Efficient Boundary Extraction from Orthogonal Pseudo-Polytopes: An Approach Based on the 𝑛D-EVM

Algorithm 4

Extracting 𝑘D boundary elements from an OPP expressed in the EVM.

Input:     An 𝑘D-OPP 𝑝

   The number k of dimensions

   A list boundaryElements containing 𝑛 lists.

   A list previousCoordinates for completing the 𝑛 coordinates of

 (𝑘−1)-coordinates points.

Procedure GetBoundaryElements(EVM 𝑝,int k,List boundaryElements,List previousCoordinates)

    if (𝑘≥2) then

    EVM FDcurr, BDcurr

    List v = GetSetVX1(𝑝)

    List identifiedBoundaryElements = boundaryElements (𝑘−1)

    List vertices = boundaryElements(0)

    {FD, BD} = GetForwardBackwardDifferences(𝑝,𝑘)

     for(int 𝑖=0,𝑖< Size(FD), 𝑖=𝑖+1)  // Forward Differences are processed.

       EVM FDcurr = FD(i)

       List compsFDcurr = ConnectedComponentsLabeling(FDcurr, 𝑘−1)

       CoordType commonCoord = v(i)

       for(int 𝑗=0,𝑗< Size(compsFDcurr), 𝑗=𝑗+1)

         EVM currComp = compsFDcurr(j)

         List verticesCurrComp

         for each Extreme Vertex eV in currComp do

           List point = getNdPoint(eV, commonCoord, previousCoordinates)

           int index = IndexOf(vertices, point)

           if(index < 0) then // A new point has been discovered.

             Append(vertices, point)

             index = Size(vertices) − 1

             end-of-if

             Append(verticesCurrComp, index)

           end-of-for

           int index = IndexOf(identifiedBoundaryElements, verticesCurrComp)

           if(index < 0) then // A new (𝑘−1)𝐷 boundary element has been found.

             Append(identifiedBoundaryElements, verticesCurrComp)

           end-of-if

           Append(previousCoordinates, commonCoord)

           // Recursive call for identifying new (𝑘−2)𝐷 boundary elements.

           GetBoundaryElements(currComp, 𝑘−1, boundaryElements, previousCoordinates)

           Remove(previousCoordinates, Size(previousCoordinates) - 1)

         end-of-for

       end-of-for

       for(int 𝑖=0,𝑖< Size(BD), 𝑖=𝑖+1)  // Backward differences are processed.

         EVM BDcurr = BD(i)

         List compsBDcurr = ConnectedComponentsLabeling(BDcurr, 𝑘−1)

         CoordType commonCoord = v(i)

         for(int 𝑗=0,𝑗< Size(compsBDcurr), 𝑗=𝑗+1)

           EVM currComp = compsBDcurr(j)

           List verticesCurrComp

           for each Extreme Vertex eV in currComp do

             List point = getNdPoint(eV, commonCoord, previousCoordinates)

             int index = IndexOf(vertices, point)

             if(index < 0) then // A new point has been discovered.

               Append(vertices, point)

               index = Size(vertices) − 1

             end-of-if

             Append(verticesCurrComp, index)

           end-of-for

           int index = IndexOf(identifiedBoundaryElements, verticesCurrComp)

           if(index < 0) then // A new (𝑘−1)𝐷 boundary element has been found.

             Append(identifiedBoundaryElements, verticesCurrComp)

           end-of-if

           Append(previousCoordinates, commonCoord)

           // Recursive call for identifying new (𝑘−2)𝐷 boundary elements.

           GetBoundaryElements(currComp, 𝑘−1, boundaryElements, previousCoordinates)

           Remove(previousCoordinates, Size(previousCoordinates) - 1)

         end-of-for

       end-of-for

     end-of-if

    end-of-procedure