REPRESENTING THE PAIRWISE INTERSESSION NETWORK CODING BY HIGH LEVEL PETRI NETS

In this paper, we introduce a new approach to the pairwise intersession network coding, which depends on using high level Petri nets to represent the butterfly network, the grail network, all networks which represent an expansion of the butterfly network and all networks which represent an expansion of the grail network, and use this representation to determine the solvability of the pairwise intersession network coding problem. AMS Subject Classification: 94A99


Introduction
The idea of network coding is introduced by Ahlswede et al. [1], Follow-up work [2] shows that linear network coding is sufficient for a single multicast session.An algebraic approach to network coding has been developed by Koetter and Medard [3].Several other related works of intra-session network coding can be found in [4]- [6].When we generalize the problem such that there is more than one session and receivers may demand different sets of information, finding the optimal network coding strategy is still an open question.The linear coding is shown to be insufficient for optimal coding in the multi-session case [7].In their paper [3] Koetter and Medard introduced an algebraic point view on network coding but the solvability decision problem is shown to involve Grobner basis computation, whose complexity may prohibit practical implementations for large problems.In our work [8] we redefine the intersession network coding (INC) as an algebraic structure.The intersession network coding on a communication network is called pairwise intersession network coding if and only if the network contains two source nodes and two sink nodes and each sink node demands the information which is generated at one source node [9,10].In this paper we introduce a new approach to the pairwise intersession network coding (PINC), which depends on using high level Petri nets (HLPNs) [11] to represent the butterfly network, the grail network, all networks which represent an expansion of the butterfly network and all networks which represent an expansion of the grail network, and use this representation to determine the solvability of pairwise intersession network problem.

Pairwise Intersession Network Coding
We assume that the network is given by a directed acyclic graph G = (V, E) where V and E are the set of all nodes and links, respectively [13], links are denoted by round brackets (v 1 , v 2 ) ∈ E and assumed to be directed, the head and tail of an link e = (v ′ , v) ∈ E are denoted by v = head(e) and v ′ = tail(e) [3], for each node v ∈ V we define In(v) = {e ∈ E; head(e) = v}, Out(v) = {e ′ ∈ E; tail(e ′ ) = v},then the sets of source and sink nodes are define as follows: If G = (V, E) contains exactly two source nodes s 1 , s 2 and exactly two sink nodes R 1 , R 2 , where R 1 demands the symbol x 1 which is generated at s 1 and R 2 demands the symbol x 2 which is generated at s 2 , then the network coding problem (NC) over G = (V, E) is called pairwise network coding problem (PINC) [9].Theorem 1. (see [13]) Define two sets of graphs, G b and G g , as follows: • G b contains the butterfly as described in Figure (1-a) and all graphs obtained from the butterfly via edge contraction, e.g. , Figure (1-b).Suppose there exists a network coding solution to the pairwise intersession network coding problem.Then one of the following two conditions must hold.

Petri Nets
Petri nets are a graphical and mathematical modeling tool applicable to many systems, they are promising tools for describing and studying information processing systems [14].

Definition 1. [8]
A net graph is a structure N G = (P, T ; F in , F out ) where: • P is a finite set of nodes, called Places.
• T is a finite set of nodes, called Transitions, disjoint from P .
• F in ⊂ T × P is a set of of directed edges called input arcs.
Figure 2 • F out ⊂ P × T is a set of of directed edges called output arcs.
• Place Types.These are non-empty sets.One type is associated with each place.
• Place Marking.A collection of elements (data items) chosen from the place's type and associated with the place.Repetition of items is allowed.The items associated with places are called tokens.
• Arc Annotations: Arcs are inscribed with expressions which may comprise constants, variables (e.g., x; y) and function images (e.g., f(x)).The variables are typed.The expressions are evaluated by assigning values to each of the variables.When an arc's expression is evaluated, it must result in a collection of items taken from the type of the arc's place.The collection may have repetitions.

Representing the Network Graph G by High Level Petri Net
Without loss of generality we assume that the output of each source node is equal to one, and assume that the input of each sink node is equal to one, so if the output of a source node s is more than one, we add a source node s ′ and an edge from s ′ to s , and if the input of a sink node R is more than one, we add a sink node R ′ and an edge from R to R ′ .

Net Graph
To represent the network graph by a net graph, we partition it into subgraphs through which the same information flows, and represent each subgraph by a place, a transition and an arc from the place to the transition, after that we represent the set of edges which connect nodes from a subgraph to nodes from another subgraph by an arc from the place which represents the first subgraph to the transition which represents the second.

Place Types
We assume that the finite field F q is the place type of all places in the high level Petri net.

Place Marking
If the place p represents the subgraph G p then the place marking of p is equal to the information symbol which flows through G p .

Arc Annotations
• The expression of the arc (p ′ , t ′ ) is equal to the place marking of p ′ .
• The expression of the arc (t, p) is called t 's operation.

Transition Condition
We assume that the transition Condition is true for all transitions in the net graph so it can be omitted.
Remark 1. we say that the transition t represents the source (sink) node s ′ ( R ′ ) if and only if it represents the subgraph which contains s ′ ( R ′ ).

The Butterfly Network
The butterfly network in Figure (1-a) can be partitioned into the subgraphs in Figure (3) through which the same information flows.
So the high level Petri net which its net gragh is depicted in Figure(4) represents the butterfly network , where the transition t 1 represents the source node s 1 , the transition t 2 represents the source node s 2 , the transition t 4 represents the sink node R 1 and the transition t 5 represents the sink node R 2  So the high level Petri net which its net gragh is depicted in Figure (6) represents the grail network, where the transition t 1 represents the source node s 1 , the transition t 2 represents the source node s 2 , the transition t 4 represents the sink node R 1 and the transition t 5 represents the sink node R 2 .

Representing the Networks which Represent an Expansion of the
Butterfly Network (The Grail Network) by High Level Petri Net In general, an expansion of a graph G is a graph resulting from the subdivision of edges in G .The subdivision of some edge e = (u, v) yields a graph containing one new vertex w , and with an edge set replacing e by two new edges, (u, w)and (w, v).

Representing an Expansion of the Butterfly Network by High Level Petri Net
The network in Figure (7), which represents an expansion of the butterfly network, can be partitioned into the subgraphs in Figure (8) through which the same information flows.So the network in Figure (7) is represented by the high level Petri which represents the butterfly network, and we note that all networks which represent Figure 7 an expansion of the butterfly network are represented by the same high level Petri net.

Representing an Expansion of the Grail Network by High Level Petri Net
The network in Figure (9), which represents an expansion of the grail network, can be partitioned into the subgraphs in Figure (10) through which the same information flows.So the network in Figure( 8) is represented by the high level Petri which represents the grail network, and we note that all networks which represent an expansion of the grail network are represented by the same high level Petri net.After we represent the butterfly network, grail network and all networks which represent an expansion of butterfly and grail networks we note that: 1.The butterfly network and all its expansions are represented by the same high level Petri net.
2. The grail network and all its expansions are represented by the same high level Petri net.

G Contains a subgraph G
′ which is represented by the high level Petri net which its net gragh is depicted in Figure (12).

G Contains a subgraph G
′ which is represented by the high level Petri net which its net gragh is depicted in Figure(4).[5] Y. Wu, K. Jain, and S.Y.Kung, A unification of network coding and tree-