On-Line Construction of the Convex Hull of a Simple Polyline

نویسنده

Avraham A. Melkman

چکیده

After McCallum and Avis [4] showed that the convex hull of a simple polygon P with n vertices can be constructed in O(n) time, several authors [1,2,3] devised simplified algorithms for this problem. Graham and Yao [2] presented a particularly simple and elegant one. After finding two points of the convex hull, their algorithm generated all other hull vertices using only one stack for intermediate storage. It is the purpose of this short article to show that a slightly modified version of their algorithm constructs, on-line, the convex hull of any simple polyline in O(n) time. In contrast, the on-line construction of a nonsimple polyline requires O(n log n) time, as shown by Preparata [5]. In the special case of a simple polygon our algorithm produces the convex hull without first identifying two of the hull vertices, as was required in [2]. The price that we pay is the use of a deque instead of a queue. After this article was submitted, we learned of a similar approach taken by Tor and Middleditch [6], who embed it in an algorithm for the convex decomposition of .a simple polygon. To keep this article short, we use where possible the definitions and notations of Graham and Yao. The polyline P = (vl , . . . , Vm) is assumed to

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