Results 1  10
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12
Many distances in planar graphs
 In SODA ’06: Proc. 17th Symp. Discrete algorithms
, 2006
"... We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilatio ..."
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Cited by 20 (3 self)
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We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs. 1
LinearSpace Approximate Distance Oracles for Planar, BoundedGenus, and MinorFree Graphs
"... Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle construct ..."
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Cited by 12 (6 self)
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Abstract. A (1 + ɛ)approximate distance oracle for a graph is a data structure that supports approximate pointtopoint shortestpathdistance queries. The relevant measures for a distanceoracle construction are: space, query time, and preprocessing time. There are strong distanceoracle constructions known for planar graphs (Thorup) and, subsequently, minorexcluded graphs (Abraham and Gavoille). However, these require Ω(ɛ −1 n lg n) space for nnode graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, boundedgenus graphs, and minorexcluded graphs we give distanceoracle constructions that require only
Exact Distance Oracles for Planar Graphs
, 2010
"... We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative arc lengths and for any ɛ> 0 there is a data structure that supports exact shortest path and d ..."
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Cited by 9 (5 self)
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We provide the first linearspace data structure with provable sublinear query time for exact pointtopoint shortest path queries in planar graphs. We prove that for any planar graph G with nonnegative 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 WulffNilsen (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 shortestpath 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
Shortest path queries in planar graphs in constant time
 In STOC’03
, 2003
"... We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so ..."
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Cited by 9 (2 self)
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We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log V ) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(V ) time provided that the constant bound on the girth is known. Our results can be easily generalized to other wide classes of graphs – for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded treewidth. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems
Submatrix maximum queries in Monge matrices and partial Monge matrices, and their applications
, 2012
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Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
, 2008
"... We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths w ..."
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Cited by 6 (1 self)
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We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our allpairs query algorithms take as input an approximation parameter ε ∈ (0,1) and a query time parameter q, in a certain range, and builds a data structure APQ(P,ε;q), which is then used for answering εapproximate distance queries in O(q) time. As a building block of the APQ(P,ε;q) data structure, we develop a single source query data structure SSQ(a;P,ε) that can answer εapproximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.
Improved distance queries in planar graphs
 In 12th International Symposium on Algorithms and Data Structures (WADS
, 2011
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More compact oracles for approximate distances in undirected planar graphs
 In SODA ’13
, 2013
"... Distance oracles are data structures that provide fast (possibly approximate) answers to shortestpath and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS‘01, Thorup int ..."
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Cited by 4 (2 self)
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Distance oracles are data structures that provide fast (possibly approximate) answers to shortestpath and distance queries in graphs. The tradeoff between the space requirements and the query time of distance oracles is of particular interest and the main focus of this paper. In FOCS‘01, Thorup introduced approximate distance oracles for planar graphs. He proved that, for any > 0 and for any planar graph on n nodes, there exists a (1 + )–approximate distance oracle using space O(n−1 logn) such that approximate distance queries can be answered in time O(−1). Ten years later, we give the first improvements on the space–query time tradeoff for planar graphs. • We give the first oracle having a space–time product with subquadratic dependency on 1/. For space Õ(n logn) we obtain query time Õ(−1) (assuming polynomial edge weights). We believe that the dependency on may be almost optimal. • For the case of moderate edge weights (average bounded by poly(logn), which appears to be the case for many realworld road networks), we hit a “sweet spot, ” improving upon Thorup’s oracle both in terms of and n. Our oracle uses space Õ(n log log n) and it has query time Õ(−1 + log log log n). (Notation: Õ(·) hides lowdegree polynomials in log(1/) and log∗(n).) ar X iv
External Data Structures for Shortest Path Queries on Planar Digraphs
"... Abstract. In this paper we present spacequery tradeoffs for external memory data structures that answer shortest path queries on planar directed graphs. For any S = Ω(N 1+ɛ)andS = O(N 2 /B), our main result is a family of structures that use S space and answer queries in O ( N2 SB) I/Os, thus obta ..."
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Cited by 1 (0 self)
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Abstract. In this paper we present spacequery tradeoffs for external memory data structures that answer shortest path queries on planar directed graphs. For any S = Ω(N 1+ɛ)andS = O(N 2 /B), our main result is a family of structures that use S space and answer queries in O ( N2 SB) I/Os, thus obtaining optimal spacequery product O(N2 /B). An S space structure can be constructed in O ( √ S · sort(N)) I/Os, where sort(N) is the number of I/Os needed to sort N elements, B is the disk block size, and N is the size of the graph. 1
Planar Reachability in Linear Space and Constant Time
, 2014
"... We show how to represent a planar digraph in linear space so that distance queries can be answered in constant time. The data structure can be constructed in linear time. This representation of reachability is thus optimal in both time and space, and has optimal construction time. The previous best ..."
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We show how to represent a planar digraph in linear space so that distance queries can be answered in constant time. The data structure can be constructed in linear time. This representation of reachability is thus optimal in both time and space, and has optimal construction time. The previous best solution used O(n log n) space for constant query time [Thorup FOCS’01].