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On the expected complexity of randomly weighted Voronoi diagrams
 In Proc. 30th Annu
"... In this paper, we provide an O(npolylogn) bound on the expected complexity of the randomly weighted Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interiordisjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, ..."
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In this paper, we provide an O(npolylogn) bound on the expected complexity of the randomly weighted Voronoi diagram of a set of n sites in the plane, where the sites can be either points, interiordisjoint convex sets, or other more general objects. Here the randomness is on the weight of the sites, not their location. This compares favorably with the worst case complexity of these diagrams, which is quadratic. As a consequence we get an alternative proof to that of Agarwal et al. [AHKS13] of the near linear complexity of the union of randomly expanded disjoint segments or convex sets (with an improved bound on the latter). The technique we develop is elegant and should be applicable to other problems. 1.
Quickly Placing a Point to Maximize Angles
 CCCG
, 2014
"... Given a set P of n points in the plane in general position, and a set of noncrossing segments with endpoints in P, we seek to place a new point q such that the constrained Delaunay triangulation of P ∪ {q} has the largest possible minimum angle. The expected running time of our (randomized) algori ..."
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Given a set P of n points in the plane in general position, and a set of noncrossing segments with endpoints in P, we seek to place a new point q such that the constrained Delaunay triangulation of P ∪ {q} has the largest possible minimum angle. The expected running time of our (randomized) algorithm is O(n2 log n) on any input, improving the nearcubic time of the best previously known algorithm. Our algorithm is somewhat complex, and along the way we develop a simpler cubictime algorithm quite different from the ones already known.
Three generalizations of Davenport–Schinzel sequences
"... We present new, and mostly sharp, bounds on the maximum length of certain generalizations of Davenport–Schinzel sequences. Among the results are sharp bounds on orders double DS sequences, for all s, sharp bounds on (double) formationfree sequences and new lower bounds on sequences avoiding zigza ..."
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We present new, and mostly sharp, bounds on the maximum length of certain generalizations of Davenport–Schinzel sequences. Among the results are sharp bounds on orders double DS sequences, for all s, sharp bounds on (double) formationfree sequences and new lower bounds on sequences avoiding zigzagging patterns.