Thesis Proposal: Multi-Splay Trees

In this thesis, we introduce multi-splay trees (MSTs) and prove several results demonstrating that MSTs have most of the useful properties different BSTs have. First, we prove a close variant of the splay tree access lemma [ST85] for multi-splay trees, a lemma that implies MSTs have the O(log n) runtime property, the static finger property, and the static optimality property. Then, we extend the access lemma by proving the remassing lemma, which is similar to the reweighting lemma for splay trees [Geo04]. The remassing lemma shows that MSTs satisfy the working set property and key-independent optimality and are competitive to parametrically balanced trees, as defined in [Geo04]. After proving the remassing lemma, we also prove that MSTs achieve the O(log log n)-competitiveness and that sequential access in MSTs costs O(n). Then we extend the static model naturally to allow insertions and deletions and show how to carry out these operations in multi-splay trees to achieve O(log log n)competitiveness, a result no other BST scheme has been proved to have. In addition, we prove that MSTs satisfy the deque property, which is still an open problem for splay trees since it was conjectured in 1985 [Tar85]. While it is easy to construct a BST that trivially satisfies the deque property, no other BST scheme satisfying other useful properties has been proved to have deque property. Together, these results show that MSTs satisfy most of the important properties satisfied by different binary search trees.