Bounds for c-Ideal Hashing

TL;DR

This paper introduces and analyzes the concept of c-ideality in hash families, providing bounds on their size and exploring implications for advice complexity, especially in high load scenarios.

Contribution

It defines c-ideality as a generalization of perfect hashing, derives bounds for c-ideal hash families, and links this concept to advice complexity in hashing.

Findings

Derived upper and lower bounds on c-ideal hash family sizes

Connected c-ideality to advice complexity, showing linearity in hash table size

Extended understanding of hashing behavior in high load conditions

Abstract

In this paper, we analyze hashing from a worst-case perspective. To this end, we study a new property of hash families that is strongly related to d-perfect hashing, namely c-ideality. On the one hand, this notion generalizes the definition of perfect hashing, which has been studied extensively; on the other hand, it provides a direct link to the notion of c-approximativity. We focus on the usually neglected case where the average load \alpha is at least 1 and prove upper and lower parametrized bounds on the minimal size of c-ideal hash families. As an aside, we show how c-ideality helps to analyze the advice complexity of hashing. The concept of advice, introduced a decade ago, lets us measure the information content of an online problem. We prove hashing's advice complexity to be linear in the hash table size.