AWLCO: All-Window Length Co-Occurrence
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Analyzing patterns in a sequence of events has applications in text analysis, computer programming, and genomics research. In this paper, we consider the all-window-length analysis model which analyzes a sequence of events with respect to windows of all lengths. We study the exact co-occurrence counting problem for the all-window-length analysis model. Our first algorithm is an offline algorithm that counts all-window-length co-occurrences by performing multiple passes over a sequence and computing single-window-length co-occurrences. This algorithm has the time complexity O(n) for each window length and thus a total complexity of O(n^2) and the space complexity O(|I|) for a sequence of size n and an itemset of size |I|. We propose AWLCO, an online algorithm that computes all-window-length co-occurrences in a single pass with the expected time complexity of O(n) and space complexity of O( √( n|I| )). Following this, we generalize our use case to patterns in which we propose an algorithm that computes all-window-length co-occurrence with expected time complexity O(n|I|) and space complexity O( √(n|I|) + e_max|I|), where e_max is the length of the largest pattern.
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