AI 中文总结
本文针对开放寻址哈希的二次探测等方案，证明固定偏移序列的开放寻址哈希在负载因子达35.74%时插入成本恒定，二次探测更是可达37.61%，核心创新为新型见证森林。
AI 中文摘要
在开放寻址哈希中，二次探测因在高引用局部性和低搜索探测次数之间取得良好平衡而颇具吸引力，但这些均为经验观察，而非理论保证。实际上，直到最近，人们仍不清楚即使使用均匀随机哈希函数，二次探测在任何正负载因子α > 0下是否具有恒定的期望插入成本。Kuszmaul与Xi（2024）取得了一项突破性成果——尽管其数值被低估——他们证明任何固定偏移序列（包括二次探测）在负载因子α ≤ 8.9%时确实具有恒定的期望插入成本，这远低于我们想要证明的结果，即二次探测在任何远离1的负载因子α < 1-ε下具有恒定插入成本。本文中，我们证明采用任何固定偏移序列的开放寻址哈希在负载因子高达35.74%时具有恒定的期望插入成本，而对于二次探测而言，我们可将负载因子提高至37.61%。我们的主要创新是一种用于记录探测序列间冲突的新型见证森林。
英文摘要
In open addressed hashing, quadratic probing is attractive for striking a nice balance between having a high locality of reference and a low number of probes per search. However, these are empirical observations, not theoretical guarantees. Indeed, until recently, it was not known whether quadratic probing had constant expected insertion cost under any positive load factor $α> 0$, even with uniformly random hash functions. In a recent breakthrough---albeit a numerically understated breakthrough---Kuszmaul and Xi (2024) proved that any fixed offset sequence (including quadratic probing) does, in fact, have constant expected insertion cost for load factors $α\leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $α< 1-ε$ bounded away from 1. In this paper, we prove that open addressed hashing with any fixed offset sequence has constant expected insertion cost for load factors up to $35.74\%$, and that for quadratic probing in particular, we can increase the load factor to $37.61\%$. Our main innovation is a new type of witness forest for recording collisions among the probe sequences.
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