A Foundation for Proving Splay is Dynamically Optimal

Consider the task of performing a sequence of searches in a binary search tree. After each search, we allow an algorithm to arbitrarily restructure the tree. The cost of executing the task is the sum of the time spent searching and the time spent optimizing the searches with restructuring operations. Sleator and Tarjan introduced this notion in 1985, along with an algorithm and a conjecture. The algorithm, Splay, is an elegant procedure for performing adjustments that move 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. We lay a foundation for proving the dynamic optimality conjecture. Central to our method is approximate monotonicity. Approximately monotone algorithms are those whose cost does not increase by more than a fixed multiple after removing searches from the sequence. As we shall see, Splay is dynamically optimal if and only if it is approximately monotone. This result extends to a weaker form of approximate monotonicity as well as insertion, deletion, and related algorithms. We prove that a lower bound on optimal execution cost is approximately monotone and outline how to adapt this proof from the lower bound to Splay, and how to overcome the remaining barriers to establishing dynamic optimality.