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.

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Fast approximation algorithms for the diameter and radius of sparse graphs

Let G = (V,E) be an undirected weighted graph on |V | = n vertices and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V,ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well-motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43-year old girth lower bound conjecture made independently by Erdos, Bollobas, and Bondy & Simonovits.

Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information, which involves growing either breadth first search trees up to t(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Preliminary version of this work appeared in the 30th International Colloquium on Automata, Languages and Programming, pages 384–396, 2003.

A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs

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).

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Better approximation algorithms for the graph diameter

Let G = (V, E) be an undirected graph on n vertices, and let δ(u, v) denote the distance in G between two vertices u and v. Thorup and Zwick showed that for any positive integer t, the graph G can be preprocessed to build a data structure that can efficiently report t-approximate distance between any pair of vertices. That is, for any u, v ∈ V, the distance reported is at least δ(u, v) and at most tδ(u, v). The remarkable feature of this data structure is that, for t≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any t-approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch t and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected O(n2) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph G = (V, E) is a subgraph (V, E), E ⊆ E, such that for any two vertices u and v in the graph, their distance in the subgraph is at most 2δ(u, v) p 1.

Approximate distance oracles for unweighted graphs in expected O(n2) time

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.

Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths

Let G(V,E) be an unweighted undirected graph on |V | = n vertices. Let δ(u,v) denote the shortest distance between vertices u,v ∈ V. An algorithm is said to compute all-pairs t-approximate shortest-paths/distances, for some t ≥ 1, if for each pair of vertices u,v ∈ V, the path/distance reported by the algorithm is not longer/greater than t · δ(u,v).

This paper presents two randomized algorithms for computing all-pairs nearly 2-approximate distances. The first algorithm takes expected O(m2/3n log n + n2) time, and for any u,v ∈ V reports distance no greater than 2δ(u,v) + 1. Our second algorithm requires expected O(n2 log3/2) time, and for any u,v ∈ V reports distance bounded by 2δ(u,v)+3.

This paper also presents the first expected O(n2) time algorithm to compute all-pairs 3-approximate distances.

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All-pairs nearly 2-approximate shortest-paths in O ( n 2 polylog n ) time

https://www.cse.iitk.ac.in/users/sbaswana/Papers-published/journal-stacs05.pdf

Vishrut Goyal

Papers

All-pairs nearly 2-approximate shortest-paths in O ( n 2 polylog n ) time

All-pairs nearly 2-approximate shortest paths in O(n2polylogn) time

All-pairs nearly 2-approximate shortest paths in $O(n^2 \mathrm polylog n)$ time