A New Path from Splay to Dynamic Optimality

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.