## Three-Dimensional Orthogonal Graph Drawing with Optimal Volume

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Citations: | 23 - 9 self |

### BibTeX

@MISC{Biedl_three-dimensionalorthogonal,

author = {Therese Biedl and Torsten Thiele and David R. Wood},

title = {Three-Dimensional Orthogonal Graph Drawing with Optimal Volume},

year = {}

}

### Years of Citing Articles

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### Abstract

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid which only possibly intersect at common endpoints. In this paper, we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where vertices have bounded aspect ratio, (2) drawings where the surface of vertices is proportional to their degree, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that match the lower bounds in all scenarios within an order of magnitude.

### Citations

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349 |
Graph Drawing: Algorithms for the Visualization of Graphs
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ...dimensional, lower bounds 1 Introduction Graph drawing is a field with a wide range of applications, for example in network visualisation, data base design and telecommunications. See the recent book =-=[10]-=- for an overview of techniques in graph drawing. Orthogonal graph drawing, where edges are routed along a rectangular grid, is a popular drawing style with applications in data flow diagrams, entity r... |

322 | P.: Ramanujan graphs
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(Show Context)
Citation Context ...V we have e(S, T ) # d|S||T | n - #(G) p |S||T | . This lemma suggests to look for graphs G with small #(G). Fortunately, such graphs (called Ramanujan graphs) have already been constructed. Lemma 3 (=-=[19, 20]-=-). If p #= q are primes, p # 1 mod 4, q # 1 mod 4, then there exists a simple n-vertex graph G p,q = (V, E) with the following properties: . G p,q is d-regular with d = p + 1. . q(q - 1)/2 # n # q(q -... |

217 |
Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators
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(Show Context)
Citation Context ...V we have e(S, T ) # d|S||T | n - #(G) p |S||T | . This lemma suggests to look for graphs G with small #(G). Fortunately, such graphs (called Ramanujan graphs) have already been constructed. Lemma 3 (=-=[19, 20]-=-). If p #= q are primes, p # 1 mod 4, q # 1 mod 4, then there exists a simple n-vertex graph G p,q = (V, E) with the following properties: . G p,q is d-regular with d = p + 1. . q(q - 1)/2 # n # q(q -... |

90 | Expanders that beat the eigenvalue bound: Explicit construction and applications, Preprint available from the authors, preliminary version presented at 25th STOC
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- 1993
(Show Context)
Citation Context ...ing this lemma, we need some background. For a graph G, denote by #(G) the second largest eigenvalue of its adjacency matrix. The following well-known inequality (see for example [3, pp. 119-125] and =-=[24]-=-) relates #(G) to the cut property we are interested in. Lemma 2. Let G = (V, E) be a d-regular n-vertex graph with second largest eigenvalue #(G). Then, for all disjoint sets S, T # V we have e(S, T ... |

44 |
Three-dimensional circuit layouts
- Leighton, Rosenberg
- 1986
(Show Context)
Citation Context ...or graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6, 8, 21, 25, 26]. Related research in three-dimensional VLSI includes =-=[1, 2, 18, 22, 23]-=-. From now on, we use the term drawing to mean a three-dimensional orthogonal boxdrawing. Furthermore, the graph-theoretic terms `vertex' and `edge' also refer to their representation in a drawing. Th... |

34 |
Three-dimensional VLSI: a case study
- Rosenberg
- 1983
(Show Context)
Citation Context ...or graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6, 8, 21, 25, 26]. Related research in three-dimensional VLSI includes =-=[1, 2, 18, 22, 23]-=-. From now on, we use the term drawing to mean a three-dimensional orthogonal boxdrawing. Furthermore, the graph-theoretic terms `vertex' and `edge' also refer to their representation in a drawing. Th... |

33 | Three-dimensional orthogonal graph drawing
- Wood
- 2000
(Show Context)
Citation Context ...x, e.g. point, line-segment, or cube, is called an orthogonal shape-drawing for each particular shape. Initial research in orthogonal drawing was mostly concerned with point-drawings; see for example =-=[9, 11, 12, 13, 21, 28, 29]-=-. However, three-dimensional orthogonal pointdrawings can only exist for graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6... |

28 | Area-ecient static and incremental graph drawings
- Biedl, Kaufmann
- 1997
(Show Context)
Citation Context ...position by Biedl [5] and Wood [25], respectively. The lifting half-edges technique developed by Biedl [5] generates drawings of simple graphs starting with a two-dimensional general position drawing =-=[7]-=- (possibly with overlapping edges). The edge routes are partitioned into sub-drawings each consisting of X -segments or Y -segments. Each sub-drawing is then assigned its own Z -plane, vertices are ex... |

25 | Three dimensional orthogonal graph drawing algorithms
- EADES, SYMVONIS, et al.
(Show Context)
Citation Context ...x, e.g. point, line-segment, or cube, is called an orthogonal shape-drawing for each particular shape. Initial research in orthogonal drawing was mostly concerned with point-drawings; see for example =-=[9, 11, 12, 13, 21, 28, 29]-=-. However, three-dimensional orthogonal pointdrawings can only exist for graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6... |

21 | Incremental orthogonal graph drawing in three dimensions
- Papakostas, Tollis
(Show Context)
Citation Context ...ing boxes. Hence vertices are possibly degenerate, in the sense that they may be represented by a rectangle or even a line-segment or a point. This is the approach taken in [6, 8, 25, 26], but not in =-=[21]-=-. (Enlarging vertices to remove this degeneracy increases the volume by a multiplicative constant.) An edge vw # E is represented by a sequence of contiguous segments of grid lines possibly bent at gr... |

17 | Fully dynamic 3-dimensional orthogonal graph drawing - Closson, Gartshore, et al. |

16 |
Three approaches to 3D-orthogonal box-drawings
- Biedl
(Show Context)
Citation Context ...s by pairwise non-intersecting boxes. Hence vertices are possibly degenerate, in the sense that they may be represented by a rectangle or even a line-segment or a point. This is the approach taken in =-=[6, 8, 25, 26]-=-, but not in [21]. (Enlarging vertices to remove this degeneracy increases the volume by a multiplicative constant.) An edge vw # E is represented by a sequence of contiguous segments of grid lines po... |

16 | A splitpush approach to 3d orthogonal drawing
- Battista, Patrignani, et al.
- 1998
(Show Context)
Citation Context ...x, e.g. point, line-segment, or cube, is called an orthogonal shape-drawing for each particular shape. Initial research in orthogonal drawing was mostly concerned with point-drawings; see for example =-=[9, 11, 12, 13, 21, 28, 29]-=-. However, three-dimensional orthogonal pointdrawings can only exist for graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6... |

15 | Bounds for orthogonal 3-D graph drawing
- BIEDL, SHERMER, et al.
- 1999
(Show Context)
Citation Context ...s by pairwise non-intersecting boxes. Hence vertices are possibly degenerate, in the sense that they may be represented by a rectangle or even a line-segment or a point. This is the approach taken in =-=[6, 8, 25, 26]-=-, but not in [21]. (Enlarging vertices to remove this degeneracy increases the volume by a multiplicative constant.) An edge vw # E is represented by a sequence of contiguous segments of grid lines po... |

14 |
Multilayer grid embeddings for VLSI
- AGGARWAL, KLAWE, et al.
- 1991
(Show Context)
Citation Context ...or graphs with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [6, 8, 21, 25, 26]. Related research in three-dimensional VLSI includes =-=[1, 2, 18, 22, 23]-=-. From now on, we use the term drawing to mean a three-dimensional orthogonal boxdrawing. Furthermore, the graph-theoretic terms `vertex' and `edge' also refer to their representation in a drawing. Th... |

14 | Multi-dimensional orthogonal graph drawing in the general position model
- Wood
- 1999
(Show Context)
Citation Context ...s by pairwise non-intersecting boxes. Hence vertices are possibly degenerate, in the sense that they may be represented by a rectangle or even a line-segment or a point. This is the approach taken in =-=[6, 8, 25, 26]-=-, but not in [21]. (Enlarging vertices to remove this degeneracy increases the volume by a multiplicative constant.) An edge vw # E is represented by a sequence of contiguous segments of grid lines po... |

13 |
A note on 3D orthogonal graph drawing
- Biedl, Chan
(Show Context)
Citation Context ...)+ 1)/2 ⌉)2 ≤ 12 · deg(v)+O (√ deg(v) ) . 248 T. Biedl, T. Thiele, and D. R. Wood Thus each vertex is 12-degree-restricted. By construction, there are at most six bends per edge route. Biedl and Chan =-=[6]-=- have developed a technique based on edge-colouring a certain bipartite graph to implement Steps 3.1 and 3.1 of Algorithm OPTIMAL VOLUME CUBE-DRAWING more efficiently. With this technique, the time co... |

12 |
Optimal three-dimensional VLSI layouts
- Preparata
- 1983
(Show Context)
Citation Context |

12 | Multilayer grid embeddings for VLSI, Algorithmica 6 - Aggarwal, Klawe, et al. - 1991 |

11 | Rectangle and box visibility graphs in 3D - Fekete, Meijer - 1999 |

9 | Minimising the number of bends and volume in threedimensional orthogonal graph drawings with a diagonal vertex layout, submitted. See
- WOOD
- 2001
(Show Context)
Citation Context |

8 | Bounded degree book embeddings and three-dimensional orthogonal graph drawing
- Wood
- 2002
(Show Context)
Citation Context ...not degree-restricted. They also construct drawings of Kn with O(n 3 ) volume and one bend per edge, but these drawings are degree-restricted only for graphs where all vertices have degree #(n). Wood =-=[27]-=- obtained drawings with one bend per edge and O(n 3/2 m) volume. Biedl [4] showed that# n 3 ) volume is required for Kn if only one bend per edge is allowed. A drawing is in general position if no two... |

8 | Graph Drawing: Algorithms for Geometric Representations of Graphs - Battista, Eades, et al. - 1998 |

7 | A new algorithm and open problems in three-dimensional orthogonal graph drawing - Wood - 1999 |

6 |
An optimal bound for two dimensional bin packing
- Kleitman, Krieger
- 1975
(Show Context)
Citation Context ...esent each vertex v # V by a square S v with side length 2 l p #deg(v)/2# + 1 m . 2. Position the squares {S v : v # V } in the (Z = 0)-plane with the square-packing algorithm of Kleitman and Krieger =-=[17]-=-. Note that squares may touch, and since all squares have even side length, we may assume that all corners of the squares have even coordinates. 3. For each vertex v # V , remove the top two rows from... |

5 |
The techniques of Kolmogorov and Barzdin for three dimensional orthogonal graph drawings
- Eades, Stirk, et al.
- 1996
(Show Context)
Citation Context |

5 | The techniques of Kolmogorov and Bardzin for three-dimensional orthogonal graph drawings - Eades, Stirk, et al. - 1996 |

4 | 1-bend 3-D orthogonal box-drawings: Two open problems solved
- Biedl
(Show Context)
Citation Context ...me and one bend per edge, but these drawings are degree-restricted only for graphs where all vertices have degree #(n). Wood [27] obtained drawings with one bend per edge and O(n 3/2 m) volume. Biedl =-=[4]-=- showed that# n 3 ) volume is required for Kn if only one bend per edge is allowed. A drawing is in general position if no two vertices are in a common grid plane. The algorithm of Papakostas and Toll... |

4 |
Cross-coloring: improving the technique by Kolmogorov and Barzdin
- Biedl, Chan
- 2000
(Show Context)
Citation Context ...face of a vertex v is 6 2 l p (deg(v) + 1)/2 m 2 # 12 deg(v) +O p deg(v) . Thus each vertex is 12-degree-restricted. By construction, there are at most six bends per edge route. Biedl and Chan [5] have developed a technique based on edge-colouring a certain bipartite graph to more e#ciently implement Steps 6 and 7 of Algorithm Optimal VolumesCube-Drawing. With this technique, the time complexi... |

4 |
Graph embedding on a three-dimensional model
- Hagihara, Tokura, et al.
- 1983
(Show Context)
Citation Context ...ch we can draw all graphs with n vertices and m edges such that each vertex v has aspect ratio at most r and surface at most # deg(v). The first lower bounds on the volume were due to Hagihara et al. =-=[16]-=- 1 . They show that, in the above notation, vol (n, m, 1, 1) =#313 {n# 2 , (n#/ log n) 3/2 }). In fact, in their construction the graphs are #-regular. Hence m = 1 2 n#, which allows us to restate the... |

3 | Graph embedding on a three-dimensional model. Systems-Comput.- Controls 14(6 - Hagihara, Tokura, et al. - 1983 |

3 |
Complexities of layouts in three-dimensional VLSI circuits
- Aboelaze, Wah
- 1991
(Show Context)
Citation Context ...s with maximum degree at most six. Overcoming this restriction has motivated recent interest in orthogonal box-drawings [5], [8], [21], [25], [26]. Related research in three-dimensional VLSI includes =-=[1]-=-, [2], [18], [22], and [23]. From now on, we use the term drawing to mean a three-dimensional orthogonal box-drawing. Furthermore, the graph-theoretic terms “vertex” and “edge” also refer to their rep... |

2 | 1-bend 3-D orthogonal drawings: two open problems solved - Biedl - 2000 |

2 | Rectangle and box visibility graphs - Fekete, Meijer |

1 |
An elementary proof of formulae of de la Vallée
- Fogel
- 1950
(Show Context)
Citation Context ...rimes p # x that satisfy p # 1 mod 4. A famous theorem by de la Vallee Poussin establishes that # 4,1 (x) = # 1 #(4)sx log x , where # is Euler's function, in particular #(4) = 2. See for example [1=-=5]-=- for a proof. Let c 1 , c 2 , x 0 be constants such that c 2 # c 1 and c 1 x log x # # 4,1 (x) # c 2 x log x for all x # x 0 . Let k = 2c 2 /c 1 . If x 1 # x 0 is so big that also log(k) # 1 4 log(x) ... |

1 |
An elementary proof of formulae of de la Vallée
- Fogels
- 1950
(Show Context)
Citation Context ...s p ≤ x that satisfy p ≡ 1 mod 4. A famous theorem by de la Vallée Poussin establishes that π4,1(x) = θ ( 1 ϕ(4) · x log x ) , where ϕ is Euler’s function, in particular, ϕ(4) = 2. See, for example, =-=[15]-=- for a proof. Let c1, c2, x0 be constants such that c2 ≥ c1 and c1 · xlog x ≤ π4,1(x) ≤ c2 · x log x 242 T. Biedl, T. Thiele, and D. R. Wood for all x ≥ x0. Let k = 2c2/c1. Choose x1 ≥ k · x0 so that ... |