## Optimal upward planarity testing of single-source digraphs (1998)

Venue: | SIAM Journal on Computing |

Citations: | 35 - 4 self |

### BibTeX

@ARTICLE{Bertolazzi98optimalupward,

author = {Paola Bertolazzi and Giuseppe Di Battista and Carlo Mannino and Roberto Tamassia},

title = {Optimal upward planarity testing of single-source digraphs},

journal = {SIAM Journal on Computing},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of single-source digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)-time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.

### Citations

244 | Algorithms for drawing graphs: An annotated bibliography - Battista, Eades, et al. - 1994 |

238 | Efficient planarity testing - Hopcroft, Tarjan - 1974 |

185 | Dividing a graph into triconnected components - Hopcroft, Tarjan - 1973 |

147 | Tidier Drawings of Trees - Reingold, Tilford - 1981 |

106 | An algorithm for planarity testing of graphs - Lempel, Even, et al. - 1966 |

86 | On the computational complexity of upward and rectilinear planarity testing - Garg, Tamassia - 1995 |

60 |
Algorithms for plane representations of acyclic digraphs
- DiBattista, Tamassia
- 1988
(Show Context)
Citation Context ...Lempel, Even, and Cederbaum [23] relate the planarity of biconnected undirected graphs to the upward planarity of stdigraphs. A combinatorial characterization of upward planar digraphs is provided in =-=[21, 9]-=-; namely, a digraph is upward planar if and only if it is a subgraph of a planar st-digraph. Di Battista, Tamassia, and Tollis [9, 12] give algorithms for constructing upward planar drawings of st-dig... |

59 |
Approximate and exact parallel scheduling with applications to list, tree, and graph problems
- Cole, Vishkin
- 1986
(Show Context)
Citation Context ...an be performed sequentially in O(n) time with straightforward methods. Regarding the parallel complexity, steps 1 and 3 take O(log n) time on a CREW PRAM with n/ log n processors, using list-ranking =-=[5]-=-. Step 2 can be executed by computing a spanning forest of the face-sink graph, which takes O(log n) time on a CRCW PRAM with n #(n)/ log n processors [5], and thus determines the parallel time comple... |

46 |
Tollis, Algorithms for drawing graphs: an annotated bibliography, Computational Geometry 4
- Battista, Eades, et al.
- 1994
(Show Context)
Citation Context ...hey can be compactly represented. Fig. 1. Examples of planar acyclic digraphs: (a) upward planar; (b) not upward planar. A survey on algorithms for planarity testing and graph drawing can be found in =-=[7]-=-. Previous work on upward planarity is as follows. Combinatorial results on upward planarity for covering digraphs of lattices were first given in [22, 26]. Further results on the interplay between up... |

44 |
Area requirement and symmetry display of planar upward drawings
- Battista, Tamassia, et al.
- 1992
(Show Context)
Citation Context ...atorial characterization of upward planar digraphs is provided in [21, 9]; namely, a digraph is upward planar if and only if it is a subgraph of a planar st-digraph. Di Battista, Tamassia, and Tollis =-=[9, 12]-=- give algorithms for constructing upward planar drawings of st-digraphs and investigate area bounds and symmetry display. Tamassia and Vitter [32] show that the above drawing algorithms can be e#cient... |

33 |
A note on optimal area algorithms for upward drawings of binary trees
- Crescenzi, Battista, et al.
- 1992
(Show Context)
Citation Context ...bounds and symmetry display. Tamassia and Vitter [32] show that the above drawing algorithms can be e#ciently parallelized. Upward planar drawings of trees and series-parallel digraphs are studied in =-=[29, 31, 6, 13, 15]-=- and [1, 2], respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial-time algorithm is given for testing t... |

31 | Planarity testing in parallel - Ramachandran, Reif - 1994 |

29 | Finding triconnected components by local replacements
- Fussell, Ramachandran, et al.
- 1993
(Show Context)
Citation Context ...e digraph G with n vertices, one can test if G is acyclic in O(log n) time with n/ log n processors on an EREW PRAM. By applying the result of Theorem 6 and various parallel techniques (in particular =-=[14, 28, 27]-=-) we can e#ciently parallelize algorithm Test. Theorem 7. Upward planarity testing of a single-source digraph with n vertices can be done in O(log n) time on a CRCW PRAM with n log log n/ log n proces... |

28 | The complexity of drawing trees nicely - Supowit, Reingold - 1983 |

25 |
Upward drawings of triconnected digraphs
- Bertolazzi, Battista, et al.
- 1994
(Show Context)
Citation Context ... a valley. Since both PK and PH are homeomorphic to a valley, then both K and H are in case 4.2; thus s(G) # K # and s(G) # H # , which is a contradiction. The proof of the next lemma can be found in =-=[3, 4]-=-. Lemma 15. Let G be digraph, let #G be a candidate planar embedding of G, and let face # # #G . Finally, let A be an assignment of the sinks and the sources of G to the faces of #G . If |A(f)| = c(f)... |

25 | Matching Theory, volume 29 of Annals of discrete mathematics - Lovasz, Plummer - 1986 |

24 |
On-Line Maintenance of Triconnected Components with SPQR-Trees
- Battista, Tamassia
- 1996
(Show Context)
Citation Context ...econd example an upward drawable digraph is considered. 2. Preliminaries. In this section we recall some terminology and basic results on upward planarity. We also review the SPQR-tree, introduced in =-=[10, 11]-=-, and the combinatorial characterization of upward planarity for embedded planar digraphs, shown in [3, 4]. We assume the reader's familiarity with planar graphs. 2.1. Drawings and embeddings. A drawi... |

24 | Parallel transitive closure and point location in planar structures - Tamassia, Vitter - 1991 |

23 |
On-line graph algorithms with SPQR-trees
- Battista, Tamassia
- 1990
(Show Context)
Citation Context ...econd example an upward drawable digraph is considered. 2. Preliminaries. In this section we recall some terminology and basic results on upward planarity. We also review the SPQR-tree, introduced in =-=[10, 11]-=-, and the combinatorial characterization of upward planarity for embedded planar digraphs, shown in [3, 4]. We assume the reader's familiarity with planar graphs. 2.1. Drawings and embeddings. A drawi... |

22 | Area-efficient upward tree drawings - Garg, Goodrich, et al. - 1993 |

21 | Planar lattices - Kelly, Rival - 1975 |

20 | Fundamentals of planar ordered sets - Kelly - 1987 |

19 | An optimal parallel algorithm for graph planarity - Ramachandran, Reif - 1989 |

17 |
How to draw a series-parallel digraph
- Bertolazzi, Cohen, et al.
- 1994
(Show Context)
Citation Context ...lay. Tamassia and Vitter [32] show that the above drawing algorithms can be e#ciently parallelized. Upward planar drawings of trees and series-parallel digraphs are studied in [29, 31, 6, 13, 15] and =-=[1, 2]-=-, respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial-time algorithm is given for testing the upward p... |

16 |
I.: Bipartite graphs, upward drawings, and planarity
- Battista, Liu, et al.
- 1990
(Show Context)
Citation Context ...[32] show that the above drawing algorithms can be e#ciently parallelized. Upward planar drawings of trees and series-parallel digraphs are studied in [29, 31, 6, 13, 15] and [1, 2], respectively. In =-=[8]-=- it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial-time algorithm is given for testing the upward planarity of digraphs w... |

15 | Upward planar drawing of single source acyclic digraphs - Hutton, Lubiw |

15 | Upward planarity testing of outerplanar dags - Papakostas - 1995 |

14 |
Minimum size h-v drawings
- Eades, Lin, et al.
- 1992
(Show Context)
Citation Context ...bounds and symmetry display. Tamassia and Vitter [32] show that the above drawing algorithms can be e#ciently parallelized. Upward planar drawings of trees and series-parallel digraphs are studied in =-=[29, 31, 6, 13, 15]-=- and [1, 2], respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial-time algorithm is given for testing t... |

12 |
On upward drawing testing of triconnected digraphs
- Bertolazzi, Battista
- 1991
(Show Context)
Citation Context ...ries-parallel digraphs are studied in [29, 31, 6, 13, 15] and [1, 2], respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In =-=[3, 4]-=- a polynomial-time algorithm is given for testing the upward planarity of digraphs with a prescribed embedding. Thomassen [33] characterizes the upward planarity of single-source digraphs in terms of ... |

11 | Planar lattices and planar graphs - Platt - 1976 |

11 | Planar acyclic oriented graphs - Thomassen - 1989 |

11 | The dimension of planar posets - Trotter, Moore - 1977 |

7 | drawing, and order - Reading - 1993 |

6 | Local reorientation, global order, and planar topology - Kao, Shannon - 1986 |