9 research outputs found
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
Compact Floor-Planning via Orderly Spanning Trees
Floor-planning is a fundamental step in VLSI chip design. Based upon the
concept of orderly spanning trees, we present a simple O(n)-time algorithm to
construct a floor-plan for any n-node plane triangulation. In comparison with
previous floor-planning algorithms in the literature, our solution is not only
simpler in the algorithm itself, but also produces floor-plans which require
fewer module types. An equally important aspect of our new algorithm lies in
its ability to fit the floor-plan area in a rectangle of size (n-1)x(2n+1)/3.
Lower bounds on the worst-case area for floor-planning any plane triangulation
are also provided in the paper.Comment: 13 pages, 5 figures, An early version of this work was presented at
9th International Symposium on Graph Drawing (GD 2001), Vienna, Austria,
September 2001. Accepted to Journal of Algorithms, 200
Orderly Spanning Trees with Applications
We introduce and study the {\em orderly spanning trees} of plane graphs. This
algorithmic tool generalizes {\em canonical orderings}, which exist only for
triconnected plane graphs. Although not every plane graph admits an orderly
spanning tree, we provide an algorithm to compute an {\em orderly pair} for any
connected planar graph , consisting of a plane graph of , and an
orderly spanning tree of . We also present several applications of orderly
spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem,
(2) the first area-optimal 2-visibility drawing of , and (3) the best known
encodings of with O(1)-time query support. All algorithms in this paper run
in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of
the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001),
Washington D.C., USA, January 7-9, 2001, pp. 506-51
The DFS-heuristic for orthogonal graph drawing☆☆Some of these result were published in the author's PhD thesis at Rutgers University; the author would like to thank her advisor, Prof. Endre Boros, for much helpful input. The results in Section 5 have been presented at the 8th Canadian Conference on Computational Geometry, Ottawa, 1996, see [1].
AbstractIn this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends
EXACT ALGORITHMS FOR SIZE CONSTRAINED CLUSTERING
This thesis investigates the following general constrained clustering problem: given a dimension , an -norm, a set , a positive integer and a finite set , find the optimal -partition of w.r.t. the -norm satisfying , .
First of all, we prove that the problem is NP-hard even if (for all ), or and (with Euclidean norm).
Moreover, we put in evidence that the problem is computationally hard if is a non-integer rational.
When , and , we design an algorithm for solving the problem in time in the case of Euclidean norm; this result relies on combinatorial geometry techniques concerning -sets and dynamic convex hulls.
Finally, we study the problem in fixed dimension with ; by means of tools of real algebraic geometry and numerical techniques for localising algebraic roots we construct a polynomial-time method for solving the constrained clustering problem with integer given in unary notation
New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity
Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity
2-Visibility Drawings of Planar Graphs
In a 2-visibility drawing the vertices of a given graph are represented by rectangular boxes and the adjacency relations are expressed by horizontal and vertical lines drawn between the boxes. In this paper we want to emphasize this model as a practical alternative to other representations of graphs, and to demonstrate the quality of the produced drawings. We give several approaches, heuristics as well as provably good algorithms, to represent planar graphs within this model. To this, we present a polynomial time algorithm to compute a bend-minimum orthogonal drawing under the restriction that the number of bends at each edge is at most 1