CS 663 : Computational Geometry
نویسندگان
چکیده
We will describe an O(n) time reduction from the set-disjointness problem to the farthest pairs problem. This will imply the Ω(n log n) lower bound. Consider an instance of the set-disjointness problem A = {a1, a2, . . . , an} and B = {b1, b2, . . . , bn}. We need to find out whether there exist i and j such that ai = bj. We map the sets A and B to the first and third quadrant of the unit circle C in the plane as follows: aj is mapped to the intersection of C with the line y = ajx in the first quadrant, while bi is mapped to the anologous intersection in the third quadrant (see figure 1). Let S be the set of these 2n intersections. It is easy to see that the diameter of S equals 2 if and only if there are two diametrically opposite points on C,that is, A ∩ B 6= Φ. Hence, the reduction.
منابع مشابه
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Two results in “computational origami” are illustrated. Computational geometry has recently been applied to solve two open problems in “origami mathematics.”
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