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.

翻译：本文从最坏情况的角度分析哈希问题。为此，我们研究了哈希族的一个新性质——c-理想性，该性质与d-完全哈希密切相关。一方面，这一概念推广了已被广泛研究的完全哈希的定义；另一方面，它为c-近似性提供了直接联系。我们聚焦于通常被忽略的、平均负载α至少为1的情况，并证明了c-理想哈希族最小规模的上界和下界参数化界限。此外，我们展示了c-理想性如何有助于分析哈希的建议复杂度。十年前引入的建议概念，使我们能够衡量在线问题的信息含量。我们证明了哈希的建议复杂度与哈希表大小呈线性关系。