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Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently
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BibTeX
@MISC{Bernholt_constrainedminkowski,
author = {Thorsten Bernholt and Friedrich Eisenbrand and Thomas Hofmeister},
title = {Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently},
year = {}
}
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Abstract
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 interval 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 logN), where N = |P|+|Q | if k is fixed. For the special case (P ⊕ Q) x1�β, where P and Q consist of points with integer x1-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 interval 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 interval problems can be modeled and solved. 1
Keyphrases
minkowski sum interval problem computational biology efficiently geometric framework linear running time constrained minkowski sum many interval problem main advantage computational biology general framework broad variety absolute value presented approach typical interval problem integer x1-coordinates running time special case convex hull inequality ax
