Sequential Composition

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();
 };