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91
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 471 (115 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Applications of parametric searching in geometric optimization
 J. Algorithms
, 1994
"... z Sivan Toledo x ..."
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Approximating Polygons and Subdivisions with MinimumLink Paths
, 1991
"... We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate object ..."
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Cited by 69 (12 self)
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We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with no selfintersections are NPhard.
Vertical decomposition of shallow levels in 3dimensional arrangements and its applications
, 2006
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New Bounds for Lower Envelopes in Three Dimensions, with Applications to Visibility in Terrains
 Geom
, 1997
"... We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect i ..."
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Cited by 60 (24 self)
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We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n 2 \Delta 2 c p log n ), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the `lower envelope' of the space of all rays in 3space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this rayenvelope is O(n 3 \Delta 2 c p log n ) for some constant c; in particular, there are at most that many rays that pass above th...
Voronoi Diagrams of Moving Points
, 1995
"... Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in ..."
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Cited by 55 (6 self)
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Consider a set of n points in ddimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) is the maximum length of a (n; s)DavenportSchinzel sequence [AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linearfactor improvement over the naive O(n d+2 ) upper bound on the number of topological events. In addition, we show that if only k points are moving (while leaving the other n \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...
On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets Under Euclidean Motion in the Plane
, 1992
"... We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al ..."
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Cited by 50 (3 self)
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We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of n points moving rigidly, has complexity O(n 2 k 2 s (k)) for some fixed s, where s (n) is the maximum length of a (n; s) DavenportSchinzel sequence. This improves the result of Aonuma et. al., who show an upper bound of O(n 3 k 4 log k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n) 6 log(mn)), where the two sets contain m and n points respectively. This work was supported in part by NSF grant IRI9057928 and matching funds from General Electric and Kodak, and in part by AFOSR under contract AFOSR910328. The second author was also supported by the Eshkol grant 0460190 from The Israeli Ministry of Science and Technology. 1. Introduction Determining the degree to ...
On ababFree and abbaFree Set Partitions
, 1996
"... These are partitions of [l] = f1; 2; : : : ; lg into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by DavenportSchinzel sequences. We summarize and extend known enumer ..."
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Cited by 45 (8 self)
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These are partitions of [l] = f1; 2; : : : ; lg into n blocks such that no four term subsequence of [l] induces the mentioned pattern and each k consecutive numbers of [l] fall into different blocks. These structures are motivated by DavenportSchinzel sequences. We summarize and extend known enumerative results for the pattern p = abab and give an explicit formula for the number p(abab; n; l; k) of such partitions. Our main tool are generating functions. We determine the corresponding generating function for p = abba and k = 1; 2; 3. For k = 2 there is a connection with the number of directed animals. We solve exactly two related extremal problems.
A Linear Upper Bound in Extremal Theory of Sequences
, 1994
"... An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) ..."
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Cited by 44 (14 self)
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An extremal problem considering sequences related to DavenportSchinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x itimes).
DavenportSchinzel Theory Of Matrices
"... Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 ..."
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Cited by 42 (2 self)
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Let C be a configuration of 1's. We define f(n; C) to be the maximal number of 1's in a 01 matrix of size n \Theta n not having C as a subconfiguration. We consider the problem of determining the order of f(n; C) for several forbidden C's. Among others we prove that f(n; i 1 1 1 1 j ) = \Theta(ff(n)n), where ff(n) is the inverse of the Ackermann function.