Approximate point set pattern matching with L_p-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 matching problem proposed in [T. Suga and S. Shimozono, Approximate point set pattern matching on sequences and planes, CPM'04]. What requested is an optimal match which minimizes the L _p -norm of the difference vector (|p₂ - p₁ - (t′₂ - t′₁)|, |p₃ - p₂ - (t′₃ - t′₂)|,..., |p _m - p _{m - 1} - (t′_m - t′_{m - 1})|), where p₁, p₂,..., p_m is the pattern and t′₁, t′₂,..., t′_m is a subsequence of the text. For p → ∞, the proposed algorithm is of time complexity O(mn), where m and n denote the lengths of the pattern and the text, respectively. For arbitrary p < ∞, the time complexity is O(mnT(p)), where T(p) is the time of evaluating x^p for x ∈ R.