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Drawing Planar Graphs Using the Canonical Ordering
 ALGORITHMICA
, 1996
"... We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for m ..."
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Cited by 65 (0 self)
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We introduce a new method to optimize the required area, minimum angle and number of bends of planar drawings of graphs on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear time and space algorithms can be designed for many graph drawing problems.  Every triconnected planar graph G can be drawn convexly with straight lines on an (2n \Gamma 4) \Theta (n \Gamma 2) grid, where n is the number of vertices.  Every triconnected planar graph with maximum degree four can be drawn orthogonally on an n \Theta n grid with at most d 3n 2 e + 4, and if n ? 6 then every edge has at most two bends.  Every 3planar graph G can be drawn with at most b n 2 c + 1 bends on an b n 2 c \Theta b n 2 c grid.  Every triconnected planar graph G can be drawn planar on an (2n \Gamma 6) \Theta (3n \Gamma 9) grid with minimum angle larger than 2 d radians and at most 5n \Gamma 15 bends, with d the maximum d...
On Finding the Rectangular Duals of Planar Triangular Graphs
 SIAM J. Comput
, 1993
"... We present a new linear time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previous known algorithm. The coordinates of the rectangular dual constructed by our algorithm are integers and have pure combinatorial meaning. This allow ..."
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Cited by 23 (3 self)
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We present a new linear time algorithm for finding rectangular duals of planar triangular graphs. The algorithm is conceptually simpler than the previous known algorithm. The coordinates of the rectangular dual constructed by our algorithm are integers and have pure combinatorial meaning. This allows us to discuss the heuristics for minimizing the size of the rectangular duals. Key words: Algorithm, Planar graph, Rectangular dual. AMS(MOS) subject classifications: 05C05, 05C38, 68Q25, 68R10. 1. Introduction Let R be a rectangle. A rectangular subdivision system of R is a partition of R into a set \Phi = fR 1 ; R 2 ; : : : ; R n g of nonintersecting smaller rectangles such that no four rectangles in \Phi meet at the same point. A rectangular dual of a graph G = (V; E) is a rectangular subdivision system \Phi and a onetoone correspondence f : V ! \Phi such that two vertices u and v are adjacent in G if and only if their corresponding rectangles f(u) and f(v) share a common boundar...
Drawings of planar graphs with few slopes and segments
 Computational Geometry Theory and Applications 38:194–212
, 2005
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2 ..."
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Cited by 18 (5 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2n segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Compact floorplanning via orderly spanning trees
 Journal of Algorithms
"... 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 no ..."
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Cited by 14 (1 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) × ⌊ ⌋
Two Algorithms for Finding Rectangular Duals of Planar Graphs
, 1992
"... We present two lineartime algorithms for computing a regular edge labeling of 4connected planar triangular graphs. This labeling is used to compute in linear time a rectangular dual of this class of planar graphs. The two algorithms are based on totally different frameworks, and both are concep ..."
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Cited by 11 (3 self)
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We present two lineartime algorithms for computing a regular edge labeling of 4connected planar triangular graphs. This labeling is used to compute in linear time a rectangular dual of this class of planar graphs. The two algorithms are based on totally different frameworks, and both are conceptually simpler than the previous known algorithm and are of independent interests. The first algorithm is based on edge contraction. The second algorithm is based on the canonical ordering. This ordering can also be used to compute more compact visibility representations for this class of planar graphs.
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 10 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
Fast Floorplanning For Effective Prediction And Construction
, 2001
"... Floorplanning is a crucial phase in VLSI Physical Design. The subsequent placement and routing of the cells/modules are coupled very closely with the quality of the floorplan. A widely used technique for floorplanning is Simulated Annealing. It gives very good floorplanning results but has major lim ..."
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Cited by 9 (0 self)
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Floorplanning is a crucial phase in VLSI Physical Design. The subsequent placement and routing of the cells/modules are coupled very closely with the quality of the floorplan. A widely used technique for floorplanning is Simulated Annealing. It gives very good floorplanning results but has major limitation in terms of running time. For more than tens of modules Simulated Annealing is not practical. Floorplanning forms the core of many synthesis applications. Designers need faster prediction of system metrics to quickly evaluate the effects of design changes. Early prediction of metrics is imperative for estimating timing and routability. In this work we propose a constructive technique for predicting floorplan metrics. We show how to modify the existing topdown partitioning based floorplanning to obtain a fast and accurate floorplan prediction. The prediction gets better as number of modules and flexibility in their shapes increases. We also explore the applicability of traditional Sizing Theorem when combining two modules based on their sizes and interconnecting wirelength. Experimental results show that our prediction algorithm can predict the area/length cost function normally within 510% of the results obtained by Simulated Annealing and is, on the average, thousand times faster.
A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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Cited by 7 (0 self)
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
ACG  Adjacent Constraint Graph for General Floorplans
, 2004
"... ACG (Adjacent Constraint Graph) is invented as a general floorplan representation. It has advantages of both adjacency graph and constraint graph of a floorplan: edges in an ACG are between modules close to each other, thus the physical distance of two modules can be measured directly in the graph; ..."
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Cited by 7 (3 self)
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ACG (Adjacent Constraint Graph) is invented as a general floorplan representation. It has advantages of both adjacency graph and constraint graph of a floorplan: edges in an ACG are between modules close to each other, thus the physical distance of two modules can be measured directly in the graph; since an ACG is a constraint graph, the floorplan area and module positions can be simply found by longest path computations. A natural combination of horizontal and vertical relations within one graph renders a beautiful data structure with full symmetry. The direct correspondence between geometrical positions of modules and ACG structures also makes it easy to incrementally change a floorplan and evaluate the result. Experimental results show the superiority of this representation.
On Touching Triangle Graphs
"... Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles ..."
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Cited by 5 (3 self)
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Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles representations is not closed under minors, making characterization difficult. We do show that pairs of vertices can only have a small common neighborhood, and we present a complete characterization of the subclass of biconnected graphs that can be represented as triangulations of some polygon.