Push-Down Trees: Optimal Self-Adjusting Complete Trees
share
Since Sleator and Tarjan's seminal work on self-adjusting lists, heaps and binary search trees, researchers have been fascinated by dynamic datastructures and the questions related to their performance over time. This paper initiates the study of another classic datastructure, self-adjusting (binary) Complete Trees (CTs): trees which do not provide a simple search mechanism but allow to efficiently access items given a global map. Our problem finds applications, e.g., in the context of warehouse optimization or self-adjusting communication networks which can adapt to the demand they serve. We show that self-adjusting complete trees assume an interesting position between the complexity of self-adjusting (unordered) lists and binary search trees. In particular, we observe that in contrast to lists, a simple move-to-front strategy alone is insufficient to achieve a constant competitive ratio. Rather, and similarly to binary search trees, an additional (efficient) tree update rule is needed. Intriguingly, while it is unknown whether the working set is a lower bound for binary search trees, we show that this holds in our model. So while finding an update rule is still an open problem for binary search trees, this paper shows that there exists a simple, random update rule for complete trees. Our main result is a dynamically optimal (i.e., constant competitive) self-adjusting CT called Push-Down Tree, on expectation against an oblivious adversary. At the heart of our approach lies a distributed algorithm called Random-Push: this algorithm approximates a natural notion of Most Recently Used (MRU) tree (essentially an approximate working set), by first performing move-to-front, but then pushing less recently accessed items down the tree using a random walk.
READ FULL TEXT