TY - GEN

T1 - On maintaining the width and diameter of a planar point-set online

AU - Janardan, Ravi

PY - 1991/1/1

Y1 - 1991/1/1

N2 - Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, 5. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(αlog2 n) time, and reports in O(αlog2 n) time an approximation, Ŵ, to the width such that Ŵ/W ≤ √l + tan2 π/4α. The algorithm for the diameter problem uses O(βn) space, supports updates in O(βlogn) time, and reports in O(βn) time an approximation, D, to the diameter such that (formula presented). Thus, for instance, even for a as small as 5, Ŵ/W ≤ 1.01, and for β as small as 11, D/D ≥ .99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.

AB - Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, 5. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(αlog2 n) time, and reports in O(αlog2 n) time an approximation, Ŵ, to the width such that Ŵ/W ≤ √l + tan2 π/4α. The algorithm for the diameter problem uses O(βn) space, supports updates in O(βlogn) time, and reports in O(βn) time an approximation, D, to the diameter such that (formula presented). Thus, for instance, even for a as small as 5, Ŵ/W ≤ 1.01, and for β as small as 11, D/D ≥ .99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.

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U2 - 10.1007/3-540-54945-5_57

DO - 10.1007/3-540-54945-5_57

M3 - Conference contribution

AN - SCOPUS:30244477460

SN - 9783540549451

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

SP - 137

EP - 149

BT - ISA 1991 Algorithms - 2nd International Symposium on Algorithms, Proceedings

A2 - Lee, R.C.T.

A2 - Hsu, Wen-Lian

PB - Springer- Verlag

T2 - 2nd Annual International Symposium on Algorithms, ISA 1991

Y2 - 16 December 1991 through 18 December 1991

ER -