A nearly optimal randomized algorithm for explorable heap selection

TL;DR

This paper introduces a nearly optimal randomized algorithm for explorable heap selection, significantly improving previous methods by reducing running time while maintaining low space complexity.

Contribution

The authors present a new randomized algorithm with improved running time and nearly optimal complexity for explorable heap selection, balancing efficiency and space usage.

Findings

New randomized algorithm with O(n log^3 n) time

Uses only O(log n) space, improving previous results

Proven near-optimality with a matching lower bound

Abstract

Explorable heap selection is the problem of selecting the $n$th smallest value in a binary heap. The key values can only be accessed by traversing through the underlying infinite binary tree, and the complexity of the algorithm is measured by the total distance traveled in the tree (each edge has unit cost). This problem was originally proposed as a model to study search strategies for the branch-and-bound algorithm with storage restrictions by Karp, Saks and Widgerson (FOCS '86), who gave deterministic and randomized $n\cdot \exp(O(\sqrt{\log{n}}))$ time algorithms using $O(\log(n)^{2.5})$ and $O(\sqrt{\log n})$ space respectively. We present a new randomized algorithm with running time $O(n\log(n)^3)$ using $O(\log n)$ space, substantially improving the previous best randomized running time at the expense of slightly increased space usage. We also show an $\Omega(\log(n)n/\log(\log(n)))$ for any algorithm that solves the problem in the same amount of space, indicating that our algorithm is nearly optimal.