Sequential access in splay trees takes linear time

Sleator and Tarjan have invented a form of self-adjusting binary search tree called the splay tree. On any sufficiently long access sequence, splay trees are as efficient, to within a constant factor, as both dynamically balanced and static optimum search trees. Sleator and Tarjan have made a much stronger conjecture; namely, that on any sufficiently long access sequence and to within a constant factor, splay trees are as efficient as any form of dynamically updated search tree. This dynamic optimality conjecture implies as a special case that accessing the items in a splay tree in sequential order takes linear time, i.e. O(1) time per access. In this paper we prove this special case of the conjecture, generalizing an unpublished result of Wegman. Our sequential access theorem not only supports belief in the dynamic optimality conjecture but provides additional insight into the workings of splay trees. As a corollary of our result, we show that splay trees can be used to simulate output-restricted deques (double-ended queues) in linear time. We pose several open problems related to our result.