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239
An InformationTheoretic Upper Bound on Planar Graphs Using WellOrderly Maps
, 2011
"... This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but a ..."
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Cited by 26 (3 self)
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This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an informationtheoretic upper bound on planar graphs, we introduce a definition of a quasicanonical embedding for planar graphs: wellorderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute wellorderly maps and implying an efficient coding of planar graphs.
Dynamic Map Labeling
, 2006
"... We address the problem of filtering, selecting and placing labels on a dynamic map, which is characterized by continuous zooming and panning capabilities. This consists of two interrelated issues. The first is to avoid label popping and other artifacts that cause confusion and interrupt navigation, ..."
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Cited by 26 (1 self)
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We address the problem of filtering, selecting and placing labels on a dynamic map, which is characterized by continuous zooming and panning capabilities. This consists of two interrelated issues. The first is to avoid label popping and other artifacts that cause confusion and interrupt navigation, and the second is to label at interactive speed. In most formulations the static map labeling problem is NPhard, and a fast approximation might have O(nlogn) complexity. Even this is too slow during interaction, when the number of labels shown can be several orders of magnitude less than the number in the map. In this paper we introduce a set of desiderata for “consistent ” dynamic map labeling, which has qualities desirable for navigation. We develop a new framework for dynamic labeling that achieves the desiderata and allows for fast interactive display by moving all of the selection and placement decisions into the preprocessing phase. This framework is general enough to accommodate a variety of selection and placement algorithms. It does not appear possible to achieve our desiderata using previous frameworks. Prior to this paper, there were no formal models of dynamic maps or of dynamic labels; our paper introduces both. We formulate a general optimization problem for dynamic map labeling and give a solution to a simple version of the problem. The simple version is based on label priorities and a versatile and intuitive class of dynamic label placements we call “invariant point placements”. Despite these restrictions, our approach gives a useful and practical solution. Our implementation is incorporated into the GVis system which is a fulldetail dynamic map of the continental USA. This demo is available through any browser.
Transversal structures on triangulations, combinatorial study and straightline drawing
, 2007
"... This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bip ..."
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Cited by 26 (5 self)
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This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bipolar orientations. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straightline drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O ( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2 ⌉ − 1) × ⌊n/2⌋.
An Algorithm to Construct Greedy Drawings of Triangulations
, 2010
"... We show an algorithm to construct a greedy drawing of every given triangulation. The algorithm relies on two main results. First, we show how to construct greedy drawings of a fairly simple class of graphs, called triangulated binary cactuses. Second, we show that every triangulation can be spanned ..."
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Cited by 24 (3 self)
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We show an algorithm to construct a greedy drawing of every given triangulation. The algorithm relies on two main results. First, we show how to construct greedy drawings of a fairly simple class of graphs, called triangulated binary cactuses. Second, we show that every triangulation can be spanned by a triangulated binary cactus. Further, we discuss how to extend our techniques in order to prove that every triconnected planar graph admits a greedy drawing. Such a result, which proves a conjecture by Papadimitriou and Ratajczak, was independently shown by Leighton and Moitra.
ThreeDimensional Grid Drawings of Graphs
, 1998
"... . A threedimensional grid drawing of a graph G is a placement of the vertices at distinct integer points so that the straightline segments representing the edges of G are pairwise noncrossing. It is shown that for any fixed r 2, every rcolorable graph of n vertices has a threedimensional grid d ..."
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Cited by 24 (0 self)
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. A threedimensional grid drawing of a graph G is a placement of the vertices at distinct integer points so that the straightline segments representing the edges of G are pairwise noncrossing. It is shown that for any fixed r 2, every rcolorable graph of n vertices has a threedimensional grid drawing that fits into a box of volume O(n 2 ). The order of magnitude of this bound cannot be improved. 1 Introduction In a grid drawing of a graph, the vertices are represented by distinct points with integer coordinates and the edges are represented by straightline segments connecting the corresponding pairs of points. Grid drawings in the plane have a vast literature [BE]. In particular, it is known that every planar graph of n vertices has a twodimensional grid drawing that fits into a rectangle of area O(n 2 ), and this bound is asymptotically tight [FP],[S]. The possibility of threedimensional representations of graphs was suggested by software engineers [MR]. The analysis of...
Lineartime succinct encodings of planar graphs via canonical orderings
 SIAM Journal on Discrete Mathematics
, 1999
"... Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, rough ..."
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Cited by 23 (6 self)
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Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 22 (10 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 22 (4 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
Compact floorplanning via orderly spanning trees
, 2003
"... Floorplanning 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 floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is n ..."
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Cited by 22 (2 self)
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Floorplanning 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 floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floorplans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floorplan area in a rectangle of size (n − 1) ×⌊(2n + 1)/3⌋. Lower bounds on the worstcase area for floorplanning any plane triangulation are also provided in the paper.