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

Algorithm 3

Connected component labeling over an 𝑛D-OPP expressed in the 𝑛D-EVM.

Input: An 𝑛D-OPP 𝑝.

    The number 𝑛 of dimensions.

Output: A list containing the EVMs which describe the 𝑛D components of 𝑝.

Procedure ConnectedComponentsLabeling(EVM 𝑝, int 𝑛)

    if(𝑛=1) then

     return ConnectedComponentsLabeling1D(𝑝) // Base case.

    else

     EVM hvl     // Current (𝑛−1)𝐷 couplet of p.

     EVM 𝑆_𝑖, 𝑆_𝑗     // Previous and next (𝑛−1)𝐷 sections about hvl.

     𝑆_𝑖 = InitEVM( )

     List components 𝑆_𝑖 // The (𝑛−1)𝐷 components of section 𝑆_𝑖.

     List indexes 𝑆_𝑖   // A list of indexes.

     List components 𝑆_𝑗  // The (𝑛−1)𝐷 components of section 𝑆_𝑗.

     List indexes 𝑆_𝑗  // A list of indexes.

     List componentsP // The 𝑛D components of input polytope p.

     CoordType currCoord = getCoordNextHvl(𝑝)

     hvl = ReadHvl(𝑝)

     CoordType nextCoord = getCoordNextHvl(𝑝)

     while(Not(EndEVM(𝑝)))

       𝑆_𝑗 = GetSection(𝑆_𝑖, hvl)

       components 𝑆_𝑗 = ConnectedComponentsLabeling (𝑆_𝑗,𝑛−1) // Recursive call.

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

         EVM 𝑐𝑆_𝑗 = components 𝑆_𝑗(𝑗)

         boolean withoutComponent = true

         for(int 𝑖=0, i < Size(components 𝑆_𝑖), 𝑖=𝑖+1)

           EVM 𝑐𝑆_𝑖 = components 𝑆_𝑖(𝑖)

           EVM c = BooleanOperation(𝑐𝑆_𝑗,𝑐𝑆_𝑖, Intersection, 𝑛−1)

           if(Not(IsEmpty(c)) and (withoutComponent = true)) then

             // CASE 1

             withoutComponent = false

             List componentToUpdate = componentsP(indexes 𝑆_𝑖(𝑖))

             Append(componentToUpdate, {𝑐𝑆_𝑗, currCood, nextCoord})

             Append(indexes 𝑆_𝑗, indexes 𝑆_𝑖(𝑖))

           else if(Not(IsEmpty(c)) and (withoutComponent = false)) then

             // CASE 2

             if(indexes 𝑆_𝑖(𝑖)≠ indexes 𝑆_𝑗(𝑗)) then

               List component1 = componentsP(indexes 𝑆_𝑖(𝑖))

               List component2 = componentsP(indexes 𝑆_𝑗(𝑗))

               for(int 𝑘=0, k < Size(component1), 𝑘=𝑘+1)

                 Append(component2, component1(𝑘))

               end-of-for

               Update(componentsP, indexes 𝑆_𝑖(𝑖), NULL)

               for(int 𝑘=0, k < Size(indexes 𝑆_𝑖), 𝑘=𝑘+1)

                 if(indexes 𝑆_𝑖(𝑘) = indexes 𝑆_𝑖(𝑖)) then

                   Update(indexes 𝑆_𝑖, k, indexes 𝑆_𝑗(𝑗))

                 end-of-if

               end-of-for

               for(int 𝑘=0, k < Size(indexes 𝑆_𝑗), 𝑘=𝑘+1)

                 if(indexes 𝑆_𝑗(𝑘) = indexes 𝑆_𝑖(𝑖)) then

                   Update(indexes 𝑆_𝑗, k, indexes 𝑆_𝑗(𝑗))

                 end-of-if

               end-of-for

               Update(indexes 𝑆_𝑖, i, indexes 𝑆_𝑗(𝑗))

             end-of-if

           end-of-if

          end-of-for

          if(withoutComponent = true) then

            // CASE 3

            List newComponent

            Append(newComponent, {𝑐𝑆_𝑗, currCoord, nextCoord})

            Append(componentsP, newComponent)

            Append(indexes 𝑆_𝑗, Size(componentsP) - 1)

          end-of-if

         end-of-for

         𝑆_𝑖=𝑆_𝑗

         components 𝑆_𝑖 = components 𝑆_𝑗

         indexes 𝑆_𝑖 = indexes 𝑆_𝑗

         currCoord = GetCoordNextHvl(𝑝)

         hvl = ReadHvl(𝑝) // Read next couplet.

nextCoord = GetCoordNextHvl(𝑝)

end-of-while

// End of sections’ processing.

int 𝑖=0

while(i < Size(componentsP))

  if(componentsP(i) = NULL) then

    Remove(componentsP, i)

    𝑖=𝑖−1

  else

    List currentListOfSections = componentsP(i)

    EVM evmComponent = getEVMforListOfSections(currentListOfSections, 𝑛)

    Update(componentsP, 𝑖, evmComponent)

  end-of-if

  𝑖=𝑖+1

  end-of-while

  return componentsP

 end-of-if

end-of-procedure