Reconstructing a Bounded-Degree Directed Tree Using Path Queries

TL;DR

This paper introduces efficient randomized algorithms for reconstructing bounded-degree directed trees using path queries, providing bounds on query complexity and extending to noisy and weighted scenarios.

Contribution

It proposes new algorithms with near-optimal query complexity for reconstructing directed trees, including variants for noisy and weighted cases.

Findings

Reconstruction algorithm uses O(dn log^2 n) path queries.

Lower bounds established: Ω(n log n) for randomized, Ω(dn) for deterministic algorithms.

Extended algorithms handle noisy and additive weighted queries efficiently.

Abstract

We present a randomized algorithm for reconstructing directed rooted trees of $n$ nodes and node degree at most $d$, by asking at most $O(dn\log^2 n)$ path queries. Each path query takes as input an origin node and a target node, and answers whether there is a directed path from the origin to the target. Regarding lower bounds, we show that any randomized algorithm requires at least $\Omega(n\log n)$ queries, while any deterministic algorithm requires at least $\Omega(dn)$ queries. Additionally, we present a $O(dn\log^3 n)$ randomized algorithm for noisy queries, and a $O(dn\log^2 n)$ randomized algorithm for additive queries on weighted trees.