M. Patrignani and F. Vargiu, 3DCube: a tool for three dimensional graph drawing. In G. Di Battista, ed., Proc. 5th International Symp. on Graph Drawing (GD '97), vol. 1353 of Lecture Notes in Comput. Sci., pp. 284-290, Springer, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....graph drawing. Proposed models for three dimensional graph drawing include straightline drawings [14,23,29] convex drawings [12,19] spline curve drawings [24] multilevel drawings of clustered graphs [18] visibility representations [2,10] and of interest in this paper orthogonal drawings [4,6,13,16,20 22,26,28,31,33,38 40]; here the edges of the graph are drawn as polygonal chains composed of axis parallel segments. This style of drawing has applications in three dimensional VLSI circuit design (see for example [1,34] The three dimensional orthogonal grM consists of grid points in three dimensional space with ....

M. Patrignani, F. Vargiu, 3DCube: a tool for three dimensional graph drawing, in: G. Di Battista (Ed.), Proc. 5th Internat. Symp. on Graph Drawing (GD'97), Lecture Notes in Computer Science, Vol. 1353, Springer, Berlin, 1998, pp. 284-290.

....v (respectively, I v ) port at a vertex v is said to be extremal if v has maximum (minimum) I coordinate taken over all vertices. Clearly, 3 D orthogonal drawings can only exist for graphs with maximum degree at most six. 3 D orthogonal drawings of maximum degree six graphs have been studied in [3, 4, 6, 10 13, 17, 19, 21, 22, 33, 35 37]. By representing a vertex by a grid box, 3 D orthogonal drawings of arbitrary degree graphs have also been considered; see for example [5, 8, 21] 3 D graph drawing has applications in VLSI circuit design [1, 2, 18, 23, 26] and software engineering [15, 16, 24, 25] for example. Note that there is ....

.... 3 1 3 3 3 3 3 21 2 3 1 2 1 3 6 2 2 3 11 14 23 8 19 2 1 5 1 3 6 17 7 12 2 3 2 1 35 14 25 Upper bounds: A number of algorithms have been proposed for 3 D orthogonal graph drawing [3, 6, 9 13, 17, 21, 22, 33, 35 37]. We now summarise the best known upper bounds for the number of bends in 3 D orthogonal drawings. The 3 BENDS algorithm of Eades et al. 13] and the INCREMENTAL algorithm of Papakostas and Tollis [21] both produce 3 bend drawings of multigraphs with maximum degree six. As discussed above ....

M. PATRIGNANI AND F. VARGIU, 3DCube: a tool for three dimensional graph drawing. In G. DI BATTISTA, ed., Proc. 5th International Symp. on Graph Drawing (GD '97), vol. 1353 of Lecture Notes in Comput. Sci., pp. 284--290, Springer, 1998.

....For our purposes LayoutShow falls short in that its design does not allow for easy extension (addition of new algorithms or models) and provides insufficient separation of concerns between layout algorithms and methods of termination. It is also limited to 2D. Related work 5 Patrignani and Vargiu [30] developed 3DCube , a tool designed for very similar purposes to ours. However, the software is not readily available and is designed primarily for orthogonal or grid based layouts. Many other implementations of experimental tools exist; 33] provides an overview of some of the better known ....

Maurizio Patrignani and Francesco Vargiu. 3DCube: A tool for three dimensional graph drawing. In Giuseppe Di Battista, editor, Graph Drawing: 5th International Symposium, GD '97, Proceedings, number 1353 in Lecture Notes in Computer Science, pages 284--290, Rome, Italy, September 1997. Springer-Verlag.

....terms vertex and edge also refer to their representation in a drawing. At a vertex v, the six directions the edges incident with v can use are called ports. Clearly, orthogonal drawings can only exist for graphs with maximum degree six. 3 D orthogonal graph drawings have been studied in [1 5, 7, 9, 10, 13, 14]. By representing a vertex by a grid box, 3 D orthogonal drawings of arbitrary degree graphs have also been considered (see [14] The bounding box of a given drawing is the minimum axis parallel box which encloses the drawing. The following aesthetic criteria are the most commonly proposed ....

M. Patrignani and F. Vargiu. 3DCube: a tool for three dimensional graph drawing. In G. Di Battista, editor, Proc. Graph Drawing: 5th International Symp. (GD'97), volume 1353 of Lecture Notes in Comput. Sci., pages 284-290, Springer, 1998.

....graph drawing. Proposed models for 3 dimensional graph drawing include straight line drawings [14, 24, 32] convex drawings [12, 20] spline curve drawings [25] multilevel drawings of clustered graphs [19] visibility representations [2, 10] and of interest in this paper orthogonal drawings [4, 8, 13, 17, 21 23, 27, 29, 34, 36, 41 43]; here the edges of the graph are drawn as polygonal chains composed of axis parallel segments. This style of drawing has applications in 3 dimensional VLSI circuit design (see for example [1, 37] The 3 dimensional orthogonal grid consists of grid points in 3 dimensional space with integer ....

M. Patrignani and F. Vargiu, 3DCube: a tool for three dimensional graph drawing. In [15], pp. 284-290.

....graph drawing. Proposed models for 3 dimensional graph drawing include straightline drawings [13, 23, 30] convex drawings [11, 19] spline curve drawings [24] multilevel drawings of clustered graphs [18] visibility representations [2, 8] and of interest in this paper orthogonal drawings [3, 5, 12, 16, 20 22, 26, 28, 32, 33, 39 41]; here the edges of the graph are drawn as polygonal chains composed of axis parallel segments. This style of drawing has applications in 3 dimensional VLSI circuit design (see for example [1, 34] The 3 dimensional orthogonal grid consists of grid points in 3 dimensional space with integer ....

M. Patrignani and F. Vargiu, 3DCube: a tool for three dimensional graph drawing. In

....I (respectively, I Gamma ) port at a vertex v is said to be extremal if v has maximum (minimum) I coordinate taken over all vertices. Clearly, orthogonal drawings can only exist for graphs with maximum degree six. 3 D orthogonal drawings of maximum degree six graphs have been studied in [7, 9 11, 15, 18, 19, 28]. By representing a vertex by a grid box, 3 D orthogonal drawings of arbitrary degree graphs have also been considered [4 6, 12, 18, 29, 30] 3 D graph visualisation has applications in VLSI circuit design [1, 2, 17, 20, 23] and software engineering [13, 14, 21, 22] for example. Drawings with ....

....degree Delta with lower bounds on the number of bends for each member of the class. Table 1 summarises the lower bounds proved in this paper for the total number of bends in 3 D orthogonal graph drawings. Upper bounds: A number of algorithms have been proposed for 3 D orthogonal graph drawing [5, 7, 9, 11, 15, 19, 28, 31] which explore the apparent tradeoff between the number of bends and the bounding box volume (see [31] We now summarise the known upper bounds for the number of bends in 3 D orthogonal drawings. The 3 Bends algorithm of Eades et al. 11] and the Incremental algorithm of Papakostas and Tollis ....

M. Patrignani and F. Vargiu, 3DCube: a tool for three dimensional graph drawing. In

....the proximity region is defined by the parameter fi. fi drawings were first introduced by Kirkpatrick and Radke [10, 17] in the computational morphology context. Recently, due to the availability of low cost workstations and of applications that require 3 dimensional representations of graphs [3, 9, 15, 18, 19], 3 dimensional proximity drawings are considered. Nevertheless, only preliminary results are available [11, 16] In the 3 dimensional space the definition of proximity is an obvious extension of that for the 2 dimensional case. In particular, we refer to Gabriel proximity region as a Gabriel ....

M. Patrignani and F. Vargiu. 3DCube: A tool for three dimensional graph drawing. In G. Di Battista, editor, Graph Drawing (Proc. GD '97), volume 1353 of Lecture Notes Comput. Sci., pages 284--290. Springer-Verlag, 1998.

....inserted. The volume of this drawing is 8 8 8 = 512 # 1.5n 3 , where n = 7. Observe that out of K 7 s 21 edges, there is one edge with no bends, 12 edges require two bends each, and the remaining eight edges are routed with three bends each. The algorithm has been implemented within 3DCube [29] and we present preliminary experimental results in the Conclusions. Papakostas and Tollis, Incremental 3 D Drawing , JGAA, 3(4) 81 115 (1999) 98 4 A Model for Vertices of High Degree In this section we describe a model to support high degree interactive three dimensional orthogonal graph ....

....the same upper bound for the number of bends per edge. It is also important to underline that our algorithm requires linear time to produce a 3 D orthogonal drawing of an n vertex graph, whereas the algorithm in [13] runs in O(n 3 2 ) time. The algorithm has been implemented within 3DCube [29] along with the algorithms of [12, 13, 21] and a new algorithm introduced in [29] The discussion here is based on the preliminary experimental results of the implementation of our algorithm sent to us by Maurizio Patrignani [30] He applied our algorithm on 1920 random, simple, connected graphs ....

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M. Patrignani and F. Vargiu, 3DCube: A Tool for Three Dimensional Graph Drawing, Proc. of GD '97, Lecture Notes in Comp. Sci. 1353, Springer-Verlag, 1997, pp. 284-290.

....involve two adjacent edges. We propose this problem as an interesting combinatorial problem per se, although it firstly arose in a practical application, when studying a new approach to three dimensional orthogonal graph drawing, based on generating the final drawing through a sequence of steps [4, 1]. Starting from a degenerate drawing, in which all vertices and edges are overlapped on the same point, at each step the drawing splits into two pieces and finds a structure more similar to its final version. The observation that a cut through non adjacent edges yields a more efficient (in ....

Maurizio Patrignani and Francesco Vargiu. 3DCube: A tool for three dimensional graph drawing. In G. Di Battista, editor, Graph Drawing (Proc. GD '97), volume 1353 of Lecture Notes Comput. Sci., pages 284--290. Springer-Verlag, 1998.

....not involve two adjacent edges. We propose this problem as an interesting combinatorial problem per se, although it firstly arose in a practical application, when studying anewapproach to three dimensional orthogonal graph drawing, based on generating the final drawing through a sequence of steps [4, 1]. Starting from a degenerate drawing, in which all vertices and edges are overlapped on the same point, at each step the drawing splits into two pieces and finds a structure more similar to its final version. The observation that a cut through non adjacent edges yields a more efficient (in ....

Maurizio Patrignani and Francesco Vargiu. 3DCube: A tool for three dimensional graph drawing. In G. Di Battista, editor, Graph Drawing(i oc. GD '97), volume 1353 of Lecture Notes Comput. Sci., pages 284--290. Springer-Verlag, 1998.

....5 Experimental Results We have implemented and performed an experimental comparison of algorithms Compact, Interactive, Kolmogorov, Reduce Forks, Slices, and Three Bends with a large test suite of graphs. The experiments have been performed on a Sun SPARC station Ultra 1 by using the 3DCube [24] system. All the algorithms have been implemented in C . The test suite was composed of 1900 randomly generated graphs having from 6 to 100 vertices, 20 graphs for each value of vertex cardinality. All graphs were connected, with maximum vertex degree 6, without multi edges and selfloops. The ....

M. Patrignani and F. Vargiu. 3DCube: A tool for three dimensional graph drawing. In G. Di Battista, editor, Graph Drawing (Proc. GD '97), volume 1353 of Lecture Notes Comput. Sci., pages 284--290. Springer-Verlag, 1997.

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M. Patrignani and F. Vargiu, 3DCube: a tool for three dimensional graph drawing. In G. Di Battista, ed., Proc. 5th International Symp. on Graph Drawing (GD '97), vol. 1353 of Lecture Notes in Comput. Sci., pp. 284-290, Springer, 1998.

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