## Shortest non-trivial cycles in directed surface graphs (2011)

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Venue: | In Proc. 27th Ann. Symp. Comput. Geom |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Erickson11shortestnon-trivial,

author = {Jeff Erickson},

title = {Shortest non-trivial cycles in directed surface graphs},

booktitle = {In Proc. 27th Ann. Symp. Comput. Geom},

year = {2011},

pages = {236--243}

}

### OpenURL

### Abstract

Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.

### Citations

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Citation Context ...finitions and results related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker [26] or Mohar and Thomassen [40] for topological graph theory, and to Hatcher =-=[28]-=- or Stillwell [43] for algebraic topology. 2.1 Surfaces and Curves A surface (more formally, a 2-manifold) Σ is a compact Hausdorff space in which every point has an open neighborhood homeomorphic to ... |

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Citation Context ...logically non-trivial. 2. BACKGROUND We begin by recalling several standard definitions and results related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker =-=[26]-=- or Mohar and Thomassen [40] for topological graph theory, and to Hatcher [28] or Stillwell [43] for algebraic topology. 2.1 Surfaces and Curves A surface (more formally, a 2-manifold) Σ is a compact ... |

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Citation Context ...lts related to surface-embedded graphs. For further background, we refer the reader to Gross and Tucker [26] or Mohar and Thomassen [40] for topological graph theory, and to Hatcher [28] or Stillwell =-=[43]-=- for algebraic topology. 2.1 Surfaces and Curves A surface (more formally, a 2-manifold) Σ is a compact Hausdorff space in which every point has an open neighborhood homeomorphic to either the plane �... |

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Citation Context ...ation of Reif’s algorithm that can find this cycle 1 in O(n log 2 n/ log log n) time; the running time of their algorithm can be improved to O(n log n) using more recent algorithms for shortest paths =-=[29, 38]-=-. The same running time can also be achieved by recent planar maximum flow algorithms [2, 20, 45]. The first results for directed graphs on surfaces of higher genus were only recently obtained by Cabe... |

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Citation Context ...e shortest non-trivial cycle in a graph embedded on an annulus. Itai and Shiloach described a simple algorithm to find this cycle in O(n 2 log n) time. Their algorithm has been improved several times =-=[42, 25]-=-, most recently by Italiano et al. [32, 33, 46], who describe an algorithm that runs in O(n log log n) time. Thomassen [44] developed the first efficient algorithm for graphs on arbitrary surfaces, wh... |

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Citation Context ...w crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and removing topological noise from surface models =-=[27, 47]-=-. In all these applications, cutting along the shortest possible cycle is preferred or even required. These and other applications have motivated a long series of results on finding shortest non-trivi... |

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Citation Context ... of a non-contractible loop that is shorter than γ. Euler’s formula implies that every system of loops has exactly 2g elements. Any system of loops is also a basis for the fundamental group π1(Σ, x0) =-=[23]-=-. We refer to the disk D = Σ QΛ as a fundamental domain. Each loop λi appears as two directed paths λ + and λ− on the boundary of D; in particular, the basepoint x i i appears as 4g different vertices... |

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Citation Context ...w crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and removing topological noise from surface models =-=[27, 47]-=-. In all these applications, cutting along the shortest possible cycle is preferred or even required. These and other applications have motivated a long series of results on finding shortest non-trivi... |

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Citation Context ...ation of Reif’s algorithm that can find this cycle 1 in O(n log 2 n/ log log n) time; the running time of their algorithm can be improved to O(n log n) using more recent algorithms for shortest paths =-=[29, 38]-=-. The same running time can also be achieved by recent planar maximum flow algorithms [2, 20, 45]. The first results for directed graphs on surfaces of higher genus were only recently obtained by Cabe... |

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Citation Context ...le in O(n 2 log n) time. Their algorithm has been improved several times [42, 25], most recently by Italiano et al. [32, 33, 46], who describe an algorithm that runs in O(n log log n) time. Thomassen =-=[44]-=- developed the first efficient algorithm for graphs on arbitrary surfaces, which runs in O(n 3 ) time and exploits his so-called 3-path condition; see also Mohar and Thomassen [40, Sect. 4.3]. Erickso... |

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Citation Context ...orithm that runs in O(n 2 log n) time [21]. This is the best algorithm known for arbitrary surface-embedded graphs, but faster algorithms are known when the genus g of the underlying surface is small =-=[4, 6, 12, 39]-=-, the fastest of which runs in O(g 2 n log n) time, where g is the genus of the underlying surface [7]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shorte... |

40 |
Matching is as easy as matrix inversion, Combinatorica 7
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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 =-=[41]-=- implies that this assumption can be enforced (with high probability) by perturbing the edge weights with random infinitesimal values [21]. Our algorithms rely the following seminal result of Klein [3... |

38 | Dynamic generators of topologically embedded graphs
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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 =-=[19]-=-. For the problems we consider, the input is actually a directed edge-weighted graph G with a cellular embedding on some surface. We use the notation u→v to denote the directed edge from vertex u to v... |

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

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Citation Context ...orithm that runs in O(n 2 log n) time [21]. This is the best algorithm known for arbitrary surface-embedded graphs, but faster algorithms are known when the genus g of the underlying surface is small =-=[4, 6, 12, 39]-=-, the fastest of which runs in O(g 2 n log n) time, where g is the genus of the underlying surface [7]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shorte... |

30 | Multiple source shortest paths in a genus g graph
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Citation Context ...orithm that runs in O(n 2 log n) time [21]. This is the best algorithm known for arbitrary surface-embedded graphs, but faster algorithms are known when the genus g of the underlying surface is small =-=[4, 6, 12, 39]-=-, the fastest of which runs in O(g 2 n log n) time, where g is the genus of the underlying surface [7]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shorte... |

29 |
Minimum s-t cut of a planar undirected network in O(n log2 n) time
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Citation Context ...e shortest non-trivial cycle in a graph embedded on an annulus. Itai and Shiloach described a simple algorithm to find this cycle in O(n 2 log n) time. Their algorithm has been improved several times =-=[42, 25]-=-, most recently by Italiano et al. [32, 33, 46], who describe an algorithm that runs in O(n log log n) time. Thomassen [44] developed the first efficient algorithm for graphs on arbitrary surfaces, wh... |

28 | Approximation algorithms via contraction decomposition
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Citation Context ...graphs into planar graphs [3, 30], drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours =-=[18]-=- and Steiner trees [1], and removing topological noise from surface models [27, 47]. In all these applications, cutting along the shortest possible cycle is preferred or even required. These and other... |

28 |
Computing crossing number in linear time
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Citation Context ...to reduce its topological complexity. Examples include algorithms for probabilistically embedding high-genus graphs into planar graphs [3, 30], drawing abstract graphs in the plane with few crossings =-=[37]-=-, testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and removing topological noise from surface models [27, 47]. In all... |

27 | Tightening non-simple paths and cycles on surfaces
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Citation Context ...ies, any system of arcs contains exactly 2g + b − 1 arcs. A simple variant of the greedy tree-cotree algorithm described in this paper constructs a greedy system of arcs in O(n log n + (g + b)n) time =-=[13, 16, 17, 22]-=-; each arc in this system is the concatenation of a directed shortest path, a directed edge, and a reversed shortest path. The rest of the algorithm and its analysis is essentially unchanged. Theorem ... |

25 | Splitting (complicated) surfaces is hard
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Citation Context ...ies, any system of arcs contains exactly 2g + b − 1 arcs. A simple variant of the greedy tree-cotree algorithm described in this paper constructs a greedy system of arcs in O(n log n + (g + b)n) time =-=[13, 16, 17, 22]-=-; each arc in this system is the concatenation of a directed shortest path, a directed edge, and a reversed shortest path. The rest of the algorithm and its analysis is essentially unchanged. Theorem ... |

23 |
Maximum flow in planar networks
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Citation Context ...possible cycle is preferred or even required. These and other applications have motivated a long series of results on finding shortest non-trivial cycles in surface-embedded graphs. Itai and Shiloach =-=[31]-=- observed that the minimum (s, t)-cut in an undirected planar graph G ∗ is dual to the shortest cycle in the dual graph that separates the dual faces s ∗ and t ∗ . Thus, computing minimum cuts in plan... |

21 |
Optimally cutting a surface into a disk. Discrete Comput
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Citation Context ...hich runs in O(n 3 ) time and exploits his so-called 3-path condition; see also Mohar and Thomassen [40, Sect. 4.3]. Erickson and Har-Peled described a faster algorithm that runs in O(n 2 log n) time =-=[21]-=-. This is the best algorithm known for arbitrary surface-embedded graphs, but faster algorithms are known when the genus g of the underlying surface is small [4, 6, 12, 39], the fastest of which runs ... |

20 |
Maximum (s, t)-flows in planar networks in O(|V | log |V |)-time
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Citation Context ...ing time of their algorithm can be improved to O(n log n) using more recent algorithms for shortest paths [29, 38]. The same running time can also be achieved by recent planar maximum flow algorithms =-=[2, 20, 45]-=-. The first results for directed graphs on surfaces of higher genus were only recently obtained by Cabello, Colin de Verdière, and Lazarus [8], who describe two algorithms. Their first algorithm, whic... |

19 | Minimum cuts and shortest homologous cycles
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Citation Context ...]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shortest non-trivial cycle crosses any shortest path at most once. For related results and extensions, see =-=[5, 9, 10, 11, 13, 15, 24]-=-. Both Thomassen’s 3-path condition [44] and Cabello and Mohar’s crossing condition [12] are consequences of the following easy observation: For any four vertices s, t, u, v in an undirected surface g... |

18 | Homology flows, cohomology cuts
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Citation Context ...th opposite orientations). Thus,like Cabello et al.[8, Section 2.3], we implicitly model directed graphs as undirected graphs with asymmetric edge weights. Duality can be extended to directed graphs =-=[14]-=-, 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,... |

17 | Graph and map isomorphism and all polyhedral embeddings in linear time
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Citation Context ...lgorithms for probabilistically embedding high-genus graphs into planar graphs [3, 30], drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus =-=[36]-=-, approximating optimal traveling salesman tours [18] and Steiner trees [1], and removing topological noise from surface models [27, 47]. In all these applications, cutting along the shortest possible... |

16 | Many distances in planar graphs
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Citation Context |

15 | Polynomial-time approximation schemes for subsetconnectivity problems in bounded-genus graphs
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Citation Context ...hs [3, 30], drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees =-=[1]-=-, and removing topological noise from surface models [27, 47]. In all these applications, cutting along the shortest possible cycle is preferred or even required. These and other applications have mot... |

14 | Finding one tight cycle
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Citation Context ...]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shortest non-trivial cycle crosses any shortest path at most once. For related results and extensions, see =-=[5, 9, 10, 11, 13, 15, 24]-=-. Both Thomassen’s 3-path condition [44] and Cabello and Mohar’s crossing condition [12] are consequences of the following easy observation: For any four vertices s, t, u, v in an undirected surface g... |

13 | Probabilistic embeddings of bounded genus graphs into planar graphs
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Citation Context ...phs is cutting a surface along a topologically interesting cycle to reduce its topological complexity. Examples include algorithms for probabilistically embedding high-genus graphs into planar graphs =-=[3, 30]-=-, drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and r... |

10 |
Minimum cut in directed planar networks
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Citation Context ...)-cut or the minimum (t, s)cut in the directed planar dual graph, whichever has smaller capacity, where s and t are the dual vertices corresponding to the boundaries of the annulus. Janiga and Koubek =-=[34]-=- describean adaptation of Reif’s algorithm that can find this cycle 1 in O(n log 2 n/ log log n) time; the running time of their algorithm can be improved to O(n log n) using more recent algorithms f... |

9 | Computing the shortest essential cycle
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Citation Context ...]. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shortest non-trivial cycle crosses any shortest path at most once. For related results and extensions, see =-=[5, 9, 10, 11, 13, 15, 24]-=-. Both Thomassen’s 3-path condition [44] and Cabello and Mohar’s crossing condition [12] are consequences of the following easy observation: For any four vertices s, t, u, v in an undirected surface g... |

8 | Randomly removing g handles at once
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Citation Context ...phs is cutting a surface along a topologically interesting cycle to reduce its topological complexity. Examples include algorithms for probabilistically embedding high-genus graphs into planar graphs =-=[3, 30]-=-, drawing abstract graphs in the plane with few crossings [37], testing isomorphism between graphs of fixed genus [36], approximating optimal traveling salesman tours [18] and Steiner trees [1], and r... |

8 | Multiple-source shortest paths in embedded graphs
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Citation Context ...s, but faster algorithms are known when the genus g of the underlying surface is small [4, 6, 12, 39], the fastest of which runs in O(g 2 n log n) time, where g is the genus of the underlying surface =-=[7]-=-. All of these faster algorithms exploit the observation by Cabello and Mohar [12] that the shortest non-trivial cycle crosses any shortest path at most once. For related results and extensions, see [... |

7 | Finding shortest contractible and shortest separating cycles in embedded graphs
- Cabello
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7 | Finding cycles with topological properties in embedded graphs
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6 | Outputsensitive algorithm for the edge-width of an embedded graph
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5 |
Parametric shortest paths and maximum flows in planar graphs
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Citation Context ...ing time of their algorithm can be improved to O(n log n) using more recent algorithms for shortest paths [29, 38]. The same running time can also be achieved by recent planar maximum flow algorithms =-=[2, 20, 45]-=-. The first results for directed graphs on surfaces of higher genus were only recently obtained by Cabello, Colin de Verdière, and Lazarus [8], who describe two algorithms. Their first algorithm, whic... |

5 |
Minimum s − t cut in undirected planar graphs when the source and the sink are close
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(Show Context)
Citation Context ... contains a subset homeomorphic to the Möbius band, and orientable otherwise. 1 Janiga and Koubek actually claim an algorithm to compute the minimum (s, t)-cut, but their algorithm has a subtle error =-=[35]-=-, which may lead to an incorrect result when the minimum (t, s)-cut is smaller than the minimum (s, t)-cut. A path in a surface Σ is a continuous function p: [0, 1] → Σ. A loop is a path whose endpoin... |

4 |
de Verdière. Shortest cut graph of a surface with prescribed vertex set
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Citation Context ...ies, any system of arcs contains exactly 2g + b − 1 arcs. A simple variant of the greedy tree-cotree algorithm described in this paper constructs a greedy system of arcs in O(n log n + (g + b)n) time =-=[13, 16, 17, 22]-=-; each arc in this system is the concatenation of a directed shortest path, a directed edge, and a reversed shortest path. The rest of the algorithm and its analysis is essentially unchanged. Theorem ... |

2 |
Shortest homologous cycles and minimum cuts via homology covers
- Erickson, Nayyeri
(Show Context)
Citation Context ...me, uses a divide-and-conquer strategy based on balanced separators. Even more recently, Erickson and Nayyeri describe an algorithm to compute the shortest non-separating cycle in 2 O(g) n log n time =-=[22]-=-. However, their approach does not imply a faster algorithm for computing shortest non-contractible cycles. This paper describes faster algorithms for computing shortest non-separating and non-contrac... |

2 |
Improved minimum cuts and maximum flows in undirected planar graphs
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- 2011
(Show Context)
Citation Context ...edded on an annulus. Itai and Shiloach described a simple algorithm to find this cycle in O(n 2 log n) time. Their algorithm has been improved several times [42, 25], most recently by Italiano et al. =-=[32, 33, 46]-=-, who describe an algorithm that runs in O(n log log n) time. Thomassen [44] developed the first efficient algorithm for graphs on arbitrary surfaces, which runs in O(n 3 ) time and exploits his so-ca... |

1 |
Min st-cut of a planar graph in O(n log log n) time
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(Show Context)
Citation Context ...edded on an annulus. Itai and Shiloach described a simple algorithm to find this cycle in O(n 2 log n) time. Their algorithm has been improved several times [42, 25], most recently by Italiano et al. =-=[32, 33, 46]-=-, who describe an algorithm that runs in O(n log log n) time. Thomassen [44] developed the first efficient algorithm for graphs on arbitrary surfaces, which runs in O(n 3 ) time and exploits his so-ca... |