## 4 Experimental Results  
In the following, we present path distributions with respect to their number of _turns_ on some cases of two-item \(\{\text{e}^{n_{1}},\text{n}^{n_{2}}\}\) multiset in Table 1. Among then, all paths of the 2D lattice \(L(3,\,4)\) generated by **G2 DLP** are presented, according to the number of _turns_, in Table 2. \begin{table}
\begin{tabular}{r r r r r r r r r r r r r r} \hline \hline \multirow{2}{*}{_Turns_} & \multicolumn{2}{c}{_n1_} & \multicolumn{2}{c}{**3**} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**5**} & \multicolumn{2}{c}{**5**} & \multicolumn{2}{c}{**5**} & \multicolumn{2}{c}{**5**} & \multicolumn{2}{c}{**5**} \\ \cline{2-13}  & \multicolumn{2}{c}{_n2_} & \multicolumn{2}{c}{**4**} & \multicolumn{2}{c}{**6**} & \multicolumn{2}{c}{7} & \multicolumn{2}{c}{**8**} & \multicolumn{2}{c}{**9**} & \multicolumn{2}{c}{**10**} & \multicolumn{2}{c}{**6**} & \multicolumn{2}{c}{7} & \multicolumn{2}{c}{**8**} & \multicolumn{2}{c}{**9**} & \multicolumn{2}{c}{**10**} & \multicolumn{2}{c}{**11**} \\ \hline
1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
2 & 5 & 8 & 9 & 10 & 11 & 12 & 9 & 10 & 11 & 12 & 13 & 14 \\
3 & 12 & 30 & 36 & 42 & 48 & 54 & 40 & 48 & 56 & 64 & 72 & 80 \\
4 & 9 & 45 & 63 & 84 & 108 & 135 & 70 & 96 & 126 & 160 & 198 & 240 \\
5 & 6 & 60 & 90 & 126 & 168 & 216 & 120 & 180 & 252 & 336 & 432 & 540 \\
6 & 1 & 40 & 75 & 126 & 196 & 288 & 100 & 180 & 294 & 448 & 648 & 900 \\
7 & 0 & 20 & 40 & 70 & 112 & 168 & 80 & 160 & 280 & 448 & 672 & 960 \\
8 & 0 & 5 & 15 & 35 & 70 & 126 & 30 & 80 & 175 & 336 & 588 & 960 \\
9 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 30 & 70 & 140 & 252 & 420 \\
10 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 21 & 56 & 126 & 252 \\ Total Paths & 35 & 210 & 330 & 495 & 715 & 1001 & 462 & 792 & 1287 & 2002 & 3003 & 4368 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The path distributions with respect to their number of _turns_. \begin{table}
\begin{tabular}{r r r r r r r r r} \hline \hline
**Path Number** & **Step 1** & **Step 2** & **Step 3** & **Step 4** & **Step 5** & **Step 6** & **Step 7** & **Number of _Turns_** \\ \hline
2 & n & n & n & e & e & e & 1 \\
3 & e & n & n & n & e & e & 2 \\
4 & e & e & n & n & n & e & 2 \\
5 & n & e & e & e & n & n & n & 2 \\
6 & n & n & e & e & e & n & 2 \\
7 & n & n & e & e & e & n & n & 2 \\
8 & e & n & e & e & n & n & n & 3  
To confirm the claim that the Algorithm **G2 DLP** is a linear time algorithm, we perform a series of experiments and present the results in Table 3 and Figure 1. The running time (h: m: s) is for generating only but without printing out the paths, by using an Acer (r) notebook with an Intel(r) Core(tm) i5-4200U CPU @ 1.6 GHz and a VBA program executed under the Microsoft Excel environment. Based on the experimental results in Table 3 and Figure 1, we confirm that Algorithm **G2 DLP** is a linear time algorithm. In order to highlight the superiority of Algorithm **G2 DLP**, we perform a series of experiments and present the running times for generating 2D lattice _L_(15, 15) paths, according to the number of _turns_, in the following Table 4. Note that we skip the two trivial paths that with number of _turns_ 1. From the results of Table 4, it is easy to know that, on average, the running time of Algorithm **G2 DLP** for generating a path is almost a constant (mean: 9\(\times\)10\({}^{-6}\) seconds, standard deviation: 7.8\(\times\)10\({}^{-6}\) seconds). This result confirms again that Algorithm **G2 DLP** is a linear time algorithm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n_{1}\) & \(n_{2}\) & Total Paths & Running Time (h:m:s) & \(n_{1}\) & \(n_{2}\) & Total Paths & Running Time (h:m:s) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Running Time of **G2 DLP**_vs_. Total Paths. Figure 1: Running Time of **G2 DLP**_vs_. Total Paths.