A relationship between self-organizing lists and binary search trees

Abstract

We establish the following relationship between the move-to-front heuristic for lists and the move-to-root heuristic for trees. Suppose we have a list s and we access an element x using the move-to-front heuristic to obtain a list s′. If we insert s into an empty binary search tree to obtain T and insert s′ into an empty binary search tree to obtain T′, then we can obtain T′ from T by accessing x using the move-to-root heuristic. We thus have a commutative diagram that relates the move-to-front and move-to-root heuristics.

Also, we show that there is no such commutative diagram relating the transposition heuristic for lists and the simple exchange heuristic for trees. But a new heuristic for trees, the conditional simple exchange heuristic, is related to the transposition heuristic by a commutative diagram. Furthermore, unlike the simple exchange heuristic, we show that the conditional simple exchange heuristic has an O(log n) asymptotic expected search time if the elements are accessed with independent and equal probabilities.

We conjecture that: if we are given an “oblivious” list heuristic that converges and we transform it into a binary search tree heuristic that results in a commutative diagram, then the tree heuristic also converges. The two examples of move-to-front and transposition support this conjecture.