#### DMCA

## Randomly Removing g Handles at Once (2009)

Citations: | 11 - 2 self |

### Citations

370 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
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Citation Context ...s that give rise to improved algorithmic solutions for numerous graph problems, if one restricts the set of possible inputs to planar graphs (see, for example, applications to maximum independent set =-=[1]-=- and computing maximum flows [2]). One natural generalization of planarity involves the genus of a graph. Informally, a graph has genus g, for some g ≥ 0, if it can be drawn without any crossings on t... |

350 | Probabilistic approximations of metric spaces and its algorithmic applications
- Bartal
- 1996
(Show Context)
Citation Context ...or quality of the solution (e.g. [3]). Unfortunately, many such extensions are complicated and based on ad-hoc techniques. Inspired by Bartal’s probabilistic approximation of general metrics by trees =-=[4]-=-, Sidiropoulos and Indyk [5] showed that every metric on a graph of genus g can be probabilistically approximated by a planar graph metric with distortion at most exponential in g. (See Section 1.1 fo... |

306 |
Lectures on discrete geometry
- Matoušek
- 2002
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Citation Context ...tic D-embedding into the family of metric graphs G. We will write F D G if we wish to emphasize the particular constant. A detailed exposition of results on metric embeddings can be found in [12] and =-=[13]-=-. 2 Random planarization We show that every metric graph of orientable or non-orientable genus g embeds into a distribution over planar graph metrics with distortion at most O(g2). 2.1 The peeling lem... |

301 | A tight bound on approximating arbitrary metrics by tree metrics
- Fakcharoenphol, Rao, et al.
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Citation Context ...ding a fixed minor. This completes the proof, except that H itself might not be planar. However, H does have small Euler number1 and so admits a probabilistic embedding into a distribution over trees =-=[9, 10]-=-. In Section 2.3, we combine these ingredients to provide a probabilistic embedding with distortion O(g2). In Section 3, we show that any such probabilistic embedding incurs at least Ω(log g) distorti... |

289 |
Graphs on surfaces
- Mohar, Thomassen
- 2001
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Citation Context ...ss otherwise stated, we restrict our attention to finite graphs. Graphs on surfaces Let us recall some notions from topological graph theory (an in-depth exposition is provided by Mohar and Thomassen =-=[11]-=-). A surface is a compact connected 2-dimensional manifold, without boundary. For a graph G we can define a one-dimensional simplicial complex C associated with G as follows: The 0-cells of C are the ... |

126 | Algorithmic applications of low-distortion geometric embeddings, in
- Indyk
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Citation Context ...a stochastic D-embedding into the family of metric graphs G. We will write F D G if we wish to emphasize the particular constant. A detailed exposition of results on metric embeddings can be found in =-=[12]-=- and [13]. 2 Random planarization We show that every metric graph of orientable or non-orientable genus g embeds into a distribution over planar graph metrics with distortion at most O(g2). 2.1 The pe... |

120 | Excluded minors, network decomposition, and multicommodity flow
- Klein, Plotkin, et al.
- 1993
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Citation Context ...of pairs of vertices in G small. The lemma requires an appropriate random partition of the shortest-path metric on G. Such a procedure is provided by the fundamental result of Klein, Plotkin, and Rao =-=[8]-=- for partitioning graphs excluding a fixed minor. This completes the proof, except that H itself might not be planar. However, H does have small Euler number1 and so admits a probabilistic embedding i... |

67 | Small distortion and volume preserving embeddings for planar and Euclidean metrics
- Rao
- 1999
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Citation Context ...c space (X, d), we write β(X,d) for the infimal β such that X admits a (β,∆)–Lipschitz random partition for every ∆ > 0, and we refer to β(X,d) as the decomposability modulus of X. The results of Rao =-=[14]-=- and Klein, Plotkin, and Rao [8] yield the following for the special case of bounded-genus metrics. (The stated quantitative dependence is due to [15]; see also [16, Cor. 3.15].) Theorem 1 (KPR Decomp... |

59 | distortion and volume preserving embeddings for planar and Euclidean metrics - Small - 1999 |

59 | Extending lipschitz functions via random metric partitions - Lee, Naor - 2005 |

55 | Optimally cutting a surface into a disk
- Erickson, Har-Peled
- 2002
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Citation Context ...oving handles one at a time incurs an exponential loss in distortion, we look for a way to remove all the handles at once. Our starting point is the minimum-length cut graph of Erickson and Har-Peled =-=[6]-=-. Given a graph G the minimum-length cut graph is (roughly speaking) a minimum-length subgraph H of G such that G \H is planar. In Section 2.2, we show that this H is nearly geodesically closed in tha... |

29 | Cuts, trees and l1embeddings of graphs - Gupta, Newman, et al. |

23 | An improved decomposition theorem for graphs excluding a fixed minor
- Fakcharoenphol, Talwar
- 2003
(Show Context)
Citation Context ... decomposability modulus of X. The results of Rao [14] and Klein, Plotkin, and Rao [8] yield the following for the special case of bounded-genus metrics. (The stated quantitative dependence is due to =-=[15]-=-; see also [16, Cor. 3.15].) Theorem 1 (KPR Decomposition). If G = (V,E) is a metric graph of orientable or non-orientable genus g ≥ 0, then β(V,dG) = O(g + 1). 3 The dilation and modulus is used in t... |

18 | Polynomial-time approximation schemes for subsetconnectivity problems in bounded-genus graphs. Algorithmica
- Borradaile, Demaine, et al.
- 2012
(Show Context)
Citation Context ... exhibit nice algorithmic properties. More precisely, algorithms for planar graphs can usually be extended to graphs of bounded genus, with a small loss in efficiency or quality of the solution (e.g. =-=[3]-=-). Unfortunately, many such extensions are complicated and based on ad-hoc techniques. Inspired by Bartal’s probabilistic approximation of general metrics by trees [4], Sidiropoulos and Indyk [5] show... |

17 | Probabilistic embeddings of bounded genus graphs into planar graphs
- Indyk, Sidiropoulos
- 2007
(Show Context)
Citation Context ...e.g. [3]). Unfortunately, many such extensions are complicated and based on ad-hoc techniques. Inspired by Bartal’s probabilistic approximation of general metrics by trees [4], Sidiropoulos and Indyk =-=[5]-=- showed that every metric on a graph of genus g can be probabilistically approximated by a planar graph metric with distortion at most exponential in g. (See Section 1.1 for a formal definition of pro... |

12 |
Exploiting Planarity for Network Flow and Connectivity Problems
- Borradaile
- 2008
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Citation Context ...orithmic solutions for numerous graph problems, if one restricts the set of possible inputs to planar graphs (see, for example, applications to maximum independent set [1] and computing maximum flows =-=[2]-=-). One natural generalization of planarity involves the genus of a graph. Informally, a graph has genus g, for some g ≥ 0, if it can be drawn without any crossings on the surface of a sphere with g ad... |

10 | On the geometry of graphs with a forbidden minor
- Lee, Sidiropoulos
- 2009
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Citation Context ... geodesic-closure property suggests that if we could randomly shift H, then the distortion of all pairs of vertices in G would be fine in expectation. We use the Peeling Lemma of Lee and Sidiropoulos =-=[7]-=- to perform the random shifting (Section 2.1). The Peeling Lemma allows one to randomly embed G into a graph consisting of copies of G \ H hanging off an isomorphic copy of H, while keeping the expect... |

6 | Lower bounds for embedding into distributions over excluded minor graph families
- Carroll, Goel
- 2004
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Citation Context ...graph obtained after replacing each edge of a K5 by a path of length g 100 . It follows that we can embed J into G with distortion Θ(1), such that the image of V (J) is contained in V (H). Lemma 1 of =-=[17]-=- states that any embedding of a graph Γ1 containing a metric copy of the 9 k-subdivision of a graph Γ2, into a graph Γ3 that excludes Γ2 as a minor, has distortion Ω(k). Therefore, any embedding of J ... |