Bottom-up rebalancing binary search trees by flipping a coin
•Space efficient binary search trees.•Randomized rebalancing schemes.•Experimental evaluartion of rebalancing heuristics. Rebalancing schemes for dynamic binary search trees are numerous in the literature, where the goal is to maintain trees of low height, either in the worst-case or expected sense....
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| Vydané v: | Theoretical computer science Ročník 1055; s. 115543 |
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| Jazyk: | English |
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09.11.2025
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| Abstract | •Space efficient binary search trees.•Randomized rebalancing schemes.•Experimental evaluartion of rebalancing heuristics.
Rebalancing schemes for dynamic binary search trees are numerous in the literature, where the goal is to maintain trees of low height, either in the worst-case or expected sense. In this paper we study randomized rebalancing schemes for sequences of n insertions into an initially empty binary search tree, under the assumption that a tree only stores the elements and the tree structure without any additional balance information. Seidel (2009) presented a top-down randomized insertion algorithm, where insertions take expected O(lg2n) time, and the resulting trees have the same distribution as inserting a uniform random permutation of n elements into a binary search tree without rebalancing. Seidel states as an open problem if a similar result can be achieved with bottom-up insertions. In this paper we fail to answer this question.
We consider two simple canonical randomized bottom-up insertion algorithms on binary search trees, assuming that an insertion is given the position where to insert the next element. The subsequent rebalancing is performed bottom-up in expected O(1) time, uses expected O(1) random bits, performs at most two rotations, and the rotations appear with geometrically decreasing probability in the distance from the leaf. For some insertion sequences the expected depth of each node is proved to be O(lgn). On the negative side, we prove for both algorithms that there exist simple insertion sequences where the expected depth is Ω(n), i.e., the studied rebalancing schemes are not competitive with (most) other rebalancing schemes in the literature. |
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| AbstractList | •Space efficient binary search trees.•Randomized rebalancing schemes.•Experimental evaluartion of rebalancing heuristics.
Rebalancing schemes for dynamic binary search trees are numerous in the literature, where the goal is to maintain trees of low height, either in the worst-case or expected sense. In this paper we study randomized rebalancing schemes for sequences of n insertions into an initially empty binary search tree, under the assumption that a tree only stores the elements and the tree structure without any additional balance information. Seidel (2009) presented a top-down randomized insertion algorithm, where insertions take expected O(lg2n) time, and the resulting trees have the same distribution as inserting a uniform random permutation of n elements into a binary search tree without rebalancing. Seidel states as an open problem if a similar result can be achieved with bottom-up insertions. In this paper we fail to answer this question.
We consider two simple canonical randomized bottom-up insertion algorithms on binary search trees, assuming that an insertion is given the position where to insert the next element. The subsequent rebalancing is performed bottom-up in expected O(1) time, uses expected O(1) random bits, performs at most two rotations, and the rotations appear with geometrically decreasing probability in the distance from the leaf. For some insertion sequences the expected depth of each node is proved to be O(lgn). On the negative side, we prove for both algorithms that there exist simple insertion sequences where the expected depth is Ω(n), i.e., the studied rebalancing schemes are not competitive with (most) other rebalancing schemes in the literature. |
| ArticleNumber | 115543 |
| Author | Brodal, Gerth Stølting |
| Author_xml | – sequence: 1 givenname: Gerth Stølting surname: Brodal fullname: Brodal, Gerth Stølting email: gerth@cs.au.dk organization: Department of Computer Science, Aarhus University, Aabogade 34, Aarhus N, 8200, Denmark |
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| Keywords | Binary search tree Random rebalancing Insertions |
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Rebalancing schemes for dynamic... |
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| Title | Bottom-up rebalancing binary search trees by flipping a coin |
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