In this paper we study the number of vertex recolorings that an algorithm needs to perform in order to maintain a proper coloring of a graph under insertion and deletion of vertices and edges. We present two algorithms that achieve different trade-offs between the number of recolorings and the number of colors used. For any d > 0, the first algorithm maintains a proper O(CdN 1/d )-coloring while recoloring at most O(d) vertices per update, where C and N are the maximum chromatic number and maximum number of vertices, respectively. The second algorithm reverses the trade-off, maintaining an O(Cd)-coloring with O(dN 1/d ) recolorings per update. The two converge when d = log N , maintaining an O(C log N )-coloring with O(log N ) recolorings per update. We also present a lower bound, showing that any algorithm that maintains a c-coloring of a 2-colorable graph on N vertices must recolor at least Ω(N 2 c(c−1) ) vertices per update, for any constant c ≥ 2.

We present a new algorithm for the widely used density-based clustering method dbscan. For a set of [Formula: see text] points in [Formula: see text] our algorithm computes the dbscan-clustering in [Formula: see text] time, irrespective of the scale parameter [Formula: see text] (and assuming the second parameter MinPts is set to a fixed constant, as is the case in practice). Experiments show that the new algorithm is not only fast in theory, but that a slightly simplified version is competitive in practice and much less sensitive to the choice of [Formula: see text] than the original dbscan algorithm. We also present an [Formula: see text] randomized algorithm for hdbscan in the plane — hdbscan is a hierarchical version of dbscan introduced recently — and we show how to compute an approximate version of hdbscan in near-linear time in any fixed dimension.

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Given a collection L of line segments, we consider its arrangement and study the problem of covering all cells with line segments of L. That is, we want to find a minimum-size set L of line segments such that every cell in the arrangement has a line from L defining its boundary. We show that the problem is NP-hard, even when all segments are axis-aligned. In fact, the problem is still NP-hard when we only need to cover rectangular cells of the arrangement. For the latter problem we also show that it is fixed parameter tractable with respect to the size of the optimal solution. Finally we provide a linear time algorithm for the case where cells of the arrangement are created by recursively subdividing a rectangle using horizontal and vertical cutting segments. * A preliminary version of this paper appeared in the proceedings of the COCOA 2015 conference [8]

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