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 $\frac1{100} \Delta n\log_\Delta n$ 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，以及一个能够返回任意两顶点$u,v\in V$之间最短路径距离的预言机，重构边集$E$需要多少次查询？我们给出了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，期望也至少需要$\frac1{100} \Delta n\log_\Delta n$次查询。此前最佳下界是信息论下界$\Omega(n\log n/\log \log n)$。我们的下界还扩展到相关查询模型，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不含长度至少为$k$的诱导环的图，使用$O_{\Delta,k}(n\log n)$次查询，这包括若干感兴趣的图类，如弦图、置换图和无AT图。由于此前已知最佳的随机化算法用于弦图时期望使用$O_{\Delta}(n\log^2 n)$次查询，我们既去除了随机性，又为弦图及其他多种图类获得了$n$的最优依赖关系。最后，我们基于Kannan、Mathieu和Zhou [ICALP, 2015]的算法，给出了一个用于重构树长为$k$的图的随机化算法，其期望查询次数为$O_{\Delta,k}(n\log^2n)$。