Approximate distance oracles
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name "oracle", and it provides faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
Abstract: Let G = (V,E) be an undirected weighted graph with vVv = n and vEv = m. Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O(kmn1/k) expected time, constructing a data structure of size O(kn1p1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k−1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdos, implies that Ω(n1p1/k) space is needed in the worst case for any real stretch strictly smaller than 2kp1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O(n1p1/k) space had a query time of Ω(n1/k).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.
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05 May 2011
TL;DR: An implementation of the labeling algorithm that is faster than any existing method on continental road networks and has the best time bounds is described.
Abstract: Abraham et al. [SODA 2010] have recently presented a theoretical analysis of several practical point-to-point shortest path algorithms based on modeling road networks as graphs with low highway dimension. They also analyze a labeling algorithm. While no practical implementation of this algorithm existed, it has the best time bounds. This paper describes an implementation of the labeling algorithm that is faster than any existing method on continental road networks.
274 citations
TL;DR: This survey reviews selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time.
Abstract: We consider the point-to-point (approximate) shortest-path query problem, which is the following generalization of the classical single-source (SSSP) and all-pairs shortest-path (APSP) problems: we are first presented with a network (graph). A so-called preprocessing algorithm may compute certain information (a data structure or index) to prepare for the next phase. After this preprocessing step, applications may ask shortest-path or distance queries, which should be answered as fast as possible.Due to its many applications in areas such as transportation, networking, and social science, this problem has been considered by researchers from various communities (sometimes under different names): algorithm engineers construct fast route planning methods; database and information systems researchers investigate materialization tradeoffs, query processing on spatial networks, and reachability queries; and theoretical computer scientists analyze distance oracles and sparse spanners. Related problems are considered for compact routing and distance labeling schemes in networking and distributed computing and for metric embeddings in geometry as well.In this survey, we review selected approaches, algorithms, and results on shortest-path queries from these fields, with the main focus lying on the tradeoff between the index size and the query time. We survey methods for general graphs as well as specialized methods for restricted graph classes, in particular for those classes with arguable practical significance such as planar graphs and complex networks.
249 citations
17 Jan 2010
TL;DR: The notion of highway dimension is introduced and it is shown how low highway dimension gives a unified explanation for several seemingly different algorithms.
Abstract: Computing driving directions has motivated many shortest path heuristics that answer queries on continental scale networks, with tens of millions of intersections, literally instantly, and with very low storage overhead. In this paper we complement the experimental evidence with the first rigorous proofs of efficiency for many of the heuristics suggested over the past decade. We introduce the notion of highway dimension and show how low highway dimension gives a unified explanation for several seemingly different algorithms.
220 citations
11 Jul 2005
TL;DR: The first deterministic linear time algorithm for constructing optimal spanners of weighted graphs is obtained by derandomizing the O(km) expected time algorithm of Baswana and Sen for constructing (2k–1)-spanners of size O(kn) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.
Abstract: Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn$^{\rm 1/{\it k}}$) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn$^{\rm 1+1/{\it k}}$) capable of returning in O(k) time an approximation $\hat{\delta}(u,v)$ of the distance δ(u,v) from u to v in G that satisfies $\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)$, for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn$^{\rm 1+1/{\it k}}$log n). We present here a deterministic O(kmn$^{\rm 1/{\it k}}$) time algorithm for constructing oracles of size O(kn$^{\rm 1+1/{\it k}}$). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor.
Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn$^{\rm 1+1/{\it k}}$) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.
202 citations
Cites background or methods or result from "Approximate distance oracles"
...The deterministic construction in [16] first computes exact APSP in Õ(mn) time, and then uses the complete distance matrix to derandomize the randomized construction algorithm....
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...We present here two independent extensions of the result of [16]....
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...We then move on to solve a major open problem raised in [16], namely the development of deterministic algorithms for constructing approximate distance oracles that are almost as efficient as the randomized ones....
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...We present here an extension of the approximate distance oracle construction of [16]....
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...Thus we get a deterministic Õ(mn) time algorithm for constructing stretch 2k − 1 approximate distance oracles of size O(kn), solving the problem from [16]....
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TL;DR: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest- Paths problem.
Abstract: We obtain the following results related to dynamic versions of the shortest-paths problem:
Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems.
A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of $\tilde {O}(m\sqrt{n})$ (we use $\tilde {O}$ to hide small poly-logarithmic factors) and a worst case query time is O(n3/4).
A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance.
200 citations
References
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to
22,704 citations
"Approximate distance oracles" refers methods in this paper
...Since this algorithm is rather complicated, we describe instead a modified version of Dijkstra’s classical SSSP algorithm [Dijkstra 1959] (see also Cormen et al....
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Book•
01 Jan 1991
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Abstract: The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.
6,594 citations
"Approximate distance oracles" refers background in this paper
...For more details, see Alon and Spencer [1992, p. 6]. (Lemma 3.6 is slightly more general than Theorem 2.2 of Alon and Spencer [1992] that assumes u = n....
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01 Feb 1956
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409
5,104 citations
Book•
01 Jan 1995TL;DR: This book introduces the basic concepts in the design and analysis of randomized algorithms and presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications.
Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.
4,412 citations