Geometric applications of a randomized optimization technique

Chan, Timothy M.

doi:10.1145/276884.276915

Cited by 36 publications

(65 citation statements)

References 79 publications

(95 reference statements)

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“…Since the vertices have fixed positions, the corresponding decision problem can be answered in O(N log N ) time by a plane-sweep method. This is enough to apply the randomized optimization technique of Chan to solve the problem in O(N log N ) time [6].…”

Section: Lemma 1 Given a Fixed Orthogonal Layout There Exists An O(nmentioning

confidence: 99%

Smooth Orthogonal Layouts

Bekos¹,

Kaufmann²,

Kobourov³

et al. 2013

JGAA

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Abstract. We study the problem of creating smooth orthogonal layouts for planar graphs. While in traditional orthogonal layouts every edge is made of a sequence of axis-aligned line segments, in smooth orthogonal layouts every edge is made of axis-aligned segments and circular arcs with common tangents. Our goal is to create such layouts with low edge complexity, measured by the number of line and circular arc segments. We show that every biconnected 4-planar graph has a smooth orthogonal layout with edge complexity 3. If the input graph has a complexity-2 traditional orthogonal layout we can transform it into a smooth complexity-2 layout. Using the Kandinsky model for removing the degree restriction, we show that any planar graph has a smooth complexity-2 layout.

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Section: Lemma 1 Given a Fixed Orthogonal Layout There Exists An O(nmentioning

confidence: 99%

Smooth Orthogonal Layouts

Bekos¹,

Kaufmann²,

Kobourov³

et al. 2013

JGAA

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“…If z does not violate the new constraint, then z does not change. Otherwise, we know that the new z is defined by this new constraint and it suffices to solve a 1-dimensional LP-type problem (analogous to finding the minimum of abstract elements, as in [7]) instead of 2-d.…”

Section: Theorem 24mentioning

confidence: 99%

Three problems about dynamic convex hulls

Chan

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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We present three results related to dynamic convex hulls:• A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small constant ε > 0. This improves the previous bound of O(log 3/2 n).• A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n + k) time with O(polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3-dimensional halfspace range reporting, the query time increases to O(log 2 n/ log log n + k).• A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n ε ). As a corollary, we can solve the following problem in O(n 1+ε ) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

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“…Then we can test whether, in a set of n red and blue points, there exists a red-blue pair at distance closer than two: compute the connected components of balls centered at the points and test whether any component contains points of both colors. A randomized reduction of Chan [6] transforms this decision algorithm into an optimization algorithm with the same complexity, for finding the closest red-blue pair, and it is known that this bichromatic closest pair problem and the Euclidean minimum spanning tree problem have asymptotically equal complexities [1,5,33].…”

Section: Lemma 47 Ifmentioning

confidence: 99%

“…Alternatively, we can imagine assigning colors to the edges in a drawing, such that each color class forms a planar drawing. A graph's thickness [32] is equal to the minimum thickness of any drawing of the given graph, while its geometric thickness is the minimum thickness of Figure 1: Geometric thickness two drawing of K 6,6 (left) and K 5,8 (right). The partitions of the drawings' edges into two non-crossing subsets are indicated by the coloring and casing of the edges.…”

Section: Introductionmentioning

confidence: 99%

Testing bipartiteness of geometric intersection graphs

Eppstein

2009

ACM Trans. Algorithms

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We show how to test whether an intersection graph of n line segments or simple polygons in the plane, or of balls in R d , is bipartite, in time O(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. Our algorithms for these problems return either a bipartition of the input or an odd cycle in its intersection graph. We also consider lower bounds for connectivity and kcolorability problems of geometric intersection graphs. For unit balls in R d , connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has the same lower bounds as Hopcroft's pointline incidence testing problem; therefore, for these problems, connectivity is unlikely to be solved as efficiently as bipartiteness. For line segments or planar disks, testing k-colorability of intersection graphs for k > 2 is NP-complete.

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Geometric applications of a randomized optimization technique

Cited by 36 publications

References 79 publications

Smooth Orthogonal Layouts

Smooth Orthogonal Layouts

Three problems about dynamic convex hulls

Testing bipartiteness of geometric intersection graphs

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