Discrete and Computational Geometry

@inproceedings{Leeuwen1998DiscreteAC,
  title={Discrete and Computational Geometry},
  author={Jan van Leeuwen and Jin Akiyama and Mikio Kano and Masatsugu Urabe and Jan van Leeuwen},
  booktitle={Lecture Notes in Computer Science},
  year={1998}
}
This talk surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized. This problem is known to be NP-complete even for k = 3. The case of k = 2, that is, bipartition problem is solvable in polynomial time. On… 
1 Citations
Equal Area Polygons in Convex Bodies
TLDR
It is shown that for a convex quadrilateral K of area 1, there exist n internally disjoint triangles of equal area such that the sum of their areas is at least $\frac {4n}{4n+1}$.

References

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We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the
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  • Mathematics, Computer Science
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  • 1998
TLDR
It is shown that, for any fixed k ≥ 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graphs on n n verticeswith no k -1 pairwise crossing edges containing at mostO(n log n) edges.
New Bounds on Crossing Numbers
The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ (n, e) the minimum of cr(G) taken over all graphs with n vertices and at
A Separator Theorem for Planar Graphs
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more
Which Crossing Number Is It Anyway?
A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the
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TLDR
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
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TLDR
It is shown that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k> 0 others, then its number of edges cannot exceed 4, and a better bound is established, (k + 3)(u− 2), which is tight for k=1 and 2.
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    Discret. Comput. Geom.
  • 1998
TLDR
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The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles. They also extend
Crossing Number is NP-Complete
In this paper we consider a problem related to questions of optimal circuit layout: Given a graph or network, how can we embed it in a planar surface so as to minimize the number of edge-crossings?
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