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Maximum line-pair stabbing problem and its variations

We study the Maximum Line-Pair Stabbing Problem: Given a planar point set S, find a pair of parallel lines within distance 6 from each others such that the number of points of S that intersect (stab) the area in between the two lines is maximized. There exists an algorithm that computes maximum stabbing in O(|S| 2 ) time and space. We give a more space-efficient solution; the time complexity increases to O(|S| 2 log |S|), but the space reduces to O(|S|). Our algorithm also extends to a dual problem where one searches for a line stabbing maximum number of variable size circles; as far as we know, this problem has previously been studied only on fixed size circles. A variant of the stabbing problem equals a onedimensional point set matching problem under translations, scalings, and errors. We study a version of this problem, where the matching has to be a oneto-one mapping. Existing techniques based on incremental maintenance of maximum matching using augmenting paths yield O((mn) 3 ) time solution, where m and n are the sizes of the point sets to be matched. Our new algorithm achieves O((mn) 2 (m + n)) time. The improvement is based on an observation that in our case the match-graph has a regular shape, and the maximum matching can be updated more efficiently.