时间团的近似线性3- spanner
Almost Linear 3-Spanners of Temporal Cliques
- University of Oxford（牛津大学）
- Università degli Studi dell’Aquila（阿奎拉大学）
- University of Warsaw（华沙大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对时间团提出简单递归算法，得到大小为n^{1+o(1)}的时间3-spanner，改进了此前上界，还给出生命周期有界时的优化结果，方法比此前构造更简单。
AI中文摘要：
时间图通过为边赋予正整数时间标签来建模动态网络，信息沿时间路径传播，其边标签按非递减顺序遍历。具有n个顶点的时间图的时间α-spanner是一个时间子图，它将每对顶点之间的最小跳数时间距离近似到α倍因子。虽然一般时间图对于任何α值都可能不存在稀疏的时间α-spanner，但已知时间团对于每个正整数k都存在大小为Õ(kn^{1+1/k})的时间(2k-1)-spanner。我们提出一种简单的递归算法，该算法针对每个具有n个顶点的时间团，计算出大小为n^{1+2/√ln n}=n^{1+o(1)}的时间3-spanner，从而改进了之前Õ(n^{3/2})的最佳上界。我们还表明，当生命周期L有界（即所有时间标签都在{1,…,L}中）时，我们算法的修改版本可计算出大小为O(nL)的时间3-spanner，从而改进了之前O(2^L n log n)的界。鉴于已知时间2-spanner的大小下界为Ω(n^2)（该下界对于生命周期L≥3的时间团已成立），上述两个结果尤为显著。这两种算法都依赖于一种新的简单递归分解，该分解仅使用O(n)条精心选择的边来证明大量源-目标对的时间连通性，并仅递归处理剩余对。除了产生显著改进的上界外，该方法比之前的构造简单得多。
英文摘要：
Temporal graphs model dynamic networks by assigning positive integer time labels to the edges, while information propagates along temporal paths, whose edge labels are traversed in nondecreasing order. A temporal $α$-spanner of a temporal graph with $n$ vertices is a temporal subgraph that approximates the minimum-hop temporal distance between every pair of vertices within a factor of $α$. While general temporal graphs may not admit sparse temporal $α$-spanners for any value of $α$, temporal cliques are known to admit temporal $(2k-1)$-spanners of size $\widetilde{\mathcal{O}}(kn^{1+1/k})$ for every positive integer $k$. We present a simple recursive algorithm that computes, for every temporal clique on $n$ vertices, a temporal $3$-spanner of size $n^{1+2/\sqrt{\ln n}}=n^{1+o(1)}$, thereby improving the previous best upper bound of $\widetilde{\mathcal{O}}(n^{3/2})$. We also show that a modified version of our algorithm computes temporal $3$-spanners of size $\mathcal{O}(nL)$ when the lifetime is bounded by $L$, i.e., all time labels are in $\{1,\ldots,L\}$, thus improving the previous bound of $\mathcal{O}(2^Ln\log n)$. Both results are particularly striking in light of the known lower bound of $Ω(n^2)$ on the size of temporal $2$-spanners, which already holds for temporal cliques of lifetime $L\geq 3$. Both algorithms rely on a new simple recursive decomposition that certifies temporal connectivity for a large collection of source-target pairs using only $\mathcal{O}(n)$ carefully selected edges and recursively processes only the remaining pairs. Besides yielding substantially improved upper bounds, this approach is significantly simpler than previous constructions.
