6,360 research outputs found
Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs
AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space
Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
In this paper we develop an integer-affine classification of
three-dimensional multistory completely empty convex marked pyramids. We apply
it to obtain the complete lists of compact two-dimensional faces of
multidimensional continued fractions lying in planes with integer distances to
the origin equal 2, 3, 4 ... The faces are considered up to the action of the
group of integer-linear transformations. In conclusion we formulate some actual
unsolved problems associated with the generalizations for n-dimensional faces
and more complicated face configurations.Comment: Minor change
Holes or Empty Pseudo-Triangles in Planar Point Sets
Let denote the smallest integer such that any set of at least
points in the plane, no three on a line, contains either an empty
convex polygon with vertices or an empty pseudo-triangle with
vertices. The existence of for positive integers ,
is the consequence of a result proved by Valtr [Discrete and Computational
Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new
results about the existence of empty pseudo-triangles in point sets with
triangular convex hulls, we determine the exact values of and , and prove bounds on and , for . By
dropping the emptiness condition, we define another related quantity , which is the smallest integer such that any set of at least points in the plane, no three on a line, contains a convex polygon with
vertices or a pseudo-triangle with vertices. Extending a result of
Bisztriczky and T\'oth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we
obtain the exact values of and , and obtain non-trivial
bounds on .Comment: A minor error in the proof of Theorem 2 fixed. Typos corrected. 19
pages, 11 figure
Relative Convex Hull Determination from Convex Hulls in the Plane
A new algorithm for the determination of the relative convex hull in the
plane of a simple polygon A with respect to another simple polygon B which
contains A, is proposed. The relative convex hull is also known as geodesic
convex hull, and the problem of its determination in the plane is equivalent to
find the shortest curve among all Jordan curves lying in the difference set of
B and A and encircling A. Algorithms solving this problem known from
Computational Geometry are based on the triangulation or similar decomposition
of that difference set. The algorithm presented here does not use such
decomposition, but it supposes that A and B are given as ordered sequences of
vertices. The algorithm is based on convex hull calculations of A and B and of
smaller polygons and polylines, it produces the output list of vertices of the
relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two
typing errors in Definition 2: has to be defined based on , and
has to be defined based on (not just using ). These errors
appeared in the text of the original conference paper, which also contained
the pseudocode of an algorithm where and appeared as correctly
define
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