## Faster shortest non-contractible cycles in directed surface graphs

Venue: | CoRR |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Fox_fastershortest,

author = {Kyle Fox},

title = {Faster shortest non-contractible cycles in directed surface graphs},

journal = {CoRR},

year = {},

pages = {2011}

}

### OpenURL

### Abstract

Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest non-contractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest non-null-homologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest non-separating cycle.

### Citations

357 |
Topological graph theory
- Gross, Tucker
- 1987
(Show Context)
Citation Context ...e shortest non-contractible cycle. 2 Preliminaries We begin by recalling several useful definitions related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker =-=[25]-=- or Mohar and Thomassen [38] for topological graph theory, and to Hatcher [27] or Stillwell [41] for surface topology and homology. We adopt the presentation of our terminology and notation directly f... |

227 |
Graphs on surfaces
- Mohar, Thomassen
(Show Context)
Citation Context ...hort non-trivial cycles is arguably one of the most natural problems for graphs embedded on a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems =-=[1, 16, 34, 38]-=- and more practical applications in areas such as graphics and graph drawing [3, 21, 26, 29, 35, 44]. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai... |

180 |
Classical topology and combinatorial group theory. Second edn. Volume 72 of Graduate Texts in Mathematics
- Stillwell
- 1993
(Show Context)
Citation Context ...ons related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker [25] or Mohar and Thomassen [38] for topological graph theory, and to Hatcher [27] or Stillwell =-=[41]-=- for surface topology and homology. We adopt the presentation of our terminology and notation directly from previous works [14, 19, 20, 22]. 2.1 Surfaces and Curves A surface (more formally, a 2-manif... |

172 | Faster shortest path algorithms for planar graphs
- Henzinger, Klein, et al.
- 1997
(Show Context)
Citation Context ...bel the boundary cycles of G as B0, B1, . . . , Bb−1. Let s be an arbitrary vertex on B0. We compute the shortest path tree T from s using Dijkstra’s algorithm in O(n log n) time or using Henzinger’s =-=[28]-=- algorithm in O(n) time if g = O(n1−ɛ ) for some constant ɛ > 0. For each index i ≥ 1, let λi be a directed path in T from B0 to Bi that contains exactly one vertex from each boundary cycle B0 and Bi.... |

126 |
Fast algorithms for shortest paths in planar networks, with applications
- Frederickson
- 1989
(Show Context)
Citation Context ...thm to find the shortest non-trivial cycle in an annulus as a subroutine for computing minimum s, t-cuts in planar graphs. Their result has seen several improvements, most recently by Italiano et al. =-=[24, 31, 40]-=-. Thomassen [42] gave the first efficient algorithm for computing non-trivial cycles on surfaces with arbitrary genus. His algorithm runs in O(n3 ) time and relies on a property of certain families of... |

105 | Topological noise removal
- Wood
- 1926
(Show Context)
Citation Context ...a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems [1, 16, 34, 38] and more practical applications in areas such as graphics and graph drawing =-=[3, 21, 26, 29, 35, 44]-=-. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai and Shiloach [30]. They give an O(n2 log n) time algorithm to find the shortest non-trivial cycle i... |

76 | Removing excess topology from isosurfaces
- Wood, Hoppe, et al.
(Show Context)
Citation Context ...a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems [1, 16, 34, 38] and more practical applications in areas such as graphics and graph drawing =-=[3, 21, 26, 29, 35, 44]-=-. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai and Shiloach [30]. They give an O(n2 log n) time algorithm to find the shortest non-trivial cycle i... |

51 |
Multiple-source shortest paths in planar graphs
- Klein
- 2005
(Show Context)
Citation Context ...es that this assumption can be enforced (with high probability) by perturbing the edge weights with random infinitesimal values [21]. Our algorithms (implicitly) rely on the following result of Klein =-=[36]-=- for planar graphs, and its generalization to higher-genus graphs by Cabello et al. [5, 6]. Lemma 2.1 (Klein [36]). Let G be a directed graph with non-negative edge weights and let f be an arbitrary f... |

44 |
Embeddings of graphs with no short noncontractible cycles
- Thomassen
- 1990
(Show Context)
Citation Context ... non-trivial cycle in an annulus as a subroutine for computing minimum s, t-cuts in planar graphs. Their result has seen several improvements, most recently by Italiano et al. [24, 31, 40]. Thomassen =-=[42]-=- gave the first efficient algorithm for computing non-trivial cycles on surfaces with arbitrary genus. His algorithm runs in O(n3 ) time and relies on a property of certain families of cycles known as... |

41 | Dynamic generators of topologically embedded graphs
- Eppstein
(Show Context)
Citation Context ..., a spanning cotree C (the dual of a spanning tree C ∗ of G ∗ ), and leftover edges L = G \ (T ∪ C). Euler’s formula implies that in any tree-cotree decomposition, the set L contains exactly 2g edges =-=[17]-=-. The definitions for dual graphs and tree-cotree decompositions given above extend to surfaces with boundary, but we do not require these extensions in this paper. For the problems we consider, the i... |

40 | Finding shortest non-separating and non-contractible cycles for topologically embedded graphs
- Cabello, Mohar
(Show Context)
Citation Context ... 1 1 Introduction There is a long line of work on computing shortest non-trivial cycles in surface embedded graphs, specifically shortest non-separating and non-contractible cycles. Cabello and Mohar =-=[11]-=- claim that finding short non-trivial cycles is arguably one of the most natural problems for graphs embedded on a surface. Additionally, finding these cycles has many benefits both for theoretical co... |

40 |
Algebraic topology, Cambridge Univ
- Hatcher
- 2002
(Show Context)
Citation Context ...al useful definitions related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker [25] or Mohar and Thomassen [38] for topological graph theory, and to Hatcher =-=[27]-=- or Stillwell [41] for surface topology and homology. We adopt the presentation of our terminology and notation directly from previous works [14, 19, 20, 22]. 2.1 Surfaces and Curves A surface (more f... |

40 |
Matching is as easy as matrix inversion,” Combinatorica 7
- Mulmuley, Vazirani, et al.
- 1987
(Show Context)
Citation Context ... require this extension. To simplify our presentation and analysis, we assume that any two vertices x and y in G are connected by a unique shortest directed path, denoted σ(x, y). The Isolation Lemma =-=[39]-=- implies that this assumption can be enforced (with high probability) by perturbing the edge weights with random infinitesimal values [21]. Our algorithms (implicitly) rely on the following result of ... |

32 | An O(n log n) algorithm for maximum st-flow in a directed planar graph - Borradaile, Klein |

32 | Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time
- Kutz
- 2006
(Show Context)
Citation Context ...trary genus. Cabello and Mohar [11] gave the first results parameterized by genus. Others have improved their results, leading to the current best running time of gO(g) n log log n by Italiano et al. =-=[5, 31, 37]-=-. For other results related to finding interesting cycles on surfaces, see [4, 8–10, 12, 14, 20, 23]. Unfortunately, all of the above results rely on the observation that two shortest paths in an undi... |

30 | Multiple source shortest paths in a genus g graph
- Chambers, Cabello
- 2007
(Show Context)
Citation Context ...trary genus. Cabello and Mohar [11] gave the first results parameterized by genus. Others have improved their results, leading to the current best running time of gO(g) n log log n by Italiano et al. =-=[5, 31, 37]-=-. For other results related to finding interesting cycles on surfaces, see [4, 8–10, 12, 14, 20, 23]. Unfortunately, all of the above results rely on the observation that two shortest paths in an undi... |

29 | Approximation algorithms via contraction decomposition
- Demaine, Hajiaghayi, et al.
- 2007
(Show Context)
Citation Context ...hort non-trivial cycles is arguably one of the most natural problems for graphs embedded on a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems =-=[1, 16, 34, 38]-=- and more practical applications in areas such as graphics and graph drawing [3, 21, 26, 29, 35, 44]. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai... |

29 |
Computing crossing number in linear time
- Kawarabayashi, Reed
- 2007
(Show Context)
Citation Context ...a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems [1, 16, 34, 38] and more practical applications in areas such as graphics and graph drawing =-=[3, 21, 26, 29, 35, 44]-=-. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai and Shiloach [30]. They give an O(n2 log n) time algorithm to find the shortest non-trivial cycle i... |

28 |
Minimum s-t cut of a planar undirected network in O(n log2 n) time
- Reif
- 1983
(Show Context)
Citation Context ...thm to find the shortest non-trivial cycle in an annulus as a subroutine for computing minimum s, t-cuts in planar graphs. Their result has seen several improvements, most recently by Italiano et al. =-=[24, 31, 40]-=-. Thomassen [42] gave the first efficient algorithm for computing non-trivial cycles on surfaces with arbitrary genus. His algorithm runs in O(n3 ) time and relies on a property of certain families of... |

25 | Splitting (complicated) surfaces is hard - Chambers, Verdière, et al. - 2006 |

23 |
Maximum flow in planar networks
- Itai, Shiloach
- 1979
(Show Context)
Citation Context ...tical applications in areas such as graphics and graph drawing [3, 21, 26, 29, 35, 44]. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai and Shiloach =-=[30]-=-. They give an O(n2 log n) time algorithm to find the shortest non-trivial cycle in an annulus as a subroutine for computing minimum s, t-cuts in planar graphs. Their result has seen several improveme... |

21 |
Maximum (s, t)-flows in planar networks in O(|V | log |V |)-time
- Weihe
- 1997
(Show Context)
Citation Context ...ycle in a directed graph embedded on an annulus as an attempt to find the minimum s, t-cut in planar graphs1 . Their result can also be achieved using recent maximum flow algorithms for planar graphs =-=[2, 18, 43]-=-. Cabello, Colin de Verdière, and Lazarus [7] gave the first efficient algorithms for computing shortest non-trivial cycles in directed surface graphs of arbitrary genus. Their algorithms run in O(n2 ... |

20 |
Optimally cutting a surface into a disk. Discrete Comput
- Erickson, Har-Peled
(Show Context)
Citation Context |

19 | Minimum cuts and shortest homologous cycles
- Chambers, Erickson, et al.
- 2009
(Show Context)
Citation Context ...ssen [38] for topological graph theory, and to Hatcher [27] or Stillwell [41] for surface topology and homology. We adopt the presentation of our terminology and notation directly from previous works =-=[14, 19, 20, 22]-=-. 2.1 Surfaces and Curves A surface (more formally, a 2-manifold with boundary) is a compact Hausdorff space in which every point has an open neighborhood homeomorphic to either the plane � 2 or a clo... |

18 | Homology flows, cohomology cuts
- Chambers, Erickson, et al.
- 2009
(Show Context)
Citation Context ...osite orientations). Thus, like Cabello et al. [7] and Erickson [19], we implicitly model directed graphs as unweighted graphs with asymmetric edge weights. Duality can be extended to directed graphs =-=[13]-=-, but the results in this paper do not require this extension. To simplify our presentation and analysis, we assume that any two vertices x and y in G are connected by a unique shortest directed path,... |

18 | Graph and Map Isomorphism and all polyhedral embeddings in linear time
- Kawarabayashi, Mohar
- 2008
(Show Context)
Citation Context ...hort non-trivial cycles is arguably one of the most natural problems for graphs embedded on a surface. Additionally, finding these cycles has many benefits both for theoretical combinatorial problems =-=[1, 16, 34, 38]-=- and more practical applications in areas such as graphics and graph drawing [3, 21, 26, 29, 35, 44]. The history of non-trivial cycles in undirected graphs goes back several years to a result of Itai... |

15 | Polynomial-time approximation schemes for subsetconnectivity problems in bounded-genus graphs
- Borradaile, Demaine, et al.
- 2009
(Show Context)
Citation Context |

14 | Finding one tight cycle - Cabello, DeVos, et al. - 2008 |

14 |
Improved algorithms for Min Cut and Max Flow in undirected planar graphs
- Italiano, Nussbaum, et al.
- 2011
(Show Context)
Citation Context ...thm to find the shortest non-trivial cycle in an annulus as a subroutine for computing minimum s, t-cuts in planar graphs. Their result has seen several improvements, most recently by Italiano et al. =-=[24, 31, 40]-=-. Thomassen [42] gave the first efficient algorithm for computing non-trivial cycles on surfaces with arbitrary genus. His algorithm runs in O(n3 ) time and relies on a property of certain families of... |

13 | Probabilistic embeddings of bounded genus graphs into planar graphs
- Indyk, Sidiropoulos
- 2007
(Show Context)
Citation Context |

12 | Quantifying homology classes II: Localization and stability
- Chen, Freedman
(Show Context)
Citation Context ...) are isomorphic.) A cycle is contractible if it is homotopic to a constant map. Homology is a coarser equivalence relation than homotopy, with nicer algebraic properties. Like several earlier papers =-=[14, 15, 19, 20, 22]-=-, we consider only one-dimensional cellular homology with coefficients in the finite field � 2. Fix a cellular embedding of an undirected graph G on a surface Σ with genus g and b boundary cycles. An ... |

11 | Minimum cuts and shortest non-separating cycles via homology covers
- Erickson, Nayyeri
- 2011
(Show Context)
Citation Context ...aphs of arbitrary genus. Their algorithms run in O(n2 log n) time and O( � gn3/2 log n) time, and rely on a variant of the 3-path condition and balanced separators, respectively. Erickson and Nayyeri =-=[22]-=- gave a 2O(g) n log n time algorithm for computing the shortest non-separating cycle that relies on computing the shortest cycle in each of 2O(g) homology classes. The latest results for these problem... |

10 | F.: Finding shortest non-trivial cycles in directed graphs on surfaces
- Cabello, Verdière, et al.
- 2010
(Show Context)
Citation Context ... attempt to find the minimum s, t-cut in planar graphs1 . Their result can also be achieved using recent maximum flow algorithms for planar graphs [2, 18, 43]. Cabello, Colin de Verdière, and Lazarus =-=[7]-=- gave the first efficient algorithms for computing shortest non-trivial cycles in directed surface graphs of arbitrary genus. Their algorithms run in O(n2 log n) time and O( � gn3/2 log n) time, and r... |

10 |
Minimum cut in directed planar networks
- Janiga, Koubek
- 1992
(Show Context)
Citation Context ... shortest non-trivial cycles in directed surface graphs. In fact, the (short) history of these results appears to coincide nicely with the history given above for undirected graphs. Janiga and Koubek =-=[32]-=- gave the first near-linear time algorithm for computing the shortest nontrivial cycle in a directed graph embedded on an annulus as an attempt to find the minimum s, t-cut in planar graphs1 . Their r... |

9 | Computing the shortest essential cycle - Erickson, Worah |

8 | Randomly removing g handles at once
- Borradaile, Lee, et al.
- 2009
(Show Context)
Citation Context |

8 | Multiple-source shortest paths in embedded graphs
- Cabello, Chambers, et al.
(Show Context)
Citation Context ...eights with random infinitesimal values [21]. Our algorithms (implicitly) rely on the following result of Klein [36] for planar graphs, and its generalization to higher-genus graphs by Cabello et al. =-=[5, 6]-=-. Lemma 2.1 (Klein [36]). Let G be a directed graph with non-negative edge weights and let f be an arbitrary face of G. We can preprocess G in O(n log n) time and O(n) space, so that the length of the... |

7 | Finding shortest contractible and shortest separating cycles in embedded graphs - Cabello |

7 | Finding cycles with topological properties in embedded graphs - Cabello, Verdière, et al. |

7 | Maximum flows and parametric shortest paths in planar graphs
- Erickson
- 2010
(Show Context)
Citation Context ...ycle in a directed graph embedded on an annulus as an attempt to find the minimum s, t-cut in planar graphs1 . Their result can also be achieved using recent maximum flow algorithms for planar graphs =-=[2, 18, 43]-=-. Cabello, Colin de Verdière, and Lazarus [7] gave the first efficient algorithms for computing shortest non-trivial cycles in directed surface graphs of arbitrary genus. Their algorithms run in O(n2 ... |

6 | Outputsensitive algorithm for the edge-width of an embedded graph - Cabello, Verdière, et al. - 2010 |

5 | Shortest non-trivial cycles in directed surface graphs
- Erickson
- 2011
(Show Context)
Citation Context ...thm for computing the shortest non-separating cycle that relies on computing the shortest cycle in each of 2O(g) homology classes. The latest results for these problems are two algorithms of Erickson =-=[19]-=-. The first algorithm computes shortest non-separating cycles in O(g 2n log n) time by computing shortest paths in several linear sized covering spaces. The second algorithm computes shortest non-cont... |

5 |
Minimum s − t cut in undirected planar graphs when the source and the sink are close
- Kaplan, Nussbaum
- 2011
(Show Context)
Citation Context ...s to either a non-separating cycle or to a cycle that separates a pair of boundaries in a finite portion of some infinite cyclic cover. 1 Unfortunately, their minimum cut algorithm has a subtle error =-=[33]-=- which may lead to an incorrect result when the minimum t, s-cut is smaller than the minimum s, t-cut.2 Faster Shortest Non-contractible Cycles in Directed Surface Graphs In addition to the above res... |

4 | Global minimum cuts in surface embedded graphs
- Erickson, Fox, et al.
- 2012
(Show Context)
Citation Context ...) are isomorphic.) A cycle is contractible if it is homotopic to a constant map. Homology is a coarser equivalence relation than homotopy, with nicer algebraic properties. Like several earlier papers =-=[14, 15, 19, 20, 22]-=-, we consider only one-dimensional cellular homology with coefficients in the finite field � 2. Fix a cellular embedding of an undirected graph G on a surface Σ with genus g and b boundary cycles. An ... |