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Register a new userAlgorithms and Hardness for Diameter in Dynamic Graphs
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Added by
Nicole Wein
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: - Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. - Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly $(3/2+epsilon)$-approximation to Diameter in directed or undirected graphs can be maintained decrementally in total time $m^{1+o(1)}sqrt{n}/epsilon^2$. This nearly matches the static $3/2$-approximation algorithm for the problem that is known to be conditionally optimal.
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We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best O(n log n) time solution [Oh and Ahn, ISAAC 2016]. Using this algorithm as a subroutine, we solve the problem of adding a shortcut to a tree so that the diameter of the new graph (which is a unicycle graph) is minimized; our algorithm takes O(n^2 log n) time and O(n) space. The previous best algorithms solve the problem in O(n^2 log^3 n) time and O(n) space [Oh and Ahn, ISAAC 2016], or in O(n^2) time and O(n^2) space [Bil`o, ISAAC 2018].
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