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(Faster) Multi-Sided 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 ($1\leq k\leq 4$) if the ports appear on $k$ different sides of $B$. Kindermann et al. ["Multi-Sided Boundary Labeling", Algorithmica, 76(1): 225-258, 2016] showed that the 1-bend three-sided and four-sided boundary labelling problems can be solved in $O(n^4)$ and $O(n^9)$ time, respectively. Bose et al. [SWAT, 12:1-12:14, 2018] improved the latter 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 $O(n^3\log n)$ and $O(n^5)$ to solve the 1-bend three-sided and four-sided boundary labelling problems, respectively.