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- Jeong-Hoon Lee
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Human-centric Computing and Information Sciences volume **11**, Article number: 13 (2021)

Cite this article **4 Accesses**

https://doi.org/10.22967/HCIS.2021.11.013

Abstract

Frequent pattern (FP) mining is useful for deriving meaningful information from data. Among related methods, FP-growth is one of the most widely used algorithms but has time and physical restrictions when applied to big data mining. FP-growth repeatedly generates conditional FP-trees (CFP-trees) on memory and mines frequent patterns. As a result, memory shortage may occur when processing big data with FP-growth. If memory overflow occurs during FP-growth, mining will no longer be performed afterward. If memory overflow is predicted in advance, the error can be handled using methods such as switching to secondary memory-based mining method or waiting until the memory is released and subsequently proceeding with the mining. In this paper, the next task size prediction method is proposed to predict whether the main memory is sufficient to operate the next task during FP-growth process. Furthermore, it can also help in the parallel and distributed processing of FP-growth. The next task size prediction method can help solve the load balancing problem by allocating the remaining memory-sized task in each workstation or appropriately sized tasks to the workstations.

Keywords

FP-Growth, PPFP, Load Balancing, Parallel FP-Growth, PNC, Big Data

Introduction

Since 1993 when the concept of the association rule [1] was introduced, the task of analyzing frequent patterns (FP) in data mining has been researched actively. The FP-growth algorithm suggested by Han et al. [2] is one of the most popular FP algorithms. Unlike the existing Apriori-based methods [3], the FP-growth algorithm compresses FP into an FP-tree. Concerning pattern extraction, this algorithm is widely used in various fields such as biology [4], commerce [5, 6], education [7], recommendation [8], healthcare [9–11], sports [12], earth science [13], and Internet of Things [14, 15].

According to the additional research conducted by Pei et al. [16], however, FP-growth still has several problems in processing big data. The FP-tree is used to compress the transaction data but is unable to do so to a maximum extent. Therefore, these data take up a large memory space. In addition, it generates a considerable volume of conditional FP-tree (CFP-tree) data and consumes considerable time and space.

Several methods were proposed to overcome these problems. In [17, 18], the authors adopted a method of pruning unnecessary areas in a tree using minimum support and confidence in traditional FP mining to reduce the mining area. In [19–22], the FP-tree size was reduced by analyzing only the patterns corresponding to the most frequent or important top-K items. Therefore, memory usage and time consumption during pattern analysis were reduced because unnecessary patterns were excluded from the analysis scope. Since the minimum support threshold and reliability required to remove the irrelevant pattern were defined based on subjective criteria proposed by an analyst manually, however, objectivity was a problem. In addition, because this method did not allow automatically extracting meaningful frequent items or eliminating irrelevant ones, considerable time and space were consumed. Under these circumstances, mining could eventually fail.

In practice, having only the physical memory is not sufficient to perform mining on very large datasets by using these algorithms, which depend on the size of the primary memory. Goethals [23], Vaarandi [24], and Buehrer et al. [25] argued that the standard physical memory available in modern CPUs was insufficient to mine association rules from real-world datasets.

Aside from the main FP-tree, the FP-growth process also generates the CFP-tree repeatedly in the main memory. When mining frequent item set from big data with FP-growth, memory overflow may occur. If the size of the next task could be predicted during the FP-growth process, however, we could predict memory overflow in advance. Subsequently, before memory overflow occurs, the mining method could be changed to secondary memory-based mining. Alternatively, it could simply wait for the main memory to be released and proceed with the main memory-based mining method.

There are several FP-growth methods based on the secondary memory: DRFP-growth [26], BFP-growth [27], and SQL-based FP-growth [28]. Previously, we also developed a novel secondary memory-based FP-growth method called the push-and-pop FP algorithm (PPFP) [29]. These secondary memory-based FP-growth methods have improved scalability. As access to secondary storage systems is computationally expensive, however, they have limited time efficiency. On the other hand, because they rarely consume the main memory space, when the memory is insufficient to mine with the main memory-based FP-growth, it is possible to continue the mining process using these secondary memory-based FP-growth methods.

In addition, predicting the size of the next task can help in solving the load balancing problem corresponding to distributed FP-growth processing. As observed previously for various FP-growth methods, it is difficult to apply FP-growth algorithms to big data in a single process because of time and memory efficiency problems [23, 24, 30]. Therefore, research dedicated to parallel processing of FP-growth has been conducted. Concerning the FP mining of big data, the FP-growth methods for parallel processing have also been investigated. For example, the parallel FP-tree (PFP), proposed in [25] is an FP-growth algorithm based on the MapReduce method [31–33]. PFP has a classic parallel form of FP-growth, and it can be used to processes mining by dividing the work into independent parallel streams. Other research studies have improved PFP on Spark and its application [34, 35]. Nonetheless, this parallel processing of FP-growth is still associated with problems such as load imbalance, etc. [36]. In parallel and distributed processing of FP-growth, the FP-growth process is divided into small tasks and simultaneously processed by multiple workstations. At this time, a load balancing problem occurs if the task cannot be allocated properly to each workstation. For example, if memory overflow occurs due to the allocation of a task in a size that cannot be processed in the memory of a specific workstation, an error recovery cost is incurred. As another example, when large-size tasks are concentrated on a specific workstation, the processing time of the workstation is longer than other workstations; therefore, the total processing time could take longer. On the other hand, the next task size prediction method proposed in this paper can help solve the load balancing problem by allocating the remaining memory-sized task to each processor or assigning appropriately sized tasks to the workstations.

Therefore, in this paper, a method of predicting the size of the next task during the FP-growth process is proposed. Using this method enables predicting the memory size needed for the next task during the FP-growth process.

The rest of this paper is organized as follows: Section 2 provides an overview of related works, focusing on FP-growth and PPFP; Section 3 introduces the next task size prediction method; Section 4 discusses the conducted performance study and presents the results of the analytical comparison using the incorrect prediction ratio; Section 5 summarizes the results of this study and outlines several future research issues.

Related Work

This section describes the original FP-growth method and FP-growth algorithm based on the secondary memory (PPFP) proposed in previous studies.

FP-Growth

The FP-growth algorithm performs FP mining in two stages. First, it generates an FP-tree by compressing frequent items from the transaction database using the tree data structure to enable efficient search. Second, FP mining is performed using an FP-tree.

At the first stage of FP-growth, the data are stored in the FP-tree’s expanded prefix tree structure, which compresses and stores the required information derived from the data in the tree structure by performing only two database scans. It determines the priority of items by calculating their frequency during the first scan, and arranges each set of items according to the priority and builds an FP-tree during the second scan.

After generating an FP-tree from the transaction database, FPs are mined by applying FP-growth, which proceeds from the bottom-most item in the FP-tree to the topmost one. The method is defined as follows: first, the algorithm obtains the path from the root to each node by examining the nodes since items in the header table are connected by links, prepares the conditional pattern base (CPB), and generates a CFP-tree just like when building an FP-tree; second, it gathers the paths from the CFP-tree generated for a node, prepares the CPB again, and generates a CFP-tree. Recursively, it extracts an FP generating the CFP-tree. The algorithm of FP-growth is provided in Pseudo Code 1.

**Pseudo Code 1.** FP-growth

Algorithm1 FP-growth(FP-tree, α)

1. if FP-tree contains a single path P

2. for each combination β of the nodes on the path P do

3. Generate pattern β∪α with support = minimum support of the nodes in β

4. end for

5. else

6. for each item ai in the header table of FP-tree do

7. Generate pattern β = ai∪α with support = ai.support

8. Construct β’s conditional pattern base and then β’s conditional FP-tree CFP-tree

9. if CFP-tree ≠ Ø then

10. Call FP-growth(CFP-tree, β)

11. end if

12. end for

13. end if

FP-growth is one of the most actively researched and widely used methods in the field. Since this method depends on memory capacity and maintains a CFP-tree generated repeatedly in the memory, however, it is difficult to mine a large volume of transaction database concerning actual data due to the limited physical memory space.1. if FP-tree contains a single path P

2. for each combination β of the nodes on the path P do

3. Generate pattern β∪α with support = minimum support of the nodes in β

4. end for

5. else

6. for each item ai in the header table of FP-tree do

7. Generate pattern β = ai∪α with support = ai.support

8. Construct β’s conditional pattern base and then β’s conditional FP-tree CFP-tree

9. if CFP-tree ≠ Ø then

10. Call FP-growth(CFP-tree, β)

11. end if

12. end for

13. end if

PPFP Mining Method

In addition to the FP-growth algorithm that mines FP by generating recursive CFP-trees, the PPFP algorithm can be used to mine FP sequentially using the push-and-pop method with a single stack. To perform mining FP with the PPFP algorithm, the FP-tree data structure is adopted.Pseudo Code 2 describes the PPFP algorithm in detail.

Algorithm 2 PPFP

Input: FP-tree table constructed based on Algorithm 1 using FP-tree, minimum support threshold ξ,

Output: FP table

Method: Call stack-growth(), which is implemented as follows.

Procedure PPFP()

1. make CFP-stack, /* CFP-stack: conditional pattern stack */

2. insert {frequent item set} into CFP-stack

3. α = POP one pattern from CFP-stack

4. ifα ≠ Ø

5. call pushAndPop(α)

Procedure pushAndPop(α)

1. Insert α into FP table /* FP table is to collect the result FPs */

2. β = collect α’s conditional pattern base

3. ifβ ≠ Ø

4. PUSH β into CFP-stack

5. α = POP one item from CFP-stack

6. ifα ≠ Ø

7. call pushAndPop (α)

As the FP-growth method based on secondary memory, PPFP can be used to enhance scalability unlike prior-memory dependent algorithms. Moreover, it is more time-efficient than the previous secondary memory-based FP-growth methods but still more time-consuming than the main memory-based methods
Input: FP-tree table constructed based on Algorithm 1 using FP-tree, minimum support threshold ξ,

Output: FP table

Method: Call stack-growth(), which is implemented as follows.

Procedure PPFP()

1. make CFP-stack, /* CFP-stack: conditional pattern stack */

2. insert {frequent item set} into CFP-stack

3. α = POP one pattern from CFP-stack

4. ifα ≠ Ø

5. call pushAndPop(α)

Procedure pushAndPop(α)

1. Insert α into FP table /* FP table is to collect the result FPs */

2. β = collect α’s conditional pattern base

3. ifβ ≠ Ø

4. PUSH β into CFP-stack

5. α = POP one item from CFP-stack

6. ifα ≠ Ø

7. call pushAndPop (α)

Next Task Size Prediction Method for FP-Growth Process

This section describes the proposed next task size prediction method during FP-growth.

The FP-growth algorithm repeatedly generates CFP-trees and mines FPs. Therefore, one CFP-tree generation is regarded as one task. To predict the next task size, the size of the CFP-tree generated is estimated. The CFP-tree consists of nodes and links. Each node must have the item name and the frequency of the item on that node. The node links to the node with the same item name in the FP-tree. In the FP-growth process, the FP-tree is explored in a bottom-up manner, generating a CPB. Therefore, a node must have a link to its parent node. As such, each node should have item name, frequency, link to the next node with the same item name, and link to the parent node. Nonetheless, the composition and size of this information vary depending on which constitutes the FP-tree. Knowing the size of the constituted nodes and the number of nodes allows predicting the size of the entire tree. Therefore, this paper presents a method of predicting the number of nodes so that the size of the node making up a program can be applied to obtain the predicted memory size required.

The proposed next task size prediction method is discussed in three steps: first, predict the maximum number of nodes (maxNC) in a CFP-tree to be generated; then, adjust the maxNC by reflecting the ratio of the actually generated node count (ANC); finally, describe how to apply the proposed node-count prediction method to FP-growth.

MaxNC: Prediction of Maximum Node Count Generated Based on N-items

Determining the maxNC in a tree to be generated with the given items requires knowing the state of the maximum number of nodes in the tree. When the FP-tree expresses all the patterns that can be generated with the given items, the maximum node count can be derived. An FP-tree with the maximum node count is denoted as the max FP-tree (maximal FP-tree).The tree structure shown in Fig. 1 is the max FP-tree using itemset {a, b, c, and d}. The item set is assumed to be sorted as {a, b, c, d} such that a has top priority and d has the lowest priority.

As shown in Fig. 1, among the child nodes in the root, the sub-tree of the node with the highest priority has the same structure as all remaining trees. In Fig. 1, node a’s sub-tree is the same as X3, which is the set of the remaining trees except node a’s sub-tree. The sub-tree of node b has X2 as a child tree.

If there is an item with higher priority than item a, X4 is added by copying it under the node corresponding to the newly added item under the root. In the max FP-tree, the structure is repeated such that a sub-tree of one item with high priority inherits the entire sub-tree model of items with low priority.

Based on this rule, for the node count generated according to the number of items, the maximum node count that can be generated is 1 with one item, 2+1with two items, and 4+(2+1)with three items and is increased to 8+(4+(2+1))with four items. According to this rule, the maximum node count (maxNC) that can be generated when the item of FP-tree is n is ∑

Equation (1) presents the formula for maxNC:

*(1)*

Fig. 2 shows the accumulated CFP-trees when FP-growth progresses with an FP-tree depicted in Fig. 1. To obtain the maximum number of nodes that can be theoretically generated, all CFP-trees are assumed to be generated by using a max CFP-tree, and the CFP-tree is generated even when there is only one item in the CPB.

When the item set is {a, b, c, d}, the maxNC in an FP-tree is 2

This way, the “abcd” pattern is saved in the FP table, and the mining process on bcd is executed. After mining of the pattern bcd is completed, bcd’s CFP-tree is deleted from the memory. The mining process then returns to the cd’s next item. As a result, FPs are mined by repeatedly generating and deleting the CFP-tree, with the total node count in the memory changed according to the creation and deletion of a CFP-tree.

The following steps correspond to the method of finding the maximum node count accumulated in memory when FP-growth is performed on n items:

The total node count of the max FP-tree (maxNC): 2^{n}

The node count of the first max CFP-tree: 2^{n-1}

The node count of the second max CFP-tree: 2^{n-2}

…………,

- The node count of the (n-2)^{th} max CFP-tree: 2^{n-(n-2)}

- The node count of the (n-1)^{th} max CFP-tree: 2^{n-(n-1)}

- The node count of the n^{th} max CFP-tree: 2^{n-n}

⇒ ∑_{x=1}^{n} 2^{n-x}

The maximum node count of an FP-tree with n items is 2n. The item set is denoted as I = {i - The node count of the (n-2)

- The node count of the (n-1)

- The node count of the n

⇒ ∑

In case of item in, the maximum node count of i

PNC: Prediction of the Node Count

It is necessary to predict the node count of the next generated CFP-tree to allocate the required memory space in the next step or distribute the load appropriately. In this section, an equation for predicting the node count in a CFP-tree to be generated in the next step is proposed.The worst case of memory usage in FP-growth with n items is when the max CFP-tree for the lowest priority item is generated continuously. In this case, the predictable total node count is ∑

To reflect the ANC on the prediction node count equation (PNC), the ANC rate corresponding to the theoretical maxNC is used. The node count rate is denoted as NCR, whose equation is presented in (2):

*(2)*

To reflect the NCR of prior-generated trees, multiply the equation denominator by the maxNC and the parent tree’s NCR value. The NCR equation’s denominator value has to be less than or equal to maxNC since the NCR denominator corresponds to the node prediction count. The PNC cannot be greater than the maximum node count. When the parent tree’s NCR is greater than 1, the child tree’s NCR denominator will be greater than the maximum node count. Therefore, limit the maximum value of NCR to 1. Consequently, it has a value greater than 0 but less than or equal to 1. When NCR is close to 1, it means the prediction is accurate. Otherwise, when NCR is close to 0, it means the prediction is inaccurate. An NCR value of 1 means that the predicted node count and the actually generated one are the same. The value after subtracting NCR from 1 is denoted as the incorrect prediction ratio (IPR). After calculating NCR, it is possible to predict the next generated CFP-tree’s node count.

To estimate the node count required to generate the next CFP-tree, the maxNC and NCR equations are applied. The equation for obtaining the predicted node count in a CFP-tree is denoted as PNC and is shown in Equation (3):

*(3)*

Implementation of PNC

Fig. 3 presents an example of PNC implementation. First, find the NCR of the main FP-tree. Fig. 3(1) shows the structure of the main FP-tree. Since the item count of the main FP-tree is 6, the main FP-tree’s maxNC is 2Before generating the first CFP-tree, calculate the PNC of the first CFP-tree. The lowest priority of an item in the main FP-tree’s header table is f. Item f’s CPB is {abd:2, bd:1, d:1}, and frequent items are {d:4, a:2, b:2}. With 3 items, max NC is 2

Following this approach, the next generated CFP-tree node count can be predicted.Pseudo Code 3 represents the PNC implementation algorithm. Fig. 4 presents a flowchart that shows the work performed by the PNC implementation.

Algorithm 3 PNC_FP-growth(FP-tree, α, ncr)

1. if FP-tree contains a single path P

2. for each combination β of the nodes on the path P do

3. Generate pattern β∪α with support = minimum support of the nodes in β

4. end for

5. else

6. for each item ai in the header table of FP-tree do

7. Generate pattern β = ai∪α with support = ai.support

8. Construct β’s conditional pattern base

9. β’s maxNC = 2^{β’s frequent items}

10. β’s pnc= ⌊β's maxNC×ncr⌋

11. Construct β’s conditional FP-tree CFP-tree

12. β’s anc= generated node count of β’s CFP-tree

13. β’s ncr = β’s anc / β’s pnc

14. if (β’s ncr > 1)

15. β’s ncr = 1

16. if CFP-tree ≠ Ø then

17. Call PNC_FP-growth(CFP-tree, β, β’s ncr)

18. end if

19. end for

20. end if

1. if FP-tree contains a single path P

2. for each combination β of the nodes on the path P do

3. Generate pattern β∪α with support = minimum support of the nodes in β

4. end for

5. else

6. for each item ai in the header table of FP-tree do

7. Generate pattern β = ai∪α with support = ai.support

8. Construct β’s conditional pattern base

9. β’s maxNC = 2

10. β’s pnc= ⌊β's maxNC×ncr⌋

11. Construct β’s conditional FP-tree CFP-tree

12. β’s anc= generated node count of β’s CFP-tree

13. β’s ncr = β’s anc / β’s pnc

14. if (β’s ncr > 1)

15. β’s ncr = 1

16. if CFP-tree ≠ Ø then

17. Call PNC_FP-growth(CFP-tree, β, β’s ncr)

18. end if

19. end for

20. end if

Results

In this section, the experiment results of the next task size prediction method proposed in this study are presented. All our experiments were conducted on a PC running on 32-bit Windows 7 with Intel i5-3210M2.5 GHz CPU and 2-GB of DDR3 RAM. To test the validity of the results of this study, three datasets denoted as T10I4D1K, T20I4D1K, and T30I4D1K were prepared using the IBM quest market basket synthetic data generator for testing the proposed algorithm for data analysis and generating testing datasets.

In TXXIYYDZZ, XX meant the average transaction length, YY was the average maximum potential frequent itemset size, and ZZ denoted the total number of transactions. The output was written into file TXXIYYDZZ. Each line of the file corresponded to a transaction. The items in each transaction were represented by item numbers and separated by spaces.

To evaluate the accuracy of a suggested equation and analyze how the equation’s accuracy varied according to the size of the tree, three datasets were generated and evaluated. The considered datasets, T10I4D1K, T20I4D1K, and T30I4D1K, had the same data generation rule under the same condition except for the average transaction length. They had an average of four maximum potential frequent itemset size, and the total number of transactions was 1K. Nonetheless, the average transaction length differed, i.e., 10, 20, and 30. An increase made a tree deeper.

Three experiments were performed here. To facilitate the experiment results’ visibility, IPR, the opposite concept of the node count prediction accuracy ratio, was calculated considering 1-NCR. In the first experiment, the average NCR during FP-growth for each dataset was calculated. Then, the IPR corresponding to the main FP-tree’s NCR and the IPR obtained before generating the CFP-tree’s NCR were compared. The second experiment checked the sequential IPR during CFP-tree generation for the dataset T30I4D1K. Here, NCRparentTree was applied until the 3^{rd} generated CFP-tree was obtained from the main FP-tree. After that, the 3^{rd} NCR value was obtained. In the third experiment, the NCRparentTree values were computed until the last depth and used to compute NCR. To ensure accurate test results, the FP-tree pruning function was disabled so that no node count was omitted.

Fig. 5 presents the results of comparing the main FP-tree NCR (NCRmainTree) and previously generated CFP-tree NCR (NCRparentTree). First, the main FP-tree maxNC and ANC were obtained. Then, NCR was computed accordingly. Next, NCRmainTree was calculated during FP-growth to obtain the NCR of the CFP-trees. The size of the main FP-tree continuously affected the sizes of the CFP-trees obtained from the main FP-tree. Therefore, NCRmainTree alone could be used to predict the node count of CFP-trees to some extent.

When NCR

As shown in Fig. 5, when NCR

The T30I4D1K itemset had a longer transaction than T10I4D3K. As a result, the main FP-tree corresponding to this data set was deeper than that of T10I3D3K. In Fig. 5, it could be seen that, when the transaction length increased, the IPR declined. When the depth of the main FP-tree was larger, it had more chances to obtain NCR

In this experiment, as shown in Fig. 6, the CFP-tree NCR was computed during the FP-growth process based on the T30I4D1K data set. In this experiment, NCR

In other words, when CFP-trees with larger depths and frequencies were generated, the PNC prediction equation could achieve higher accuracy. In addition, no IPRs exceeded those of the main FP-tree. In other words, the error between the total node count generated while the FP-growth was executed on T30I4D1K and the one expected to be generated through the PNC prediction equation occurred within the maximum threshold of 0.4%. This indicates that the PNC equation’s prediction accuracy could be considered appropriate in predicting the total node count that could be generated within the meaningful error range in analyzing a dataset of a large scale.

In the second experiment, computations were performed only until the 3

When a CFP-tree was generated during FP-growth, if the last depth of the CFP-tree was reached, it was returned to the previous depth. After each time when the link returned to the previous depth and a CFP-tree was generated, the CFP-tree’s IPR increased, and the IPR was augmented as well.

Conclusion

The experiment results confirm that the proposed next task prediction method achieves high prediction accuracy. In the case of the T30I4D1K dataset, the error between the total node count generated and the total node count expected to be generated using the PNC prediction equation is within a maximum of 0.4%. This indicates that the PNC equation’s prediction accuracy is appropriate for predicting the total node count that can be generated within the reasonable error range in analyzing data of a large scale. Moreover, prediction accuracy is shown to have a tendency to increase when the depth of a tree increases.

Our next task is to study how to apply PNC.

First, a hybrid FP-growth method is being studied so that FP mining can proceed in case of insufficient memory during FP-growth. To improve FP-growth scalability, we previously proposed the FP-growth method based on the secondary memory. Although this method allows improving scalability, it is less time-efficient than the original FP-growth algorithm. Therefore, we will focus on developing a hybrid FP-growth method that applies the original FP-growth method to mine FPsif the main memory is sufficient or the secondary memory-based method if the main memory is insufficient. Using the PNC described in this paper, it can predict memory shortage before a memory overflow occurs and switch to the secondary memory-based FP-growth method accordingly.

Next, the application of the distributed and parallel processing of the FP-growth will be studied. In the case of big data FP mining, it is difficult to perform data mining through a single process. Accordingly, many studies on distributed parallel processing for FP-growth have been conducted. Nonetheless, the load balancing problem may still occur concerning distributed processing. Using the next task size prediction method proposed in this paper can mitigate this problem by predicting the size of the next task and establishing efficient task allocation. After completing the research on the hybrid FP-growth method without memory overflows, we plan to develop a method of allocating tasks efficiently by applying PNC to the task allocation in distributed parallel FP-growth.

Acknowledgements

The author would like to thank Mi-Jin Lee (University of Dongguk) for discussing the issue related to frequent pattern mining.

Author’s Contributions

The author performed the design and implementation of the proposed method and read and approved the final manuscript.

Funding

None.

Competing Interests

The author declares no competing interests.

Availability of Data and Materials

The datasets generated and analyzed during the current study are available in the IBM data generator, https://github.com/zakimjz/IBMGenerator.

Author Biography

**Name : Jeong-Hoon Lee**

**Affiliation :** Department of Liberal Arts & Science, Gachon University, Seongnam, Korea

**Biography :**

2005Bachelor of Computer Engineering from Dongguk University in Seoul, Korea

2007 Master of Computer Engineering from Dongguk University in Seoul, Korea

2011 Ph.D. of Computer Engineering from Dongguk University in Seoul, Korea

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Jeong-Hoon Lee^{*}, Next Task Size Prediction Method for FP-Growth Algorithm, Article number: 11:13 (2021) Cite this article 4 Accesses

**Recived**19 March 2020**Accepted**22 December 2020**Published**31 March 2021

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