Computing simple circuits from a set of line segments

We address the problem of connecting line segments to form the boundary of a simple polygon—a simple circuit. However, not every set of segments can be so connected. We present anO(n logn)-time algorithm to determine whether a set of segments, constrained so that each segment has at least one endpoint on the boundary of the convex hull of the segments, admits a simple circuit. Furthermore, this technique can also be used to compute a simple circuit of minimum perimeter, or a simple circuit that bounds the minimum area, with no increase in computational complexity.

[1]  David Avis,et al.  Computing Monotone Simple Circuits in the Plane , 1988 .

[2]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[3]  Robert E. Tarjan,et al.  An O(n log log n)-Time Algorithm for Triangulating a Simple Polygon , 1988, SIAM J. Comput..

[4]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[5]  Ronald L. Graham,et al.  An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set , 1972, Inf. Process. Lett..

[6]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[7]  David S. Johnson,et al.  The Planar Hamiltonian Circuit Problem is NP-Complete , 1976, SIAM J. Comput..

[8]  David Rappaport.  The complexity of computing simple circuits in the plane , 1987 .

[9]  Michael Ian Shamos,et al.  Geometric complexity , 1975, STOC.

[10]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[11]  Michael Ian Shamos,et al.  Geometric intersection problems , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[12]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[13]  David Rappaport,et al.  Computing simple circuits from a set of line segments , 1987, SCG '87.

[14]  Godfried T. Toussaint,et al.  A historical note on convex hull finding algorithms , 1985, Pattern Recognit. Lett..

[15]  Godfried T. Toussaint,et al.  On the ultimate convex hull algorithm in practice , 1985, Pattern Recognit. Lett..

[16]  Nathan Friedman,et al.  Some Results on the Effect of Arithmetics on Comparison Problems , 1972, SWAT.