**Computing Concept: **

Let TSG(v, w) and TSG(w, z) be transfer subgraphs in G = (V, E) of depth d1 and d2, respectively. We let + denote the sequential composition of two transfer subgraphs. TSG(v, z) = TSG(v, w) + TSG(w, z).

It is clear that the composite transfer subgraph TSG(v, z) has a starting vertex v, ending vertex z, depth d1 + d2, and the set of its arcs is the union of the sets of arcs of TSG(v, w) and TSG(w, z).

Let G = (V, E). Let TSG_Container(i) be the collection of transfer subgraphs in G of depth i, where i >= 0. TSG_Container(0) contains the trivial transfer subgraph of depth 0 and empty set of arcs. We adopt the representation you have used in your implementation to demonstrate the composition of simple paths:

// define the type for the TSG containers // this definition corresponds to what you have already used typedef map< int, vector<tsgTYPE> > container_t; // declare the containers container_t STGcontainers;

Undefined TSG

In the previous blog, we have introduced a default constructor for tsgTYPE class and vUNDEFINED was introduced. An undefined TSG is simply tsgTYPE(vUNDEFINED, vUNDEFINED). It can be also obtained by creating a TSG object using the default constructor.

**A Simple Algorithm**

Finding all the combinations of starting and ending vertices in the TSG containers, held in your map representations, is a daunting challenge. We adopt “don’t care” strategy and combine any two transfer subgraphs and produce the resulting transfer subgraph if the rules of sequential composition (see above) apply; *otherwise *the resulting transfer subgraph is *undefined*.

// create an instance of an undefined TSG // for later comparisons tsgTYPE undefined_tsg; // declare the map of containers container_t STGcontainers; // suppose that TSG subgraphs of depth 1, 2, ...., k (k > 0) have been generated // produce TSG subgraphs of depth k + 1 for (int i = 1; i <= k; i++) { vector<tsgTYPE> NextContainer; for each tsgx in the vector retrieved from STGContainers[i] { for each tsgy in the vector retrieved from STGContainers[k+1-i] { tsgTYPE newTSG; newTSG = tsgx + tsgy; if (newTSG != undefined_tsg) NextContainer.push_back(newTSG); } } }

The proof of correctness follows directly from TSG definitions and the implementation of new operators on transfer subgraphs: operator= (assignment), binary operator+ (composition), checking for equality == and inequality !=. These operators shall be defined for tsgTYPE class. Accordingly, the new specification of tsgTYPE class becomes:

class tsgTYPE { private: VKEY _v; // starting vertex VKEY _w; // ending vertex ESET_TYPE E; // set of arcs of TSG(v, w) static VSET_TYPE _V; unsigned int _depth; // the length of the longest path in TSG(v, w) public: // basic constructors // other constructors may be added later for convenience tsgTYPE() { _v = _w = vUNDEFINED; } tsgTYPE(VKEY v, VKEY w) { _v =v; _w = w;} // getters and setters void setArcs(ESET_TYPE E) { _E = E; } void setStartingVertex(VKEY v) { _v = v; } void setEndingVertex(VKEY w) { _w = w; } void setDepth(unsigned int d} { _depth = d; } VKEY getStartingVertex() const { return _v; } VKEY getEndingVertex() const { return _w;} ESET_TYPE getArcs() const { return _E} unsigned int getDepth() const ( return _depth; } // static member function static void vertex_universe(VSET_TYPE V); // Added functionality is specified below // copy constructor tsgTYPE(const tsgTYPE& rhs); // overloading of the assignment operator tsgTYPE& operator=(const tsgTYPE& rhs); // overloading of the + operator (composition) tsgTYPE operator+(const tsgTYPE& rhs) const; // overloading of operators: == and != bool operator==(const tsgTYPE& rhs) const; bool operator!=(const tsgTYPE& rhs) const; // destructor ~tsgTYPE(); };