## Exact Distance Oracles for Planar Graphs (2010)

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Citations: | 3 - 2 self |

### BibTeX

@MISC{Mozes10exactdistance,

author = {Shay Mozes and Christian Sommer},

title = {Exact Distance Oracles for Planar Graphs},

year = {2010}

}

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### Abstract

We provide the first linear-space data structure with provable sublinear query time for exact point-topoint shortest path queries in planar graphs. We prove that for any planar graph G with non-negative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and distance queries in G with the following properties: the data structure can be created in time O(n lg(n) lg(1/ɛ)), the space required is O(n lg(1/ɛ)), and the query time is O(n 1/2+ɛ). Previous data structures by Fakcharoenphol and Rao (JCSS’06), Klein, Mozes, and Weimann (TransAlg’10), and Mozes and Wulff-Nilsen (ESA’10) with query time O(n 1/2 lg 2 n) use space at least Ω(n lg n / lg lg n). We also give a construction with a more general tradeoff. We prove that for any integer S ∈ [n lg n, n 2], we can construct in time Õ(S) a data structure of size O(S) that answers distance queries in O(nS −1/2 lg 2.5 n) time per query. Cabello (SODA’06) gave a comparable construction for the smaller range S ∈ [n 4/3 lg 1/3 n, n 2]. For the range S ∈ (n lg n, n 4/3 lg 1/3 n), only data structures of size O(S) with query time O(n 2 /S) had been known (Djidjev, WG’96). Combined, our results give the best query times for any shortest-path data structure for planar graphs with space S = o(n 4/3 lg 1/3 n). As a consequence, we also obtain an algorithm that computes k–many distances in planar graphs in time O((kn) 2/3 (lg n) 2 (lg lg n) −1/3 + n(lg n) 2 / lg lg n). 1

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Citation Context ... structure, assuming that storing the graph is free. It is known that, for approximate distances, a query algorithm can run efficiently using a data structure that occupies sublinear additional space =-=[KKS11]-=-. For exact distances and sublinear space, nothing better than the linear-time SSSP algorithm [HKRS97] is known. 6 Distance Oracles with Query Time Quasi-Proportional to the Shortest-Path Length We us... |

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Citation Context ... in a planar graph between a set of nodes that lie on a constant number of faces. It does not rely on any other properties of the r–division. FR-Dijkstra can be extended to the following setting (cf. =-=[BSWN10]-=-). Let J be a planar graph. Let n ′ denote the number of nodes of J ∪ H. We can compute shortest paths in H ∪ J in O(|H| lg 2 |H| + n ′ lg n ′ ) time. The edges of H are relaxed using FR-Dijkstra, whi... |

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Citation Context ...csom@csail.mit.edu 1 Asymptotic notation as in Õ(·) suppresses polylogarithmic factors in the number of nodes n. nodes. Indeed, shortest-path query processing is an integral part of many applications =-=[Som10]-=-, in particular in Geographic Information Systems (GIS), intelligent transportation systems [JHR96]. These systems may help individuals in finding fast routes or they may also assist companies in impr... |

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Citation Context ...ts as small as possible, i.e. linear in the size of the input. This is the first linear-space data structure with provably sublinear query time for exact pointto-point shortest-path queries. Nussbaum =-=[Nus11]-=- has simultaneously obtained a similar result. Theorem 1.3. For any directed planar graph G with non-negative arc lengths and for any constant ɛ > 0, there is a data structure that supports exact dist... |

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