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An efficient fast algorithm for discovering closed+ high utility itemsets

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Abstract

In recent years, high utility itemsets (HUIs) mining from the transactional databases becomes one of the most emerging research topic in the field of data mining due to its wide range of applications in online e-commerce data analysis, identifying interesting patterns in biomedical data and for cross marketing solutions in retail business. It aims to discover the itemsets with high utilities efficiently by considering item quantities in a transaction and profit values of each item. However, it produces a tremendous number of HUIs, which imposes further burden in analysis of the extracted patterns and also degrades the performance of mining methods. Mining the set of closed + high utility itemsets (CHUIs) solves this issue as it is a loss-less and condensed representation of all HUIs. In this paper, we aim to present a new algorithm for finding CHUIs from a transactional database, called the CHUM (Closed + High Utility itemset Miner), which is scalable and efficient. The proposed mining algorithm adopts a tricky aimed vertical representation of the database in order to speed up the execution time in generating itemset closures and compute their utility information without accessing the database. The proposed method makes use of the item co-occurrences strategy in order to further reduce the number of intersections needed to be performed. Several experiments are conducted on various sparse and dense datasets and the simulation results clearly show the scalability and superior performance of our algorithm as compared to those for the existing state-of-the-art CHUD (Closed + High Utility itemset Discovery) algorithm.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Kharagpur, 721 302, India

    Jayakrushna Sahoo & A. Goswami

  2. Center for Security, Theory and Algorithmic Research, International Institute of Information Technology, Hyderabad, 500 032, India

    Ashok Kumar Das

Authors
  1. Jayakrushna Sahoo
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  2. Ashok Kumar Das
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  3. A. Goswami
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Corresponding author

Correspondence to A. Goswami.

Appendices

Appendix A: Proof of Theorem 3

Let Y be an extension of X. Then, X ⊂ Y. If T i d l i s t(X) denotes the set of tid’s of g U L(X), and T i d l i s t(Y) for the set of tid’s of g U L(Y), T i d l i s t(Y) ⊆T i d l i s t(X). Thus, we have, ∀t⊇Y, \(\widetilde {Y/X}\subseteq \widetilde {t/X}\). Furthermore, we have,

$$\begin{array}{@{}rcl@{}} u(Y,t) &=& u(X,t)+u(\widetilde{Y/X},t)=u(X,t)+\underset{i\in \widetilde{Y/X}}{\sum}u(i,t)\\ &\leq& u(X,t)+\underset{i\in \widetilde{t/X}}{\sum}u(i, t)\\ &=& u(X,t)+\widetilde{ru}(X,t). \end{array} $$

Hence,

$$\begin{array}{@{}rcl@{}} u(Y) &=& \underset{t\in Tidlist(Y)}{\sum}{u(Y,t)}\\ &\leq& \underset{t\in Tidlist(Y)}{\sum}{u(X,t)+\widetilde{ru}(X,t)}< min\_util. \end{array} $$

As a result, the itemset Y is not a high utility itemset and the theorem follows.

Appendix B: Proof of Theorem 4

Let Y be an extension of X. Clearly, X ⊂ Y. If T i d l i s t(X) denotes the set of tid’s of g U L(X), we denote T i d l i s t(Y) is the set of tid’s of g U L(Y). Then, T i d l i s t(Y) ⊆T i d l i s t(X). Thus, the promising utility value of Y in \(\mathcal {D}\) is given by

$$\begin{array}{@{}rcl@{}} PU(Y) &=& \underset{t\in Tidlist(Y)}{\sum}(u(Y, t)+\widetilde{ru}(Y, t))\\ &=& \underset{t\in Tidlist(Y)}{\sum}(u(X, t)+u(Y\setminus X, t)+\widetilde{ru}(Y, t))\\ &&\leq \underset{t\in Tidlist(Y)}{\sum}(u(X, t)+\widetilde{ru}(X, t)),\\ && \text{since} \, POST\_SET(Y)\subseteq POST\_SET(X)\\ &&\leq \underset{t\in Tidlist(X)}{\sum}u(X, t)+\widetilde{ru}(X, t)\\ & = & PU(X). \end{array} $$

Hence, the promising utility value of the itemset Y is less or than equal to the promising utility value of X.

Appendix C: Proof of Theorem 6

We prove this theorem by the method of contradiction. If possible, let i be added to the PRE-SET of j. This means for any generator g e n = ∅ ∪j, we have to test T i d l i s t(g e n)⊆T i d l i s t(i) for the order preserving property. If T i d l i s t(g e n)⊆T i d l i s t(i), a candidate closed itemset, say Y ′, which is an extension of i. Then, P U(Y ′) ≥ m i n_u t u l. But P U(i) ≥ P U(Y ′) by Theorem 4. So, we have P U(i) ≥ m i n_u t i l, which contradicts the hypothesis that P U(i)<m i n_u t i l. Hence, the theorem follows.

Appendix D: Proof of Theorem 7

We also prove the theorem by method of contradiction. Let T i d l i s t(g e n ′)⊆T i d l i s t(i). Then, T i d l i s t(g e n ′∪i) = T i d l i s t(g e n ′) and hence, we have P U(g e n ′∪i) = P U(g e n ′). Again, Y ′ is an extension of Y. So, the P O S T_S E T (g e n ′) ⊂ P O S T_S E T (gen). Since Y⊆Y ′, we can decompose g e n ′ as g e n ′ = Y∪Z∪i ′ such that Z = Y ′∖Y and for all j∈Z, j∈ P O S T_S E T (Y). We then have

$$\begin{array}{@{}rcl@{}} PU(gen^{\prime}) &=& PU(gen^{\prime}\cup i)\\ &=& \underset{t\in Tidlist(gen^{\prime})}{\sum}PU(gen^{\prime}\cup i, t)\\ &=& \underset{t\in Tidlist(gen^{\prime})}{\sum}(u(gen^{\prime}\cup i, t)\\ &&+\widetilde{ru}(gen^{\prime}\cup i, t))\\ &=& \underset{t\in Tidlist(gen^{\prime})}{\sum}(u(Y, t)+u(Z, t)\\ &&+u(i^{\prime}, t)+u(i, t) + \quad\widetilde{ru}(gen^{\prime}\cup i, t))\\ &&\quad\leq \underset{t\in Tidlist(gen^{\prime})}{\sum}(u(Y, t)+u(i, t)\\ &&+ \widetilde{ru}(Y\cup i, t)),\\ &&\quad\text{since} \, \textnormal { POST\_SET(\textit{Y}) \(\supseteq \)POST\_SET(\(Y^{\prime}\))}\\ &&\quad\leq \sum\limits_{t\in Tidlist(Y\cup i)}u(Y\cup i, t)+\widetilde{ru}(Y\cup i, t)\\ &=& PU(Y\cup i) = PU(gen) < min\_util. \end{array} $$

This is a contradiction to the hypothesis that P U(g e n ′) ≥ m i n_u t i l. Hence, the proof.

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Sahoo, J., Das, A.K. & Goswami, A. An efficient fast algorithm for discovering closed+ high utility itemsets. Appl Intell 45, 44–74 (2016). https://doi.org/10.1007/s10489-015-0740-4

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