Covering polygons with minimum number of rectangles
In the fabrication of masks for integrated circuits, it is desirable to replace the polygons comprising the layout of a circuit with as small as possible number of rectangles. Let Q be the set of all simple polygons with interior angles ≥ 90 degrees. Given a polygon P ε Q, let ϑ(P) be the minimum number of (possibly overlapping) rectangles lying within P necessary to cover P, and let r(P) be the ratio between the length of the longest edge of P and the length of the shortest edge of P. For every natural n ≥ 5, and k, a uniform polygon Pn,k with n corners is constructed such that r(Pn,k) ≥ k and ϑ(Pn) ≥ ω(nloglog(r(Pn,k))). On the other hand, by modifying a known heuristic it is shown that for all convex polygons P in Q with n vertices ϑ(P) ≤ O(nlog(r(P))).
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