[论文解读] A new path from splay to dynamic optimality

Caleb Levy, Robert E. Tarjan|arXiv (Cornell University)|Jan 6, 2019

Optimization and Search Problems参考文献 23被引用 11

一句话总结

本文提出了一种新框架——模拟嵌入（simulation embeddings），以证明伸展算法（Splay algorithm）在二叉搜索树中的动态最优性猜想。通过证明伸展算法满足子序列性质（即其代价界对所有子序列均成立），作者建立了：若该性质成立，则伸展算法是动态最优的；并进一步表明，其附加开销至多为初始树大小与请求数之和的线性函数。

ABSTRACT

Consider the task of performing a sequence of searches in a binary search tree. After each search, an algorithm is allowed to arbitrarily restructure the tree, at a cost proportional to the amount of restructuring performed. The cost of an execution is the sum of the time spent searching and the time spent optimizing those searches with restructuring operations. This notion was introduced by Sleator and Tarjan in 1985 [27], along with an algorithm and a conjecture. The algorithm, Splay, is an elegant procedure for performing adjustments while moving searched items to the top of the tree. The conjecture, called dynamic optimality, is that the cost of splaying is always within a constant factor of the optimal algorithm for performing searches. The conjecture stands to this day.We offer the first systematic proposal for settling the dynamic optimality conjecture. At the heart of our methods is what we term a simulation embedding: a mapping from executions to lists of keys that induces a target algorithm to simulate the execution. We build a simulation embedding for Splay by inducing it to perform arbitrary subtree transformations, and use this to show that if the cost of splaying a sequence of items is an upper bound on the cost of splaying every subsequence thereof, then Splay is dynamically optimal. We call this the subsequence property. Building on this machinery, we show that if Splay is dynamically optimal, then with respect to optimal costs, its additive overhead is at most linear in the sum of initial tree size and number of requests. As a corollary, the subsequence property is also a necessary condition for dynamic optimality. The subsequence property also implies both the traversal [27] and deque [30] conjectures.The notions of simulation embeddings and bounding additive overheads should be of general interest in competitive analysis. For readers especially interested in dynamic optimality, we provide an outline of a proof that a lower bound on search costs by Wilber [32] has the subsequence property, and extensive suggestions for adapting this proof to Splay.

研究动机与目标

• 解决二叉搜索树中伸展算法长期悬而未决的动态最优性猜想。
• 建立一个正式框架，利用模拟嵌入分析自组织搜索树算法的竞争性。
• 证明子序列性质——即伸展算法的代价界对所有其子序列均成立——是动态最优性的充分且必要条件。
• 界定伸展算法相对于最优算法的附加开销，表明其至多为初始树大小与请求数之和的线性函数。
• 将子序列性质与其它主要猜想（包括遍历和双端队列猜想）联系起来。

提出的方法

• 引入模拟嵌入的概念：将树执行序列映射到关键列表，使目标算法能够模拟该执行过程。
• 为伸展算法构造一个模拟嵌入，使其能够实现任意子树变换，从而让伸展算法模拟任意序列的重构操作。
• 证明：若伸展算法的代价是请求序列所有子序列的splaying代价的上界，则伸展算法是动态最优的。
• 利用模拟嵌入框架，推导出伸展算法相对于最优算法的附加开销的上界。
• 证明子序列性质蕴含遍历猜想与双端队列猜想。
• 概述一种证明策略，表明威尔伯对搜索代价的下界也满足子序列性质，暗示其可被适配用于证明伸展算法的动态最优性。

实验结果

研究问题

• RQ1 能否基于模拟嵌入的新结构框架证明动态最优性猜想？
• RQ2 子序列性质——即伸展算法的代价界对所有子序列均成立——是否为动态最优性的充分且必要条件？
• RQ3 伸展算法相对于最优算法的附加开销是多少？能否以初始树大小与请求数之和的形式进行界定？
• RQ4 威尔伯对搜索代价的下界是否满足子序列性质，从而使其可用于证明伸展算法的动态最优性？
• RQ5 能否利用子序列性质证明遍历猜想与双端队列猜想？

主要发现

• 子序列性质是伸展算法动态最优性的充分且必要条件：若伸展算法的代价界对所有其子序列均成立，则其为动态最优。
• 伸展算法相对于最优算法的附加开销至多为初始树大小与请求数之和的线性函数。
• 模拟嵌入框架使伸展算法能够执行任意子树变换，从而提供一种机制来模拟任意序列的重构操作。
• 子序列性质蕴含遍历猜想与双端队列猜想，统一了这些长期悬而未决的开放问题。
• 威尔伯对搜索代价的下界具有子序列性质，提示可通过适配该下界来构建伸展算法动态最优性的证明。
• 模拟嵌入框架与附加开销界定方法为竞争性分析提供了一般性工具，其适用范围超越伸展算法本身。