224 research outputs found
On Semantic Word Cloud Representation
We study the problem of computing semantic-preserving word clouds in which
semantically related words are close to each other. While several heuristic
approaches have been described in the literature, we formalize the underlying
geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this
model each word is associated with rectangle with fixed dimensions, and the
goal is to represent semantically related words by ensuring that the two
corresponding rectangles touch. We design and analyze efficient polynomial-time
algorithms for some variants of the WRAC problem, show that several general
variants are NP-hard, and describe a number of approximation algorithms.
Finally, we experimentally demonstrate that our theoretically-sound algorithms
outperform the early heuristics
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
Weakly Chordal Graphs: An Experimental Study
Graph theory is an important field that enables one to get general ideas about graphs and their properties. There are many situations (such as generating all linear layouts of weakly chordal graphs) where we want to generate instances to test algorithms for weakly chordal graphs. In my thesis, we address the algorithmic problem of generating weakly chordal graphs. A graph G=(V, E), where V is its vertices and E is its edges, is called a weakly chordal graph, if neither G nor its complement G\u27, contains an induced chordless cycle on five or more vertices. Our work is in two parts. In the first part, we carry out a comparative study of two existing algorithms for generating weakly chordal graphs. The first algorithm for generating weakly chordal graphs repeatedly finds a two-pair and adds an edge between them. The second-generation algorithm starts by constructing a tree and then generates an orthogonal layout (also weakly chordal graph) based on this tree. Edges are then inserted into this orthogonal layout until there are edges. The output graphs from these two methods are compared with respect to several parameters like the number of four cycles, run times, chromatic number, the number of non-two-pairs in the graphs generated by the second method. In the second part, we propose an algorithm for generating weakly chordal graphs by edge deletions starting from an arbitrary input random graph. The algorithm starts with an arbitrary graph to be able to generate a weakly chordal graph by the basis of edge deletion. The algorithm iterates by maintaining weak chordality by preventing any hole or antihole configurations being formed for any successful deletion of an edge
On Layered Area-Proportional Rectangle Contact Representations
A pair of graphs admits a mutual witness proximity
drawing when: (i) represents
, and (ii) there is an edge in if and only if there is
no vertex in that is ``too close'' to both and
(). In this paper, we consider infinitely many definitions of closeness
by adopting the -proximity rule for any and study
pairs of isomorphic trees that admit a mutual witness -proximity
drawing. Specifically, we show that every two isomorphic trees admit a mutual
witness -proximity drawing for any . The
constructive technique can be made ``robust'': For some tree pairs we can
suitably prune linearly many leaves from one of the two trees and still retain
their mutual witness -proximity drawability. Notably, in the special
case of isomorphic caterpillars and , we construct linearly separable
mutual witness Gabriel drawings.Comment: Appears in the Proceedings of the 18th International Conference and
Workshops on Algorithms and Computation (WALCOM 2024
Chordal Graphs and Their Relatives: Algorithms and Applications
While the problem of generating random graphs has received much attention, the problem of generating graphs for specific classes has not been studied much. In this dissertation, we propose schemes for generating chordal graphs, weakly chordal graphs, and strongly chordal graphs. We also present semi-dynamic algorithms for chordal graphs and strongly chordal graphs. As an application of a completion technique for chordal graphs, we also discuss a 1-round algorithm for approximate point placement in the plane in an adversarial model where the distance query graph presented to the adversary is chordal. The proposed generation algorithms take the number of vertices, n, and the number of edges, m, as input and produces a graph in a given class as output. The generation method either starts with a tree or a complete graph. We then insert additional edges in the tree or delete edges from the complete graph. Our algorithm ensures that the graph properties are preserved after each edge is inserted or deleted. We have also proposed algorithms to generate weakly chordal graphs and strongly chordal graphs from an arbitrary graph as input. In this case, we ensure the graph properties will be achieved on the termination of the conversion process. We have also proposed a semi-dynamic algorithm for edge-deletion in a chordal graph. To the best of our knowledge, no study has been done for the problem of dynamic algorithms for strongly chordal graphs. To address this gap, we have also proposed a semi-dynamic algorithm for edge-deletions and a semi-dynamic algorithm for edge-insertions in strongly chordal graphs
Experiments with Point Placement Algorithms and Recognition of Line Rigid Graphs
The point placement problem is to determine the position of n distinct points on a line, up to translation and reflection by fewest possible pairwise adversarial distance queries. This masters thesis focusses on two aspects of point placement problem. In one part we focusses on an experimental study of a number of deterministic point placement algorithms and an incremental randomized algorithm, with the goal of obtaining a greater insight into the behavior of these algorithms, particularly of the randomize algorithm. The pairwise distance queries in the point placement problem creates a type of graph, called point placement graph. A point placement graph G is dened as line rigid graph if and only if the vertices of G has unique placement on a line. The other part of this thesis focusses on recognizing line rigid graph of certain class based on structural property of an arbitrarily given graph. Layer graph drawing and rectangular drawing are used as key idea in recognizing line rigid graphs
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