Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the all-pairs-shortest-path problem in unweighted undirected graphs
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication)
SIAM Journal on Computing
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
All-Pairs Almost Shortest Paths
SIAM Journal on Computing
Journal of Algorithms
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Diameter determination on restricted graph families
Discrete Applied Mathematics
Center and diameter problems in plane triangulations and quadrangulations
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
All pairs shortest paths using bridging sets and rectangular matrix multiplication
Journal of the ACM (JACM)
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Solving 3-Satisfiability in Less Then 1, 579n Steps
CSL '92 Selected Papers from the Workshop on Computer Science Logic
A Linear-Time Algorithm for Finding a Central Vertex of a Chordal Graph
ESA '94 Proceedings of the Second Annual European Symposium on Algorithms
A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
An improved exponential-time algorithm for k-SAT
Journal of the ACM (JACM)
Computing almost shortest paths
ACM Transactions on Algorithms (TALG)
Efficient algorithms for center problems in cactus networks
Theoretical Computer Science
A New Combinatorial Approach for Sparse Graph Problems
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
All-pairs nearly 2-approximate shortest paths in O(n2polylogn) time
Theoretical Computer Science
Solving satisfiability in less than 2n steps
Discrete Applied Mathematics
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the possibility of faster SAT algorithms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs
SIAM Journal on Computing
Fast and simple approximation of the diameter and radius of a graph
WEA'06 Proceedings of the 5th international conference on Experimental Algorithms
Multiplying matrices faster than coppersmith-winograd
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
All-pairs shortest paths for unweighted undirected graphs in o(mn) time
ACM Transactions on Algorithms (TALG)
On Problems as Hard as CNF-SAT
CCC '12 Proceedings of the 2012 IEEE Conference on Computational Complexity (CCC)
Faster approximation of distances in graphs
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Faster Algorithms for Rectangular Matrix Multiplication
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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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 ε0 there is an O(m2-ε) time (3/2-ε)-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 δ0, 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-ε) time algorithm that gives a (3/2-ε) approximation for constant ε0. This is interesting since the diameter approximation problem is hardest to solve for small D.