## 6 Computation  
Computation leads to the delivery of a computational tool based on the modified linear programming method for finding a minimum set of axial lines. The problem-solution procedure follows several steps. To begin with, the s-lines and all-line axial maps are generated for a given plan. This paper uses the cases shown in Figures 4 and 5. For each combination of an axial line and an s-line, whether they have a crossing point is checked. The checking results then are used to populate matrix \(\mathbf{A}\) in Formulations 3. Next, each pair of axial lines is checked to determine whether they have a crossing point, and these checking results populate matrix \(\mathbf{C}\) in Formulation 3. The column vectors \(\mathbf{B}\) and \(\mathbf{D}\) in Formulations 3 are assigned values of 1 and then the formulations solve the given linear programming problem. The axial lines whose corresponding variables have values of 0 are eliminated. A minimal set of axial lines is generated from this procedure to get everywhere in a building.  
For the computations in this paper, Syntax2D was used to calculate an all-line axial map, s-lines, and intersections among them. Syntax2D is a computer program for Space Syntax analysis that was developed at the University of Michigan, which is available at the sourceforge website ([[https://sourceforge.net/projects/syntax2](https://sourceforge.net/projects/syntax2) dnew]([https://sourceforge.net/projects/syntax2](https://sourceforge.net/projects/syntax2) dnew)) [14]. To solve the linear programming problem in this paper, a free linear programming solver library, lp_solve 5.5, that is available at the sourceforge website ([[http://lpsolve.sourceforge.net/5.5/](http://lpsolve.sourceforge.net/5.5/)]([http://lpsolve.sourceforge.net/5.5/](http://lpsolve.sourceforge.net/5.5/))) is integrated into Syntax2D. The solutions are calculated with all pre-solve options of lp_solve active. Figure 6 illustrates minimal sets and numbers of axial lines in the collected buildings and urban layouts.  
Figure 5: Cases of Buildings and Urban Layout. (**a**) Plan of Building 1 (Source [2]) (**b**) Plan of Building 2 (Source [2]) (**c**) Plan of Building 3 (Source [2]) (**d**) Plan of Gassin (Source [2]) (**e**) Plan of Apt (Source [2]). (The authors, S.K.J. and Y.K., redraw original figures in [2] with permission of the authors, [PERSON] et al. [2]) and the publisher, SAGE publications).  
It should be pointed out that the solution time may greatly vary depending on the complexity of problem, which affects the internal business of the solver library. Table 1, summarizing the computation time of the sample cases describes such an effect: the number of axial lines of Apt is similar to that of Gassin, but its solution time is far longer. A 3.6 GHz single core of a multi-core CPU was used for the computations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multirow{2}{*}{**Number of Axial Lines**} & \multicolumn{2}{c}{**Data Preparation and**} \\  & & **Problem Formulation (ms)** & **Solution (ms)** \\ \hline Building 1 & 692 & 110 & 30 \\ Building 2 & 184 & 10 & \textless{}1 \\ Building 3 & 225 & 10 & 10 \\ Gassin & 1124 & 532 & 1722 \\ Apt & 1296 & 740 & 124,389 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computation time of the sample cases.  
Figure 6: Minimal sets and numbers of axial lines in building and urban layouts. (**a**) Plan of Building 1, n = 15 (**b**) Plan of Building 2, n = 13 (**c**) Plan of Building 3, n = 7 (**d**) Plan of Gassin, n = 38 (**e**) Plan of Apt, n = 63.