan ChakrabortyTechno India College of Technology

Kolkata, India




Email: [email protected]

Debajyoti MukhopadhyayMaharastra Institute of Technology

Pune, India




Email: [email protected]

Abstract—Current paper introduces a Hypergraph-Graph(HG(2)) model of data storage which can be represented as ahybrid data structure based on Hypergraph and Graph. The pro-posed data structure is claimed to realize complex combinatorialstructures. The formal definition of the data structure is presentedalong with the proper justification from real world scenarios.The paper reports some elementary concepts of Hypergraphand presents theoretical aspects of the proposed data structureincluding the concepts of Path, Cycle etc. The detailed analysisof weighted HG(2) is presented along with discussions on Costinvolved with HG(2) paths. Keywords—ypergraph, Graph, Hyper-path, Hyperedges, HG(2), Cost of HG(2) Pathypergraph, Graph,Hyperpath, Hyperedges, HG(2), Cost of HG(2) PathH

I. INTRODUCTION

Hypergraphs, a generalization of Graph, is being widelyand deeply investigated since last few decades as a successfultool to represent and model complex concepts and structures invarious areas of Computer Science and Discrete Mathematics.Graphs on the other hand have enjoyed the spotlight ofheterogeneous research as an important and implementabletool to represent and analyze real world complex scenarios.

Hence, both Hypergraphs and Graphs are the well suitedcandidates for representing complex structures having logicalor real world relations with certain advantages over eachother. While Hypergraphs have more potential than graphs inrepresenting complex scenarios, it lacks readability, simplicityand easeness in conceptualization and physical representation.

In the current paper, a new data structure ”Hypergraph- Graph” (HG(2)) has been introduced, which takes theadvantages of both the data structures of Graph andHypergraph. This data structure can model a problem spaceinto two logical partitions with different levels of complexities.The more complex level can be represented by Hypergraphin order to take the leverages of its power to designate therelationships among a group of objects. The other level whichis characterized by comparatively lesser complexity and betterorderedness, could be represented by Graph in order to takethe advantage of its naturalness and simplicity.

With a high objective of introducing HG(2), the presentpaper has been organized as follows -Section 2 proposes HG(2) and related theoretical aspects

formally with definitions and illustrative examples. Section 3reports the concepts of paths within HG(2) with some co re-lated issues, followed by discussions on the impact of weightson HG(2) and computation of Cost of any HG(2) Path inSection 4. Lastly, Section 5 concludes the current investigationemphasizing the level of contribution , importance and futureresearch scope related with study of HG(2).

II. HYPERGRAPH - GRAPH (HG(2)) DATA STRUCTURE

Hypergraph - Graph data structure denoted as HG(2) isconceptualized as a model to represent a complex problemspace based on certain criteria. The criteria could be formalizedas follows -The problem space (PS) must logically be divided into twolevels with different complexities, one (PSG) with relativelylesser complexities, better orderdness and bounded by for-malized set of rules, and another (PSH) which could becharacterized by greater complexities and absence of orderedrule sets.The some or all interrelationship between objects of PSHmust be dictated by the objects of PSG and even the behaviorof some or all objects of PSH must be defined by the objectsof PSG. Here the term ”object” is being used informally andmust not necessarily indicate any Object Oriented paradigm.As the complex real life combinatorial structures are not rareat all, proposed HG(2) has an intrinsic objective to representPSH with Hypergraphs and PSG with Graphs. The interdependencies of PSH and PSG form the basis of evolutionof the behavior of HG(2) as a whole.

The theories behind the Hypergraph Data Structure arepresented in [5] and due to space constraint, it is omitted in thecurrent discussion. On this background, following subsectionpresents the theories of Hypergraph - Graph (HG(2)) datastructure directly.

A. Introducing HG(2)

A Hypergraph-graph data structure HG(2) is a tripledenoted as HG(2) = (H,G,C) where H is a Hypergraph, Gis a graph and C is a set of connectors.

H is a Hypergraph defined as H = (V h, Eh), where V h

= vh1 , vh2 , · · · , vhn, n = 2, 3, · · ·; and Eh = Eh

1 , Eh2 , · · · , Eh

m,

2013 International Conference on Cloud & Ubiquitous Computing & Emerging Technologies

978-0-4799-2235-2/13 $26.00 © 2013 IEEE

DOI 10.1109/CUBE.2013.45

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978-0-4799-2235-2/13 $26.00 © 2013 IEEE

DOI 10.1109/CUBE.2013.45

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978-0-4799-2235-2/13 $31.00 © 2013 IEEE

DOI 10.1109/CUBE.2013.45

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978-0-4799-2235-2/13 $31.00 © 2013 IEEE

DOI 10.1109/CUBE.2013.45

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m = 2, 3, · · · where each Ehi ⊆ V h

G is a Graph defined as G = (V g, Eg), where V g =vg1 , v

g2 · · · vgp , p = 2, 3, · · ·; and Eg = eg1, e

g2 · · · egq , q = 2, 3, · · ·

where each egi could be expressed in the form of egxy whichconnects vgy from vgx.

C is a set of connectors, which could be conceptualizedas a set of all the dependencies between PSH and PSG(as described earlier) which are characterized by H and Grespectively.

here we define two types of connectors:(a) cvxy which connects a node in the Graph vgy from a

node in the Hypergraph vhx . It is to note that the behavioraldependency of an object of PSH on an object of PSG getsrealized through cvxy; and(b) cexy which connects a node in the Graph vgy from a

Hyperedge Ehx . Here cexy realizes the dependencies of

collective behavior (bound with a specific relation) of theobjects of PSH on the objects of PSG.

The set of all cvxys is noted as Cv while Ce representsall cexys. Hence on the basis of ongoing discussion, it couldbe concluded that C = (Cv, Ce). For the rest of the paper,it is assumes for simplicity that the dependency flows fromPSH to PSG. That is the Hypergraph layer is dependent onthe Graph layer. No dependency flows through a connectorbackward from the Graph layer to the Hypergraph layer.Above discussion can be illustrated with the example as

Fig. 1. Illustration of an HG(2) data structure

shown in Fig 1. In this figure, an HG(2) is shown tohave a Hypergraph H and a Graph G. The H and the Gare connected with connectors C. The Hypergraph H iscomposed of (V h, Eh) where V h consists nodes 1, 2, 3, 4, 5,6 & 7 and Eh consists E1, E2, E3 & E4. Further, the GraphG is composed of nodes a, b, c, d, e & f . For simplicity theedges of the G are not highlighted.

It is to be noted that E1 is composed of nodes 1, 2, 3, E2

is composed of nodes 3, 4, 5, 6, E3 is composed of nodes4, 5, 7 and finally E4 is composed of nodes 5, 6 and 7.

Following is the identification of head and tail nodes for eachHyperedge:H(E1) = 1, 2 and T (E1) = 3H(E2) = 3, 4 and T (E2) = 5, 6H(E3) = 4, 5 and T (E3) = 7H(E4) = 5, 6 and T (E4) = 7

The Fig 1 further illustrates that there are three connectors(marked with dashed square ended bi-directional arrow)that connect Hyperedge nodes and Graph nodes. They arecv1a =c1, cv6b =c2 and cv2d =c3. Moreover there are threeconnectors (marked with bold bidirectional arrow) thatconnect a Hyperedge with a Graph node. They are identifiedas ce1c = d1, ce3e = d2 and ce4f = d3. Hence formally C whichis a pair (Cv, Ce) holds the following:Cv = c1, c2 and c3; and Ce = d1, d2 and d3.

Though the diagram shows the individual connectors withbidirectional arrow, the earlier assumption still holds that thedirection of the dependency is from Hypergraph to Graphlayer. The bidirectional arrows have been given for easydiagrammatic identification of connectors out of many differentedges.

However, Fig 1 shows the entire problem space to belogically divided (by a dashed horizontal divider) into PSGand PSH which are represented by the Graph and Hypergraphrespectively.

On the basis of current discussion, next section reportscritical issues related with concept of Path in a HG(2).

III. CONCEPT OF PATH IN HG(2)

The study of Paths in HG(2) and the related algorithmsdemand formalization of certain essential terminologies whichare presented next.

A node dependent pair or simply the node pair is definedas the pair of Hypergraph and graph nodes which share adependency relation through cvxy and is denoted as vhx(v

gy). In

Fig 1, 1(a), 6(b) and 5(d) could be identified as three nodepairs. It is to be noted that, in context to a Hypergraph nodenot involved in Cv , the node pair is symbolized as vhx( ). Thenode pair of a Hypergraph node n is designated as NPn.

Similarly a edge dependent pair or simple edge pair isdefined as the pair of Hyperedges and graph nodes which sharea dependency relation through cexy and is denoted as Eh

x (vgy).

Any edge pair in context to a Hyperedge not involved in Ce

is symbolized as Ehx ( ). In Fig 1, three edge pairs could be

identified as E1(c), E3(e) and E4(f). However, the edge pairof a Hyperedge n is designated as EPn

Based on the terminologies discussed above,

A path PHG(2)st in HG(2) (known as HG(2) path

of length q could be defined as a sequence

of node pair and edge pair. That is PHG(2)st =

{NP vh1=s, EPEh

1 , NP vh2 , EPEh

2 · · ·EPEhq , NP vh

q+1=t};where following conditions are all true;

i) PHG(2)st is a valid Hyperpath considering only the

Hyperedges in Hii) each NP vh

i = vhx(vgy) and each EPEh

i = Ehz (v

gw); i and x

= 2, 3, · · ·

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iii) for any NP vhi , EPEh

i pair, there exists a valid path fromvgy to vgw in G

iv) for any EPEhi , NP vh

i pair, there exists a valid path fromvgw to vgy in G

A path in Graph G traced by the Hyperpath in HypergraphH is defined as Graph Path or simple as GPath. A GPath isformed by the graph nodes involved in constituent node pairsand edge pairs (in sequence) of a HG(2) path.

An HG(2) path PHG(2)st is said to contain a Graph Loop or

simply GLoop if there exists a cycle in GPath

An HG(2) path PHG(2)st is said to contain a Hypergraph

Loop or simply HLoops if there exists a cycle in Hyperpathconsidering the Hypergraph only.

An HG(2) path PHG(2)st containing GLoop must not

necessarily contain HLoop and the vice-versa is also true.

Even the Gloop and Hloop may co-exist in a single PHG(2)st .

An HG(2) path PHG(2)st is qualified as Elementary if

all the nodes in the Hyperpath are distinct, and the same istermed as Simple if all the Hyperedges are distinct.

However, the idea regarding HG(2) path, that evolved upfrom the ongoing study, could be illustrated with the exampleshown in Fig 1.

Let us consider an HG(2) path PHG(2)17 . This path

starts from Hypergraph node 1 (V h1 = 1) and reaches

another Hypergraph node 7 (V h7 = G). P

HG(2)17 follows two

Hyperpaths The nodes and edges of the first and secondHyperpaths result sequences of node pair and edge pair asshown below:{1(a), E1(c), 3( ), E2( ), 5(d), E3(e), 7( )} and;{1(a), E1(c), 3( ), E2( ), 6(b), E4(f), 7( )}

Both these HG(2) paths ensure that the every followinggraph node in the sequence are connected to its previousnode. For example in {a, c, d, e}, c is connected to a, d isconnected to c (through b) and e is connected to d That is{a, c, b, d, e} and {a, c, b, f} form two valid paths in GraphG. Hence these two graph paths can be termed as GPath.

There exists another Hyperpath {1, E1, 3, E2, 5, E4, 7}that can be followed for P

HG(2)17 . The corresponding

node pair and edge pair sequence is generated as{1(a), E1(c), 3( ), E2( ), 5(d), E4(f), 7( )} and the sequenceof Graph nodes corresponding the node pair and edge pairsequence is generated as {a, c, d, f}. It is to be noted here thatnode f is not connected to node d and hence this sequencecan not form a valid path in the Graph G.Hence {1(a), E1(c), 3( ), E2( ), 5(d), E4(f), 7( )} is not avalid HG(2) path.

A path in a HG(2), thus, as seen in the above discussion,a valid Hyperpath with some additional constraints. Theseadditional constraints can be attributed to the GPath. Henceit is to be considered that an HG(2) path can have greaterpotential than a Hypergraph to model sequence of complexrelationships.

On this background, the next section discusses the notionsof Weighted HG(2) and consequently the cost involved inHG(2) paths.

IV. COST ANALYSIS OF PATH ON A WEIGHTED HG(2)

A Weighted HG(2) is composed of a weighted Hypergraph,a weighted Graph and weighted Connectors. In this inves-tigation, the weight simply means the weight of edges thatmeans the cost of connecting nodes within individual Graph,Hypergraph, nodes of Graph an Hypergraph and nodes ofGraph and edges of Hypergraph. The weight of graph edgesthat represents the cost of connecting two Graph nodes vgi andvgj is denoted by Ceg

ijor Ceg

kwhere egk connects node i and

node j. Similarly, the weight of hyperedge that represents thecost of connecting vha , v

hb , v

hc · · · related by hyperedge Eh

i isdenoted by CEh

i. The weight of node to node connector cvij is

the cost of connecting a node of Hypergraph vhi and a node ofgraph vgj , and it is denoted as Ccv

ij. Hence Cce

ijwill determine

the weight of edge to node connector ceij which connects Ehi

to cvj . Based on this discussion, the cost associated with a pathcan be viewed as follows.

Before proceeding any further, it is to be noted that antHG(2) Path may give rise to multiple valid Hyperpath routes.

For example, HG(2) Path PHG(2)17 in the HG(2) shown in Fig 3

gives rise three valid Hyperpath routes {1, E1, 3, E2, 5, E3, 7}, {1, E1, 3, E2, 6, E4, 7} and {1, E1, 3, E2, 5, E4, 7}. Conse-quently we get two valid HG(2) route out of these threeHyperpath routes as :{1(a), E1(c), 3( ), E2( ), 5(d), E3(e), 7( )} and;{1(a), E1(c), 3( ), E2( ), 6(b), E4(f), 7( )}Each of these valid HG(2) routes will be denoted by R1

PHG(2)17

and R2

PHG(2)17

. Hence in general, Rk

PHG(2)st

denotes the kth route

for HG(2) path PHG(2)st .

It is to be further noted that for each individual valid HG(2)route, there may exist multiple GPaths. Each of these GPathsare the valid connectors among the subgraph of G which hasbeen induced by corresponding cvij and ceij of the chosen HG(2)

route Rk

PHG(2)st

. Hence the notation GRk

PHG(2)st

will denote the

subgraph of G induced by the valid HG(2) route Rk

PHG(2)st

of

HG(2) Path PHG(2)st . An ith GPath GPpq on GRk

PHG(2)st

denotes

a path from graph node p to graph node q where p and q arethe first and last graph nodes encountered tracing a HG(2) PathP

HG(2)st . The ith GPath for graph node pair p and q is thus

denoted by GP ipq

The simplified concept of HG(2) Path presented in theprevious section could be represented by single or multipleHG(2) routes where each such HG(2) routes can be tracedwith single or multiple GPaths. The cost of HG(2) Path hencewill be dependent on cost of HG(2) routes and cost of GPath.

However Cost of HG(2) route is the summation of cost

of Hyperedges that is CRk

PHG(2)st

=

n∑

i=0

CEhi

where Eh0 to Eh

n

are the Hyperedges that constitute the Hyperpath of the HG(2)

207206206206

route. Similarly the cost of GPath CGP ipq

=

n∑

i=0

Cegi

where egi

to egn constitute the GPath.

This paragraph will investigate the effects of cost of con-nectors into the computation of cost of HG(2) Path. While con-sidering GP i

pq , there might be some nodes which are connectedwith Hypernodes and Hyperedges for the Hyperpath selected,and at the same time ther might exist some graph nodes vgi swho are not connected with the any of the Hypernodes orHyperedges, rather they got traced in order to establishing avalid path between two consecutive graph nodes of node pairand edge pair (discussed in previous section).Let us consider the case as shown in Fig 3. For the Hg(2) PathP

HG(2)st and for G(2) route R1

PHG(2)17

, node b got selected to

establish the path between node c and node d. Here node cand node d are directly connected to the Hyperedge E1 andHypernode 5 respectively, but node b has no connection withthe selected Hyperpath. While the nodes connected with theselected Hyperpath are referred to as Participating Nodes, theother types of graph nodes are referred to as Auxiliary Nodes.

The characterization of Auxiliary Nodes results that theremight be three cases as follows:(i) A particular Auxiliary Node has no connection at all. Neitherit participates in any cvij nor in any ceij(ii) A particular Auxiliary Node has connections out of thescope of the selected Hyperpath; and(ii) A particular Auxiliary Node has connections within thescope of the selected Hyperpath.The example shown just above represents the case (ii).

The engagement of Case (i) clearly has no effect incomputing Cost of HG(2) Path, while engagement of Case(ii) and case (iii) induces some dependency of Graph layerto Hypergraph layer. But as stated earlier in Section 3, forsimplicity, there exists no dependency from Graph layer toHypergraph layer, so even the existence of Case (ii) and case(iii) does not make any difference in computation of Costs ofHG(2) Path. Had there been a scope of mutual dependencybetween Graph and Hypergraph layer, the existence of Case(ii) and Case (iii) had have severe effect in HG(2) Path cost,but in present research, this is beyond the scope of any furtherdiscussions.

On this background, for all the Participating Nodes, thereare wighted connectors and only the connectors connected tothe Participating Nodes will contribute the cost in computingHG(2) Path. Hence the cost of connectors for a set of specificRx

PHG(2)st

and specific GP ypq , which is denoted as Cxy , could

be given by the relation:

Cxy =

n∑

r=0

Ccvr +

n∑

r=0

Ccer where each cvr = cvij and each cer = ceij

denote node to node and edge to node connector respectivelywhich are connected to each of the participating node j for aspecific selected GPw

pq (w = 1, 2 · · ·)All the preceding discussions identify the cost expression

of a HG(2) Path PHG(2)st for a specific HG(2) route Ru

PHG(2)st

and specific GPath GP zpq as

CP

HG(2)st

(u, z) = CRu

PHG(2)st

+ CGP zpq

+ Cuz .

For a given PHG(2)st there will be multiple set of (u, z) pair

and the min(CHG(2)stP (u, z)) will denote the least cost.

V. CONCLUSION

The present research has introduced a data structure whichhas been denoted as HG(2). This proposed data structurehas a great potential in modeling and resolving complexcombinatorial problems. The basic Hypergraph and Graphbased architecture of this data structure has been presented.The theoretical concepts of Paths in HG(2) has been discussed.Cost Analysis of HG(2) Path has been thoroughly presented.

Within the current discussion, the future scope of HG(2)research has aptly been identified to focus on generation ofMinimum Spanning Tree out of HG(2), Identification of LeastCost Path, HG(2) Path Traversal algorithms and modeling ofdifferent real life problems (like RDF integration in SemanticWeb).

REFERENCES

[1] Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973.

[2] Boullie, F.: Un modle universel de banque de donnes, simultanmentpartageable, portable et rpartie. Thse de Doctorat d’Etat es SciencesMathmatiques (mention Informatique), Universit Pierre et Marie Curie,1977.

[3] Berge, C.: Minimax theorems for normal hypergraphs and balancedhypergraphs - a survey. Annals of Discrete Mathematics, 21 (1984), 3-19.

[4] Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North-Holland,Amsterdam, 1989.

[5] Gallo, G., Longo, G., Nguyen, S., Pallot-tino, S.: Directed Hypergraphs and Applications.www.cis.upenn.edu/ lhuang3/wpe2/papers/gallo92directed.pdf

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