Distance-based Learning of Hypertrees

TL;DR

This work defines orderly hypertrees—hypertrees whose skeleton graphs form a tree and fit between known acyclicity classes in the Fagin hierarchy—and shows they can be learned efficiently with shortest-path queries. It develops an online SP-query learning algorithm that inserts vertices incrementally, achieving $O(n\Delta\log_\Delta m)$ query complexity (with an $h^2$ overhead for the initial one-edge phase) and proves this is asymptotically optimal via adversarial lower bounds. The paper also analyzes offline learning (removing the $h^2$ cost) and extends the framework to bounded-distance queries, deriving tight bounds for various $d$-bounded settings. Skeleton graphs enable a concise representation and efficient reconstruction of orderly hypertrees, and the results offer practical insights for streaming or partially revealed data in domains like phylogenetics and database theory.

Abstract

We study the problem of learning hypergraphs with shortest-path queries (SP-queries), and present the first provably optimal online algorithm for a broad and natural class of hypertrees that we call orderly hypertrees. Our online algorithm can be transformed into a provably optimal offline algorithm. Orderly hypertrees can be positioned within the Fagin hierarchy of acyclic hypergraph (well-studied in database theory), and strictly encompass the broadest class in this hierarchy that is learnable with subquadratic SP-query complexity. Recognizing that in some contexts, such as evolutionary tree reconstruction, distance measurements can degrade with increased distance, we also consider a learning model that uses bounded distance queries. In this model, we demonstrate asymptotically tight complexity bounds for learning general hypertrees.

Figures (5)

• Figure 1: (Left) Two hypertrees with identical distances between corresponding vertices; vertices displayed in the same position in the left and right trees are assumed to be identical. (Right) Any class $\mathcal{H}$ containing all hypertrees isomorphic to this hypertree, with diameter 3, is hard to learn with $\mathit{SP}$-queries. (Here $*_{n-6}$ denotes a cluster of $n-6$ vertices).
• Figure 2: Orderly hypertree (left) and its skeleton graph (right)
• Figure 3: Sub-skeletons of the orderly hypertree from Figure \ref{['fig:hypertree+skeleton']} induced on vertex sets $\{v_1, v_{2} \}$ (a), $\{ v_1, v_{2}, v_3 \}$ (b), $\{ v_1, v_{2}, v_3, v_4 \}$ (c), $\{v_1, \ldots, v_5 \}$ (d) and $\{v_1, \ldots, v_6 \}$ (e).
• Figure 4: Hyperpath $P_n$ that requires $\Theta(n^2)$$\mathrm{dist}_{\le 1}$-queries.
• Figure 5:

Theorems & Definitions (42)

• definition 1
• remark 1
• definition 2
• remark 2
• remark 3
• remark 4
• lemma 1
• proof
• definition 3
• Claim 1