One-dimensional approximate point set pattern matching with LpLp-norm

Given two sets of points, the text and the pattern, determining whether the pattern “appears” in the text is modeled as the point set pattern matching problem. Applications usually ask for not only exact matches between these two sets, but also approximate matches. In this paper, we investigate a one-dimensional approximate point set pattern matching problem proposed in [19]. We generalize the measure between approximate matches from L1L1-norm to LpLp-norm. More specifically, what requested is an optimal match which minimizes the LpLp-norm of the difference vector (|p2−p1−(t2′−t1′)|,|p3−p2−(t3′−t2′)|,…,|pm−pm−1−(tm′−tm−1′)|), where p1,p2,…,pmp1,p2,…,pm is the pattern and t1′,t2′,…,tm′ is a subsequence of the text. For p→∞p→∞, we propose an O(mn)O(mn)-time algorithm, where m and n   denote the lengths of the pattern and the text, respectively. For arbitrary p<∞p<∞, we propose an algorithm which runs in O(mnT(p))O(mnT(p)) time, where T(p)T(p) is the time of evaluating xpxp for x∈Rx∈R.