Dynamic Optimality Refuted -- For Tournament Heaps

TL;DR

This paper demonstrates a fundamental separation between offline and online algorithms for tournament heaps, showing that no online algorithm can achieve a competitive ratio better than roughly the square root of the logarithm of the number of elements, even with multiple fingers.

Contribution

It establishes a lower bound on the competitive ratio of online algorithms for tournament heaps, highlighting the limitations of self-adjusting heaps with finger access.

Findings

Offline algorithms can handle any sequence with logarithmic cost relative to the number of fingers.

Online algorithms cannot achieve a competitive ratio better than o(√log n) for tournament heaps.

Fingers provide more power than static servers, enabling more efficient offline handling of access sequences.

Abstract

We prove a separation between offline and online algorithms for finger-based tournament heaps undergoing key modifications. These heaps are implemented by binary trees with keys stored on leaves, and intermediate nodes tracking the min of their respective subtrees. They represent a natural starting point for studying self-adjusting heaps due to the need to access the root-to-leaf path upon modifications. We combine previous studies on the competitive ratios of unordered binary search trees by [Fredman WADS2011] and on order-by-next request by [Mart\'inez-Roura TCS2000] and [Munro ESA2000] to show that for any number of fingers, tournament heaps cannot handle a sequence of modify-key operations with competitive ratio in $o(\sqrt{\log{n}})$. Critical to this analysis is the characterization of the modifications that a heap can undergo upon an access. There are $\exp(\Theta(n \log{n}))$ valid heaps on $n$ keys, but only $\exp(\Theta(n))$ binary search trees. We parameterize the modification power through the well-studied concept of fingers: additional pointers the data structure can manipulate arbitrarily. Here we demonstrate that fingers can be significantly more powerful than servers moving on a static tree by showing that access to $k$ fingers allow an offline algorithm to handle any access sequence with amortized cost $O(\log_{k}(n) + 2^{\lg^{*}n})$.