## Tentative Prune-and-Search for Computing Fixed-Points with Applications to Geometric Computation (1995)

Venue: | Fundamenta Informaticae |

Citations: | 18 - 4 self |

### BibTeX

@ARTICLE{Kirkpatrick95tentativeprune-and-search,

author = {David Kirkpatrick and Jack Snoeyink},

title = {Tentative Prune-and-Search for Computing Fixed-Points with Applications to Geometric Computation},

journal = {Fundamenta Informaticae},

year = {1995},

volume = {22},

pages = {353--370}

}

### OpenURL

### Abstract

. Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixed-point of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking \Theta(log 2 n) time. Other cases can be solved in logarithmic time by using a pruneand -search technique that may make tentative discards and later revoke or certify them. This work finds application in optimal subroutines that compute approximations to convex polygons, dense packings, and Voronoi vertices for Euclidean and polygonal distance functions. 1 Introduction Several fundamental problems in computational geometry can be expressed as a search for special k-tuple with one element drawn from each of k lists. Examples with k equal to 2 or 3 include common tangents to two convex polygons, special chords in a polygon, and Voronoi vertices (circles or polygons tangent to three given points or polygons). Many of these p...

### Citations

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Citation Context ...problem by performing some computation and pruning the amount of input data by a constant fraction. Many (perhaps most) prune-and-search algorithms, for example Megiddo's linear programming algorithm =-=[12]-=-, perform computation on the entire input to discard a fraction. Nevertheless, computations such as binary search and Overmars and van Leeuwen's common tangent algorithm [5, 14, 15] can be viewed as p... |

159 |
Leeuwen, "Maintenance of Configurations in the Plane
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Citation Context ...ear programming algorithm [12], perform computation on the entire input to discard a fraction. Nevertheless, computations such as binary search and Overmars and van Leeuwen's common tangent algorithm =-=[5, 14, 15]-=- can be viewed as prune-and-search algorithms that look at local information in O(1) time to discard a fraction of the input. In section 2.5 we develop the tentative prune-and-search techniquesfor sea... |

106 | Determining the Separation of Preprocessed Polyhedra - A Unified Approach
- Dobkin, Kirkpatrick
- 1990
(Show Context)
Citation Context ...ese algorithms improve the running time of algorithms that approximate convex polygons by rectangles [1, 17]. Let us assume that ff is vertical and polygon P is given by a hierarchical representation =-=[3], which he-=-re means that "middle points" of a polygonal chain and their tangents are available. Hierarchical representations can be implemented by storing the vertices of P in order in an array or stor... |

95 |
A Kinetic Framework for Computational Geometry
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Citation Context ...nuous when b is a vertex. We can avoid these problems by making A and B smooth spline curves and taking the limit as these splines approach the polygon. An alternative is to use the kinetic framework =-=[7]-=- to parameterize point/tangent pairs as we will do in the next section.) Clearly, a fixed-point a 2 A such that g(f(a)) = a gives points of A and B with the same x-coordinate and tangent slope. It is ... |

78 | Fixed Point Theorems with Applications to Economics and Game Theory - Border - 1985 |

74 |
An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments
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(Show Context)
Citation Context ...one obtains the "largest triangle homothet" problem. Motivated by retraction-based motion planning, Levin and Sharir [10] extended Yap's algorithm for computing the Voronoi diagram of n line=-= segments [19]-=- to compute Voronoi diagrams under convex distance functions. When the distance function is defined by a m-gon (for m constant), their algorithm runs in O(n log n) time and O(n) space. (To be precise,... |

49 |
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(Show Context)
Citation Context ...ear programming algorithm [12], perform computation on the entire input to discard a fraction. Nevertheless, computations such as binary search and Overmars and van Leeuwen's common tangent algorithm =-=[5, 14, 15]-=- can be viewed as prune-and-search algorithms that look at local information in O(1) time to discard a fraction of the input. In section 2.5 we develop the tentative prune-and-search techniquesfor sea... |

46 |
Computing the extreme distances between two convex polygons
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Citation Context ...[13]. This improves algorithms for approximating convex polygons by rectangles [1, 17]. In section 3.2 we cast an old problem of computing the separation distance between two disjoint convex polygons =-=[4]-=- as a fixed-point problem. In section 3.3 we solve the inscribed triangle homothet problem: given a convex polygon P , find the largest homothet of a given triangle T that can be inscribed in P . (A h... |

26 | A compact piecewise-linear Voronoi diagram for convex sites
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- 1996
(Show Context)
Citation Context ...4, we compute Voronoi vertices: given three disjoint convex polygons, find the points that are equidistant from all three (where distance is measured by the Euclidean metric). With Michael McAllister =-=[11]-=-, we use this as a primitive to compute an approximate Voronoi diagram of convex objects. One can obtain a piecewise-linear diagram for retraction-based motion planning whose complexity is linear in t... |

24 |
Approximation of convex polygons
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Citation Context ...eta(log n) time to compute chords of a given length and direction in a convex n-gon, answering a question posed by Mount [13]. This improves algorithms for approximating convex polygons by rectangles =-=[1, 17]-=-. In section 3.2 we cast an old problem of computing the separation distance between two disjoint convex polygons [4] as a fixed-point problem. In section 3.3 we solve the inscribed triangle homothet ... |

12 | Maintaining the approximate width of a set of points in the plane
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- 1993
(Show Context)
Citation Context ...hout using a specific separating line, computing the shortest segment that joins two convex polygons while avoiding two others [9], and approximate a local minimum or maximum for convex polygon width =-=[16]-=-. For our purposes, prune-and-search is a candidate elimination strategy, involving a collection of a k lists, where k is a constant. A candidate is a k-tuple with one element (the local candidate) dr... |

10 |
Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane
- Edelsbrunner, Maurer
- 1984
(Show Context)
Citation Context ...ear programming algorithm [12], perform computation on the entire input to discard a fraction. Nevertheless, computations such as binary search and Overmars and van Leeuwen's common tangent algorithm =-=[5, 14, 15]-=- can be viewed as prune-and-search algorithms that look at local information in O(1) time to discard a fraction of the input. In section 2.5 we develop the tentative prune-and-search techniquesfor sea... |

7 |
Compact interval trees: a data structure for convex hulls
- Guibas, Hershberger, et al.
- 1991
(Show Context)
Citation Context ...gle intervals then we can find a fixed-point of the composition of the basic functions defined on these intervals. In the remainder of this section, we generalize a lower bound proof of Guibas et al. =-=[6]-=- to prove that in the worst case\Omega\Gamma k log 2 n) search steps are required for a fixed-point problem. Let k be even and let n = m 2 for an integer m. We describe an adversary that chooses k mon... |

7 |
The densest double-lattice packing of a convex polygon
- Mount
- 1991
(Show Context)
Citation Context ...eral examples in section 3. In section 3.1 we sketch an algorithm that uses \Theta(log n) time to compute chords of a given length and direction in a convex n-gon, answering a question posed by Mount =-=[13]-=-. This improves algorithms for approximating convex polygons by rectangles [1, 17]. In section 3.2 we cast an old problem of computing the separation distance between two disjoint convex polygons [4] ... |

6 |
An algorithm for computing compacted Voronoi diagrams defined by convex distance functions
- Kao, Mount
- 1991
(Show Context)
Citation Context ...e largest homothet of a given triangle T that can be inscribed in P . (A homothet of T is a scaled and translated copy of T .) This problem arises in the following manner in Kao and Mount's algorithm =-=[8]-=- for computing generalized Voronoi diagrams. A convex polygon P that contains the origin in its interior defines a convex distance function d : E 2 \Theta E 2 ! R by d(p; q) = lim supfff : ff(q \Gamma... |

6 | Approximation of convex figures by pairs of rectangles
- Schwarzkopf, Fuchs, et al.
- 1998
(Show Context)
Citation Context ...eta(log n) time to compute chords of a given length and direction in a convex n-gon, answering a question posed by Mount [13]. This improves algorithms for approximating convex polygons by rectangles =-=[1, 17]-=-. In section 3.2 we cast an old problem of computing the separation distance between two disjoint convex polygons [4] as a fixed-point problem. In section 3.3 we solve the inscribed triangle homothet ... |

4 |
Planning a purely translational motion for a convex polygonal object in two dimensional space using generalized Voronoi diagrams
- Leven, Sharir
- 1987
(Show Context)
Citation Context ...f one instead fixes P and scales and translates the triangle defined by the sites, one obtains the "largest triangle homothet" problem. Motivated by retraction-based motion planning, Levin a=-=nd Sharir [10]-=- extended Yap's algorithm for computing the Voronoi diagram of n line segments [19] to compute Voronoi diagrams under convex distance functions. When the distance function is defined by a m-gon (for m... |

2 |
Computing constrained segments: Butterfly wingspans in logarithmic time
- Kirkpatrick, Snoeyink
- 1993
(Show Context)
Citation Context ...que has been used to compute the common tangents of disjoint polygons without using a specific separating line, computing the shortest segment that joins two convex polygons while avoiding two others =-=[9]-=-, and approximate a local minimum or maximum for convex polygon width [16]. For our purposes, prune-and-search is a candidate elimination strategy, involving a collection of a k lists, where k is a co... |

1 |
Complexity of fixed point computations
- Vavasis
- 1989
(Show Context)
Citation Context ...sider here the complexity of finding fixed-points in higher dimensions or under other models of function evaluation. There is a large body of literature on such questions in mathematics and economics =-=[2, 18]-=-. 2 Fixed-points of the composition of functions In this section we study the complexity of finding a fixed-point of the composition of k monotone, continuous functions. We define the problem for gene... |