稠密可剥随机一致超图
Dense Peelable Random Uniform Hypergraphs
摘要：
本文描述了一类新的具有独立随机边的$k$ -一致超图。超图有很高的概率是可剥离的，即不允许最小度$2$的子超图，即使当边密度（顶点上的边数）接近$1$。在我们的构造中，顶点集被划分为线性排列的段，每条边与$k$连续段的随机顶点关联。令人惊讶的是，线性几何允许我们的图“从外向内”剥离。给出了超图（$f_k$，$f_4 \approx 0.977$，$f_5 \approx 0.992$，.）远超过相应的阈值（$c_3 \approx 0.818$、$c_4 \approx 0.772$、$c_5 \approx 0.702$、.）标准$k$ -均匀随机超图的一个实例。为了掌握$f_k$，我们分析了我们的超图族的随机弱极限上的理想化剥离过程。该过程可以用函数上的运算符来描述，并且$f_k$可以链接到与运算符相关的阈值。然后，这些阈值是易于处理的数值方法。随机超图是基于散列的各种数据结构的基础。这些数据结构经常依赖于超图的可剥离性或可剥离性允许简单的线性时间算法。为了证明我们的构造的有用性，我们使用我们的$3$ -均匀超图作为Botelho等人在检索数据结构中的标准$3$ -均匀超图的直接替代。这将内存使用从$1.23m$位减少到$1.12m$位（$m$是输入大小），运行时间几乎没有变化。
英文摘要：
We describe a new family of $k$-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree $2$, even when the edge density (number of edges over vertices) is close to $1$. In our construction, the vertex set is partitioned into linearly arranged segments and each edge is incident to random vertices of $k$ consecutive segments. Quite surprisingly, the linear geometry allows our graphs to be peeled "from the outside in". The density thresholds $f_k$ for peelability of our hypergraphs ($f_3 \approx 0.918$, $f_4 \approx 0.977$, $f_5 \approx 0.992$, ...) are well beyond the corresponding thresholds ($c_3 \approx 0.818$, $c_4 \approx 0.772$, $c_5 \approx 0.702$, ...) of standard $k$-uniform random hypergraphs. To get a grip on $f_k$, we analyse an idealised peeling process on the random weak limit of our hypergraph family. The process can be described in terms of an operator on functions and $f_k$ can be linked to thresholds relating to the operator. These thresholds are then tractable with numerical methods. Random hypergraphs underlie the construction of various data structures based on hashing. These data structures frequently rely on peelability of the hypergraph or peelability allows for simple linear time algorithms. To demonstrate the usefulness of our construction, we used our $3$-uniform hypergraphs as a drop-in replacement for the standard $3$-uniform hypergraphs in a retrieval data structure by Botelho et al. This reduces memory usage from $1.23m$ bits to $1.12m$ bits ($m$ being the input size) with almost no change in running time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
