TY - GEN

T1 - Translational polygon containment and minimal enclosure using Linear Programming Based Restriction

AU - Milenkovic, Victor J.

N1 - Funding Information:
l University of Miami, Department of Math and Computer Science. Email: vjm@cs.miami.edu. This research was funded by the Textile/Clothing Technology Corporation from funds awarded to them by the Alfred P. Sloan Foundation, by NSF CCR-91-157993, and by a subcontract of a National Textile grant to Auburn University, Department of Consumer Affairs.

PY - 1996/7/1

Y1 - 1996/7/1

N2 - We introduce and analyze a new technique Linear Programming Based Restriction (LP Restriction) for solving translational containment and enclosure problems. The containment task is to translate k m-gons into a n-gon container. The enclosure task is to translate k m-gons into a minimum area n-gon which is convex with fixed orientation edges. All running times are based on an assumption of fixed k and are asymptotic in m and n. Lower bounds are proved for containment in a nonconvex container: Ω((mn)2k) for nonconvex polygons and Ω(nk) for convex polygons. LP restriction achieves upper bound O((m2 + mn)2k log n) for nonconvex polygons and O((mn)k log n) for convex polygons. The former almost matches the lower bound, and the latter does also for constant m. For arbitrary m and k > 3, the latter is faster than any other known algorithm for translational containment of convex polygons. A proof is given that the area function for fixed-angle convex polygons is Lorentzian. Enclosure algorithms based on this result have running times O((m2 + n)2k-2(n + log m)) for nonconvex polygons and O((m + n)2k(n + log m)) or O(mk-1(n2k+1 + log m)) for convex polygons. We are not aware of any previous running times for k-polygon minimum area enclosure. Other containment and enclosure results are given. LP restriction demonstrates a useful combination of mathematical programming and computational geometry which may be a paradigm for solving other tasks. We show that the containment and enclosure algorithms are useful in industry. Software based on LP restriction has been licensed to industry, and it is the fastest available solution to translational containment and enclosure for nonconvex polygons.

AB - We introduce and analyze a new technique Linear Programming Based Restriction (LP Restriction) for solving translational containment and enclosure problems. The containment task is to translate k m-gons into a n-gon container. The enclosure task is to translate k m-gons into a minimum area n-gon which is convex with fixed orientation edges. All running times are based on an assumption of fixed k and are asymptotic in m and n. Lower bounds are proved for containment in a nonconvex container: Ω((mn)2k) for nonconvex polygons and Ω(nk) for convex polygons. LP restriction achieves upper bound O((m2 + mn)2k log n) for nonconvex polygons and O((mn)k log n) for convex polygons. The former almost matches the lower bound, and the latter does also for constant m. For arbitrary m and k > 3, the latter is faster than any other known algorithm for translational containment of convex polygons. A proof is given that the area function for fixed-angle convex polygons is Lorentzian. Enclosure algorithms based on this result have running times O((m2 + n)2k-2(n + log m)) for nonconvex polygons and O((m + n)2k(n + log m)) or O(mk-1(n2k+1 + log m)) for convex polygons. We are not aware of any previous running times for k-polygon minimum area enclosure. Other containment and enclosure results are given. LP restriction demonstrates a useful combination of mathematical programming and computational geometry which may be a paradigm for solving other tasks. We show that the containment and enclosure algorithms are useful in industry. Software based on LP restriction has been licensed to industry, and it is the fastest available solution to translational containment and enclosure for nonconvex polygons.

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U2 - 10.1145/237814.237840

DO - 10.1145/237814.237840

M3 - Conference contribution

AN - SCOPUS:0029703214

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 109

EP - 118

BT - Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996

PB - Association for Computing Machinery

T2 - 28th Annual ACM Symposium on Theory of Computing, STOC 1996

Y2 - 22 May 1996 through 24 May 1996

ER -