In this paper, we introduce the notion of a constrained Minkowski sum which for two (finite) point-sets P, Q ⊆ R 2 and a set of k inequalities Ax ≥ b is defined as the point-set (P ⊕Q) Ax≥b = {x = p+q | p ∈ P, q ∈ Q, Ax ≥ b}. We show that typical subsequence problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O (N log N ), where N = |P | + |Q| if k is fixed. For the special case (P ⊕ Q) x 1 ≥β , where P and Q consist of points with integer x 1 -coordinates whose absolute values are bounded by O(N ), we even achieve a linear running time O(N ). We thereby obtain a linear running time for many subsequence problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of subsequence problems can be modeled and solved. This includes objective functions and constraints which are even more complex than the ones considered before.
Standard median filters preserve abrupt shifts (edges) and remove impulsive noise (outliers) from a constant signal but they deteriorate in trend periods. FIR median hybrid (FMH) filters are more flexible and also preserve shifts, but they are much more vulnerable to outliers. Application of robust regression methods, in particular of the repeated median, has been suggested for removing subsequent outliers from a signal with trends. A fast algorithm for updating the repeated median in linear time using quadratic space is given in Bernholt and Fried (2003). We construct repeated median hybrid filters to combine the robustness properties of the repeated median with the edge preservation ability of FMH filters. An algorithm for updating the repeated median is presented which needs only linear space. We also investigate analytical properties of these filters and compare their performance via simulations.
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