Dynamic Maintanance of Low-Stretch Probabilistic Tree Embeddings with Applications

We give the first non-trivial fully dynamic probabilistic tree embedding algorithm for weighted graphs undergoing edge insertions and deletions. We obtain a trade-off between amortized update time and expected stretch against an oblivious adversary. At the two extremes of this trade-off, we can maintain a tree of expected stretch $ O (\log^4 n) $ with update time $ m^{1/2 + o(1)} $ or a tree of expected stretch $ n^{o(1)} $ with update time $ n^{o(1)} $ (for edge weights polynomial in $ n $). A guarantee of the latter type has so far only been known for maintaining tree embeddings with average (instead of expected) stretch [Chechik/Zhang, SODA '20]. Our main result has direct implications to fully dynamic approximate distance oracles and fully dynamic buy-at-bulk network design. For dynamic distance oracles, our result is the first to break the $ O (\sqrt{m}) $ update-time barrier. For buy-at-bulk network design, a problem which also in the static setting heavily relies on probabilistic tree embeddings, we give the first non-trivial dynamic algorithm. As probabilistic tree embeddings are an important tool in static approximation algorithms, further applications of our result in dynamic approximation algorithms are conceivable. From a technical perspective, we obtain our main result by first designing a decremental algorithm for probabilistic low-diameter decompositions via a careful combination of Bartal's ball-growing approach [FOCS '96] with the pruning framework of Chechik and Zhang [SODA '20]. We then extend this to a fully dynamic algorithm by enriching a well-known 'decremental to fully dynamic' reduction with a new bootstrapping idea to recursively employ a fully dynamic algorithm instead of a static one in this reduction.