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Minimum co-operative guards in spiral polygons
Abstract
The Minimum Co-operative guard problem is an interesting Computational Geometry problem which has military as well as civilian applications. Given an n-vertex polygon P, the problem is to compute the minimum number of guards that should be posted in P such that each point in the polygon is visible to at least one guard and each guard is seen by at least one other guard. This problem has been proved to be NP-hard. There is no known solution for the Minimum Co-operative Guard problem in k-spiral polygons, where k $>$ 2. Also, there is no known bounds on the number of minimum co-operative guards needed for a k-spiral polygon. Liaw et al have solved the problem for 1-spiral polygons. They have also given a partial solution for the case of 2-spirals. We first classify 2-spiral polygons into seven types and then extend Liaw et al's algorithm to cover all these seven cases of 2-spiral polygons. Structural properties of 3-spiral polygons are studied and 3-spiral polygons are classified into three different types based on structural properties. Greedy algorithms to solve the minimum co-operative guard problem for each of these cases is presented. The algorithm for Type 1 takes O(n) time. The worst case time complexity of the algorithm for Types 2 and 3 is $O(n\sp2$). In the worst case, the solution presented may need three additional guards as compared to an optimal solution. An upper bound on the number of minimum co-operative guards for k-spiral polygons is also established.
Subject Area
Computer science|Electrical engineering|Mathematics
Recommended Citation
Sarasamma, Suseela T, "Minimum co-operative guards in spiral polygons" (1996). ETD collection for University of Nebraska-Lincoln. AAI9715982.
https://digitalcommons.unl.edu/dissertations/AAI9715982