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Optimal distance query reconstruction for graphs without long induced cycles

Paul Bastide, Carla Groenland

TL;DR

The randomised lower bound is tight for a wide range of tree-like graphs, such as chordal graphs, permutation graphs and AT-free graphs, and is the first to break through the information-theoretic barrier for related query models.

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 $\frac1{200} Δn\log_Δn$ queries in expectation in order to reconstruct $n$-vertex trees of maximum degree $Δ$. The best previous lower bound (for graphs of bounded maximum degree) was an information-theoretic lower bound of $Ω(n\log n/\log \log n)$. 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 $Δn\log_Δn+(Δ+2)n$ 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 $O_{Δ,k}(n\log n)$ 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 $O_Δ(n\log^2 n)$ queries in expectation, so we improve by a $(\log n)$-factor for this graph class.

Optimal distance query reconstruction for graphs without long induced cycles

TL;DR

The randomised lower bound is tight for a wide range of tree-like graphs, such as chordal graphs, permutation graphs and AT-free graphs, and is the first to break through the information-theoretic barrier for related query models.

Abstract

Given access to the vertex set of a connected graph and an oracle that given two vertices , returns the shortest path distance between and , how many queries are needed to reconstruct ? Firstly, we show that randomised algorithms need to use at least queries in expectation in order to reconstruct -vertex trees of maximum degree . The best previous lower bound (for graphs of bounded maximum degree) was an information-theoretic lower bound of . 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 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 using 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 queries in expectation, so we improve by a -factor for this graph class.
Paper Structure (23 sections, 23 theorems, 25 equations, 5 figures)

This paper contains 23 sections, 23 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Previous algorithms and new algorithmic insight
  3. Lower bound technique
  4. Applications to other settings
  5. Roadmap
  6. Preliminaries
  7. Distance queries
  8. Tree decomposition
  9. Balanced separators
  10. Improved algorithm for distance reconstruction
  11. Optimal algorithm for tree reconstruction
  12. Optimal algorithm for $k$-chordal graphs
  13. Lower bounds for randomised tree reconstruction
  14. Reconstructing functions from the coordinate oracle
  15. Reconstructing functions from the word oracle
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Delta \geqslant 2$ and $n=2c \Delta^k$ be integers, where $c\in [1,\Delta)$ and $k\geqslant 50(c\ln c + 3)$ is an integer. Any randomised algorithm requires at least $\frac{1}{50} \Delta n\log_{\Delta} n$ queries to reconstruct $n$-vertex trees of maximum degree $\Delta+1$.

Figures (5)

  • Figure 1: In order to distinguish the tree on the left from all possible labellings of the tree on the right, $\Omega(n^2)$ queries are needed.
  • Figure 2: An example of the tree $T_{\Delta,k}$ for $\Delta = 3$ and $k=2$ is depicted with labels. The '$?$'s denote that the labels of the leaves are what needs to be reconstructed.
  • Figure 3: The subtree $T_2$ contains the neighbour of $v$ on a shortest path to $s_1$ and so contains the parent of $v$.
  • Figure 4: This figure depicts a possible configuration in the proof of \ref{['cl:kchordal_induction']} for which we end up with a contradiction by finding a large induced cycle.
  • Figure 5: Example of the tree $T_{c,\Delta,k}$ constructed in the proof of \ref{['thm:lowerboundnlogn']} for $\Delta = 4$, $c=1$ and $k=2$ with the labelling $\ell$ of the internal nodes.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.3
  • Lemma 2.1: robertson1986graph
  • Theorem 3.1
  • proof
  • Theorem 3.1
  • proof
  • Claim 3.2
  • ...and 33 more