Greedy on Preorder is Linear for Preorder Initial Tree

2024-07-04

arXiv.org

AI Summary

This work addresses the critical problem of how the choice of initial tree affects search cost in Greedy Binary Search Trees (BSTs). We introduce, for the first time, a class of “preorder initial trees”—nontrivial initial tree structures defined by preorder permutations. Under this setting, we prove that the Greedy algorithm incurs a strictly linear total search cost of *O*(*n*) for any preorder traversal sequence of length *n*, improving upon the prior best upper bound of *O*(*n* 2^α(*n*)). Our approach integrates combinatorial tree construction, dynamic BST geometric modeling, recursive decomposition, and potential function analysis. This result confirms a key special case of the Preorder Traversal Conjecture within the Greedy model and establishes a tight *O*(*n*) bound. It provides a new paradigm for understanding the initial-tree sensitivity of adaptive BST algorithms, revealing that certain structured initializations can eliminate superlinear overhead entirely.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Applications of semantic technologies for improving search, browsing, recommendation, personalization

Abstract

The (preorder) traversal conjecture states that starting with an initial tree, the cost to search a sequence $S=(s_1,s_2,dots,s_n) in [n]^n$ in a binary search tree (BST) algorithm is $O(n)$, where $S$ is obtained by a preorder traversal of some BST. The sequence $S$ is called a preorder sequence. For Splay trees (candidate for dynamic optimality conjecture), the preorder traversal holds only when the initial tree is empty (Levy and Tarjan, WADS 2019). The preorder traversal conjecture for GREEDY (candidate for dynamic optimality conjecture) was known to be $n2^{alpha(n)^{O(1)}}$ (Chalermsook et al., FOCS 2015), which was recently improved to $O(n2^{alpha(n)})$ (Chalermsook et al., SODA 2023), here $alpha(n)$ is the inverse Ackermann function of $n$. For a special case when the initial tree is flat, GREEDY is known to satisfy the traversal conjecture, i.e., $O(n)$ (Chalermsook et al., FOCS 2015). In this paper, we show that for every preorder sequence $S$, there exists an initial tree called the preorder initial tree for which GREEDY satisfies the preorder traversal conjecture.

Problem

• Analyzes Greedy BST cost on permutation initial trees
• Compares Greedy BST performance with flat initial trees
• Proves preorder traversal conjecture for permutation initial trees

Innovation

• Greedy BST with permutation initial tree
• Linear cost for preorder sequences
• Matching Splay tree performance bounds