Let $G=(V,E)$ be an $n$-vertex connected graph of maximum degree $\Delta$. Given access to $V$ and an oracle that given two vertices $u,v\in V$, returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? We give a simple deterministic algorithm to reconstruct trees using $\Delta n\log_\Delta n+(\Delta+2)n$ distance queries and show that even randomised algorithms need to use at least $\frac1{200} \Delta n\log_\Delta n$ queries in expectation. The best previous lower bound was an information-theoretic lower bound of $\Omega(n\log n/\log \log n)$. Our lower bound also extends to related query models including distance queries for phylogenetic trees, membership queries for learning partitions and path queries in directed trees. We extend our deterministic algorithm to reconstruct graphs without induced cycles of length at least $k$ using $O_{\Delta,k}(n\log n)$ queries, which includes various graph classes of interest such as chordal graphs, permutation graphs and AT-free graphs. Since the previously best known randomised algorithm for chordal graphs uses $O_{\Delta}(n\log^2 n)$ queries in expectation, we both get rid off the randomness and get the optimal dependency in $n$ for chordal graphs and various other graph classes. Finally, we build on an algorithm of Kannan, Mathieu, and Zhou [ICALP, 2015] to give a randomised algorithm for reconstructing graphs of treelength $k$ using $O_{\Delta,k}(n\log^2n)$ queries in expectation.

翻译：设 $G=(V,E)$ 是一个最大度为 $\Delta$ 的 $n$ 顶点连通图。给定对顶点集 $V$ 的访问权限，以及一个对于任意两个顶点 $u,v\in V$ 返回它们之间最短路径距离的预言机，重构边集 $E$ 需要多少次查询？我们提出一个简单的确定性算法，使用 $\Delta n\log_\Delta n+(\Delta+2)n$ 次距离查询来重构树，并证明即使是随机化算法，期望下也至少需要 $\frac1{200} \Delta n\log_\Delta n$ 次查询。先前的最佳下界是信息论下界 $\Omega(n\log n/\log \log n)$。我们的下界也适用于相关的查询模型，包括系统发育树的距离查询、学习划分的成员查询以及有向树中的路径查询。我们将确定性算法扩展到重构无长度至少为 $k$ 的诱导圈的图，使用 $O_{\Delta,k}(n\log n)$ 次查询，这涵盖了多种感兴趣的图类，如弦图、置换图和无AT图。由于先前弦图的最佳已知随机化算法期望使用 $O_{\Delta}(n\log^2 n)$ 次查询，我们不仅去除了随机性，还在弦图及其他多种图类中获得了关于 $n$ 的最优依赖性。最后，我们基于 Kannan、Mathieu 和 Zhou 的算法 [ICALP, 2015]，给出了一个使用 $O_{\Delta,k}(n\log^2 n)$ 次期望查询来重构树宽为 $k$ 的图的随机化算法。