Frequent pattern mining is an effective approach for spatiotemporal association analysis of mobile trajectory big data in datadriven intelligent transportation systems. While existing parallel algorithms have been successfully applied to frequent pattern mining of large-scale trajectory data, two major challenges are how to overcome the inherent defects of Hadoop to cope with taxi trajectory big data including massive small files and how to discover the implicitly spatiotemporal frequent patterns with MapReduce. To conquer these challenges, this paper presents a MapReduce-based Parallel Frequent Pattern growth (MR-PFP) algorithm to analyze the spatiotemporal characteristics of taxi operating using large-scale taxi trajectories with massive small file processing strategies on a Hadoop platform. More specifically, we first implement three methods, that is, Hadoop Archives (HAR), CombineFileInputFormat (CFIF), and Sequence Files (SF), to overcome the existing defects of Hadoop and then propose two strategies based on their performance evaluations. Next, we incorporate SF into Frequent Pattern growth (FP-growth) algorithm and then implement the optimized FP-growth algorithm on a MapReduce framework. Finally, we analyze the characteristics of taxi operating in both spatial and temporal dimensions by MR-PFP in parallel. The results demonstrate that MR-PFP is superior to existing Parallel FP-growth (PFP) algorithm in efficiency and scalability.

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To determine the spectrum of Steiner quadruple systems (SQS) admitting a specific automorphism group is of great interest in design theory. We consider a strictly cyclic SQS which is invariant under the affine group, called an AsSQS. For the applications of designs of experiments, group testing, filing schemes, authentication codes, and optical orthogonal codes for CDMA communication, etc., a larger automorphism group containing the cyclic group may work efficiently for the procedures of generating and searching blocks in a design with less storage and time. In this paper, constructions and a necessary condition for the existence of an AsSQS are investigated. For a prime p ≡ 1 (mod 4), Direct Construction A establishes an AsSQS(2 p), provided that a 1-factor of a graph exists, where the graph is defined by using a system of generators of the projective special linear group PSL(2, p). Direct Construction B gives an AsSQS(2 p) which is 2-chromatic, provided that a rainbow 1-factor of a specific hypergraph exists. Accordingly, by proposing two recursive constructions of an AsSQSs(2 p m ) for a positive integer m, we prove that an AsSQS(2 p m ) exists, if the criteria developed for an AsSQS(2 p) are satisfied. We verified the claim and found that an AsSQS(2 p m ) exists for every prime p ≡ 1 (mod 4) with p < 10 5 and any positive integer m.

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