Algorithms and Recursive Functions by A.I. Mal'cev

By A.I. Mal'cev

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The Voronoi vertices of the nearest-site Voronoi diagram are the candidates for the placement of the industrial zone. The vertex with the maximum distance to a site is the sought solution. 1 Related Work The Voronoi diagram of point sites has been studied extensively in the literature. In the Euclidean metric, the combinatorial complexity of both the nearest- and farthest-site Voronoi diagram is O(n), where n is the number of point sites [8] or disjoint line segments [1,16] in the diagram. These diagrams can be constructed in optimal O(n log n) time.

Banik Let B be the subset of points of B1 having y coordinate less than yrm . Define the left staircase Bl to be the subset of B , such that for any point r2 ∈ B , r2 belongs to Bl if xr2 < xp and r1 ∈ B such that xr2 < xr1 < xp and yr2 ≥ yr1 (see Fig. 1). Similarly, we define the right staircase Br as follows. For any r2 of B , r2 ∈ Br if xq < xr2 and r1 ∈ B such that xq < xr1 < xr2 and yr1 ≤ yr2 (see Fig. 1). Also let Ba = Bl ∪ Br ∪ {rm }. Now we have the following observation. Observation 3. For any rectangle T ∈ C , T contains p, q in one of its sides and each of the other sides of T either contains a point of Ba or is unbounded.

Every Voronoi region N VP (si ) in N V DP (S ) is simply connected. Let s1 , s2 be two line segments in the plane, and B(s1 , s2 ) and BP (s1 , s2 ) be the bisectors of s1 and s2 with respect to the Euclidean distance and the polygon-offset distance DP , respectively. In the sequel, we will use the term polyline to denote a piecewise-simple curve all of whose elements are described by low-degree polynomials, like line segments and parabolic arcs. It is well known, then, that B(s1 , s2 ) is a polyline.

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