TY - GEN

T1 - Computing shortest transversals

AU - Bhattacharya, Binay

AU - Toussaint, Godfried

N1 - Funding Information:
The research reported here was supported by the Natural Sciences & Engineering Research Council of Canada, FCAR in Quebec, and the British Columbia Advanced Systems Institute. The research was carried out while the second author was an Advanced Systems Institute Fellow at Simon Fraser University during the fall of 1988. The authors are grateful to David Dobkin, David Kirkpatrick, Victor Klee and Slawomir Pilarski for fruitful discussions on this topic. Finally we thank an anonimous referee for providing an elegant proof that l(x) in lemma 2.1 is in fact a convex function.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1991.

PY - 1991

Y1 - 1991

N2 - We present an 0(n log2 n) time and 0(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in 0(n log k) time, where k is the combinatorial complexity of the space of transversals and k ≤ 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which 0(n) time algorithms are also given. AU the algorithms are based on the solution to a new fundamental geometric optimization problem that is of independent interest and should find application in different contexts as well.

AB - We present an 0(n log2 n) time and 0(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in 0(n log k) time, where k is the combinatorial complexity of the space of transversals and k ≤ 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which 0(n) time algorithms are also given. AU the algorithms are based on the solution to a new fundamental geometric optimization problem that is of independent interest and should find application in different contexts as well.

UR - http://www.scopus.com/inward/record.url?scp=69949154918&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69949154918&partnerID=8YFLogxK

U2 - 10.1007/3-540-54233-7_171

DO - 10.1007/3-540-54233-7_171

M3 - Conference contribution

AN - SCOPUS:69949154918

SN - 9783540542339

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 649

EP - 660

BT - Automata, Languages and Programming - 18th International Colloquium, Proceedings

A2 - Albert, Javier Leach

A2 - Artalejo, Mario Rodriguez

A2 - Monien, Burkhard

PB - Springer Verlag

T2 - 18th International Colloqulum on Automata, Languages, and Programming, ICALP 1991

Y2 - 8 July 1991 through 12 July 1991

ER -