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Daoqi Wang Problem 41 from Open Problems Project http://maven.smith.edu/~orourke/TOPP/

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Sorting X + Y where X + Y = {x + y | x ∈ X, y ∈ Y} (Minkowski sum) Ex: X = {1, 2, 3}, Y = {4, 5, 6} {5, 6, 6, 7, 7, 7, 8, 8, 9} X1X1 X2X2 X3X3 Y1Y1 1 + 42 + 43 + 4 Y2Y2 1 + 52 + 53 + 5 Y3Y3 1 + 62 + 63 + 6

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Michael Fredman, 1976 Algorithm found by William Steiger and Ileana Streinu in 1995 with O(n^2logn) running time with O(n^2) comparisons Question: is it o(n^2logn)?

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Antonio Herrera, 1996 ◦ Presented problems that are at least as hard as sorting X + Y ◦ Tracing the intersections of a ray ◦ Enumerating distances (K shortest distances) O(nlogn + klogn)

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Timothy Chan, 2006 ◦ Necklace alignment (L 1, L 2, L ∞ )

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Timothy Chan, 2006 ◦ Convolution (max, min, range)

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Status ◦ Open ◦ o(n^2logn) Importance ◦ 3SUM(a + b + c = 0) ◦ Topological Sorting ◦ Sperm Sorting

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Tangents and Normals We’re going to talk about tangents and normals to 3-D surfaces such as x 2 + y 2 + z 2 = 4 It’s useful to think of these surfaces.

Tangents and Normals We’re going to talk about tangents and normals to 3-D surfaces such as x 2 + y 2 + z 2 = 4 It’s useful to think of these surfaces.

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