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## Many distances in planar graphs (2006)

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Venue: | In SODA ’06: Proc. 17th Symp. Discrete algorithms |

Citations: | 19 - 3 self |

### Citations

456 | A separator theorem for planar graphs
- Lipton, Tarjan
- 1977
(Show Context)
Citation Context ...y may have too many vertices. The algorithm of Frederickson [9] and Goodrich [11] do not have control over the number of boundary walks because they rely on the separator theorem of Lipton and Tarjan =-=[19]-=-. Our algorithm to construct an r-decomposition with a few holes is a slight modification of the approach by Frederickson [9]. Namely, weachievecontrol over thetotal numberofboundarywalksusingthecycle... |

199 | S.: Faster shortest-path algorithms for planar graphs
- Henzinger, Klein, et al.
- 1997
(Show Context)
Citation Context ...og 2 n) preprocessing time. Using this data structure, the many distances problem can be solved in O(nlog 2 n + k √ nlog 2 n) time. Our approach is better for k = ω((n/logn) 5/6 ). • Henzinger et al. =-=[13]-=- show that the single source shortest path problem can be solved in linear time. Applyingthis k times gives the best known solution for the many distances problem when k = O(log 2 n). • Chen and Xu [5... |

148 | Fast algorithms for shortest paths in planar graphs, with applications
- Frederickson
- 1987
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Citation Context ... two vertices of ∂B that is otherwise disjoint from ∂B is contained in B or in one of its holes H(B,W j ). 2.3 Decompositions The following definition is a variation on the definition by Frederickson =-=[9]-=-, where we require that each piece is connected and also consider the number of boundary walks. Definition 1 Given a parameter r ∈ (0,n) and a graph G, an r-decomposition with a few holes consists of ... |

145 | Spanning trees and spanners
- Eppstein
- 2000
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Citation Context ...e |·| denotes the Euclidean distance. This parameter measures how well the distances in the graph resemble the Euclidean distances, and is the key parameter for the construction of geometric spanners =-=[7]-=-. Narasimhan and Smid have shown the following result. Theorem 15 [21] Given a Euclidean graph G in R d , and a parameter 3 ≥ ε > 0, computing a value t such that t ≤ tG ≤ (1 + ε)t takes O(nlogn) time... |

135 |
Design of Dynamic Data Structures
- Overmars
- 1987
(Show Context)
Citation Context ...ma 11 shows that it takes O( √ rlogn) = O((n/ √ S) √ lognlogn) = O((n/ √ S)log 3/2 n) time to answer a query. □ □ 5 Many distances in planar graphs Using Theorem 12 and standard rebuilding techniques =-=[22]-=- we obtain the following result. Theorem 13 Let G be a planar graph of size n. The distance between k pairs of vertices in G that are given online can be computed in O(k 2/3 n 2/3 logn+n 4/3 log 1/3 n... |

104 |
Compact Oracles for Reachability and Approximate Distances in Planar Digraphs
- Thorup
- 2001
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Citation Context ...urowski[16] and references therein. The distance between all pairs of vertices in planar graphs was first solved optimally by Frederickson [10]. For approximate distances in planar graphs, see Thorup =-=[24]-=- and references therein. Finally, our result relies heavily on the recent result by 3(a) (b) (c) Figure 1: (a) A plane graph B with a cycle C in bold. The gray region is the interior of C. (b) The gr... |

99 |
Finding small simple cycle separators for 2-connected planar graphs
- Miller
- 1986
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Citation Context ...alksusingthecycle-separator 5Figure 3: From left to right: a face in a piece B; the face in the supergraph B ′ ; a possible cycle used to split B ′ ; how the split affects the face. result of Miller =-=[20]-=-, instead of the separator theorem of Lipton and Tarjan [19]. While our construction needs O(nlog(n/r)) time, the original algorithm of Goodrich [11] achieves linear time using several data structures... |

67 | Planar graphs, negative weight edges, shortest paths, and near linear time
- Fakcharoenphol, Rao
- 2001
(Show Context)
Citation Context ... structure, the many distances problem can be solved in O(n 5/3 +nk 1/2 log 1/2 n) time if k < n 3/2 /logn or O(kn 1/4 logn) time otherwise. Our approach is better for any k. • Fakcharoenphol and Rao =-=[8]-=-, with the logarithmic improvement by Klein [15], give a data structure that answers distance queries in O( √ nlog 2 n) time per query after O(nlog 2 n) preprocessing time. Using this data structure, ... |

67 | Exact and approximate distances in graphs - a survey
- Zwick
- 2001
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Citation Context ...hs can be arbitrary real, non-negative values and we only compare values that come from sums of edge-lengths. In the RAM model of computation, some logarithmic improvements may be possible; see Zwick =-=[25]-=- for a discussion. For the last application, computing the stretch factor of geometric graphs, we have to compare sums of square roots. Therefore, we assume a Real RAM model of computation, as it is s... |

62 |
Multiple-source shortest paths in planar graphs
- Klein
- 2005
(Show Context)
Citation Context ...olved in O(n 5/3 +nk 1/2 log 1/2 n) time if k < n 3/2 /logn or O(kn 1/4 logn) time otherwise. Our approach is better for any k. • Fakcharoenphol and Rao [8], with the logarithmic improvement by Klein =-=[15]-=-, give a data structure that answers distance queries in O( √ nlog 2 n) time per query after O(nlog 2 n) preprocessing time. Using this data structure, the many distances problem can be solved in O(nl... |

54 | Planar separators and parallel polygon triangulation
- Goodrich
- 1995
(Show Context)
Citation Context ...dary walks of Bi. In the following we describe an algorithm to construct an r-decomposition with a few holes in O(nlog(n/r)) time. Similar approaches have been described by Frederickson [9], Goodrich =-=[11]-=-, and Henzinger et al. [13], but they do not achieve all the properties listed in our definition of r-decomposition with a few holes. The algorithm of Henzinger et al. [13] produces pieces whose bound... |

45 | Approximating the stretch factor of Euclidean graphs
- Narasimhan, Smid
(Show Context)
Citation Context ...l the distances in the graph resemble the Euclidean distances, and is the key parameter for the construction of geometric spanners [7]. Narasimhan and Smid have shown the following result. Theorem 15 =-=[21]-=- Given a Euclidean graph G in R d , and a parameter 3 ≥ ε > 0, computing a value t such that t ≤ tG ≤ (1 + ε)t takes O(nlogn) time plus the time to compute the distance between O(ε −d n) pairs of vert... |

44 | Planar spanners and approximate shortest path queries among obstades
- Arikati, Chen, et al.
- 1996
(Show Context)
Citation Context ...∈ [n 3/2 ,n 2 ] there is a data structure of size O(S) that answers distance queries in O(n 2 /S) time per query after O(S) preprocessing time. The case S = n 3/2 was also described by Arikati et al. =-=[1]-=-. Our data structure is better when S = o(n 2 /log 3 n). Using this data structure, the many distances problem can be solved in O(n 3/2 + nk 1/2 ) time. Our approach is better when k = o(n 2 /log 6 n)... |

41 | All-pairs shortest paths with real weights in O (n 3/log n) time
- Chan
- 2005
(Show Context)
Citation Context ...known because we currently do not know how to find such an apex in linear time. We will need to compute the distance between all pairs of vertices in general graphs. The following result is from Chan =-=[4]-=-. 7Theorem 6 Given a graph G with n vertices, we can compute the values dG(u,u ′ ) for all pairs of vertices (u,u ′ ) ∈ V(G)×V(G) in O(n 3 /logn) time and space. Finally, we will also use the followi... |

39 | N.: Planar graph decomposition and all pairs shortest paths
- Frederickson
- 1991
(Show Context)
Citation Context ...log 2 n). • Chen and Xu [5] give data structures that depend on a parameter measuring the minimum number of faces over the planar embeddings of the graph, a parameter first introduced by Frederickson =-=[10]-=-. In the worst case, this parameter is linear, and for any value S ∈ [n 4/3 ,n 2 ], Chen and Xu give a data structure of size O(S) requiring O(n 3 /S) preprocessing time if S ≤ n 3/2 and O(n √ S) prep... |

35 | Multiple source shortest paths in a genus g graph
- Cabello, Chambers
- 2007
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Citation Context ...evious algorithms for finding a shortest non-contractible cycle in a graph embedded in a surface of bounded genus, orientable or not. For orientable surfaces, these results have been already improved =-=[3, 17]-=- using different techniques. The full version of [3] shall also treat non-orientable surfaces. 1.1 Our results and roadmap Let G be a planar graph with n vertices and non-negative edge-lengths. Our ma... |

35 | Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time
- Kutz
- 2006
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Citation Context ...evious algorithms for finding a shortest non-contractible cycle in a graph embedded in a surface of bounded genus, orientable or not. For orientable surfaces, these results have been already improved =-=[3, 17]-=- using different techniques. The full version of [3] shall also treat non-orientable surfaces. 1.1 Our results and roadmap Let G be a planar graph with n vertices and non-negative edge-lengths. Our ma... |

31 | Shortest paths in directed planar graphs with negative lengths: A linear-space O(n log 2 n)-time algorithm
- Klein, Mozes, et al.
(Show Context)
Citation Context ... this, we just convert the problem to a problem involving positive weights by computing all the distances from an arbitrary source in O(nlog 2 n) time and then defining a feasible price function; see =-=[14]-=- or [8]. The main open problem concerns the complexity of the many distances problem. In particular, can it be solved in roughly O(n+k) time? Near-linear time answers are known only when k = O( √ n) o... |

23 |
Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation
- Tazari, Müller-Hannemann
(Show Context)
Citation Context ...ime [19]. Apex graphs are closed under taking subgraphs: a subgraph of an apex graph is an apex graph. Thus, the algorithm by Henzinger et al. [13] implies the result. See Tazari and Müller-Hannemann =-=[23]-=- for an alternative approach. □ It is unclear if this result holds when the apex is unknown because we currently do not know how to find such an apex in linear time. We will need to compute the distan... |

12 | Approximate distance oracles for geometric spanners
- Gudmundsson, Levcopoulos, et al.
(Show Context)
Citation Context ... running time of O(n 3/2 ) [21]. On the other hand, if we restrict ourselves to families of graphs whose dilations tG are boundedby a constant, and we also consider ε to beconstant, Gudmundssonet al. =-=[12]-=- have shown how to compute in O(nlogn) time a value t such that t ≤ tG ≤ (1+ε)t. This latter result is applicable to arbitrary graphs. |uv| } , 7 Discussion We have presented data structures and algor... |

12 | Shortest path queries in planar graphs
- Chen, Xu
- 2000
(Show Context)
Citation Context ...al. [12] show that a distance in a planar graph can be computed in linear time. Applying this k times gives the best known solution for the k-many distances problem when k = O(log 2 n). • Chen and Xu =-=[5]-=- give data structures that depend on a parameter measuring the minimum number of faces over the planar embeddings of the graph, a parameter first introduced by Frederickson [10]. In the worst scenario... |

8 | M.: Short path queries in planar graphs in constant time
- Kowalik, Kurowski
- 2003
(Show Context)
Citation Context ...ur data structure has a better trade-off between construction and query time when many distance queries have to be answered. Fordistancesinplanargraphswithsmallintegeredge-lengthsseeKowalikandKurowski=-=[16]-=- and references therein. The distance between all pairs of vertices in planar graphs was first solved optimally by Frederickson [10]. For approximate distances in planar graphs, see Thorup [24] and re... |

7 |
Efficient algorithms for shortest path problems on planar digraphs
- Djidjev
(Show Context)
Citation Context ...g time then this data structure can be used to solve the many distances problem in T +kQ time. When T and Q depend on a parameter, we choose it so as to minimize T +kQ as a function of k. 2• Djidjev =-=[6]-=- uses planar separators to give the following data structures: – For a parameter S ∈ [n 3/2 ,n 2 ] there is a data structure of size O(S) that answers distance queries in O(n 2 /S) time per query afte... |

1 |
Chenand J.Xu. Shortestpathqueries inplanar graphs. InSTOC ’00
- Z
- 2000
(Show Context)
Citation Context ...3] show that the single source shortest path problem can be solved in linear time. Applyingthis k times gives the best known solution for the many distances problem when k = O(log 2 n). • Chen and Xu =-=[5]-=- give data structures that depend on a parameter measuring the minimum number of faces over the planar embeddings of the graph, a parameter first introduced by Frederickson [10]. In the worst case, th... |