Fast generation of sequential patterns with item constraints from concise representations

Abstract

Constraint-based frequent sequence mining is an important and necessary task in data mining since it shows results very close to the requirements and interests of users. Most existing algorithms for performing this task are based on a traditional approach that mines patterns directly from a sequence database (SDB). However, in fact, SDBs are often very large. The algorithms thus often exhibit poor performance because the number of generated candidates and the search space are enormous, especially for low minimum support thresholds. In addition, these algorithms must read an SDB again when a constraint is changed by the user. In the context of frequently varied constraints, repeatedly scanning SDBs consume much time. To address this issue, we propose a novel approach for generating frequent sequences with various constraints from the two sets of frequent closed sequences (${{\mathcal{F}}{\mathcal{C}}{\mathcal{S}}}$) and frequent generator sequences (${{\mathcal{F}}{\mathcal{G}}{\mathcal{S}}}$), which are the concise representations of the set ${{\mathcal{F}}{\mathcal{S}}}$ of all frequent sequences. The proposed approach is based on novel theoretical results that show an explicit relationship between ${{\mathcal{F}}{\mathcal{S}}}$ and these two sets and have been strictly proved. The approach is then used to develop an efficient algorithm named MFS-IC for quickly generating frequent sequences with item constraints, a task that has many real-life applications. Extensive experiments on real-life and synthetic databases show that the proposed MFS-IC algorithm outperforms state-of-the-art algorithms, which directly mine frequent sequences with constraints from an SDB, in terms of runtime, memory usage and scalability.

Appendix

1.1 Appendix 1: Proof of Proposition 1

• (i) “${{\mathcal{F}}{\mathcal{S}}}(\sigma , \gamma ) \subseteq {{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma ,\gamma )$”: Consider any $\alpha \in {{\mathcal{F}}{\mathcal{S}}}(\sigma , \gamma ):\gamma \sqsubseteq \alpha \sqsubseteq \sigma$. Without loss of generality, we can assume that $\gamma = E_{1} \to E_{2} \to \cdots \to E_{p}$, $\alpha = F_{1} \to F_{2} \to \cdots \to F_{q}$, $\sigma = E_{1}^{{\prime }} \to E_{2}^{{\prime }} \to \cdots \to E_{q}^{{\prime }}$, $F_{i} \subseteq E_{i}^{{\prime }} ,\;\forall i = 1, \ldots ,q$ and $\exists lp = \{ j_{1} ,j_{2} , \ldots ,j_{p} \} \in LP(\alpha ,\gamma ) \subseteq LP(\sigma ,\gamma )$, with $1 \le j_{1} < j_{2} < \cdots < j_{p} \le q: E_{k} \subseteq F_{jk} \subseteq E_{jk}^{{\prime }} ,\;\forall k = 1, \ldots ,p$. Set $d_{i} \mathop = \limits^{\text{def}} \left\{ {\begin{array}{*{20}l} {F_{i} , } \hfill & {{\text{if}}\;i \notin lp} \hfill \\ {F_{i} \backslash E_{k} , } \hfill & { {\text{if}}\;i = j_{k} \in lp} \hfill \\ \end{array} } \right.$, $D_{i} \mathop = \limits^{\text{def}} \left\{ {\begin{array}{*{20}l} {E^{{\prime }}_{i} , } \hfill & {{\text{if}}\; i \notin lp} \hfill \\ {E^{{\prime }}_{i} \backslash E_{k} ,} \hfill & {{\text{if}}\;i = j_{k} \in lp} \hfill \\ \end{array} } \right.$, $\delta (lp)\mathop = \limits^{\text{def}} d_{1} \to d_{2} \to \cdots \to d_{q}$, $E_{i}^{\sim} \mathop = \limits^{\text{def}} \left\{ {\begin{array}{*{20}l} {\emptyset ,} \hfill & {{\text{if}}\; i \notin lp} \hfill \\ {E_{k} ,} \hfill & {{\text{if}}\;i = j_{k} \in lp} \hfill \\ \end{array} } \right.$, $\forall i = 1, \ldots ,q,\;Ex(\gamma ,lp)\mathop = \limits^{\text{def}} E_{1}^{\sim} \to E_{2}^{\sim} \to \cdots \to E_{q}^{\sim}$, then $d_{i} \subseteq D_{i}$ and $F_{i} = E_{i}^{\sim} + d_{i} ,\;\forall i = 1, \ldots ,q$, so $\alpha = Ex(\gamma ,lp) \oplus \delta (lp) \in {{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma ,\gamma )$.

• (ii) “${{\mathcal{F}}{\mathcal{S}}}(\sigma , \gamma ) \supseteq {{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma ,\gamma )$”: $\forall \alpha \in {{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma ,\gamma )$, $\alpha = Ex(\gamma , \, lp) \oplus \delta (lp) = F_{1} \to F_{2} \to \cdots \to F_{q}$, then $F_{i} = E_{i}^{\sim} + d_{i}$, $d_{i} \subseteq D_{i}$, $E_{i}^{\sim} \subseteq F_{i} \subseteq E_{i}^{\sim} + D_{i} \subseteq E_{i}^{{\prime }} ,\;\;\forall i = 1, \ldots ,q$. Thus, $\gamma \sqsubseteq \alpha \sqsubseteq \sigma$, $\alpha \in {{\mathcal{F}}{\mathcal{S}}}(\sigma ,\gamma )$.□

1.2 Appendix 2: Proof of Theorem 1

• (a) First, it is found that $\forall \alpha \in {{\mathcal{F}}{\mathcal{S}}},\exists \sigma \in CloSet(\alpha )$ such that $ro\mathop = \limits^{\text{def}} \rho (\alpha ) = \rho (\sigma )$, so $\sigma \in {{\mathcal{F}}{\mathcal{C}}{\mathcal{S}}}$ and $ro \in RO$, then $FS(ro) = [\sigma ]$. Since $\sim$ is an equivalence relation, then different equivalence classes $[\sigma ]$ or $FS(ro)$ are disjoint. Thus, the assertion ${{\mathcal{F}}{\mathcal{S}}\ominus } = \mathop \sum \nolimits_{ro \in RO} FS(ro)$ is obvious.

• (b) First, by contradiction, it is easy to prove that all different subsets ${{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} ),\forall \sigma_{i} \in CS(ro)$, $\forall \gamma_{j} \in GenSet(\sigma_{i} )$, are disjoint. Indeed, assume conversely that $\exists \sigma_{m} ,\;\sigma_{i} \in CS(ro),\exists \gamma_{k} \in GenSet(\sigma_{m} )$, $\exists \gamma_{j} \in GenSet(\sigma_{i} ):\;(\sigma_{m} \ne \sigma_{i} \;{\text{or}}\;\gamma_{k} \ne \gamma_{j} )$ and $\exists \alpha \in {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{m} ,\gamma_{k} ) \cap {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} )\;( \ne \emptyset )$. Indeed, it is found that two cases $\sigma_{m} \ne \sigma_{i}$ or ($\sigma_{m} \equiv \sigma_{i}$ and $\gamma_{k} \ne \gamma_{j}$) always lead to contradicting the condition $not(DCondC(\alpha ,\sigma_{i} ,CS(\rho (\sigma_{i} ))))$ in ${{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma )$ or $not(DCondG(\alpha ,\sigma_{i} ,\gamma_{j} ))$ in ${{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} )$. Second, we will prove that $FS(ro) = \mathop \sum \nolimits_{{\sigma_{i} \in CS(ro)}} \mathop \sum \nolimits_{{\gamma_{j} \in GenSet(\sigma_{i} )}} {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} , \gamma_{j} ),\;\;\forall ro \in RO$.

  • (i) “$\subseteq$”: $\forall \alpha \in FS(ro)$, then $\alpha \in {{\mathcal{F}}{\mathcal{S}}}$ and $\rho (\alpha ) = ro$. Since $CloSet(\alpha ) \ne \emptyset$ and $GenSet(\alpha ) \ne \emptyset , \, [\exists \sigma_{i} \in CloSet(\alpha ),\exists \gamma_{j} \in GenSet(\alpha ):\sigma_{i} \in {{\mathcal{C}}{\mathcal{S}}},\;\gamma_{j} \in {{\mathcal{G}}{\mathcal{S}}},\;\gamma_{j} \sqsubseteq \alpha \sqsubseteq \sigma_{i} ,\;\rho (\sigma_{i} ) = \rho (\gamma_{j} ) = \rho (\alpha )]^{(*)}$, so $\gamma_{j} \in GenSet(\sigma_{i} ),\;supp(\sigma_{i} ) = supp(\gamma_{j} ) = supp(\alpha ) \ge ms$, $\sigma_{i} \in {{\mathcal{F}}{\mathcal{C}}{\mathcal{S}}}$ and $\sigma_{i} \in CS(ro)$. Hence, $\alpha \in {{\mathcal{F}}{\mathcal{S}}}(\sigma_{i} ,\gamma_{j} )$. Without loss of generality, we can select $\sigma_{i}$ and $\gamma_{j}$ that are, respectively, the first closed and generator sequences satisfying the condition ^(*). Thus, $FS(ro) \subseteq \mathop \sum \nolimits_{{\sigma_{i} \in CS(ro)}} \mathop \sum \nolimits_{{\gamma_{j} \in GenSet(\sigma_{i} )}} {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} , \gamma_{j} )$.

  • (ii) “$\supseteq$”: $\forall \sigma_{i} \in CS(ro)$, $\forall \gamma_{j} \in GenSet(\sigma_{i} )$, $\forall \alpha \in {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} )$, we have $\alpha \in FS(ro)$. Thus, ${{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} , \gamma_{j} ) \subseteq FS(ro)$ and $FS(ro) \supseteq \mathop \sum \nolimits_{{\sigma_{i} \in CS(ro)}} \mathop \sum \nolimits_{{\gamma_{j} \in GenSet(\sigma_{i} )}} {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} , \gamma_{j} )$□

1.3 Appendix 3: Proof of Theorem 2

• (a) It is clear that ${{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma ) \subseteq { {\mathcal{F}}{\mathcal{S}}}^{{\prime }} \left( {\sigma , \gamma } \right)$ In the proof of the above three pruning cases, $\forall \alpha \in {{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma , \gamma )$, if $\exists i = 1, \, 2, \, 3 :\;DCond_{i}$ is true, then $\alpha$ is a duplicate of some previously generated sequences. Thus, ${{\mathcal{F}}{\mathcal{S}}}^{{\prime }} (\sigma , \gamma ) \subseteq {{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$ and ${{\mathcal{F}}{\mathcal{S}}}^{{\prime }} \text{(}\sigma , \gamma \text{)} = {{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$.

• (b) We will prove the first assertion by contradiction. Assume conversely that there are two sequences $\alpha = Ex(\gamma ,lp_{m} ) \oplus \delta (lp_{m} )$ and $\beta = Ex(\gamma ,lp_{n} ) \oplus \delta (lp_{n} )$ (in ${{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$) according to two different position lists in $LP(\sigma ,\gamma ):lp_{m} \ne lp_{n}$, but $\alpha \equiv \beta$, i.e., there exist two $lp_{m} = \{ j_{1} ,j_{2} , \ldots ,j_{p} \}$, $lp_{n} = \{ i_{1} ,i_{2} , \ldots ,i_{p} \} \in LP(\sigma ,\gamma )$ with $m < n, \, 1 \le j_{1} < j_{2} < \cdots < j_{p} \le q$, $1 \le i_{1} < i_{2} < \cdots < i_{p} \le q$, $d(lp_{m} )_{i} \mathop = \limits^{\text{def}} \left\{ {\begin{array}{*{20}l} {E^{{\prime }}_{i} , } \hfill & {{\text{if}}\; i \notin lp_{m} } \hfill \\ {E^{{\prime }}_{i} \backslash E_{k} ,} \hfill & {{\text{if}}\;i = j_{k} \in lp_{m} } \hfill \\ \end{array} } \right.$, $d_{i} \mathop = \limits^{\text{def}} d(lp_{n} )_{i} \mathop = \limits^{\text{def}} \left\{ {\begin{array}{*{20}l} {E_{i}^{'} , } \hfill & { {\text{if}}\; i \notin lp_{n} } \hfill \\ {E^{{\prime }}_{i} \backslash E_{k} , } \hfill & {{\text{if}}\; i = i_{k} \in lp_{n} } \hfill \\ \end{array} } \right.$, $F_{i} \subseteq d(lp_{m} )_{i}$, $F^{{\prime }}_{i} \subseteq d_{i} ,\;\forall i = 1, \ldots ,q$, and two corresponding sequences $\alpha = F_{1} \to F_{2} \to \cdots \to F_{q}$ and $\beta = F_{1}^{{\prime }} \to F_{2}^{{\prime }} \to \cdots \to F^{{\prime }}_{q}$ that belong to ${{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$ such that $\alpha \equiv \beta$ and $DCond_{k} (lp_{m} )$, $DCond_{k} (lp_{n} )$ are false, $\forall k = 1, \, 2, \, 3$.

After deleting all empty itemsets in $\alpha$ and $\beta$, we obtain $\alpha = F_{{u_{1} }} \to F_{{u_{2} }} \to \cdots \to F_{{u_{N} }}$, $\beta = F^{{\prime }}_{{v_{1} }} \to F^{{\prime }}_{{v_{2} }} \to \cdots \to F^{{\prime }}_{{v_{N} }}$ with $N = size(\alpha ) = size(\beta )$, $1 \le u_{1} < u_{2} < \cdots < u_{N} \le q, \, 1 \le v_{1} < v_{2} < \cdots < v_{N} \le q$ and $F_{{u_{i} }} = F^{{\prime }}_{{v_{i} }} \ne \emptyset ,\;\;\forall i = 1, \ldots ,N$ (because $\alpha \equiv \beta$). We set $i_{0} = j_{0} = u_{0} = v_{0} = k_{0} \equiv 0,k_{p + 1} = N$ and $1 \le k_{1} < k_{2} < \cdots < k_{p} \le N:j_{r} = u_{{k_{r} }}$ (so $F_{{u_{{k_{r} }} }} = F_{{j_{r} }} \supseteq E_{r} \subseteq F^{{\prime }}_{{j_{r} }}$), $\forall r = 1,..,p$.

For $\forall r = 1,..,p + 1$, $\forall k = (k_{r - 1} + 1),.., k_{r}$, then $u_{{k_{r - 1} }} = j_{r - 1} < u_{k} \le j_{r} \le i_{r}$. By induction, we can prove that if $\{ (v_{h} = u_{h} ,\forall h = \, 0,..,k - 1),\;(v_{k - 1} < i_{r} )\;{\text{and}}\;(\forall r^{{\prime }} = 0,..,r - 1,\;j_{{r^{{\prime }} }} = i_{{r^{{\prime }} }} = u_{{k_{{r^{{\prime }} }} }} = v_{{k_{{r^{{\prime }} }} }} )\}$, then $\{ v_{k} = u_{k} ,\;{\text{and}}\;{\text{if}}(r \le p)\;{\text{then}}[(v_{k} < i_{r} ,\;{\text{if}}\;k < k_{r} )\;{\text{and}}(j_{r} = i_{r} = u_{{k_{r} }} = v_{{k_{r} }} )]\}$.

Finally, under the hypothesis$\alpha \equiv \beta$, then $\forall r = 1, \ldots ,p + 1,\forall k = (k_{r - 1} + 1), \ldots ,k_{r}$, we always have $(v_{k} = u_{k} ),\;(j_{r} = i_{r} = u_{{k_{r} }} = v_{{k_{r} }} ,\;{\text{if}}\;r \le p)$. Therefore, $(\forall k^{{\prime }} = 1,..,N,\;v_{{k^{{\prime }} }} = u_{{k^{{\prime }} }} )^{(*)}$ and $(\forall r = 1,..,p,j_{r} = i_{r} )^{(**)}$.

• If $lp_{m} = lp_{n}$, then $j_{r} = i_{r} ,\;\forall r = 1,..,p$. Since $\alpha$ and $\beta$ are generated from two different position lists in ${{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma ),\exists i_{0} = 1, \ldots ,N$ such that $u_{{i_{0} }} \ne v_{{i_{0} }}$. This contradicts ^(*).

• If $lp_{m} \ne lp_{n}$, i.e., $\exists m_{0} = 1,..,p$ such that $i_{{m_{0} }} \ne j_{{m_{0} }}$. This also contradicts ^(**). In other words, the hypothesis$\alpha \equiv \beta$ always yields a contradiction.

Thus, $\alpha \ne \beta$, i.e., all sequences in ${{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$ are distinctly generated.

From Proposition 1 and Theorem 1.a, we have ${{\mathcal{F}}{\mathcal{S}}}(\sigma ,\gamma ) = {{\mathcal{F}}{\mathcal{S}}}^{'} (\sigma ,\gamma )= {{\mathcal{F}}{\mathcal{S}}}^{**} (\sigma ,\gamma )$ and the remaining assertions are deduced from Theorem 1.□

1.4 Appendix 4: Proof of Theorem 3

• (a) First, based on the equivalence relation $\sim$, it is easily found that ${{\mathcal{F}}{\mathcal{S}}}^{E} = \mathop \sum \nolimits_{{ro \in RO^{E} }} {{\mathcal{F}}{\mathcal{S}}}^{E} (ro)$.

• (b)

  • “$\subseteq$”: $\forall \alpha \in {{\mathcal{F}}{\mathcal{S}}}^{E}$, according to Theorem 2, we have ${\mathcal{C}}_{I} (\alpha )$ and $\exists ro \in RO$, $\sigma_{i} \in CS(ro) \subseteq {{\mathcal{C}}{\mathcal{S}}}$ and $\gamma_{j} \in GenSet(\sigma_{i} ):\alpha \in {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} )$, so $\gamma_{j} \sqsubseteq \alpha \sqsubseteq \sigma_{i}$, ${\mathcal{C}}_{I} (\gamma_{j} )$ and $\gamma_{j} \in GenSet^{E} (\sigma_{i} )$, $ro = \rho (\sigma_{i} ) \in RO^{E}$, $\sigma_{i} \in CS^{E} (ro)$ and $\alpha \in {{\mathcal{F}}{\mathcal{S}}}^{*E} (\sigma_{i} ,\gamma_{j} )$. All the subsets ${{\mathcal{F}}{\mathcal{S}}}^{*E} (\sigma_{i} ,\gamma_{j} )$ are disjoint because $GenSet^{E} (\sigma_{i} ) \subseteq GenSet(\sigma_{i} ),\;{{\mathcal{F}}{\mathcal{S}}}^{*E} (\sigma_{i} ,\gamma_{j} ) \subseteq {{\mathcal{F}}{\mathcal{S}}}^{*} (\sigma_{i} ,\gamma_{j} )$.

  • “$\supseteq$”: This inclusion is clearly because $GenSet^{E} (\sigma_{i} ) \subseteq GenSet(\sigma_{i} )$ and ${{\mathcal{F}}{\mathcal{S}}}^{*E} (\sigma_{i} ,\gamma_{j} ) \subseteq {{\mathcal{F}}{\mathcal{S}}}^{E} (\sigma_{i} ,\gamma_{j} )$.

  □