Fully dynamic (2 + epsilon) approximate all-pairs shortest paths with fast query and close to linear update time
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For any fixed 1 > [epsilon] > 0 we present a fully dynamic algorithm for maintaining (2 + [epsilon])-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the expected amortized update time. Our query time O(log log log n). The update time is O[over ~](mnO(1/[sqrt](log n)) log (nR)), where R is the ratio between the heaviest and the lightest edge weight in the graph (so R = 1 in unweighted graphs). Unfortunately, the update time does have the drawback of a super-polynomial dependence on e. it grows as (3/[epsilon])[sqrt]log n/log(3/[epsilon]) = n [sqrt]log (3/[epsilon])/log n. Our algorithm has a significantly faster update time than any other algorithm with sub-polynomial query time. For exact distances, the state of the art algorithm has an update time of O[over ~](n[superscript 2]). For approximate distances, the best previous algorithm has a O(kmn[superscript 1/k]) update time and returns (2 k - 1) stretch paths. Thus, it needs an update time of O(m[sqrt](n)) to get close to our approximation, and it has to return O([sqrt](log n)) approximate distances to match our update time.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
50th Annual IEEE Symposium on Foundations of Computer Science, 2009. FOCS '09
Bernstein, Aaron. “Fully Dynamic (2 + Epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time.” IEEE, 2009. 693–702.
Final published version
INSPEC Accession Number: 11207109
shortest paths, graph algorithms, dynamic algorithms, approximation algorithms