L. Danzer and B. Grunbaum. Intersection properties of boxes in R d . Combinatorica, 2(3):237--246, 1982. Home/Search Document Not in Database Summary Related Articles Check |
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Algorithms for Maximum Independent Set Applied to Map.. - Strijk, Verweij, Aardal (2000) (Correct)
....in GQ , and let S = Q v Q # C be the rectangular regions in Q that correspond to the nodes in C. Because C is a clique, all rectangular regions in S intersect pairwise. This implies that the region R = T Q#S Q is non empty by the following argument (see e.g. Danzer and Grunbaum [11]) The intersection of a set of axis parallel rectangular regions is non empty if and only if the intersection of the projections of the 7 rectangular regions on the axis are non empty. The projections of the rectangular regions on the axes are sets of line segments in R. By Helly s theorem, a ....
L. Danzer and B. Grunbaum. Intersection properties of boxes in R d . Combinatorica, 2(3):237--246, 1982.
Constrained Square-Center Problems - Katz, Kedem, Segal (1998) (1 citation) (Correct)
....then R is 1 pierceable, and we check whether it is also 1 constrained pierceable by checking whether P has a point in R. If R is 1 constrained pierceable then we are done, so assume that it is not. If R was not found to be 1 pierceable, then we apply the linear time algorithm of [20] see also [5]) to check whether R is 2 pierceable. If R is neither 1 pierceable nor 2 pierceable, then obviously R is not 2 constrained pierceable and we are done. Assume therefore that R is 2 pierceable (or 1 pierceable) Assume R is 2 constrained pierceable, and let p 1 ; p 2 2 P be a pair of piercing ....
L. Danzer and B. Grunbaum, "Intersection properties of boxes in R d ", Combinatorica 2(3) (1982), 237--246.
On Piercing Sets of Objects - Katz, Nielsen (1995) (7 citations) (Correct)
....to 1 (infinity) Let Q k denote the k pierceability property and C d be the class of d dimensional convex objects. Then Helly s theorem states that h(C d ; Q 1 ) d 1. There are many Helly type theorems in convex geometry (see [GW93] for an up to date survey) Danzer and Grunbaum [DG82] studied the case of d dimensional (axis parallel) boxes B d and obtained Helly type theorems whenever they exist. They prove in particular that h(B d ; Q k ) 1, for d; k 3. They conclude their paper with the following conjecture: h(T d (K) Q 2 ) 1 if and only if K is a convex ....
....d dimensional (axis parallel) boxes and k points, that is, h(d; k) is the smallest integer (if such exists) such that, if every subset of cardinality h(d; k) of a set B of d boxes is k pierceable, then B is k pierceable. If such an integer does not exist, we set h(d; k) 1. Danzer and Grunbaum [DG82] have proven the following theorem, which we present as a table: Theorem 1 (Danzer and Grunbaum) h(d; k) k = 1 k = 2 k = 3 k 4 d = 1 2 3 4 k 1 d = 2 2 5 16 1 d 3 2 ae 3d if d is odd 3d Gamma 1 if d is even 1 1 Their proof for the case of d dimensional boxes, d 2, and two points is ....
[Article contains additional citation context not shown here]
L. Danzer and B. Grunbaum. Intersection properties of boxes in R d . Combinatorica, 2(3):237--246, 1982.
Constrained Square-Center Problems - Katz, Kedem, Segal (1998) (1 citation) (Correct)
....then R is 1 pierceable, and we check whether it is also 1 constrained pierceable by checking whether P has a point in R. If R is 1 constrained pierceable then we are done, so assume that it is not. If R was not found to be 1 pierceable, then we apply the linear time algorithm of [22] see also [6]) to check whether R is 2 pierceable. If R is neither 1 pierceable nor 2 pierceable, then obviously R is not 2 constrained pierceable and we are done. Assume therefore that R is 2 pierceable (or 1 pierceable) Assume R is 2 constrained pierceable, and let p 1 ; p 2 2 P be a pair of piercing points ....
L. Danzer and B. Grunbaum. "Intersection properties of boxes in R d ". Combinatorica, 2(3), 237--246, 1982.
Fast Stabbing of Boxes in High Dimensions - Nielsen (1996) (4 citations) (Correct)
....for future research. 2 An optimal algorithm for stabbing intervals In this section, we consider the case of intervals, i.e. 1 dimensional boxes. Let S be a set of n intervals. 2. 1 Principle Finding the minimum value c so that S can be stabbed with c points is easy and already known in [DG82, HM84] Consider the interval I that has the rightmost left endpoint p. I must be stabbed by a point and clearly, the best place to stab it is on the left endpoint p. We then remove all the intervals stabbed by p and loop until all the intervals are stabbed. We thus obtain a minimal size set of ....
....boxes S, i.e. to each box corresponds a node and there is an edge between two nodes ioe their corresponding boxes intersect. Isothetic boxes have nice combinatorial properties. For example, a set of boxes have a nonempty intersection ioe they intersect pairwise (this is an Helly type theorem [DG82, HD60] Therefore, nding a minimum size set of stabbing points can be done by rst computing G in quadratic time and then, nding a minimum clique partition of G, i.e. a set of cliques (complete subsets of G) whose union covers the vertices of G. There is a one to one correspondence between ....
L. Danzer and B. Gr#nbaum. Intersection Properties of Boxes in R d . Combinatorica, 2(3):237 246, 1982.
Algorithms for Maximum Independent Set Applied to Map - Tycho Strijk Bram (2000) (Correct)
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L. Danzer and B. Grunbaum. Intersection properties of boxes in R d . Combinatorica, 2(3):237--246, 1982.
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