Sphendamnœ: a proof that k-splay fails to achieve logk N behaviour

• Authors: D. J. McClurkin;G. F. Georgakopoulos
• Venue: PCI'01 Proceedings of the 8th Panhellenic conference on Informatics
• Year: 2001

The "splay" technique was introduced to cope with biasedness or changeability in data access frequencies, achieving optimality within a small constant factor w.r.t. static trees, without sacrificing log N worst-case behaviour, in an amortized sense. It achieves this through a series of local transformations, starting from the search node and propagating upwards along the search path to the root. It seems plausible that a similar policy, suitably adapted to multi-way trees, could achieve logk N amortized performance (nodes visited), where k is the degree of the tree. Sherk's k-splay, a generalization of Sleator and Tarjan's splay technique to multi-way trees, proved to exhibit amortized log2 N behaviour, could be considered the most likely candidate for a multiway splay-like self-adjusting policy with logk N amortized complexity. We construct a family of k-ary trees having depth k and containing 2k nodes and we show that Sherk's k-splay applied to the deepest node of any tree in this family always produces another tree in the family. A fractal-like process of replacing leaf-nodes of trees with copies of themselves then allows us to create such trees of arbitrary size for any given k. This provides us with a family of counterexamples for which Sherk's k-splay always visits log2 N nodes when applied to the deepest node in the tree.