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12
Distance oracles for sparse graphs
 IN PROCEEDINGS OF THE 50TH IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
"... Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answer queries i ..."
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Thorup and Zwick, in their seminal work, introduced the approximate distance oracle, which is a data structure that answers distance queries in a graph. For any integer k, they showed an efficient algorithm to construct an approximate distance oracle using space O(kn 1+1/k) that can answer queries in time O(k) with a distance estimate that is at most α = 2k − 1 times larger than the actual shortest distance (α is called the stretch). They proved that, under a combinatorial conjecture, their data structure is optimal in terms of space: if a stretch of at most 2k−1 is desired, then the space complexity is at least n 1+1/k. Their proof holds even if infinite query time is allowed: it is essentially an “incompressibility ” result. Also, the proof only holds for dense graphs, and the best bound it can prove only implies that the size of the data structure is lower bounded by the number of edges of the graph. Naturally, the following question arises: what happens for sparse graphs? In this paper we give a new lower bound for approximate distance oracles in the cellprobe model. This lower bound holds even for sparse (polylog(n)degree) graphs, and it is not an “incompressibility ” bound: we prove a threeway tradeoff between space, stretch and query time. We show that, when the query time is t, and the stretch is α, then the space S must be S ≥ n 1+Ω(1/tα) / lg n. (1) This lower bound follows by a reduction from lopsided set disjointness to distance oracles, based on and motivated by recent work of Pǎtras¸cu. Our results in fact show that for any highgirth regular graph, an approximate distance oracle that supports efficient queries for all subgraphs of G must obey Eq. (1). We also prove some lemmas that count sets of paths in highgirth regular graphs and highgirth regular expanders, which might be of independent interest.
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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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
Balancing degree, diameter and weight in Euclidean spanners
 In Proc. of 18th ESA
, 2010
"... Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) ..."
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Abstract. In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R d and any ɛ>0, provides a(1+ɛ)spanner with diameter O(log n), weight O(log 2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ɛ)spanner with O(n) edges and diameter α(n), where α stands for the inverseAckermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(log n)andweightO(log n)w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1+ɛ)spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n log n)w(MST(S)), and O(n) edges.Fork = O(1) this gives rise to maximum degree O(1), diameter O(log n) andweightO(log 2 n)w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n 1/α(n) (log n)α(n))w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log 2 n, but the diameter is allowed to grow beyond log n.
Fast pruning of geometric spanners
 In Proc. 22nd International Symposium on Theoretical Aspects of Computer Science
, 2005
"... Abstract. Let S be a set of points in R d. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)spanner of G with O(S) edges in time O(E  + S  log S). Previous algorithms require a preliminary step in which the e ..."
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Abstract. Let S be a set of points in R d. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)spanner of G with O(S) edges in time O(E  + S  log S). Previous algorithms require a preliminary step in which the edges are sorted in nondecreasing order of their lengths and, thus, have running time Ω(E  log S). We obtain our result by designing a new algorithm that finds the pair in a wellseparated pair decomposition separating two given query points. Previously, it was known how to answer such a query in O(log S) time. We show how a sequence of such queries can be answered in O(1) amortized time per query. 1
An OptimalTime Construction of Sparse Euclidean Spanners with Tiny Diameter
, 2011
"... In STOC’95 [5] Arya et al. showed that for any set of n points in Rd, a (1 + ǫ)spanner with diameter at most 2 (respectively, 3) and O(nlog n) edges (resp., O(nlog log n) edges) can be built in O(nlog n) time. Moreover, Arya et al. [5] conjectured that one can build in O(nlog n) time a (1+ǫ)spanne ..."
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Cited by 3 (2 self)
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In STOC’95 [5] Arya et al. showed that for any set of n points in Rd, a (1 + ǫ)spanner with diameter at most 2 (respectively, 3) and O(nlog n) edges (resp., O(nlog log n) edges) can be built in O(nlog n) time. Moreover, Arya et al. [5] conjectured that one can build in O(nlog n) time a (1+ǫ)spanner with diameter at most 4 and O(nlog ∗ n) edges. Since then, this conjecture became a central open problem in this area. Nevertheless, very little progress on this problem was reported up to this date. In particular, the previous stateoftheart subquadratictime construction of (1 + ǫ)spanners with o(nlog log n) edges due to Arya et al. [5] produces spanners with diameter 8. In addition, general tradeoffs between the diameter and number of edges were established [5, 26]. Specifically, it was shown in [5, 26] that for any k ≥ 4, one can build in O(n(log n)2kαk(n)) time a (1+ǫ)spanner with diameter at most 2k and O(n2kαk(n)) edges. The function αk is the inverse of a certain Ackermannstyle function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α0(n) = ⌈n/2⌉,α1(n) = ⌈ √ n ⌉,α2(n) = ⌈log n⌉,α3(n) = ⌈log log n⌉,α4(n) = log ∗ n,α5(n) = ⌊ 1 2 log ∗ n⌋,..., etc. It is also known [26] that if one allows quadratic time then these bounds can be improved. Specifically, for any k ≥ 4, a (1 + ǫ)spanner with diameter at most k and O(nkαk(n)) edges can be constructed in O(n2) time [26]. A major open question in this area is whether one can construct within time O(nlog n+nkαk(n)) a (1+ǫ)spanner with diameter at most k and O(nkαk(n)) edges. This question in the particular case of k = 4 coincides with the aforementioned conjecture of Arya et al. [5]. In this paper we answer this longstanding question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any k ≥ 4, a (1 + ǫ)spanner with diameter at most k and O(nαk(n)) edges can be built in optimal time
Fast, precise and dynamic distance queries
"... We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε> 0, the oracle supports (1 + ε)approximate distance queries in (universal) constant time, occupies space [ε −O(λ) + 2 O(λ log λ)]n, and can be constructed in [2 O(λ) log 3 n+ε −O(λ) +2 ..."
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We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε> 0, the oracle supports (1 + ε)approximate distance queries in (universal) constant time, occupies space [ε −O(λ) + 2 O(λ log λ)]n, and can be constructed in [2 O(λ) log 3 n+ε −O(λ) +2 O(λ log λ)]n expected time. This improves upon the best previously known constructions, presented by HarPeled and Mendel [13]. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2O(λ) log n+ε−O(λ) O(λ log λ) +2 update time. This is the first fully dynamic (1 + ε)distance oracle. 1
Practical Route Planning Under Delay Uncertainty: Stochastic Shortest Path Queries
"... Abstract—We describe an algorithm for stochastic path planning and applications to route planning in the presence of traffic delays. We improve on the prior state of the art by designing, analyzing, implementing, and evaluating data structures that answer approximate stochastic shortestpath queries ..."
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Abstract—We describe an algorithm for stochastic path planning and applications to route planning in the presence of traffic delays. We improve on the prior state of the art by designing, analyzing, implementing, and evaluating data structures that answer approximate stochastic shortestpath queries. For example, our data structures can be used to efficiently compute paths that maximize the probability of arriving at a destination before a given time deadline. Our main theoretical result is an algorithm that, given a directed planar network with edge lengths characterized by expected travel time and variance, precomputes a data structure in quasilinear time such that approximate stochastic shortestpath queries can be answered in polylogarithmic time (actual worstcase bounds depend on the probabilistic model). Our main experimental results are twofold: (i) we provide methods to extract traveltime distributions from a large set of heterogenous GPS traces and we build a stochastic model of an entire city, and (ii) we adapt our algorithms to work for realworld road networks, we provide an efficient implementation, and we evaluate the performance of our method for the model of the aforementioned city. I.
Approximate Distance Oracles for Graphs with Dense Clusters
, 2004
"... Let H1 = (V, F1) be a collection of N pairwise vertex disjoint O(1)spanners where the weight of an edge is equal to the Euclidean distance between its endpoints. Let H2 = (V, F2) be a graph on V with M edges of nonnegative weight. The union of the two graphs is denoted G = (V, F1 ∪ F2). We prese ..."
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Let H1 = (V, F1) be a collection of N pairwise vertex disjoint O(1)spanners where the weight of an edge is equal to the Euclidean distance between its endpoints. Let H2 = (V, F2) be a graph on V with M edges of nonnegative weight. The union of the two graphs is denoted G = (V, F1 ∪ F2). We present a data structure of size O(M 2 + V  log V) that answers (1 + ε)approximate shortest path queries in G in constant time, where ε> 0 is constant.