Finding large sticks and potatoes in polygons
Olaf Hall-Holt,Matthew J. Katz,Piyush Kumar,Joseph S. B. Mitchell,Arik Sityon +4 more
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This work studies a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato").Abstract:
We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n7) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1 - ∈)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ∈)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).read more
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References
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Matrix multiplication via arithmetic progressions
Don Coppersmith,Shmuel Winograd +1 more
TL;DR: In this article, a new method for accelerating matrix multiplication asymptotically is presented, based on the ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product.
Proceedings ArticleDOI
Matrix multiplication via arithmetic progressions
Don Coppersmith,Shmuel Winograd +1 more
TL;DR: A new method for accelerating matrix multiplication asymptotically is presented, by using a basic trilinear form which is not a matrix product, and making novel use of the Salem-Spencer Theorem.
Journal Article
Triangulating a simple polygon in linear time
TL;DR: A deterministic algorithm for triangulating a simple polygon in linear time is given, using the polygon-cutting theorem and the planar separator theorem, whose role is essential in the discovery of new diagonals.
Journal ArticleDOI
Triangulating a simple polygon in linear time
TL;DR: In this paper, a deterministic algorithm for triangulating a simple polygon in linear time is presented. But the main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals.