All-pairs nearly 2-approximate shortest paths in O(n2polylogn) time
01 Jan 2009-Theoretical Computer Science (Elsevier Science Publishers Ltd.)-Vol. 410, Iss: 1, pp 84-93
TL;DR: This paper presents two extremely simple randomized algorithms for computing all-pairs nearly 2-approximate distances and reports a distance bounded by 2@d(u,v)+3.
About: This article is published in Theoretical Computer Science.The article was published on 2009-01-01 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Shortest-path tree.
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01 Jun 2013
TL;DR: This paper presents the first improvement over the diameter approximation algorithm of Aingworth et.
Abstract: The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of ~O(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] presented an algorithm that computes in ~O(m√ n + n2) time an estimate D for the diameter D, such that ⌊ 2/3 D ⌋ ≤ ^D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years.Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et. al, producing an algorithm with the same estimate but with an expected running time of ~O(m√ n). We thus show that for all sparse enough graphs, the diameter can be 3/2-approximated in o(n2) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node.We also provide strong evidence that our diameter approximation result may be hard to improve. We show that if for some constant e>0 there is an O(m2-e) time (3/2-e)-approximation algorithm for the diameter of undirected unweighted graphs, then there is an O*( (2-δ)n) time algorithm for CNF-SAT on n variables for constant δ>0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false.Motivated by this negative result, we give several improved diameter approximation algorithms for special cases. We show for instance that for unweighted graphs of constant diameter D not divisible by 3, there is an O(m2-e) time algorithm that gives a (3/2-e) approximation for constant e>0. This is interesting since the diameter approximation problem is hardest to solve for small D.
276 citations
Cites methods from "All-pairs nearly 2-approximate shor..."
...[3] presented an algorithm for unweighted undirected graphs with an expected running time of O(mn log n+ n(2)) that computes an approximation of all distances with a multiplicative error of 2 and an additive error of 1....
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05 Jan 2014
TL;DR: Two algorithms are deterministic, and thus the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs is presented.
Abstract: The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem.In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in O (n2 + m√n) time an estimate D for the diameter D in directed graphs with nonnegative edge weights, such that [EQUATION], where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to O (m√n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O (n2-e) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large.In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in O (m3/2) time, and one running in O (mn2/3). time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs.In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D such that D -- c ≤ D ≤ D. An extremely simple O (mn1-e) time algorithm achieves an additive ne-approximation; no better results are known. We show that for any e > 0, getting an additive ne-approximation algorithm for the diameter running in O (n2-e) time for any δ > 2e would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely.Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in O (m√n) time, one can compute for each v e V in an undirected graph, an estimate e(v) for the eccentricity e (v) such that max{R, 2/3 · e(v)} ≤ e (v) ≤ min {D, 3/2 · e(v)} where R = minv e (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates e' (v) with 3/5 · e (v) ≤ e' (v) ≤ e (v).
116 citations
Cites methods from "All-pairs nearly 2-approximate shor..."
...[3] presented an algorithm for unweighted undirected graphs with an expected running time of O(mn log n + n(2)) that computes an approximation of all distances with a multiplicative error of 2 and an additive error of 1....
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23 Jan 2005
TL;DR: 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.
Abstract: An (α, β)-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u, v: δH (u, v) ≤ αδG(u, v) + β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) (2k - 1, 0)-spanners of size O(n1+1/k) and that there exist (purely additive) (1, 2)-spanners of size O(n3/2). However no other (1, O(1))-spanners are known to exist.In this paper we develop a couple new techniques for constructing (α, β)-spanners. The first result is a purely additive (1, 6)-spanner of size O(n4/3). Our 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. This general approach should lead to new spanner constructions.The second result is a truly simple linear time construction of (k, k - 1)-spanners with size O(n1+1/k). In a distributed network the algorithm terminates in a constant number of rounds and has expected size O(n1+1/k). The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.
74 citations
TL;DR: A simple, novel, and generic scheme for all-pairs approximate shortest paths and answers an open question posed by Thorup and Zwick in their seminal paper.
Abstract: Let $G=(V,E)$ be a weighted undirected graph having nonnegative edge weights. An estimate $\hat{\delta}(u,v)$ of the actual distance $\delta(u,v)$ between $u,v\in V$ is said to be of stretch $t$ if and only if $\delta(u,v)\leq\hat{\delta}(u,v)\leq t\cdot\delta(u,v)$. Computing all-pairs small stretch distances efficiently (both in terms of time and space) is a well-studied problem in graph algorithms. We present a simple, novel, and generic scheme for all-pairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for all-pairs $t$-stretch distances for a whole range of stretch $t$, and we also answer an open question posed by Thorup and Zwick in their seminal paper [J. ACM, 52 (2005), pp. 1-24].
46 citations
01 Jan 2012
TL;DR: This paper proposes a fast algorithm which utilize the preiously-calculated results to accelerate the Dijkstra's algorithm and shows that the time complexity is reduced to about O(n 2.4 ) in scalefree complex networks.
Abstract: Finding shortest paths is a fundamental problem in graph theory, which has a large amount of applications in many areas like computer science, operations research, network routing and network analysis. Although many exact and approximate algorithms have been proposed, it is still a time-consuming task to calculate shortest paths for large-scale networks with tremendous volume of data available in recent years. In this paper, we find that the classic Dijkstra's algorithm can be improved by simple modification. We propose a fast algorithm which utilize the preiously-calculated results to accelerate the latter calculation. Simple optimization strategies are also proposed with consideration of characteristics of scale-free complex networks. Our experimental results show that the average running time of our algorithm is lower than the Dijkstra's algorithm by a factor relating to the connection probability in random networks of ER model. The performance of our algorithm is significantly better than the Dijkstra's algorithm in scale-free networks generated by the AB model. The results show that the time complexity is reduced to about O(n 2.4 ) in scalefree complex networks. When the optimization strategies are applied, the algorithm performance is further improved slightly in scale-free networks.
29 citations
Cites background or methods from "All-pairs nearly 2-approximate shor..."
...[3] propose a randomized algorithm for computing all-pairs nearly 2-approximate distances with time complexity of O(n2log3/2n)....
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...Many approximating shortest-path algorithms have been proposed to meet the requirements of large-scale complex network analysis [3][4]....
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References
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TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
Abstract: t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for invert ing a matr ix of order n, solving a system of n linear equations in n unknowns, comput ing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] tha t Gaussian elimination for solving a system of l inearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note tha t WlNOGRAD [21 modifies the usual algorithms for matr ix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. I t is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which mult iply matrices of order m2 ~, by induction on k: ~ , 0 is the usual algorithm, for matr ix multiplication (requiring m a multiplications and m 2 ( m t) additions), e~,k already being known, define ~ , ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write
2,581 citations
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..., (+,×) product of two matrices, Strassen [17] gave the first subcubic algorithm, and many faster algorithms followed this algorithm....
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TL;DR: In this article, a new method for accelerating matrix multiplication asymptotically is presented, based on the ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product.
2,454 citations
"All-pairs nearly 2-approximate shor..." refers methods in this paper
...The fastest known algorithm for matrix multiplication due to Coppersmith and Winograd [11] 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
"All-pairs nearly 2-approximate shor..." refers background or methods in this paper
...This scheme, using a new idea, builds upon the earlier work of Thorup and Zwick [19, 20] which deals with the computation of all pairs approximate distances with stretch ≥ 3....
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...An important construct from [20] which we shall use is a Ball which is defined as follows....
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...Now we present the notations, definitions and important Lemmas (most of them from [20]) to be used in the rest of the paper....
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...The scheme may appear similar to the 3-approximate distance oracle of Thorup and Zwick [20] except the third step....
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...2 [20] Given a graph G = (V,E), the expected size of Ball(v,X, Y ) is at most 1/p if the set Y is constructed by either of the following two sampling methods : (i) Y contains each vertex from set X independently with probability p....
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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
"All-pairs nearly 2-approximate shor..." refers background or methods in this paper
...We shall employ the random sampling scheme given by Thorup and Zwick [8] to compute the desired sample R as follows....
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...It is shown in [8] that in every iteration, the size of V ′ decreases by a factor of 2 with probability at-least 1/2....
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