Optimal distance query reconstruction for graphs without long induced cycles

Abstract

Given access to the vertex set $V$ of a connected graph $G=(V,E)$ 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 $E$? Firstly, we show that randomised algorithms need to use at least $\frac1{200} \Delta n\log_\Delta n$ queries in expectation in order to reconstruct $n$-vertex trees of maximum degree $\Delta$. The best previous lower bound (for graphs of bounded maximum degree) was an information-theoretic lower bound of $\Omega(n\log n/\log \log n)$. Our randomised lower bound is also the first to break through the information-theoretic barrier for related query models including distance queries for phylogenetic trees, membership queries for learning partitions and path queries in directed trees. Secondly, we provide a simple deterministic algorithm to reconstruct trees using $\Delta n\log_\Delta n+(\Delta+2)n$ distance queries. This proves that our lower bound is optimal up to a multiplicative constant. We extend our algorithm to reconstruct graphs without induced cycles of length at least $k$ using $O_{\Delta,k}(n\log n)$ queries. Our lower bound is therefore tight for a wide range of tree-like graphs, such as chordal graphs, permutation graphs and AT-free graphs. The previously best randomised algorithm for chordal graphs used $O_{\Delta}(n\log^2 n)$ queries in expectation, so we improve by a $(\log n)$-factor for this graph class.