## 1 Introduction  
The seminal work for the frequent itemsets (FIs) discovery is named as the Apriori algorithm [1], and its successor procedures are defined to find frequently occurring items in the datasets qualifying user-tuned support threshold parameter. The support threshold is a fundamental parameter which is needed by the FIs' discovery processes before the initiation of mining operations. However, it is a challenging task to select a suitable support threshold parameter for a dataset depending on the characteristics of the dataset [2, 3]. Moreover, it is not clear to users how many numbers of patterns could be returned in response to a particular support threshold parameter [4]. The classical frequent itemsets mining (FIM) methods could return many redundant or/and insignificant patterns as itemsets if a low support threshold parameter is provided by users. In addition, providing a low support threshold parameter could result in less pruning of itemsets and high support threshold parameter could prune large number of itemsets [5]. Therefore, several interest-based pattern mining techniques have been proposed to cut down the size of the resultant itemset [6, 7, 8, 9, 10, 11]. One of the criterions for the interestingness of the patterns is to consider the periodicity or regularity of the patterns along with their frequency [7]. The periodicity or regularity of a pattern present the occurrence behavior of a pattern (i.e., whether it is found regularly, irregularly, or if it is present in a particular time interval in a given dataset) [7; 12]. A pattern is termed as a periodic frequent pattern if its frequency is greater than the support threshold parameter and its periodicity is less than the maximum periodicity [7]. The discovery of periodic frequent patterns is useful in many applications. For example, the analysis of a customer's purchase behavior patterns can result in a better marketing strategy [13] and analysis of the Web users' regular access patterns can help in efficiently browsing a particular website [6]. Similarly, the regular and frequent stock market indices patterns can be pertinent for companies and individuals for future investment [7; 14; 15]. A variety of methods are proposed by the researchers, motivated by the importance and applications of periodic frequent patterns, such as mining the approximate periodic frequent patterns [11], mining rare periodic frequent patterns [8], using efficient approaches to mine periodic frequent patterns from transactional datasets [9], periodic frequent pattern mining from big data [10], and periodic frequent pattern mining with maximum items support constraints [16]. Moreover, periodicity of patterns is noticed by the data mining researchers in sequential pattern mining, such as discovering periodic high-utility itemsets in a discrete sequences [17] and an efficient mining of the periodic high-utility sequential patterns [18]. However, all methods need support threshold parameter to be defined by users, and the mining operation could produce large number of patterns, most of which would not be of any interest to a user. To address the issues, a simple approach could be to ask a user about the number of top-most frequent regular patterns, which are required initially in place of a support threshold. Following this notion, a number of top-K frequent regular pattern mining methods are introduced in [2; 11; 12; 19]. All the top-K frequent regular patterns classically perform breadth-first traversal of the search space to generate the required patterns. The top-K frequent regular patterns maintain a top-K list to record the topmost-K frequent regular patterns in descending support order during exploration of a search space. In top-K frequent regular pattern mining, different support calculation procedures are adopted, such as the top-K frequent regular pattern transaction identifiers (TIDs) intersection method [2], compressed TIDs [12; 19], using dataset partitioning and support estimation [19; 20], and a partitioned dynamic bit vector [21].