A $(1+\varepsilon)$-stretch tree cover of an edge-weighted $n$-vertex graph $G$ is a collection of trees, where every pair of vertices has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al. [STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with a constant number of trees, where the constant depends on $\varepsilon$ and the dimension $d$. This result was generalized for arbitrary doubling metrics by Bartal et. al. [ICALP'19]. While the total number of edges in the tree covers of Arya et. al. and Bartal et. al. is $O(n)$, all known tree cover constructions incur a total lightness of $\Omega(\log n)$; whether one can get a tree cover of constant lightness has remained a longstanding open question, even for 2-dimensional point sets. In this work we resolve this fundamental question in the affirmative, as a direct corollary of a new construction of $(1+\varepsilon)$-stretch spanning tree cover for doubling graphs; in a spanning tree cover, every tree may only use edges of the input graph rather than the corresponding metric. To the best of our knowledge, this is the first constant-stretch spanning tree cover construction (let alone for $(1+\varepsilon)$-stretch) with a constant number of trees, for any nontrivial family of graphs. Concrete applications of our spanning tree cover include a $(1+\varepsilon)$-stretch light tree cover, a compact $(1+\varepsilon)$-stretch routing scheme in the labeled model, and a $(1+\varepsilon)$-stretch path-reporting distance oracle, for doubling graphs. [...]

翻译：边赋权$n$顶点图$G$的$(1+\varepsilon)$-拉伸树覆盖是指一个树集合，其中任意顶点对在某一棵树中都存在一条$(1+\varepsilon)$-拉伸路径。Arya等人[STOC'95]提出的著名哑铃定理指出：$d$维欧几里得空间中任意$n$点集都允许具有常数棵树的$(1+\varepsilon)$-拉伸树覆盖，该常数依赖于$\varepsilon$和维度$d$。Bartal等人[ICALP'19]将这一结果推广到任意加倍度量空间。尽管Arya等人和Bartal等人的树覆盖总边数为$O(n)$，但所有已知树覆盖构造的总轻量度均为$\Omega(\log n)$；能否获得常数轻量度的树覆盖长期以来始终是悬而未决的根本性问题，即使对于二维点集亦然。在本工作中，我们通过为加倍图构造新的$(1+\varepsilon)$-拉伸生成树覆盖，直接推论出该基本问题的肯定解答；在生成树覆盖中，每棵树仅能使用输入图的边而非对应度量空间的边。据我们所知，这是首个针对任何非平凡图族、具有常数棵树且保持常数拉伸的生成树覆盖构造（对于$(1+\varepsilon)$-拉伸情形更是如此）。我们提出的生成树覆盖的具体应用包括：针对加倍图的$(1+\varepsilon)$-拉伸轻量树覆盖、标记模型下的紧凑型$(1+\varepsilon)$-拉伸路由方案，以及$(1+\varepsilon)$-拉伸路径报告距离预言机。[...]