## Orderly Spanning Trees with Applications to Graph Encoding and Graph Drawing (2001)

Venue: | In 12 th Symposium on Discrete Algorithms (SODA |

Citations: | 34 - 6 self |

### BibTeX

@INPROCEEDINGS{Chiang01orderlyspanning,

author = {Yi-ting Chiang and Ching-Chi Lin and Hsueh-i Lu},

title = {Orderly Spanning Trees with Applications to Graph Encoding and Graph Drawing},

booktitle = {In 12 th Symposium on Discrete Algorithms (SODA},

year = {2001},

pages = {506--515},

publisher = {ACM-SIAM}

}

### Years of Citing Articles

### OpenURL

### Abstract

The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a linear-time algorithm that obtains an orderly pair (H

### Citations

617 |
Text Compression
- Bell, Cleary, et al.
- 1990
(Show Context)
Citation Context ...he k-th position of S. Clearly, if k = select(S, i, ✷), then i = rank(S, k, ✷). An auxiliary string χ of S is a binary string with |χ| = o(|S|) that is obtainable from S in O(|S|) time. Fact 3.1 (see =-=[1,7]-=-) For any strings S1, S2, . . . , Sk with k = O(1), there is an auxiliary string χ0 such that given the concatenation of χ0, S1, S2, . . . , Sk as input, the index of the first symbol of any given Si ... |

348 |
Universal codeword sets and representations of the integers
- Elias
- 1975
(Show Context)
Citation Context ...he k-th position of S. Clearly, if k = select(S, i, ✷), then i = rank(S, k, ✷). An auxiliary string χ of S is a binary string with |χ| = o(|S|) that is obtainable from S in O(|S|) time. Fact 3.1 (see =-=[1,7]-=-) For any strings S1, S2, . . . , Sk with k = O(1), there is an auxiliary string χ0 such that given the concatenation of χ0, S1, S2, . . . , Sk as input, the index of the first symbol of any given Si ... |

262 | Edgebreaker: Connectivity compression for triangle meshes - Rossignac - 1999 |

196 |
Embedding planar graphs on the grid
- SCHNYDER
- 1990
(Show Context)
Citation Context ...remarks Our orderly-pair algorithm appears to be a fundamental graph-algorithmic tool. Besides the applications shown in Sections 4 and 5, our algorithm gives an alternative way to compute a realizer =-=[19]-=- of a plane triangulation, which gives the best known straight-line drawing of planar graphs on the grids. Compared to the proof given in [19], ours is quite simple: Given a plane triangulation G, let... |

170 |
Space-efficient static trees and graphs
- JACOBSON
- 1989
(Show Context)
Citation Context ... a linear-space encoding of a k-connected G, obtainable in O(n 4 m4 k /k 2 ) time, from which one can determine in O(1) time whether any two nodes are connected by k + 1 node-disjoint paths. Jacobson =-=[14]-=- gave an Θ(n)-bit encoding for a connected and simple planar G that supports traversal in Θ(log n) time per node visited. Under the word model of computation [2,4, 9,21, 22,24], where operations such ... |

156 |
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
- Fredman, Willard
- 1994
(Show Context)
Citation Context ...y k + 1 node-disjoint paths. Jacobson [14] gave an Θ(n)-bit encoding for a connected and simple planar G that supports traversal in Θ(log n) time per node visited. Under the word model of computation =-=[2,4, 9,21, 22,24]-=-, where operations such as read, write, and add on O(log n) consecutive bits take O(1) time, an encoding S of G is weakly convenient [3] if it takes (a) O(m + n) time to encode G and decode S; (b) O(1... |

151 |
How to draw a planar graph on a grid
- Fraysseix, Pach, et al.
- 1990
(Show Context)
Citation Context ...anning trees yields (i) the best known encoding of G with query support, and (ii) the first area-optimal 2-visibility drawing of G. 1 Introduction The canonical orderings of triconnected plane graphs =-=[6,10,16,17]-=- are crucial in designing several graphdrawing and graph-encoding algorithms [3,8,12]. This paper introduces the concept of orderly pair for connected planar graphs, which extends that of canonical or... |

140 | Succinct Representation of Balanced Parentheses and Static Trees
- Munro, Raman
(Show Context)
Citation Context ...node in G. If the degree of a node can be determined from a weakly convenient S in O(1) times, then S is convenient [3]. For a planar G that may have multiple edges but no self-loops, Munro and Raman =-=[18]-=- gave the first 1nontrivial convenient encoding of G that has 2m + 8n + o(m + n) bits. Their result is based on the four-page decomposition of planar graphs [25] and an auxiliary string that encodes ... |

111 | A census of planar triangulations - Tutte - 1962 |

92 | Compact pat trees - CLARK - 1998 |

71 | Implicit representation of graphs
- Kannan, Naor, et al.
- 1988
(Show Context)
Citation Context ...e-d node. Without accounting for the time required to read the labels, the adjacency of two nodes can be answered from their encoding in O(1) time. For certain graph families, Kannan, Naor and Rudich =-=[15]-=- gave schemes that encode each node with O(log n) bits and support O(log n)time testing of adjacency between two nodes. Cohen, Di Battista, Kanevsky, and Tamassia [5] gave a linear-space encoding of a... |

70 | On RAM priority queues - Thorup - 1996 |

65 | Drawing planar graphs using the canonical ordering
- Kant
- 1996
(Show Context)
Citation Context ...anning trees yields (i) the best known encoding of G with query support, and (ii) the first area-optimal 2-visibility drawing of G. 1 Introduction The canonical orderings of triconnected plane graphs =-=[6,10,16,17]-=- are crucial in designing several graphdrawing and graph-encoding algorithms [3,8,12]. This paper introduces the concept of orderly pair for connected planar graphs, which extends that of canonical or... |

63 |
Membership in constant time and almostminimum space
- Brodnik, Munro
- 1999
(Show Context)
Citation Context ...y k + 1 node-disjoint paths. Jacobson [14] gave an Θ(n)-bit encoding for a connected and simple planar G that supports traversal in Θ(log n) time per node visited. Under the word model of computation =-=[2,4, 9,21, 22,24]-=-, where operations such as read, write, and add on O(log n) consecutive bits take O(1) time, an encoding S of G is weakly convenient [3] if it takes (a) O(m + n) time to encode G and decode S; (b) O(1... |

60 | How to Draw a Planar Graph on a Grid. Combinatorica - Fraysseix, Pach, et al. - 1990 |

58 |
Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems, Theor
- Kant, He
- 1997
(Show Context)
Citation Context ...anning trees yields (i) the best known encoding of G with query support, and (ii) the first area-optimal 2-visibility drawing of G. 1 Introduction The canonical orderings of triconnected plane graphs =-=[6,10,16,17]-=- are crucial in designing several graphdrawing and graph-encoding algorithms [3,8,12]. This paper introduces the concept of orderly pair for connected planar graphs, which extends that of canonical or... |

47 | Compact encodings of planar graphs via canonical orderings and multiple parentheses
- Chuang, Garg, et al.
- 1443
(Show Context)
Citation Context ...st area-optimal 2-visibility drawing of G. 1 Introduction The canonical orderings of triconnected plane graphs [6,10,16,17] are crucial in designing several graphdrawing and graph-encoding algorithms =-=[3,8,12]-=-. This paper introduces the concept of orderly pair for connected planar graphs, which extends that of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar ... |

36 | Algorithms for Drawing Planar Graphs - Kant - 1993 |

29 | Succinct representation of general unlabeled graphs - Naor - 1990 |

25 | Drawing planar graphs using the canonical ordering, Algorithmica – Special Issue on Graph Drawing 16 - Kant - 1996 |

23 | A fast general methodology for information-theoretically optimal encodings of graphs
- He, Kao, et al.
- 2000
(Show Context)
Citation Context ...atches that of the best known convenient encoding for an outerplanar graph [18]. Besides the orderly-pair algorithm, our results are also based on an improved auxiliary string for a folklore encoding =-=[3, 11,18]-=- of a rooted tree T. With the auxiliary string of Munro and Raman [18], the degree of a degree-d node in T can be answered in O(d) time. In this paper, we give a nontrivial auxiliary string that suppo... |

22 | Representation of graphs - Itai, Rodeh - 1982 |

21 | Linear-time succinct encodings of planar graphs via canonical orderings
- He, Kao, et al.
- 1999
(Show Context)
Citation Context ...st area-optimal 2-visibility drawing of G. 1 Introduction The canonical orderings of triconnected plane graphs [6,10,16,17] are crucial in designing several graphdrawing and graph-encoding algorithms =-=[3,8,12]-=-. This paper introduces the concept of orderly pair for connected planar graphs, which extends that of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar ... |

13 | An Algorithm for Straight-Line Drawing of Planar Graphs
- Harel, Sardas
- 1995
(Show Context)
Citation Context |

7 |
Reinventing the wheel: An optimal data structure for connectivity queries (extended abstract
- Cohen, Battista, et al.
- 1993
(Show Context)
Citation Context ...amilies, Kannan, Naor and Rudich [15] gave schemes that encode each node with O(log n) bits and support O(log n)time testing of adjacency between two nodes. Cohen, Di Battista, Kanevsky, and Tamassia =-=[5]-=- gave a linear-space encoding of a k-connected G, obtainable in O(n 4 m4 k /k 2 ) time, from which one can determine in O(1) time whether any two nodes are connected by k + 1 node-disjoint paths. Jaco... |

3 | A census of planar maps, Canadian Journal of Mathematics 15 - Tutte - 1963 |

2 | Compact encodings of planar graphs via canonical orderings and multiple parentheses - Lu - 1998 |

2 | Short encodings of planar graphs and maps. Discrete Applied Mathematics 58, 239–252 - Keeler, Westbrook, et al. - 1995 |

1 |
Compact implicit representation of graphs (extended abstract
- Talamo, Vocca
(Show Context)
Citation Context ...ber of coding schemes with different trade-offs have been proposed in the literature. The widely useful adjacency-list encoding of an n-node m-edge graph G requires 2m⌈log 2 n⌉ bits. Talamo and Vocca =-=[20]-=- gave an encoding, obtainable in O(n 3 ) time, that assigns an O(d log 3 n)bit label to each degree-d node. Without accounting for the time required to read the labels, the adjacency of two nodes can ... |

1 | 25] M. Thorup. Undirected single source shortest paths in linear time - Springer, Republic - 1998 |