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Improved algorithms for finding level ancestors in dynamic trees. (English) Zbl 0973.68665
Montanari, Ugo (ed.) et al., Automata, languages and programming. 27th international colloquium, ICALP 2000, Geneva, Switzerland, July 9-15, 2000. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1853, 73-84 (2000).
Summary: Given a node $$x$$ at depth $$d$$ in a rooted tree $$\text{LevelAncestor} (x,i)$$ returns the ancestor to $$x$$ in depth $$d-i$$. We show how to maintain a tree under addition of new leaves so that updates and level ancestor queries are being performed in worst case constant time. Given a forest of trees with $$n$$ nodes where edges can be added, $$m$$ queries and updates take $$O(m\alpha (m,n))$$ time. This solves two open problems. In a tree with node weights, $$\min(x,y)$$ report the node with minimum weight on the path between the nodes $$x$$ and $$y$$. We can substitute the LevelAncestor query with min, without increasing the complexity for updates and queries. Previously such results have been known only for special cases.
For the entire collection see [Zbl 0941.00034].

##### MSC:
 68W05 Nonnumerical algorithms 68P05 Data structures 68W40 Analysis of algorithms 05C05 Trees