Research ArticleAn Efficient Algorithm for Extracting High-Utility HierarchicalSequential Patterns

Chunkai Zhang , Zilin Du , and Yiwen Zu

School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen, China

Correspondence should be addressed to Chunkai Zhang; ckzhang@hit.edu.cn

Received 18 March 2020; Revised 8 June 2020; Accepted 23 June 2020; Published 6 July 2020

Academic Editor: Huimin Lu

Copyright © 2020 Chunkai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

High-utility sequential pattern mining (HUSPM) is an emerging topic in data mining, where utility is used to measure theimportance or weight of a sequence. However, the underlying informative knowledge of hierarchical relation between differentitems is ignored in HUSPM, which makes HUSPM unable to extract more interesting patterns. In this paper, we incorporate thehierarchical relation of items into HUSPM and propose a two-phase algorithm MHUH, the first algorithm for high-utilityhierarchical sequential pattern mining (HUHSPM). In the first phase named Extension, we use the existing algorithm FHUSpanwhich we proposed earlier to efficiently mine the general high-utility sequences (g-sequences); in the second phase namedReplacement, we mine the special high-utility sequences with the hierarchical relation (s-sequences) as high-utility hierarchicalsequential patterns from g-sequences. For further improvements of efficiency, MHUH takes several strategies such as Reduction,FGS, and PBS and a novel upper bounder TSWU, which will be able to greatly reduce the search space. Substantial experimentswere conducted on both real and synthetic datasets to assess the performance of the two-phase algorithm MHUH in terms ofruntime, number of patterns, and scalability. Conclusion can be drawn from the experiment that MHUH extracts moreinteresting patterns with underlying informative knowledge efficiently in HUHSPM.

1. Introduction

Sequential pattern mining (SPM) [1–3] is an interesting andcritical research area in data mining. According to the prob-lem definition [4], a large database of customer transactionshas three fields, i.e., customer-id, transaction-time, and theitems bought. Each transaction corresponds to an itemset,and all the transactions from a customer are ordered byincreasing transaction-time to form a sequence called cus-tomer sequence. The support of a sequence is the numberof customer sequences that contains it. If the support of asequence is larger than a user-specified minimum support,we call it a frequent sequence. The sequential pattern miningalgorithm will discover the frequent sequences called sequen-tial patterns among all sequences. In a word, the purposeof sequential pattern mining is to discover all frequentsequences as sequential patterns, which reflect the potentialconnections within items, from a sequence database underthe given minimum support. An example of such a sequentialpattern is that customers typically buy a phone, then a phone

shell, and then a phone charger. Customers who buy someother commodities in between also support this sequentialpattern. In the past decades, many algorithms [1, 5] have beenproposed for sequential pattern mining, which makes it bewidely applied in many realistic scenarios (e.g., consumerbehavior analysis [6] and web usage mining [7]). However,sequential pattern mining has two apparent limitations.

Firstly, frequency does not fully reveal the importance(i.e., interest) in many situations [8–12]. In fact, many rarebut important patterns may be missed under the frequency-based framework. For example, in retail selling, a phone usu-ally brings more profit than selling a bottle of milk, while thequantity of phones sold is much lower than that of milk [9],and the decision-maker tends to emphasize the sequencesconsisting of high-profit commodities, instead of those fre-quent commodity sequences. This issue leads to the emer-gence of high-utility sequential pattern mining (HUSPM)[8, 12–15]. To represent the relative importance of patterns,each item in the database is associated with a value calledexternal utility (e.g., indicating the unit profit of the item

HindawiWireless Communications and Mobile ComputingVolume 2020, Article ID 8816228, 12 pageshttps://doi.org/10.1155/2020/8816228

purchased by a customer). In addition, each occurrence ofthe item is associated with a quantity called internal utility(e.g., indicating the number of units of the item purchasedby a customer in a transaction). The utility of a sequence iscomputed by applying a utility function on all sequences inthe database where it appears. The task of high-utilitysequential pattern mining is to discover all high-utilitysequential patterns (HUSPs, the sequences with high utility)from the quantitative sequence database with a predefinedminimum utility threshold. Many high-utility sequential pat-tern mining algorithms have been proposed in the pastdecades [13, 16–20], and high-utility sequential patternscan be extracted more efficiently with a series of novel datastructures and pruning strategies proposed. In addition,high-utility sequential pattern mining has many practicalapplications including web log data [21], mobile commerceenvironments [22], and gene regulation data.

Secondly, in sequential pattern mining, the hierarchicalrelation (e.g., product relation and semantic relation) betweendifferent items is ignored, so some underlying knowledgemaybe missed. In general, the individual items of the inputsequences are naturally arranged in a hierarchy [23]. Forexample, suppose both the sequence S1 : <phone, mobilepower pack > and the sequence S2 : <phone, bluetoothheadset > are infrequent, then it seems that there is no asso-ciation between the three commodities. However, we mayfind that the sequence S3 : <phone, phone accessory > is fre-quent from the perspective of product hierarchy, indicatingthat customers usually buy a phone first, then buy a phoneaccessory (including “mobile power pack” and “bluetoothheadset”). That is to say, products in sequences of customertransactions can be arranged in a product hierarchy, wheremobile power pack and bluetooth headset can generalizephone accessory. Another example is that the individualword in a text can form a semantic hierarchy. The wordsdrives and driven can generalize to their common lemmadrive, which in turn generalize to their respective part-of-speech tag verb. The concept of hierarchy (is a taxonomy)provides the deciders with a different perspective to analyzesequential patterns. More informative patterns can beextracted through the hierarchy-based methodology. Besides,although the information reveled from the hierarchicalperspective may be relatively fuzzy, it reduces the loss ofunderlying knowledge to a certain extent. Particularly, thehierarchical relation between different items is sometimesinherent to the application (e.g., hierarchies of directoriesor web pages) or they are constructed in a manual or auto-matic way (e.g., product relation) [23]. Figure 1 shows a sim-ple example of a taxonomy of biology in the real application.Sequential pattern mining with hierarchical relation can betraced back to the article [6] where the hierarchy manage-ment was incorporated into sequential pattern mining, andthe GSP algorithm was proposed to extract sequential pat-terns according to different levels of hierarchy. Later, sequen-tial pattern mining with hierarchical relation has beenstudied extensively in the literature [24, 25]. Efficient algo-rithms were proposed in a wide range of real-world applica-tions, such as customer behavior analysis [6, 26] andinformation extraction [27].

However, to the best of our knowledge, there is no relatedwork taking consideration of both two limitations. In thispaper, given a quantitative sequence database D (with anexternal utility table), a user-defined minimum utility thresh-old, and a series of taxonomies denoting the hierarchical rela-tion, we are committed to finding all high-utility sequencesconsisting of items with the hierarchical relation (i.e., high-utility hierarchical sequential patterns). In fact, mining suchpatterns is more complicated than high-utility sequentialpattern mining and sequential pattern mining with hierar-chical relation. Firstly, compared with high-utility sequentialpattern mining, the introduction of hierarchical relationleads to consumption of a large amount of memory and haslong execution times due to the combinatorial explosion ofthe search space. Secondly, the methods of mining sequentialpattern mining with hierarchical relation cannot be directlyapplied, for the download closure property (also known asthe Apriori property) [28] is not held under the utility-based framework.

To address the above issues, we propose a new algorithmcalled MHUH (mining high-utility hierarchical sequentialpatterns) to mine high-utility hierarchical sequential patterns(to be defined later) by taking several strategies. The majorcontributions of this paper are as follows.

Firstly, we introduce the concepts of hierarchical relationinto high-utility sequential pattern mining and formulate theproblem of high-utility hierarchical sequential pattern min-ing (HUHSPM). Especially, important concepts and compo-nents of HUHSPM are defined.

Secondly, we propose a two-phase algorithm namedMHUH (mining high-utility hierarchical sequential pat-terns), the first algorithm for high-utility hierarchical sequen-tial pattern mining. So that the underlying informativeknowledge of hierarchical relation between different itemswill not be missed and to improve efficiency of extracting

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HUHPs, several strategies (i.e., FGS, PBS, and Reduction)and a novel upper bounder TSWU are proposed.

Thirdly, substantial experiments were conducted on bothreal and synthetic datasets to assess the performance of thetwo-phase algorithm MHUH in terms of runtime, numberof patterns, and scalability. In particular, the experimentalresults demonstrate that MHUH can extract more interestingpatterns with underlying informative knowledge efficientlyin HUHSPM.

The rest of this paper is organized as follows. Relatedwork is briefly reviewed in Section 2. We describe the relateddefinitions and problem statement of HUHSPM in Section 3.The proposed algorithm is presented in Section 4, and anexperimental evaluation assessing the performance of theproposed algorithm is shown in Section 5. Finally, a conclu-sion is drawn and future work is discussed in Section 6.

2. Related Work

In this section, related work is discussed. The section brieflyreviews (1) the main approaches for sequential pattern min-ing, (2) the previous work of high-utility sequential patternmining, and (3) state-of-the-art algorithms for sequentialpattern mining with hierarchical relation.

2.1. Sequential Pattern Mining. Agrawal et al. [28] firstpresented a novel algorithm Apriori holding the downloadclosure property for association rule mining. The proposedApriori algorithm is based on a candidate generationapproach that repeatedly scans the database to generate andcount candidate sequential patterns and prunes those infre-quent. They then defined the problem of sequential patternmining over a large database of customer transactions andproposed the efficient algorithms AprioriSome and Aprior-iAll [4]. Srikant and Agrawal then proposed GSP, which issimilar to AprioriAll in the execution process but greatlyimproves performance over AprioriAll. As Apriori’s succes-sor, adopting several technologies including time constraints,sliding time windows, and taxonomies, GSP uses a multiple-pass, candidate generation-and-test method to find sequen-tial patterns [6]. Zaki [1] proposed the efficient SPADE whichonly needs three database scans. SPADE utilizes combinato-rial properties to decompose the original problem intosmaller subproblems, which can be independently solved inmain memory using efficient lattice search techniques andusing simple join operations. Later, SPAM was proposed byAyres et al. [29], which applies to the situation that sequentialpatterns in the database are very long. In order to deal withthe problems of large search spaces and the ineffectivenessin handling dense datasets, Yang et al. [30] proposed a novelalgorithm LAPIN with a simple idea that the last position isthe key to judging whether to extend the candidate sequentialpatterns or not. Then, they developed the LAPIN-SPAMalgorithm by combining SPAM, which outperforms SPAMup to three times on all kinds of dataset in experiments.Notably, the property, used in SPAM, LAPIN, LAPIN-SPAM, and SPADE, that the support of superpatterns isalways less than or equal to the support of its support pat-terns is different from the Apriori property used in GSP.

Summarizing all algorithmsmentioned above, they all belongto Apriori-based algorithms [2, 3].

It is known that database scans will be time-consumingwhen discovering sequential patterns. For this reason, a setof pattern growth sequential pattern mining algorithms thatare able to avoid recursively scanning the input data wereproposed. For example, Han et al. [31] proposed a novel, effi-cient algorithm FreeSpan that uses projected sequential data-bases to confine the search and growth of subsequences. Theprojected database can greatly reduce the size of a database.Pei et al. [7] designed a novel data structure called web accesspattern tree, or WAP-tree in short, in their algorithm.WAP-tree stores highly compressed and critical informa-tion, and it makes mine access patterns from web logs effi-ciently. Then, Han et al. [32] proposed PrefixSpan withtwo kinds of database projections level-by-level projectionand bilevel projection. PrefixSpan projects only their corre-sponding postfix subsequences into the projected database,so it runs considerably faster than both GSP and FreeSpan.Using a preorder linked, position-coded version of WAP-tree and providing a position code mechanism, the PLWAPalgorithm was proposed by Ezeife et al. based on WAP-tree[33]. Recently, Sequence-Growth, the parallelized version ofthe PrefixSpan algorithm, was proposed by Liang and Wu[34], which adopts a lexicographical order to generate candi-date sequences that avoid exhaustive search over the transac-tion databases.

There are some drawbacks to pattern growth sequentialpattern mining algorithms. Obviously, it is time-consumingto build projected databases. Consequently, some algo-rithms with early pruning strategies were developed toimprove efficiency. Chiu et al. [35] designed an efficientalgorithm called DISC-all. Different with previous algo-rithms, DISC-all adopts the DISC strategy to prune thenonfrequent sequences according to the other sequenceswith the same length instead of the frequent sequences withshorter lengths. Recently, a more fast algorithm called Clo-FAST was proposed for mining closed sequential patternsusing sparse and vertical id-lists. CloFAST combines anew data representation of the dataset, whose theoreticalproperties are studied in order to fast count the support ofsequential patterns, with a novel one-step technique bothto check sequence closure and to prune the search space[36]. It is more efficient than previous approaches. Moredetails about the background of sequential pattern miningcan be found in [2, 3].

2.2. High-Utility Sequential Pattern Mining. To address theproblem that frequency does not fully reveal the importancein many situations, utility-oriented pattern mining frame-works, for example, high-utility itemset mining (HUIM),have been proposed and extensively studied [12, 37].Although HUIM algorithms can extract interesting patternsin many real-life applications, they are not able to handlethe sequence database where the timestamp is embedded ineach item. Many high-utility sequential pattern mining algo-rithms have been proposed in the past decades [9, 13, 16, 18,38], and high-utility sequential patterns can be extractedmore efficiently with a series of novel data structures and

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pruning strategies proposed. Ahmed et al. [13] first definedthe problem of mining high-utility sequential patterns andproposed a novel framework for mining high-utility sequen-tial patterns. They presented two new algorithms UL and USto find all high-utility sequential patterns. The UL algorithm,which is simpler and more straightforward, follows the can-didate generation approach (based on breadth-first search),while the US algorithm follows the pattern growth approach(based on depth-first search). They can both be regarded astwo-phase algorithms. In the first phase, they find a set ofhigh-SWU sequences. In the second phase, they calculatethe utility of sequences by scanning the sequence databaseto output high-SWU sequences, only those whose utility isno less than the threshold minutil.

The two-phase algorithms mentioned above have twoimportant limitations, especially for low minutil values [8].One limitation is that the set of high-SWU sequences discov-ered in the first phase needs a considerable amount of mem-ory. The other one is that computing the utility of candidatesequences can be very time-consuming when scanning thesequence database. Instead of dividing the algorithm intotwo phases, Shie et al. [22] proposed a one-phase algorithmnamed UM-Span for high-utility sequential pattern mining.It improves efficiency by using a projected database-basedapproach to avoid additional scans of databases to checkactual utilities of patterns. Similarly, a one-phase algorithmnamed PHUS was proposed by Lan and Hong [39], whichadopted an effective upper bound model and an effectiveprojection-based pruning strategy. Furthermore, the index-ing strategy is also developed to quickly find the relevantsequences for prefixes in mining, and thus, unnecessarysearch time can be reduced.

Yin et al. then enriched the related definitions and con-cepts of high-utility sequential pattern mining. Two algo-rithms, USpan [9] and TUS [17], were proposed by Yinet al. for mining high-utility sequential patterns and top-khigh-utility sequential patterns, respectively. In USpan, theyintroduced the lexicographic quantitative sequence tree torepresent the search space and designed concatenationmechanisms for calculating the utility of a node and its chil-dren with two effective pruning strategies. The width prun-ing strategy avoids constructing unpromising patterns intothe LP-Tree, while the depth pruning strategy stops USpanfrom going deeper by identifying the leaf nodes in the tree.Based on USpan, Alkan and Karagoz and Wang et al.,respectively, proposed HuspExt [16] and HUS-Span [38]to increase efficiency of the mining process. Zhang et al.[18] proposed an efficient algorithm named FHUSpan(named HUS-UT in the paper), which adopts a novel datastructure named Utility-Table to store the sequence databasein the memory and the TRSU strategy to reduce searchspace. Recently, Gan et al. proposed two efficient algorithmsnamed ProUM [40] and HUSP-ULL [41], respectively, toimprove mining efficiency. The former utilizes the projec-tion technique in generating utility array, while the latteradopts a lexicographic q-sequence- (LQS-) tree and autility-linked-list structure to quickly discover HUSPs. Morecurrent development of HUSPM can be referred to in liter-ature reviews [8, 14].

2.3. Sequential Pattern Mining with Hierarchical Relation.Sequential pattern mining with hierarchical relation can betraced back to article [6] where the hierarchies were incorpo-rated into the mining process, and the GSP algorithm wasproposed to extract sequential patterns according to differentlevels of hierarchy. There are two key strategies to improveefficiency in GSP. The first one is precomputing the ancestorsof each item and dropping ancestors which are not in any ofthe candidates before making a pass over the data. The sec-ond strategy is to not count sequential patterns with an ele-ment containing both item and its ancestor. However, thedepth of the hierarchy limits the efficiency of the algorithmbecause it increases the size of the sequence database. To rep-resent the relationships among items in a more complete andnatural manner, Chen and Huang [25] sketched the idea offuzzy multilevel sequential patterns and presented the FMSMalgorithm and the CROSS-FMSM algorithm based on GSP.Each item in hierarchies can have more than one parent withdifferent degrees of confidence in their paper.

Plantevit et al. [24] incorporated the concept of hierarchyinto a multidimensional database and proposed the two-phase algorithm HYPE extending their preceding M2SPapproach to extract multidimensional h-generalized sequen-tial patterns. Firstly, the maximally specific items areextracted. Secondly, the multidimensional h-generalizedsequences are mined in a further step. As the accessor ofHYPE, they then proposedM3SP [42] to extract multidimen-sional and multilevel sequential patterns based onM2SP. Theapproaches are not incomplete; in other words, they do notmine all frequent sequences. Similarly applying fuzzyconcepts to the hierarchy, Huang [43] later presented adivide-and-conquer strategy based on the pattern growthapproaches to mine such fuzzy multilevel patterns. Recently,Egho then presented MMISP to extract heterogeneous multi-dimensional sequential patterns with hierarchical relationand applied it to analyze the trajectories of care for colorectalcancer. Beedkar et al. [23], who were inspired by MG-FSM,designed the first parallel algorithm named LASH for effi-ciently mining sequential patterns with hierarchical relation.MG-FSM first partitions the data and subsequently mineseach partition independently and in parallel. Drawing lessonsfrom the basic strategy of MG-FSM, Lash adopts a novel,hierarchy-aware variant of item-based partitioning, opti-mized partition construction techniques and an efficientspecial-purpose algorithm called pivot sequence miner(PSM) for mining each partition. As we know, the sequencedatabase contains not only rich features (e.g., occurrencequantity, risk, and profit) but also multidimensional auxiliaryinformation, which is partly associated with the concept ofhierarchy. Recently, Gan et al. [44] proposed a novel frame-work named MDUS to extract multidimensional utility-oriented sequential useful patterns.

There are also several hierarchical frequent itemset min-ing algorithms, which are more or less similar to sequentialpattern mining with hierarchical relations. For example,Kiran et al. [45] proposed a hierarchical clustering algorithmusing closed frequent itemsets that useWikipedia as an exter-nal knowledge to enhance the document representation. InPrajapati and Garg’s research [46], the transactional dataset

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is generated from a big sales dataset; then, the distributedmultilevel frequent pattern mining algorithm (DMFPM) isimplemented to generate level-crossing frequent itemsetusing the HadoopMapreduce framework. And then, themul-tilevel association rules are generated from frequent itemset.

3. Preliminaries and Problem Formulation

3.1. Definitions. Let I be a set of items. A nonempty subsetX ⊆ I is called an itemset, and the symbol ∣X ∣ denotes thesize of X. A sequence S : <X1, X2,⋯, Xn > is an ordered listof itemsets, where Xk ⊆ I (1 ≤ k ≤ n). The length of S is ∑n

k=1∣ Xk ∣ , and the size of S is n. A sequence with the length of lis called an l-sequence. T : <Z1, Z2,⋯, Zm > is the subse-quence of S, if there exists m integers: 1 ≤ k1 < k2 <⋯<km ≤n so that ∀1 ≤ v ≤m, Zv ⊆ Xkv

. For example, <fag fb cg > isthe subsequence of <fa bg fa b cg fbg > .

A q-item (quantitative-item) is an ordered tuple ði, qÞ,where i ∈ I and q is a positive real number representing thequantity of i. A q-itemset with n q-items is denoted as fði1,q1Þði2, q2Þ⋯ ðin, qnÞg. A q-sequence, denoted as <Y1, Y2,⋯,Ym > , is an ordered list ofmq-itemsets. A q-sequence databaseD (e.g., Figure 2(a)) consists of a collection of tuple <ID,Q > ,where ID is the identifier and Q is a q-sequence.

The hierarchical relation of different items is representedin the form of taxonomy which is a tree consisting of items indifferent abstraction levels. We assume that each item is onlyassociated with one taxonomy. Figure 2(b) shows a simpleexample of taxonomies. In a taxonomy, if an item i is anancestor of item j, we say that i is more general than j/j ismore specific than i, denoted as i≺I j/j≻I i. We distinguishthree different types of items: leaf items (most specific, nodescendants), root items (most general, no ancestors), andintermediate items. The complete set consisting of descen-dants of item i is denoted as downðiÞ. For example, inFigure 2(b), A is a root item, a2 is an intermediate item, a2,1is a leaf item, and downða2Þ = fa2,1 a2,2 a2,3g. In this paper,we assume that different items belonging to the same item-set/q-itemset belong to different taxonomies.

Given two itemsets X and Y , we say that X is more spe-cific than or equal to Y/Y is more general than or equal toX (denoted as X≽ISY/Y≼ISX), if f ∣ X ∣ = ∣Y ∣ and ∀i ∈ X, ∃j∈ Y so that i≻I j or i = j. For example, in Figure 2(b), fa1,1b1 Cg≽ISfa1 BCg; fABg≽ISfABg. Similarly, given twosequences with the size of m, S, and T , we say that S is morespecific than or equal to T/T is more general than or equal to(denoted as S≽ST/T≼SS); if ∀1 ≤ k ≤m, we have X≽ISY ,where X is the kth itemset of S and Y is the kth itemset ofT . In particular, if S≽ST and S ≠ T , we say that S is more spe-cific than T/T is more general than S, denoted as S≻ST/T≺SS.For example, in Figure 2(b), <fa1,1 b1 Cg fABg > ≻S < fa1 BCg fABg > .

3.2. Utility Calculation. Each item i is associated with anexternal utility (represented as euðiÞ) which is a positive realnumber representing the weight of i. For a nonleaf item i,it should meet the condition that euðiÞ ≥max feuðjÞ ∣ j ∈

downðiÞg. The external utility of each item i ∈ I is recordedin an external utility table (e.g., Figure 2(c)).

The utility of a q-item ði, qÞ is defined as q × euðiÞ. Theutility of a q-itemset/q-sequence/q-sequence database is thesum of the utility of the q-items/q-itemset/q-sequence it con-tains. For example, in Figure 2, the utility of a2,1 in the 1stitemset of Q4 is 6 (1 × 6); the utility of the 1st itemset of Q4is 16 (6 + 10); the utility of Q4, represented as uðQ4Þ, is 44(16 + 4 + 20 + 4); and the utility of DE, represented as uðDEÞ,is 228 (74 + 39 + 71 + 44).

Given an itemset X : fi1, i2,⋯, img and a q-itemset :fðj1, q1Þðj2, q2Þ⋯ ðjn, qnÞgðm ≤ nÞ, we say that X occursin Y , denoted as X⊂ISY , iff there exist m distinct integers:1 ≤ k1, k2,⋯, km ≤ n so that ∀1 ≤ v ≤m, iv = jkv or iv≺I jkv .The utility of X in Y is defined as uðX, YÞ =∑m

v=1qkv × euðjkvÞ if X⊂ISY ; otherwise, uðX, YÞ = 0. For example, inFigure 2, let Y1 = fða2, 2ÞðC, 2Þg, fa2g⊂ISY1; fACg⊂ISY1;fABg⊄ISY1; uðfAg, Y1Þ = 14ð2 × 7Þ; uðfACg, Y1Þ = 32ð14+ 18Þ; and uðfABg, Y1Þ = 0.

Given a sequence S : <X1, X2,⋯, Xm > and a q-sequenceQ : <Y1, Y2,⋯, Yn > where m ≤ n, we make the followingdefinitions. We say that S occurs in Q (denoted as S⊂SQ) atposition p: <k1, k2,⋯, km > iff there exist m integers: 1 ≤ k1< k2 <⋯<km ≤ n so that ∀1 ≤ v ≤m, Xv⊂ISYkv

. The utilityof S in Q at p, denoted as uðS, p,QÞ, is defined as uðS, p,QÞ=∑m

v=1uðXv, YkvÞ. For example, in Figure 2, S1 : <fAg fAC

g > occurs in Q1 at position <1, 3 > ; uðS1,<1, 3>,Q1Þ = 8 +32 = 40.

Obviously, Smay occur in Qmany times. The utility of Sin Q, denoted as uðS,QÞ, is defined as uðS,QÞ =max fuðS, p,QÞ ∣ p ∈ PðS,QÞg, where the symbol PðS,QÞ denotes thecomplete set containing all positions of S in Q. The utilityof S in a q-sequence database D, denoted as uðSÞ, is definedas uðSÞ =∑Q∈D∧S⊂SQ

uðS,QÞ. For example, in Figure 2, S2 : <fa1g fCg > occurs in Q1 three times; PðS2,Q1Þ = f<1, 3>,<1, 4>,<1, 5 > g ; uðS2,<1, 3>,Q1Þ = 8 + 18 = 26; uðS2,Q1Þ =max f26,13,16g = 26; uðS2Þ = 26 + 31 = 57. More detailsabout the methods of utility calculation can be foundin [8].

Given a minimum utility ξ, we say that sequence W ishigh-utility if uðWÞ ≥ ξ. In particular, W is the most specificpattern, denoted as s-sequence, iff uðWÞ ≥ ξ and ∀T≻SW, uðTÞ < ξ. Similarly, sequence S is the most general pattern,denoted as g-sequences, iff uðSÞ ≥ ξ and ¬∃T≺SS. The s-sequences contain the underlying informative knowledgeof hierarchical relations between different items, which coverless meaningless information compared with those sequenceshighly generalized. Therefore, we define these s-sequencesas high-utility hierarchical sequential patterns (HUHSPs)to be extracted.

3.2.1. Problem Statement.Given a minimum utility ξ, a utilityhierarchical sequence database including a quantitativesequential database D, a set of taxonomies, and an externalutility table, the utility-driven mining problem of high-utility hierarchical sequential pattern mining (HUHSPM)consists of enumerating all HUHSPs whose overall utility

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values in this database are no less than the prespecified min-imum utility account ξ.

4. Proposed HUHSPM Algorithm: MHUH

In this section, we present the proposed algorithm MHUHfor HUHSPM. We incorporate the hierarchical relation ofitems into high-utility sequential pattern mining, whichmakes MHUH able to find the underlying informativeknowledge of hierarchical relation between different itemsignored in high-utility sequential pattern mining. In otherwords, MHUH can extract more interesting patterns. Themining process of MHUH mainly includes two phasesnamed Extension and Replacement. MHUH finds high-utility sequences by the existing algorithm FHUSpan (alsonamed HUS-UT) which we proposed earlier based on theprefix-extension approach in the first phase. For a g-sequence S, we then generate all sequences that are morespecific than S by progressive replacement and store s-sequences with a collection G in the second phase. Thework we need to do in the two phases can be observed visu-ally from the two names. The mining process with twophases ensures that the underlying informative knowledgeof hierarchical relation between different items will not bemissed. At the same time, it can increase efficiency when dis-covering HUHSPs.

Without the loss of generality, in this section, we for-malize the theorems under the context of a minimumutility ξ and a utility hierarchical sequence database(includes a q-sequence database D, taxonomies, and exter-nal utility table).

4.1. Reduction: Remove Useless Items. Before mining thesequential patterns, MHUH adopts the Reduction strategyin the data preprocessing procedure, which removes uselessitems to reduce search space in advance. It mainly consistsof two points, removing the unpromising items from the q-sequence database and removing the redundant items fromthe taxonomies.

An item is unpromising if any sequence containing thisitem is not high-utility. Here, we propose a novel upperbound TSWU (Taxonomy Sequence-Weighted Utility) basedon SWU [13] to filter out the unpromising items.

Definition 1. Given an item i, we define TSWUðiÞ as∑Q∈D∧<frg>⊂SQ

uðQÞ, where r is the root item of the taxonomycontaining i.

For example, in Figure 2, TSWUða1,1Þ = TSWUðAÞ = 228;TSWUðb1Þ = TSWUðb2Þ = TSWUðBÞ = 39; TSWUðc2Þ =TSWUðCÞ = 189.

Theorem 2. Given a q-sequence Q, two sequences S1 and S2,where S1≻SS2, PðS1,QÞ ⊆ PðS2,QÞ.

Proof. Let S1 : <X1, X2,⋯, Xm>,S2 : <Z1, Z2,⋯, Zm > . Wehave ∀1 ≤ v ≤m, Zv≼ISXv . For a q-sequence Q : <Y1, Y2,⋯,Yn > ðm ≤ nÞ, ∀p : <k1, k2,⋯, km>∈PðS1,QÞ, Zv≼ISXv⊂ISY jvð1 ≤ v ≤mÞ, so p ∈ PðS2,QÞ. Further, PðS1,QÞ ⊆ PðS2,QÞ.

Theorem 3. For any sequence S that contains item i, uðSÞ < ξif TSWUðiÞ < ξ.

Proof. From Theorem 3, we know that Pð<fig>,QÞ ⊆ Pð<frg>,QÞ, so fQ ∣Q ∈D∧<fig > ⊂SQg ⊆ fQ ∣Q ∈D∧<frg >⊂SQg. If TSWUðiÞ < ξ, uðSÞ =∑Q∈D∧S⊂SQ

uðS,QÞ ≤∑Q∈D∧S⊂SQ

uðQÞ ≤∑Q∈D∧<fig>⊂SQuðQÞ ≤∑Q∈D∧<frg>⊂SQ

uðQÞ = TSWUðiÞ< ξ.

For a given ξ, we can remove items satisfying TSWUðiÞ< ξ safely according to the above theorem. For example, inFigure 2, when ξ = 40, b1 and b2 can be safely removed fromFigure 2(a).

We say that an item is redundant if it (1) appears in tax-onomy but does not appear in q-sequence database and (2)has at most one child in taxonomy. For example, inFigure 2, a1 and c1 are redundant items. In terms of utility,removing these items has no effect on correctness, which willbe proved in Subsection 4.3. Thus, we can safely removethese items.

4.2. Extension (Phase I): Find g-Sequences. In the first phasenamed Extension, we use the existing algorithm FHUSpan[18] which we proposed earlier to efficiently mine the generalhigh-utility sequences (g-sequences). The main tasks of thisphase are improving efficiency greatly of MUHU and extractg-sequences preparing for the next phase.

ID q-sequence<{(a1,1, 1)} {(a2, 3)} {(a2, 2)(C, 2)} {(c1,1, 1)} {(c2, 4)}>Q1

Q2

Q3Q4

<{(a1,1, 2)} {(a2, 1)} {(b2, 2)} {(b1, 4)}><{(a1,1, 2)} {(c1,1, 3)} {(A, 4)}>

<{(a2,1, 1)(c1,1, 2)} {(a2,2, 2)}{(c1,1, 4)} {(a2,3, 4)}>

(a) Quantitative-sequence database DE

A B C

a1 b1c1 c2b2

a1,1 a2,1 a2,2 a2,3 c1,1

a2

(b) Taxonomies

8 6 2 1 8 7 10 3 2 3 5 5 2 9

Item

eu

a1,1 a2,1 a2,2 a2,3 a1 a2 A b1 b2 B c1,1 c1 c2 C

(c) External utility table

Figure 2: An example of a hierarchical utility sequence database.

6 Wireless Communications and Mobile Computing

In fact, no s-sequences will be missed based on the FGS(From General to Special) strategy. To prove the correctnessof this conclusion, we need to prove two points: (1) theredoes not exist s-sequence S that cannot be discovered bythe FGS strategy and (2) the correctness of the algorithm thatfinds s-sequence is based on a given g-sequence. Here, weprove the correctness of (1), and the proof about (2) is illus-trated in the next subsection.

Theorem 4. Given two sequences S1 and S2 where S1≻SS2,uðS1Þ ≤ uðS2Þ.

Proof. We first prove that uðS1,QÞ ≤ uðS2,QÞ. From Section4.2 (euðiÞ ≥max feuðjÞ ∣ j ∈ downðiÞg) and Theorem 3, uðS2,QÞ =max fuðS2, p,QÞ ∣ p ∈ PðS2,QÞg ≥max fuðS1, p,QÞ ∣ p∈ PðS1,QÞg = uðS1,QÞ. Then, we prove uðS1Þ ≤ uðS2Þ. Wehave D1 ⊆D2 if S1≻SS2, where Dkðk = 1, 2Þ = fQ ∣Q ∈D ∧Sk⊂SQg. uðS2Þ =∑Q∈D2\D1

uðS2,QÞ +∑Q∈D1uðS2,QÞ ≥∑Q∈D1

uðS2,QÞ ≥∑Q∈D1uðS1,QÞ = uðS1Þ.

Corollary 5. Given a g-sequence S, ∀T≻SS, uðTÞ ≤ uðSÞ.

Theorem 4 and Corollary 5 reveal the correctness of (1).We assume that S is a s-sequence that cannot be discoveredby the FGS strategy. In fact, we can always find (replace itemwith the item’s ancestor) the sequence Sg where Sg≼SS and¬∃T≺SS. Because Sg is not g-sequence, uðSgÞ < ξ or ∃T≺SS.So, uðSgÞ < ξ. We then draw a contraction that Sg≼SS ∧ uðSgÞ < uðSÞ. Therefore, the assumption does not hold, whichensures the correctness of (1).

Theorem 6. All items contained in g-sequence are root items.

Proof. Given a g-sequence S, we assume that the ith item of Sis not the root item. Then, we can find a sequence T whereT≺SS. We then draw a contraction that S is not g-sequence.Therefore, the theorem holds.

We then introduce how to find g-sequences. Theorem 6shows that we merely need to consider the root items in theprocess of finding g-sequences. Thus, we can transform theq-sequences into another form so that we can ignore thehierarchical relation in this phase. We illustrate this transfor-mation through an example. Consider Q3 in Figure 2, wetransform it into <fA½16�g fC½15�g fA½40�g > , where thevalue in the bracket is utility (eu × iu). Obviously, with thistransformation, mining g-sequence is equivalent to mininghigh-utility sequences. So, we use the existing high-utilitysequential pattern mining algorithm FHUSpan [18], whichwe proposed earlier to find g-sequences.

Here, we briefly introduce the mining process of FHU-Span, which finds high-utility sequences based on theprefix-extension approach. It first finds all appropriate items(only the sequence starting with these items may be high-utility). Then, for each appropriate item, it constructs asequence containing only this item and extends the sequencerecursively until all sequences starting with the item are

checked. In particular, two extension approaches are used, S-Extension (appending an itemset containing only one itemto the end of the current sequence) and I-Extension (append-ing an item to the last itemset of the current sequence). It isbased on the algorithm HUS-Span which uses two pruningstrategies, PEU (Prefix Extension Utility) strategy and RSU(Reduced Sequence Utility) strategy to reduce the searchspace. The novel data structure named Utility-Table andthe pruning are used to terminate extension in FHUSpan sothat it can efficiently discover high-utility sequences.

4.3. Replacement (Phase II): Find s-Sequence. In the secondphase named Replacement, we mine the special high-utilitysequences with the hierarchical relation (s-sequences) fromg-sequences by the PBS strategy. The main task of this phaseis to extract s-sequences efficiently.

For a g-sequence S, we then generate all sequences thatare more specific than S by progressive replacement andstore s-sequences with a collection G. In particular, foreach replacement, we replace the kth item of S with a childitem. For example, in Figure 2, we replace the first item ofS3 : <fAgfAgfCg > with the child items of A, and one spe-cific sequence is S4 : <fa2gfAgfCg > .

Algorithm 1 shows the progressive replacement startingfrom the kth item of a sequence, which is based on DFS.Firstly, it checks if the current sequence S has been visitedto avoid repeated utility calculation (line 1). If uðSÞ < ξ, wehave ∀S′≻SS, uðS′Þ < ξ according to Theorem 4, so we termi-nate search (lines 2-3). Otherwise, it adds S into G andremoves the sequences that are more general than S from G(line 5). Then, it generates the more specific sequences basedon S, which follows the order from top to bottom (line 9), leftto right (lines 10-12). In detail, it first finds RðS, kÞ which isthe set containing all child items for replacement. For eachr ∈ RðS, kÞ, it replaces the ith item of S with r to generate S′.It then checks the sequences that are more specific than S′(line 9, from top to bottom). After that, it checks thesequences that are more specific than S′ from left to right(lines 10-12), where l is the length of S′.

We also use a strategy, PBS (Pruning Before Search), toreduce search space before Algorithm 1. The main ideabehind this strategy is considering only the items in the cur-rent index. In other words, we generate and check the morespecific sequences in one direction (from top to bottom) toreduce the size of taxonomies.

We illustrate this strategy through an example under thecontext of Figure 2. Let ξ = 45, the sequence S3 : <fAgfAgfCg > is a g-sequence (uðS3Þ = 77 > ξ). We construct copiesof taxonomy, denoted as T1, T2, and T3, for the kth(k = 1, 2, 3) item of S3. Then, we reduce the size of the threetaxonomies. For T1, we have RðS3, 1Þ = fa1,1, a2g (a1 is aredundant item and was removed). Then, we generate S5: <fa1,1gfAgfCg > by replacing the first A with a1,1, and uðS5Þ = 47 > ξ. So, we retain a1,1. Because RðS5, 1Þ =∅, we thenconsider a2 and generate S6 : <fa2gfAgfCg > . We alsoretain a2, for uðS6Þ = 73 > ξ. Then, we continue to check thechild items of a2. Such a procedure will continue until allitems belonging to downðAÞ have been checked. Finally, we

7Wireless Communications and Mobile Computing

remove a2,1, a2,2, a2,3 from T1, for uð<fa2,1gfAgfCg > Þ =30 < ξ, uð<fa2,2gfAgfCg > Þ = uð<fa2,3gfAgfCg > Þ = 0 < ξ.We then continue the above procedure for T2 and T3, andthe processed taxonomies are shown in the right ofFigure 3. In addition, note that in Algorithm 1, RðS, kÞ isobtained from the processed taxonomies instead of the orig-inal taxonomies.

In the above example, the max count of sequences thatare more specific than S reduces from 107 (6 × 6 × 3 − 1) to11 (3 × 2 × 2 − 1). In fact, for a l-sequence S, this countreduces from

Qlk=1ðd1k + 1Þ − 1 to

Qlk=1ðd2k + 1Þ − 1, where

d1k and d2k are the sizes of downðikÞ in the original and proc-essed taxonomies, respectively, and ik is the kth item of S.

In the rest of this subsection, we prove the conclusion leftbefore. We first prove that removing redundant items has noeffect on correctness.

Proof. For a sequence S, we assume that the kth item of S, ik, isa redundant item. Firstly, if ik is a leaf item, we can safelyremove it, because for each sequence W that contains ik, wehave uðWÞ = 0. Secondly, if ik has one child, we generatesequence W by replacing ik with its child. Then, we have uðWÞ = uðSÞ according to the related utility definition (theutility of X in Y). Therefore, removing redundant items doesnot change the utility of related sequences, which means thatit has no effect on the correctness.

Then, we prove the conclusion the correctness of the algo-rithm which finds s-sequence based on a given g-sequence.

Proof. Firstly, the PBS strategy does not ignore the underlyings-sequences. Suppose we cannot find a s-sequence S from thetaxonomies processed by PBS strategy, then we have ∃Sg≺S

S, uðSgÞ < uðSÞ, which violates Theorem 4. So, the assump-tion does not hold. Secondly, Algorithm 1 does not ignoreany s-sequences. Algorithm 1 is based on the DFS frame-work, which ensures the completeness of the algorithm.Besides, Algorithm 1 terminates search in advance based on

Theorem 4, so it does not ignore any s-sequences. In sum-mary, the conclusion holds.

5. Experiments

We performed experiments to evaluate the proposed MHUHalgorithm which was implemented in Java. All experimentswere carried out on a computer with Intel Core i7 CPU of3.2GHz, 8GB memory, and Windows 10.

5.1. Datasets. Five datasets, including three real datasets andtwo synthetic datasets, were used in the experiments. DS1 isthe conversion of Bible where each word is an item. DS2 isthe conversion of the classic novel called Leviathan. DS3 isa click-stream dataset called BMSWebView2. The three data-sets can be obtained from the SPMF website [47]. DS4 andDS5 are two synthetic datasets. The characteristic of themis summarized in Table 1. The values of parameters inTable 1 are as follows: #S is the number of sequences, #I isthe number of distinct items, M is the max length ofsequences, and A is the average length of sequences.

Note that these datasets do not contain taxonomies. So,for each dataset, we generated taxonomies based on the itemsit contains. The max depth and degree of these taxonomiesare 3, which indicates that the max number of leaf items con-tained in taxonomy is 27. The datasets and source code willbe released at the author’s Github after the acceptance forpublication.

5.2. Performance Evaluation. We evaluated the performanceof the proposed algorithm on different datasets when varyingξ. For the sake of simplicity, here, we calculate ξ as δ × uðDÞ,where δ is a decimal between 0 and 1, and uðDÞ is the utilityof the q-sequence database (see the concepts in Subsection3.2). In addition, we also tested the effect of the PBS strategy,and the modified MHUH algorithm which does not take thePBS strategy is denoted as MHUH_base.

The execution times of MHUH and MHUH_base onDS1 to DS3 are shown in Figure 4. When δ/ξ increases, bothof the two algorithms take less execution time since thesearch space reduces. The results prove that the PBS strategyeffectively decreases the execution time, for it greatly reducesthe search space on these datasets. Besides, the results alsoshow that the MHUH algorithms can efficiently extract s-sequences under a low ξ.

Figure 5 shows the distribution of discovered patterns byMHUH on DS1 to DS3. It shows that the number of patternsper length increases with the decrease of ξ. In particular, it isinteresting that some longer patterns may disappear as ξincreases, which indicates that the shorter patterns may havehigher utility.

5.3. Utility Comparison with High-Utility Sequential PatternMining.We conducted this experiment to evaluate the utilitydifference between the patterns discovered by MHUH andthat discovered by the existing algorithm FHUSpan [18]which we proposed earlier.

Figure 6 shows the sum utility of top # (depends on util-ity) patterns discovered by FHUSpan andMHUH from threedatasets. The X-axis refers to the value of #, and the Y-axis

Search for specific.Input: S: the sequence, k: start index, visited, G1: if visitedisfalse then2: if u(S) < ξ then3: return4: end if5: G ⟵ Filter(G ∪ S)6: end if7: for all r ∈ R(S,k) do8: S’⟵ Replace(S,k,r)9: SearchForSpecific(S ,k,false,G)10: for v = k +1 ⟶ l do11: SearchForSpecific(S ,v,true,G)12: end for13: end for

Algorithm 1

8 Wireless Communications and Mobile Computing

represents the sum utility of top # patterns. For example, onDS1, the sum utility of top 1000 patterns extracted byMHUH is higher than the sum utility of that discovered byFHUSpan. Figure 7 shows that the average utility per lengthof top # patterns on DS1 to DS3 (# is set to 1000, 700, and600, respectively). The X-axis refers to the length of patterns,

and the Y-axis denotes the average utility of patterns with thesame length. For example, on DS1, in terms of the top 1000patterns, the average utility of patterns with length of 8 dis-covered by MHUH is higher than the average utility of thatdiscovered by FHUSpan. From these two figures, we knowthat MHUH can discover higher utility patterns compared

T1

A

T2

A

T3 T3T2T1

C CAA

c1,1 c2a2a1,1a2a1,1

a2,1 a2,2 a2,3 a2,1 a2,2 a2,3

a1,1 a2 a2 c1,1

Figure 3: Reduce the size of T1, T2, and T3 through the PBS strategy.

Table 1: Characteristic of datasets.

Dataset #S #I M A Type

DS1 36,369 13,905 100 21.64 Real (text)

DS2 5,834 9,162 100 33.81 Real (text)

DS3 77,512 6,120 161 4.62 Real (click stream)

DS4 10,000 4,000 40 20.54 Synthetic

DS5 60,000 5,000 20 10.50 Synthetic

00.0125 0.015 0.0175

MHUH_base

MHUH

0.02 0.0225 0.025 0.0275 0.045 0.0475 0.05 0.0525 0.055 0.0575 0.060 0

100

200

300

400

500

700

600

200400600800

100012001400160018002000

1000

2000

3000

4000

5000

6000

Exec

utio

n tim

e (se

c)

Exec

utio

n tim

e (se

c)

Exec

utio

n tim

e (se

c)

7000

8000DS1 DS2 DS3

𝛿 𝛿

0.004 0.005 0.006 0.007 0.008 0.009 0.01𝛿

Figure 4: Execution time on three datasets when varying δ.

DS1

00 1 2 3 4 5

Length6 7 8 9 2 3 4 5 6 7

100

200

300

400

Num

ber

500

0

100

200

300

400

Num

ber

500600

0

50

100

150

200

Num

ber

250

300 DS2 DS3

0.01250.0150.01750.02

0.02250.0250.0275

Length

0.0450.04750.050.0525

0.0550.05750.06

20 4 6 8 10Length

0.0040.0050.0060.007

0.0080.0090.01

Figure 5: Distribution of discovered patterns on three datasets when varying δ.

9Wireless Communications and Mobile Computing

with FHUSpan, indicating that more informative knowledgecan be found by MHUH.

5.4. Scalability. We conducted experiments to evaluateMHUH’s performance on large-scale datasets. For each data-

set, we increased its data size through duplication and per-formed the MHUH algorithm with different δ. Figure 8shows the experimental results. We know from the figurethat the MHUH algorithm has well scalability on the twodatasets, for the execution time is almost linear with the data

DS1

0M0 200 400 600

Top #

FHUSpan

MHUH

800 1000 0 100 200 300 400 500 600 700Top #

0 100 200 300 400 500 600Top #

20M

40M

60M

80M

100M

Sum

util

ity o

f top

#

0M

5M

10M

15M

20M

25M

30M

35M

Sum

util

ity o

f top

#

0M

1M

2M

3M

4M

5M

6M

Sum

util

ity o

f top

#

120M

140M

160M

180MDS2 DS3

Figure 6: Sum utility of top # patterns discovered from three datasets.

0K

50K

100K

150K

Ave

rage

util

ity p

er le

ngth

200K

250KDS1 DS2 DS3

0K

10K

FHUSpan

20K

30K

Ave

rage

util

ity p

er le

ngth

40K

50K

0K

2K

4K

6K

Ave

rage

util

ity p

er le

ngth

8K

10K

12K

1 2 3 4 5Length

6 7 8 9 1 2 3 4 5Length

6 7 8 91 2 3 4 5Length

6 7 8

MHUH

Figure 7: Average utility per length of top # patterns.

DS5

10K 20K 30K 40KData size

0.00060.00070.0008

Data size50K 60K

00

10

15

20

25

30

35

40

45 DS4

50

100

150

200

Exec

utio

n tim

e (se

c.)

Exec

utio

n tim

e (se

c.)

250

300

120K 180K 240K 300K

0.00030.00040.0005

0.000250.00030.00035

0.00010.000150.0002

Figure 8: Scalability test on two datasets.

10 Wireless Communications and Mobile Computing

size. For example, the execution time of MHUH (δ = 0:0003)on DS4 almost linearly increases when the data size (thenumber of q-sequences it contains) changes from 10K to50K. It also shows that MHUH can efficiently identify thedesired patterns from the large-scale dataset with a low ξ.For example, in terms of DS5, MHUH costs 300 s when thedata size is 300K and δ = 0:0001.

6. Conclusion and Future Work

In this paper, we incorporate the hierarchical relation ofitems into high-utility sequential pattern mining and proposea two-phase algorithm MHUH, the first algorithm for high-utility hierarchical sequential pattern mining (HUHSPM).In the first phase named Extension, we use the existing algo-rithm FHUSpan which we proposed earlier to efficientlymine the general high-utility sequences (g-sequences); inthe second phase named Replacement, we mine the specialhigh-utility sequences with the hierarchical relation (s-sequences) from g-sequences. The proposed MHUH algo-rithm takes several novel strategies (e.g., Reduction, FGS,and PBS) and a new upper bound TSWU, so it will be ableto greatly reduce the search space and discover the desiredpattern HUHSPs efficiently. A conclusion can be drawn fromthe experiment that MHUH extracts more interesting pat-terns with underlying informative knowledge efficiently inHUHSPM.

In the future, we will generalize the proposed algorithmbased on the more complete concepts. Besides, several exten-sions of the proposed MHUH algorithm can be consideredsuch as improving the efficiency of the MHUH algorithmbased on better pruning strategies, efficient data structures[40, 41], and the multithreading technology [2].

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors have declared that no conflict of interest exists.

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province, China (Grant No. 2020A1515010970) and Shenzhen Research Council (Grant No. GJHZ20180928155209705).

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