Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths
21 Oct 2006-pp 591-602
TL;DR: An O(n2 log n) algorithm is presented to construct a data structure of size O(kn1+1k/) for all integers k ges 2 and a new generic scheme for all-pairs approximate shortest paths is used for these results.
Abstract: Let G = (V,E) be a weighted undirected graph with |V | = n and |E| = m. An estimate \hat \delta \left( {u,v} \right) of the distance \delta \left( {u,v} \right) in G between u, v \in V is said to be of stretch t iff \delta \left( {u,v} \right) \leqslant \hat \delta \left( {u,v} \right) \leqslant t ? \delta \left( {u,v} \right). The most efficient algorithms known for computing small stretch distances in G are the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k \geqslant 1, Thorup and Zwick in [16] gave an O(kmn^{1/k}) algorithm to construct a data structure of size O(kn^{1+1/k}) which, given a query (u, v) \in V ? V , returns in O(k) time, a 2k - 1 stretch estimate of \delta \left( {u,v} \right). But for small values of k, the time to construct the oracle is rather high. Here we present an O(n^2 log n) algorithm to construct such a data structure of size O(kn^{1+1/k}) for all integers k \geqslant 2. Our query answering time is O(k) for k \ge 2 and \Theta (log n) for k = 2. We use a new generic scheme for all-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for allpairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n^2 log n) time.
Citations
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TL;DR: This article develops a couple of new techniques for constructing (α, β)-spanners and presents an additive (1,6)-spanner of size O, an economical agent that assigns costs and values to paths in the graph, and shows that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs.
Abstract: An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k−1, 0)-spanner of size O(n1+1/k) and an (additive) (1,2)-spanner of size O(n3/2). However no other additive spanners are known to exist.In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1,6)-spanner of size O(n4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any k, a (k,k−1)-spanner with size O(kn1+1/k) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
123 citations
Cites background from "Faster Algorithms for Approximate D..."
...…India; e-mail: sbaswana@cse.iitk.ac.in; T. Kavitha, Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India; e-mail: kavitha@cse.iisc.ernet.in; K. Mehlhorn, Max-Planck-Institut f¨ur Informatik, Stuhlsatzenhausweg 61, 66123 Saarbr¨ucken, Germany;…...
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...…application of spanners is the construction of labeling schemes and distance oracles [Thorup and Zwick 2005; Baswana and Sen 2006; Roditty et al. 2005; Baswana and Kavitha 2006; Baswana et al. 2008], which are data structures that can report approximately accurate distances in constant time....
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TL;DR: By assembling connection schemes in different ways the authors can recreate the additive 2- and 6-spanners of Aingworth et al.
Abstract: A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u,v at distance d in G are at distance at most f(d) in H. There is clearly some trade-off between the sparsity of H and the distortion function f, though the nature of the optimal trade-off is still poorly understood.In this article we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. [1999] and Baswana et al. [2009], and give spanners whose multiplicative distortion quickly tends toward 1. Our results rival the simplicity of all previous algorithms and provide substantial improvements (up to a doubly exponential reduction in edge density) over the comparable spanners of Elkin and Peleg [2004] and Thorup and Zwick [2006].
117 citations
Cites background from "Faster Algorithms for Approximate D..."
...A recent application of spanners is in the design of approximate distance oracles and labeling schemes for arbitrary metrics; see [23, 4] for further references....
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...A recent application of spanners is in the design of approximate distance oracles and labeling schemes [Thorup and Zwick 2005; Baswana and Sen 2007; Roditty et al. 2005; Baswana and Kavitha 2006] for arbitrary metrics....
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22 Jul 2013
TL;DR: This work gives a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log ) bits, and demonstrates its utility for various routing problems, exact and approximate diameter computation, and spanner construction.
Abstract: Given a simple graph G=(V,E) and a set of sources S ⊆ V, denote for each node ν e V by Lν(∞) the lexicographically ordered list of distance/source pairs (d(s,v),s), where s ∈ S. For integers d,k ∈ N∪{∞}, we consider the source detection, or (S,d,k)-detection task, requiring each node v to learn the first k entries of Lν(∞) (if for all of them d(s,v) ≤ d) or all entries (d(s,v),s) ∈ Lν(∞) satisfying that d(s,v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadth-first search (BFS) tree constructions. For example, the special case of k=∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d=∞ and S=V requires constructing a partial BFS tree comprising at least k nodes from every node in V.In this work, we give a simple, near-optimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d+k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions.
110 citations
23 Oct 2010
TL;DR: It is shown that a 2-approximate distance oracle requires space $\tOmega(n^2 / \sqrt{\alpha})$ and this implies a space lower bound to achieve approximation $2d+1$.
Abstract: We give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size $O(n^{5/3}) = O(n^{1.66\cdots})$ and, when queried about vertices at distance $d$, returns a path of length $2d+1$. For weighted graphs with $m=n^2/\alpha$ edges, our distance oracle has size $O(n^2 / \sqrt[3]{\alpha})$ and returns a factor 2 approximation. Based on a plausible conjecture about the hardness of set intersection queries, we show that a 2-approximate distance oracle requires space $\tOmega(n^2 / \sqrt{\alpha})$. For unweighted graphs, this implies a $\tOmega(n^{1.5})$ space lower bound to achieve approximation $2d+1$.
103 citations
09 Jul 2012
TL;DR: A distributed algorithm that computes the diameter of the network in O(n) rounds and two distributed approximation algorithms that almost match their lower bounds for constant diameter and for constant girth.
Abstract: This paper considers the problem of computing the diameter D and the girth g of an n-node network in the CONGEST distributed model. In this model, in each synchronous round, each vertex can transmit a different short (say, O(logn) bits) message to each of its neighbors. We present a distributed algorithm that computes the diameter of the network in O(n) rounds. We also present two distributed approximation algorithms. The first computes a 2/3 multiplicative approximation of the diameter in $O(D\sqrt n \log n)$ rounds. The second computes a 2−1/g multiplicative approximation of the girth in $O(D+\sqrt{gn}\log n)$ rounds. Recently, Frischknecht, Holzer and Wattenhofer [11] considered these problems in the CONGEST model but from the perspective of lower bounds. They showed an $\tilde{\Omega}(n)$ rounds lower bound for exact diameter computation. For diameter approximation, they showed a lower bound of $\tilde{\Omega}(\sqrt n)$ rounds for getting a multiplicative approximation of. Both lower bounds hold for networks with constant diameter. For girth approximation, they showed a lower bound of $\tilde{\Omega}(\sqrt n)$ rounds for getting a multiplicative approximation of on a network with constant girth. Our exact algorithm for computing the diameter matches their lower bound. Our diameter and girth approximation algorithms almost match their lower bounds for constant diameter and for constant girth.
101 citations
References
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IBM1
TL;DR: A new method for accelerating matrix multiplication asymptotically is presented, by using a basic trilinear form which is not a matrix product, and making novel use of the Salem-Spencer Theorem.
Abstract: We present a new method for accelerating matrix multiplication asymptotically. This work builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent is 2.376.
1,413 citations
"Faster Algorithms for Approximate D..." refers methods in this paper
...The fastest known algorithm for matrix multiplication due to Coppersmith and Winograd [9] implies ω < 2....
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TL;DR: A data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time and is easy to implement.
Abstract: A data structure for representing a set of n items from a umverse of m items, which uses space n + o(n) and accommodates membership queries m constant time is described. Both the data structure and the query algorithm are easy to ~mplement.
943 citations
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.
618 citations
06 Jul 2001
TL;DR: The most impressive feature of the data structure is its constant query time, hence the name ``oracle', which 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 |V|=n and |E|=m. Let k\ge 1 be an integer. We show that G=(V,E) can be preprocessed in O(kmn^{1/k}) expected time, constructing a data structure of size O(kn^{1+1/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, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k-1. We show that a 1963 girth conjecture of Erd{\H{o}}s, implies that ω(n^{1+1/k}) space is needed in the worst case for any real stretch strictly smaller than 2k+1. 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(n^{1+1/k}) space had a query time of ω(n^{1/k}) and a slightly larger, non-optimal, stretch. 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.}
563 citations
"Faster Algorithms for Approximate D..." refers background or methods in this paper
...This algorithm is based on the following important property of A(S, k) (proved in Theorem 3.1 below) : For any two vertices u, v ∈ S, the scheme A(S, k) stores a (2, ω)-approximate distance between them....
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...This scheme also enables us to design faster algorithms for allpairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n2 log n) time....
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...However, the complexity of the fastest known algorithm for the APSP problem in a graph with m edges and n vertices with real non-negative edge weights is O(mn + n2 log log n) [14]....
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03 Jul 2001
TL;DR: Several compact routing schemes for general weighted undirected networks are described, which achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch.
Abstract: We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extremely short. The routing decision at each node takes constant time. Yet, the stretch of these routing schemes, i.e., the worst ratio between the cost of the path on which a packet is routed and the cost of the cheapest path from source to destination, is a small constant. Our schemes achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch. More specifically, we obtain: A routing scheme that uses only O (n 1/2) bits of memory at each node of an n-node network that has stretch 3. The space is optimal, up to logarithmic factors, in the sense that every routing scheme with stretch n2), and every routing scheme with stretch n3/2). The headers used are only (1 + O(1)) log2> n-bits long and each routing decision takes constant time. A variant of this scheme with [log2 n] -bit headers makes routing decisions in O(log log n) time. Also, for every integer k > 2, a general handshaking based routing scheme that uses O (n1/k) bits of memory at each node that has stretch 2k - 1. A conjecture of Erdos from 1963, settled for k = 3, 5, implies that the routing tables are of near-optimal size relative to the stretch. The handshaking is similar in spirit to a DNS lookup in TCP/IP. Headers are O(log2 n) bits long and each routing decision takes constant time. Without handshaking, the stretch of the scheme increases to 4k - 5. One ingredient used to obtain the routing schemes mentioned above, may be of independent practical and theoretical interest: A shortest path routing scheme for trees of arbitrary degree and diameter that assigns each vertex of an n-node tree a (1 + O(1)) log2 n-bit label. Given the label of a source node and the label of a destination it is possible to compute, in constant time, the port number of the edge from the source that heads in the direction of the destination. The general scheme for k > 2 also uses a clustering technique introduced recently by the authors. The clusters obtained using this technique induce a sparse and low stretch tree cover of the network. This essentially reduces routing in general networks into routing problems in trees that could be solved using the above technique.
560 citations