A new approach to all-pairs shortest paths on real-weighted graphs
Theoretical Computer Science - Special issue on automata, languages and programming
Efficient data structures for range-aggregate queries on trees
Proceedings of the 12th International Conference on Database Theory
On Cartesian Trees and Range Minimum Queries
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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We consider the problem of preprocessing an edge-weighted tree T in order to quickly answer queries of the following type: does a given edge e belong in the minimum spanning tree of T \cup \{ e\}? Whereas the offline minimum spanning tree verification problem admits a lovely linear time solution, we demonstrate an inherent inverse-Ackermann type tradeoff in the online MST verification problem. In particular, any scheme that answers queries in t comparisons must invest \Omega (n\log \lambda _t (n)) time preprocessing the tree, where \lambda _tis the inverse of the tth row of Ackermann's function. This implies a query lower bound of \Omega (\alpha (n)) for the case of linear preprocessing time. We also show that our lower bound is tight to within a factor of 2 in the t parameter.