Greedy BST on Permutation Initial Tree
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909728688308224 |
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| author | Pareek, Akash |
| author_facet | Pareek, Akash |
| contents | The Greedy binary search tree (BST) algorithm, like the Splay tree, is a prominent candidate for the \emph{dynamic optimality conjecture}. While Greedy satisfies many desirable properties of BST, its cost and analysis to execute a search sequence $S$ is known to depend heavily on the choice of the \emph{initial tree} configuration. Most prior analyses assume a flat (empty) initial tree, under which several tight bounds are established.
In this work, we introduce the notion of a \emph{permutation initial tree}, a specific class of non-flat initial tree and prove that for any permutation search sequence $S=(s_1,s_2,\dots, s_n)$, there exists a permutation initial tree $I_p$ such that the cost of Greedy on $I_p$ is same as its cost on the flat initial tree.
As an application of our result, we show that the \emph{preorder traversal conjecture} holds for Greedy when the initial tree is a permutation initial tree. While it was previously known that Greedy achieves an $O(n)$ cost on preorder sequences for flat initial tree (Chalermsook et al., FOCS 2015), our result demonstrates that the same linear bound holds when the initial tree is a permutation initial tree. This result also matches the $O(n)$ bound for Splay tree on preorder sequence when the initial tree aligns with the traversal order (Chaudhuri and Höft, SIGACT 1993). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Greedy BST on Permutation Initial Tree Pareek, Akash Data Structures and Algorithms The Greedy binary search tree (BST) algorithm, like the Splay tree, is a prominent candidate for the \emph{dynamic optimality conjecture}. While Greedy satisfies many desirable properties of BST, its cost and analysis to execute a search sequence $S$ is known to depend heavily on the choice of the \emph{initial tree} configuration. Most prior analyses assume a flat (empty) initial tree, under which several tight bounds are established. In this work, we introduce the notion of a \emph{permutation initial tree}, a specific class of non-flat initial tree and prove that for any permutation search sequence $S=(s_1,s_2,\dots, s_n)$, there exists a permutation initial tree $I_p$ such that the cost of Greedy on $I_p$ is same as its cost on the flat initial tree. As an application of our result, we show that the \emph{preorder traversal conjecture} holds for Greedy when the initial tree is a permutation initial tree. While it was previously known that Greedy achieves an $O(n)$ cost on preorder sequences for flat initial tree (Chalermsook et al., FOCS 2015), our result demonstrates that the same linear bound holds when the initial tree is a permutation initial tree. This result also matches the $O(n)$ bound for Splay tree on preorder sequence when the initial tree aligns with the traversal order (Chaudhuri and Höft, SIGACT 1993). |
| title | Greedy BST on Permutation Initial Tree |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.03666 |