Faster Multi-sided One-Bend Boundary Labelling.

A 1-bend boundary labelling problem consists of an axis-aligned rectangle B , n points (called sites ) in the interior, and n points (called ports ) on the labels along the boundary of B . The goal is to find a set of n axis-aligned curves (called leaders ), each having at most one bend and connecting one site to one port, such that the leaders are pairwise disjoint. A 1-bend boundary labelling problem is k -sided ( 1 ≤ k ≤ 4 ) if the ports appear on k different sides of B . Kindermann et al. [Algorithmica, 76(1): 225–258, 2016] showed that the 1-bend 4-sided and 3-sided boundary labelling problems can be solved in O ( n 9 ) and O ( n 4 ) time, respectively. Bose et al. [SWAT, 12:1–12:14, 2018] improved the former running time to O ( n 6 ) by reducing the problem to computing maximum independent set in an outerstring graph. In this paper, we improve both previous results by giving new algorithms with running times O ( n 5 ) and O ( n 3 log n ) to solve the 1-bend 4-sided and 3-sided boundary labelling problems, respectively.