压缩静态函数的一致性方法
Consensus for Compressed Static Functions
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中文总结 AI 辅助
本文提出Consensus-CSF，将一致性技术应用于压缩静态函数，通过任务插入实现接近经验熵的空间开销，并给出新的一致性推理框架。
中文摘要 AI 辅助
一致性（Consensus）技术在最小完美哈希函数（MPHFs）的构建中实现了突破，在构建时间与相对于最优情况的空间开销之间达到了线性权衡。一致性技术提供了一种巧妙的方案，用于在随机数据结构中搜索并编码任务的种子。我们将一致性技术应用于压缩静态函数（CSFs）这一相关领域。这些数据结构存储一个函数 $f: S \to \Sigma$，使得查询键 $x \in S$ 返回 $f(x)$，而查询 $x \notin S$ 返回任意值。CSFs 不需要存储键 $S$，且仅需接近值多重集合的零阶经验熵的空间。通常，某些值比其他值更为常见。在这些情况下，CSFs 可以比其非压缩对应物使用更少的空间。CSFs 是实用的构建模块，例如在数据库设计和生物信息学中。我们引入了 Consensus-CSF，它可以在假设值分布的某些参数为常数时，以 $n \exp(\tilde{\cal{O}} (\sqrt{1 / \delta}))$ 的构建时间达到任意接近经验熵 $n H_0$ 的空间使用，即空间开销为 $n H_0 (1 + \delta)$。这种权衡优于先前实现的方法，后者只能达到熵下界之上的某个固定阈值。我们通过引入任务插入，在结构不如 MPHFs 规则的 CSFs 环境中启用了一致性技术。我们的方法将键随机分布到一位一致性任务中，然后在构建否则会卡住的位置策略性地插入额外任务。我们提供了算法的实现版本，其在空间开销上达到了与竞争对手相同的数量级，但在实践中不具备竞争力。除了这些结果，我们还提出了一种思考和推理一致性技术的新方式，这种方式也可能应用于其他问题。
英文摘要
The Consensus technique marked a breakthrough in the construction of minimal perfect hash functions (MPHFs), reaching a linear tradeoff between construction time and space overhead relative to the optimum. Consensus provides a clever scheme to search for and encode seeds of tasks in random data structures. We apply Consensus to the related field of compressed static functions (CSFs). These data structures store a function $f: S \to Σ$ such that querying a key $x \in S$ returns $f(x)$ and querying $x \not \in S$ returns an arbitrary value. CSFs do not need to store the keys $S$ and only need space close to the zeroth-order empirical entropy of the multiset of values. Often, some values are much more common than others. In these cases, CSFs can use less space than their non-compressed counterparts. CSFs are a useful building block, for example in database design and bioinformatics. We introduce Consensus-CSF, which can reach arbitrarily close to the empirical entropy $n H_0$, with a construction time of $n \exp(\tilde{\cal{O}} (\sqrt{1 / δ}))$ for space usage of $n H_0 (1 + δ)$ when assuming some parameters of the value distribution to be constants. This tradeoff beats previously implemented approaches that can only reach some fixed threshold above the entropy lower bound. We enable Consensus in the setting of CSFs, which is less structured than MPHFs, with the introduction of task insertions. Our approach randomly distributes the keys into one-bit Consensus tasks and then strategically inserts additional tasks in places where the construction would get stuck otherwise. We provide an implemented version of our algorithm which reaches the same order of magnitude in space overhead as competitors but is not competitive in practice. Beyond these results, we present a new way to think and reason about Consensus, which may also be applied to other problems.
发表机构
- Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。
