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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)).
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.
Ultrametric Subsets with Large . . .
"... 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 ..."
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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.
Distance Oracles for Stretch Less Than 2 Abstract
"... We present distance oracles for weighted undirected graphs that return distances of stretch less than 2. For the realistic case of sparse graphs, our distance oracles exhibit a smooth threeway tradeoff between space, stretch and query time — a phenomenon that does not occur in dense graphs. In par ..."
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We present distance oracles for weighted undirected graphs that return distances of stretch less than 2. For the realistic case of sparse graphs, our distance oracles exhibit a smooth threeway tradeoff between space, stretch and query time — a phenomenon that does not occur in dense graphs. In particular, for any positive integer t and for any 1≤α ≤ n, our distance oracle is of size O(m+n 2 /α) and returns distances of stretch at most(1+ 2 t+1) in time O((αµ)t), whereµ=2m/n is the average degree of the graph. The query time can be further reduced to O((α+µ) t) at the expense of a small additive stretch. 1
Prioritized Metric Structures and Embedding
"... Metric data structures (distance oracles, distance labeling schemes, routing schemes) and lowdistortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [LLR95], online algorithms [BBMN11], distributed algorithms [KKM+12] and fo ..."
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Metric data structures (distance oracles, distance labeling schemes, routing schemes) and lowdistortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [LLR95], online algorithms [BBMN11], distributed algorithms [KKM+12] and for computing sparsifiers [ST04]. However, this methodology appears to have a limitation: the worstcase performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices/points should enjoy a better service (i.e., stretch/distortion, label size/dimension) than that given by the worstcase guarantee. In this paper we alleviate this limitation by devising a suit of prioritized metric data structures and embeddings. We show that given a priority ranking (x1, x2,..., xn) of the graph vertices (respectively, metric points) one can devise a metric data structure (respectively, embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex xj will depend on the rank j of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on j. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) simultaneously. The worstcase performance of our metric data structures and embeddings is typically asymptotically no worse than of their nonprioritized counterparts.
Distance Oracles for TimeDependent Networks
"... Abstract. We present the first approximate distance oracle for sparse directed networks with timedependent arctraveltimes determined by continuous, piecewise linear, positive functions possessing the FIFO property. Our approach precomputes (1 + ε)−approximate distance summaries from selected la ..."
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Abstract. We present the first approximate distance oracle for sparse directed networks with timedependent arctraveltimes determined by continuous, piecewise linear, positive functions possessing the FIFO property. Our approach precomputes (1 + ε)−approximate distance summaries from selected landmark vertices to all other vertices in the network, and provides two sublineartime query algorithms that deliver constant and (1+σ)−approximate shortesttraveltimes, respectively, for arbitrary origindestination pairs in the network. Our oracle is based only on the sparsity of the network, along with two quite natural assumptions about traveltime functions which allow the smooth transition towards asymmetric and timedependent distance metrics. 1
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"... Abstract. 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 ..."
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Abstract. 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.