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Exploring Generic
 Proceedings of the IFIP TC2 Working Conference on Generic Programming, Schloss Dagstuhl
, 2004
"... doi:10.1182/blood200902204800 ..."
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
Improved distance oracles and spanners for vertexlabeled graphs
 of Lecture Notes in Computer Science
, 2012
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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.
Models and Techniques for Proving Data Structure Lower Bounds
, 2013
"... In this dissertation, we present a number of new techniques and tools for proving lower bounds on the operational time of data structures. These techniques provide new lines of attack for proving lower bounds in both the cell probe model, the group model, the pointer machine model and the I/Omodel. ..."
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In this dissertation, we present a number of new techniques and tools for proving lower bounds on the operational time of data structures. These techniques provide new lines of attack for proving lower bounds in both the cell probe model, the group model, the pointer machine model and the I/Omodel. In all cases, we push the frontiers further by proving lower bounds higher than what could possibly be proved using previously known techniques. For the cell probe model, our results have the following consequences: • The first Ω(lg n) query time lower bound for linear space static data structures. The highest previous lower bound for any static data structure problem peaked at Ω(lg n / lg lg n). • An Ω((lg n / lg lg n) 2) lower bound on the maximum of the update time and the query time of dynamic data structures. This is almost a quadratic improvement over the highest previous lower bound of Ω(lg n). In the group model, we establish a number of intimate connections to the fields of combinatorial discrepancy and range reporting in the pointer machine
Faster Approximate Distance Queries and Compact Routing in Sparse Graphs
, 2012
"... A distance oracle is a compact representation of the shortest distance matrix of a graph. It can be queried to retrieve approximate distances and corresponding paths between any pair of vertices. A lower bound, due to Thorup and Zwick, shows that a distance oracle that returns paths of worstcase st ..."
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A distance oracle is a compact representation of the shortest distance matrix of a graph. It can be queried to retrieve approximate distances and corresponding paths between any pair of vertices. A lower bound, due to Thorup and Zwick, shows that a distance oracle that returns paths of worstcase stretch (2k−1) must require spaceΩ(n 1+1/k) for graphs over n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degreeΩ(n 1/k). We present distance oracles that, for sparse graphs, substantially break the lower bound barrier at the expense of higher query time. For any 1≤α ≤ n, our distance oracles can return stretch 2 paths using O(m+ n 2 /α) space and stretch 3 paths using O(m+n 2 /α 2) space, at the expense of O(αm/n) query time. By setting appropriate values ofα, we get the first distance oracles that have size linear in the size of the graph, and return constant stretch paths in nontrivial query time. The query time can be further reduced to O(α), by using an additional O(mα) space for all our distance oracles, or at the cost of a small constant additive stretch. We use our stretch 2 distance oracle to design a compact routing scheme that requires Õ(n 1/2) memory at each node and, after a handshaking phase, routes along paths with worstcase stretch 2. Moreover, supported by largescale simulations on graphs including the ASlevel Internet graph, we argue that our stretch2 scheme would be simple and efficient to implement as a distributed compact routing protocol. An earlier version of this paper appeared in INFOCOM 2011[1]. The extended version presents results that improve upon the results presented in the conference version; significantly more simplified presentation and proofs for the results in the conference version; and in addition, distance oracles for unweighted graphs.
A LinearSize Logarithmic Stretch PathReporting Distance Oracle for General Graphs
, 2014
"... In a seminal paper [27] for any nvertex undirected graph G = (V,E) and a parameter k = 1, 2,..., Thorup and Zwick constructed a distance oracle of size O(kn1+1/k) which upon a query (u, v) constructs a path Π between u and v of length δ(u, v) such that dG(u, v) ≤ δ(u, v) ≤ (2k−1)dG(u, v). The que ..."
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In a seminal paper [27] for any nvertex undirected graph G = (V,E) and a parameter k = 1, 2,..., Thorup and Zwick constructed a distance oracle of size O(kn1+1/k) which upon a query (u, v) constructs a path Π between u and v of length δ(u, v) such that dG(u, v) ≤ δ(u, v) ≤ (2k−1)dG(u, v). The query time of the oracle from [27] is O(k) (in addition to the length of the returned path), and it was subsequently improved to O(1) [29, 11]. A major drawback of the oracle of [27] is that its space is Ω(n · log n). Mendel and Naor [18] devised an oracle with space O(n1+1/k) and stretch O(k), but their oracle can only report distance estimates and not actual paths. In this paper we devise a pathreporting distance oracle with size O(n1+1/k), stretch O(k) and query time O(n), for an arbitrarily small > 0. In particular, for k = log n our oracle provides logarithmic stretch using linear size. Another variant of our oracle has linear size, polylogarithmic stretch, and query time O(log log n). For unweighted graphs we devise a distance oracle with multiplicative stretch O(1), additive stretch O(β(k)), for a function β(), space O(n1+1/k · β), and query time O(n), for an arbitrarily small constant > 0. The tradeoff between multiplicative stretch and size in these oracles is far below Erdős’s girth conjecture threshold (which is stretch 2k − 1 and size O(n1+1/k)).
Logarithmic Cell Probe Lower Bounds for NonDeterministic Static Data Structures
"... In this paper, we present a new technique for proving static cell probe lower bounds. Our technique takes the field of static cell probe lower bounds one step further, by yielding the highest lower bound to date for any explicit problem, namely t = Ω((w −lg n) / lg(S/n)), where w is the cell size in ..."
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In this paper, we present a new technique for proving static cell probe lower bounds. Our technique takes the field of static cell probe lower bounds one step further, by yielding the highest lower bound to date for any explicit problem, namely t = Ω((w −lg n) / lg(S/n)), where w is the cell size in bits, n the input size, S the space of the data structure in number of cells, and t the cell probes needed to answer a query. Thus for linear space data structures we achieve t = Ω(lg n) when the cell size is just w ≥ (1 + ε) lg n for any constant ε> 0. Furthermore, our bounds also apply to nondeterministic static data structures, providing the first nontrivial lower bounds in the most natural setting of cell size w = Θ(lg n). Finally we believe our new technique sheds much new light on the seemingly inherent lower bound barrier of Ω(w), and we hope our results eventually may inspire ways of overcoming the barrier.
Ultrametric Subsets with . . .
"... It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) � (1 − ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via ..."
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It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) � (1 − ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.