Applications of a Numbering Scheme for Polygonal Obstacles in the Plane

We present efficient algorithms for the problems of matching red and blue disjoint geometric obstacles in the plane and connecting the matched obstacle pairs with mutually nonintersecting paths that have useful geometric properties. We first consider matching n red and n blue disjoint isothetic rectangles and connecting the n matched rectangle pairs with nonintersecting monotone rectilinear paths; each such path consists of O(n) segments and is not allowed to touch any rectangle other than the matched pair that it is linking. Based on a numbering scheme for certain geometric objects and on several useful geometric observations, we develop an O(n log n) time, O(n) space algorithm that produces a desired matching for isothetic rectangles. If an explicit printing of all the n paths is required, then our algorithm takes O(n log n+λ) time and O(n) space, where λ is the total size of the desired output. We then extend these matching algorithms to other classes of red/blue polygonal obstacles. The numbering scheme also finds applications to other problems.

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