## 3 Materials and Methods
Given that  
\[H\Big{(}I(E_{k}),I\Big{(}E_{k_{y}}^{y}\Big{)}\Big{)}<\lambda_{H}, \tag{15}\]  
where \(\lambda_{H}\in[0,1]\) is a predetermined parameter value, with probability one; it can be stated that the index element with number \(k\) is similar to an index element with a number \(k_{y}\) tables with number \(y\). Consequently, the table with the number \(y\) can be similar to an input table with a threshold \(\lambda_{H}\), that is, it contains a near duplicate. We construct a finite numerical sequence \(N=\{n_{1},n_{2},\ldots,n_{v}\}\) input table B set of subsequences \(K_{1},K_{2},\ldots,K_{v-g+1}\). We also consider that for each of the tables \(B_{1},B_{2},\ldots,B_{p}\)based on their numerical data sequences \(\mathrm{N}^{y}=\{n_{1}^{y},n_{2}^{y},\ldots,n_{\mathrm{v}}^{y}\}\), \(y=\overline{1,p}\), subsequences are constructed as \(\mathrm{K}_{1}^{y},\mathrm{K}_{2}^{y},\ldots,\mathrm{K}_{\mathrm{v}-g+1}^{y}\) by the sliding window method:  
\[ \begin{array}{c}\mathrm{K}_{1}^{y}=\{n_{1}^{y},n_{2}^{y},\ldots,n_{\mathrm{g} }^{y}\},\\ \mathrm{K}_{2}^{y}=\left\{n_{2}^{y},n_{3}^{y},\ldots,n_{\mathrm{g}+1}^{y}\right\},\\ \ldots\\ \mathrm{K}_{\mathrm{v}-g+1}^{y}=\left\{n_{\mathrm{v}-g+1}^{y},n_{\mathrm{v}-g+ 2}^{y},\ldots,n_{\mathrm{v}-1}^{y},n_{\mathrm{v}}^{y}\right\}.\end{array}  \tag{16}\]  
If we represent the constructed subsequences \(\mathrm{K}_{1},\mathrm{K}_{2},\ldots,\mathrm{K}_{\mathrm{v}-g+1}\) and \(\mathrm{K}_{1}^{y},\mathrm{K}_{2}^{y},\ldots,\mathrm{K}_{\mathrm{v}-g+1}^{y}\) in the form of tuples, then their similarity measures are determined based on the Euclidean distance, the urban metric, or the Minkowski distance. Then for each table a distance matrix will be:  
\[\rho_{1}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{r}}^{y}\big{)}= \sqrt{\sum_{j=\mathrm{r}}^{g+r-1}\left(n_{j+u-r}-n_{j}^{y}\right)^{2}}, \tag{17}\]  
\[\rho_{2}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{r}}^{y}\big{)}= \sum_{j=\mathrm{r}}^{g+r-1}\left|n_{j+u-r}-n_{j}^{y}\right|, \tag{18}\]  
\[\rho_{3}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{r}}^{y}\big{)}= \left(\sum_{j=\mathrm{r}}^{g+r-1}\left|n_{j+u-r}-n_{j}^{y}\right|^{t}\right)^{ \frac{1}{t}}, \tag{19}\]  
\(y=\overline{1,p}\), \(u=\overline{1,v-g+1}\), \(r=\overline{1,v-g+1}\), \(t\) is the Minkowski distance parameter. Then for each \(y=\overline{1,p}\) we find the minimum values for each row of matrices \(\rho_{\mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{r}}^{y} \big{)}\), and we get the distances:  
\[\xi_{\mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{min}}^ {y}\big{)}=\min_{r=\overline{1,v-g+1}}\{\rho_{\mathrm{\tau}}\big{(}\mathrm{K}_ {\mathrm{u}},\mathrm{K}_{\mathrm{r}}^{y}\big{)}\}, \tag{20}\]  
\(u=\overline{1,v-g+1}\) for fixed \(\tau=\overline{1,3}\). If the condition is met \(\xi_{\mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{min}}^ {y}\big{)}<\lambda_{p}\) for a predetermined parameter value \(\lambda_{p}\in[0,1]\), then it can be asserted with probability one that the vector with number \(u\) is similar to a vector of a table with a number \(y\), that is, the table contains a near duplicate. The larger the value \(\lambda_{p}\), the more stringent requirements are imposed on the search for near duplicates. The distance for \(y=\overline{1,p}\), \(u=\overline{1,v-g+1}\) is normalized:  
\[\xi_{\mathrm{\tau}}^{\mathrm{N}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{ \mathrm{min}}^{y}\big{)}=\frac{\xi_{\mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{ u}},\mathrm{K}_{\mathrm{min}}^{y}\big{)}-\min_{u=\overline{1,v-g+1}}\{\xi_{ \mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{min}}^{y} \big{)}\}}{\max_{u=\overline{1,v-g+1}}\{\xi_{\mathrm{\tau}}\big{(}\mathrm{K}_ {\mathrm{u}},\mathrm{K}_{\mathrm{min}}^{y}\big{)}\}-\min_{u=\overline{1,v-g+1} }\{\xi_{\mathrm{\tau}}\big{(}\mathrm{K}_{\mathrm{u}},\mathrm{K}_{\mathrm{min}}^ {y}\big{)}\}}. \tag{21}\]  
Figure 1 illustrates how the hybrid model works for detecting similar text fragments and tables. First, the system receives input data.