Optimal Bounds for Open Addressing Without Reordering

TL;DR

The paper resolves long-standing questions about open-addressing hash tables without reordering by introducing Elastic Hashing, which achieves amortized $O(1)$ and worst-case $O(\log \delta^{-1})$ expected probes (and insertion time) for $n - \lfloor \delta n \rfloor$ insertions, under $\delta > O(1/n)$ and $\delta^{-1}$ a power of two. It also shows that nonreordering greedy schemes can attain $O(\log^2 \delta^{-1})$ worst-case expected probes with Funnel Hashing, while proving matching lower bounds that certify tightness; in particular, Yao's conjecture about uniform hashing being near-optimal for greedy schemes is contradicted. The key ideas include a non-greedy, multi-level construction that decouples insertion cost from search cost via a two-dimensional probe sequence mapped by an injection $\phi$, and a funnel-structure greedy scheme that leverages the power-of-two-choices paradigm. Together, these results clarify the role of reordering in achieving small probe costs and establish tight benchmarks for open-addressing without reordering in heavily loaded tables.

Abstract

In this paper, we revisit one of the simplest problems in data structures: the task of inserting elements into an open-addressed hash table so that elements can later be retrieved with as few probes as possible. We show that, even without reordering elements over time, it is possible to construct a hash table that achieves far better expected search complexities (both amortized and worst-case) than were previously thought possible. Along the way, we disprove the central conjecture left by Yao in his seminal paper ``Uniform Hashing is Optimal''. All of our results come with matching lower bounds.

Theorems & Definitions (39)

• Theorem 1
• Theorem 2
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4