Quasi-Linear Distance Query Reconstruction for Graphs of Bounded Treelength

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices u and v, the oracle returns the shortest path distance between u and v in the graph. The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree ∆ and treelength k in O<inf>k</inf><inf>∆</inf>(nlog² n) queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.