I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. Proc. 3rd Intl. Symp. Graph Drawing (GD '95), Springer LNCS 1027, pp. 162--165, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Introduction The recent interest in three dimensional graph drawing has been motivating studies on how to extend two dimensional techniques to 3D space. Work in this direction includes extensions of simulated annealing techniques, spring embedder techniques, and incremental techniques (see e.g. [5, 9, 15, 22, 23, 28]) However, while a rich body of literature is devoted to three dimensional orthogonal drawings (see e.g. 3, 8, 14, 16, 17, 21, 23, 29] little is known on the challenging task of extending to 3D the well known topology shapemetrics approach [26] The topology shape metrics approach for ....

I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes Comput. Sci., pages 162-165. Springer-Verlag, 1996.

....have positive volume. We are interested in polyline drawings with small volume. The volume of straight line drawings has been widely studied [21, 26, 31, 37, 41, 42, 51, 103, 107, 125] Three dimensional graph drawings in which the vertices are allowed real coordinates have also been studied [23, 28, 29, 34, 43, 59, 78 81, 93, 102]. Aesthetic criteria besides volume that have been considered include symmetry [78 81] aspect ratio [29, 59] angular resolution [29, 59] edge separation [29, 59] and convexity [28, 29, 43] Table 2 summarises the best known upper bounds on the volume of polyline drawings, including those ....

I. F. CRUZ AND J. P. TWAROG, 3D graph drawing with simulated annealing. In [22], pp. 162--165.

....so that 2D drawings have positive volume. We are interested in 3D drawings with small volume. The volume of 3D drawings has been widely studied [3, 6, 9, 11, 14, 16, 19, 36 38, 40] Three dimensional graph drawings in which the vertices are allowed real coordinates have also been studied [5, 7, 8, 10, 17, 21, 26 29, 32, 35]. Aesthetic criteria besides volume which have been considered include symmetry [26 29] aspect # Research supported by NSERC and FCAR. Completed at the Departament de Matem atica Aplicada II, Universitat Polit ecnica de Catalunya, Barcelona, Spain. ratio [8, 21] angular resolution [8, 21] ....

I. F. CRUZ AND J. P. TWAROG, 3D graph drawing with simulated annealing. In [4], pp. 162--165.

....to the graph theoretic distance between the two 11 vertices. Fruchterman and Reingold [Fruchterman and Reingold, 1991] use a slightly different heuristic for the spring embedder to provide a faster algorithm, while others have investigated simulated annealing [Davidson and Harel, 1996, Cruz and Twarog, 1996] and genetic algorithms [Kosak and Marks, 1991, Branke, 1996, Rosete, 1997] for a more general energy function minimisation strategy. Sugiyama and Misue [Sugiyama and Misue, 1995] look at applying a magnetic field in a physical simulation, where forces act on each edge in a graph in order to ....

Cruz, I. F. and Twarog, J. P. (1996). 3-d graph drawing with simulated annealing. In Lecture Notes in Computer Science (Proc. GD '95), 1027, pages 162---165. Springer-Verlag.

....[11] proved that series parallel graphs with maximum degree three have three dimensional drawings with linear volume. Note that three dimensional drawings with the vertices having real coordinates have been studied by Bru and Frick [4] Chilakamarri et al. 6] Chrobak et al. 7] Cruz and Twarog [9], Eades and Garvan [15] Garg et al. 18] Hong [22] Hong and Eades [23, 24] Hong et al. 25] Monien et al. 27] and Ostry [29] Aesthetic criteria besides volume which have been considered include symmetry [22 25] aspect ratio [7, 18] angular resolution [7, 18] edge separation [7, 18] ....

I. F. CRUZ AND J. P. TWAROG, 3D graph drawing with simulated annealing. In [3], pp. 162--165.

....drawings with O(n ) volume, and that this bound can be improved to O(n ) and O(n) in certain special cases. Note that three dimensional drawings with the vertices having real coordinates have been studied by Bru and Frick [4] Chilakamarri et al. 6] Chrobak et al. 7] Cruz and Twarog [9], Eades and Garvan [13] Garg et al. 16] Hong [19] Hong and Eades [20, 21] Hong et al. 22] Monien et al. 24] and Ostry [26] Aesthetic criteria besides volume which have been considered include symmetry [19 22] aspect ratio [7, 16] angular resolution [7, 16] edgeseparation [7, 16] and ....

I. F. CRUZ AND J. P. TWAROG, 3D graph drawing with simulated annealing. In [3], pp. 162--165.

.... of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5] Drawing general graphs in 3D using the force directed approach is studied by Fruchterman and Reingold [19] and Monien, Ramme, and Salmen [31] Other recent 3D drawing algorithms include Bru and Frick [4] Cruz and Twarog [7], and Ostry [33] In the context of orthogonal drawings, 3D point drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34] 3D orthogonal box drawings were studied by Biedl [1] and multi dimensional orthogonal graph drawings are presented by Wood ....

I. F. Cruz and J. P. Twarog. 3d graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes Computer Science, pages 162--165, 1996.

.... for the representation of graphs in the plane, and the most sophisticated of them also allow some three dimensional representation, either as an additional presentation feature of basically 2D results (see GMB [13] and PLUM [17] or as the result of an actual 3D drawing algorithm (see 3DSA [4], COMAIDE [6] the ffGraph library [9] GEM 3D [2] GIOTTO3D [11] GOLD [10] GOVE [18] and PARSA [14] The tool we describe in this paper offers to the user a general frame for the representation of graphs entirely devoted to 3D drawing algorithms, and especially to their evaluation, ....

I.F. Cruz and J.P. Twarog. 3D Graph Drawing with Simulated Annealing. Proc. GD '95, LNCS 1027, pp. 162-165, Springer-Verlag, 1995.

.... of graphs, for example the algorithms of Cohen, Eades, Lin, and Ruskey [5] Drawing general graphs in 3D using the force directed approach is studied by Fruchterman and Reingold [19] and Monien, Ramme, and Salmen [31] Other recent 3D drawing algorithms include Bru and Frick [4] Cruz and Twarog [7], and Ostry [33] In the context of orthogonal drawings, 3D point drawing algorithms were developed by Eades, Symvonis, and Whitesides [15] and Papakostas and Tollis [34] 3D orthogonal box drawings were studied by Biedl [1] and multi dimensional orthogonal graph drawings are presented by Wood ....

I. F. Cruz and J. P. Twarog. 3d graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes Computer Science, pages 162-165, 1996.

....formally capture the quality of a layout, defined by local criteria such as uniform vertex distribution, uniform edge lengths, and su#cient distance between vertices and those edges they are not incident to. The corresponding terms of our objective function (see Table 1) which is very similar to [5], are called potentials, since the objective function itself can be interpreted as an energy measuring layout distortion with respect to the above criteria. The additional potentials account for the contest graph being directed and, most of the time, disconnected. To avoid that disconnected ....

I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. F. J. Brandenburg, editor. Proc. 3rd Intl. Symp. Graph Drawing (GD '95), Lecture Notes in Computer Science vol. 1027, pages 162--165. Springer, 1996.

....It uses a physical analogy, where a graph is seen as a system of bodies with forces acting among them. These algorithms seek a configuration of the system with, possibly local, minimal energy. Force directed approaches have been exploited in 3D graph drawing to devise the algorithms presented in [4, 6, 9, 19, 26, 14, 20]. Further, the research on straight line drawings stimulated a deep investigation on theoretical bounds. Examples of bounds on the volume of a straight line drawing can be found in [5, 21] Namely, in [5] it is shown that a graph can be drawn in an n 2n 2n volume, which is asymptotically ....

I. F. Cruz and J. P. Twarog. 3d graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes Comput. Sci., pages 162--165. Springer-Verlag, 1996.

....knowledge about the structural properties of the graph, because layout rules have to be specified in terms of structural decomposition. The second class of declarative layout techniques consists of universal stochastic global optimization techniques, in particular it comprises simulated annealing [DH89, DH96, CT95, BHR95] and genetic algorithms [KMS94, Mas92, FKV96] These methods do not require to specify any knowledge about graph structures. Instead, the layout objectives are defined in terms of global visual criteria such as uniform edge length or minimal number of edge crossings. The drawback of declarative ....

....above cases suggest that the ISOM works well in 3D space, a critical assessment is in place. It is not really clear whether the optimization of the layout objectives derived for 2D space also always leads to good 3D drawings. The same has been observed by other authors in their work on 3D layouts [CT95]. In fact, it is not all that well understood what the aesthetic criteria 6 The boxes framing the 3D layouts are artifacts generated by this implementation to provide a better spatial orientation. The small solid flying cube in the pictures is an indicator for the location of the current input ....

I.F. Cruz and J.P. Twarog. 3D graph drawing with simulated annealing. In F.J. Brandenburg, editor, Graph Drawing (GD'95), pages 162--165, Passau, Germany, September 1995.

....research. The original approach to graph layout by SA in [6] uses a cost function which sums over (1) the number of edge crossings, 2) the node distribution, 3) the edge length, 4) the node edge distances, and (5) the borderlines, but other cost functions are also possible. For example, in [5] a generalization of the SA approach to three dimensional layout is presented. Recently, genetic algorithms which can be regarded as another generic stochastic search and global optimization method have also been applied to graph layout [21, 23, 10] However, their performance is ....

I. Cruz and J. Twarog. 3D graph drawing with simulated annealing. In [3], pages 162--165.

....In the case of [5] the system is based on the spring embedder paradigm [19] Spring embedders use a physical model based on vertices treated as currents exerting a repulsive force, while edges are modeled as forces attracting the vertices they combine. Simulated Annealing has also been used [8] to produce straight line 3 D drawings of graphs. The idea here is that there is a predefined cost associated with the current 3 D drawing of the graph, and the system moves to drawings of lower costs (while sometimes accepting higher cost drawings if they look nice ) until no further ....

....located on grid points, and no two edges cross. In the same paper, a technique to convert an orthogonal 2 D drawing of area HV to a 3 D straight line drawing of volume # # H## # H#V is also presented. Naturally, orthogonal drawing in three dimensional space has also received attention recently [4, 5, 8, 12, 13, 17, 32]. A 3 D orthogonal drawing typically has the following properties: Vertices are points with integer coordinates in three dimensional space. Each edge is a polyline sequence of consecutive straight line segments; Papakostas and Tollis, Incremental 3 D Drawing , JGAA, 3(4) 81 115 (1999) 83 ....

I. Cruz and J. Twarog, 3D Graph Drawing with Simulated Annealing, Proc. of Workshop GD '95, Lecture Notes in Comp. Sci. 1027, Springer-Verlag, 1995, pp. 162-165.

....may appear to cross on a 2D computer screen. The advantage, however, of the 3D representation is that through our preattentive depth perception we will automatically perceive the display correctly. There has been previous research into 3D network displays; for some of the early papers see [8] [6]. We build on this work and differentiate our results in three important ways. For geographic context we concentrate on 3D networks layouts that maintain geographic context. As we indicated above, for many information spaces involving spatial locations, maintaining the geography in the ....

I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. In GD'95 Proceedings Lecture Notes in Computer Science, volume 1027, pages 162-- 165. Springer-Verlag, 1996.

....In the case of [5] the system is based on the spring embedder paradigm [19] Spring embedders use a physical model based on vertices treated as currents exerting a repulsive force, while edges are modeled as forces attracting the vertices they combine. Simulated Annealing has also been used [8] to produce straight line 3 D drawings of graphs. The idea here is that there is a predefined cost associated with the current 3 D drawing of the graph, and the system moves to drawings of lower costs (while sometimes accepting higher cost drawings if they look nice ) until no further ....

I. Cruz and J. Twarog, 3D Graph Drawing with Simulated Annealing, Proc. of Workshop GD '95, Lecture Notes in Comp. Sci. 1027, Springer-Verlag, 1995, pp. 162-165.

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I. F. Cruz and J. P. Twarog. 3D graph drawing with simulated annealing. Proc. 3rd Intl. Symp. Graph Drawing (GD '95), Springer LNCS 1027, pp. 162--165, 1996.

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I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing. In [41], pp. 162-165.

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I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing. In

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I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing. In

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I. F. CRUZ AND J. P. TWAROG, 3D graph drawing with simulated annealing. In [12], pp. 162--165.

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I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing, in Brandenburg

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I. F. Cruz and J. P. Twarog. 3d graph drawing with simulated annealing. Graph Drawing (Proc. GD '95). LNCS, 1027:162--165, Springer-Verlag. 1996.

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I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing. In [41], pp. 162-165.

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I. F. Cruz and J. P. Twarog. 3d graph drawing with simulated annealing. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes Computer Science, pages 162--165, 1996.

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