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Embedding Graphs as Euclidean kNN-Graphs

T. Schibler, S. Suri, J. Xue

TL;DR

This work addresses the problem of realizing a directed $k$-regular graph as a Euclidean $k$NN-graph. It provides an efficient, dimension-aware strategy: for fixed $d\ge 2$, an efficient algorithm yields a $(1-\varepsilon)$-approximate realization or proves non-realizability, using a separator-based edge-splitting approach and per-component realizations; for $d=1$, it offers a linear-time decision procedure and, if realizable, a realization via a line-embedded linear program with $O(n)$ variables, solvable in $O(n^{2.5}\mathrm{polylog}(n))$ time. The results establish an efficient polynomial-time approximation scheme (EPTAS) in higher dimensions and a tight, fast 1D algorithm, advancing the understanding of embedding ordinal neighbor relations into Euclidean space. The methods hinge on graph separators, quasi-realizations on small components, and a constructive concatenation across components, with potential extensions to ranked neighbor data and related proximity graphs. These findings facilitate practical embedding of proximity graphs while clarifying the computational limits of exact realizability in low and higher dimensions.

Abstract

Let G = (V, E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p1, p2, ..., pn} in R^d along with a one-to-one mapping phi: V -> P such that for any u, v in V, u is an out-neighbor of v in G if and only if phi(u) is one of the k nearest neighbors of phi(v); we call the map phi a "kNN realization" of G in R^d. The kNN realization problem, which aims to compute such a mapping in R^d, is known to be NP-hard already for d = 2 and k = 1 (Eades and Whitesides, Theoretical Computer Science, 1996), and to the best of our knowledge, has not been studied in dimension d = 1. The main results of this paper are the following: (1) For any fixed dimension d >= 2, we can efficiently compute an embedding realizing at least a (1 - epsilon) fraction of G's edges, or conclude that G is not kNN-realizable in R^d. (2) For d = 1, we can decide in O(kn) time whether G is kNN-realizable and, if so, compute a realization in O(n^{2.5} * polylog(n)) time.

Embedding Graphs as Euclidean kNN-Graphs

TL;DR

This work addresses the problem of realizing a directed -regular graph as a Euclidean NN-graph. It provides an efficient, dimension-aware strategy: for fixed , an efficient algorithm yields a -approximate realization or proves non-realizability, using a separator-based edge-splitting approach and per-component realizations; for , it offers a linear-time decision procedure and, if realizable, a realization via a line-embedded linear program with variables, solvable in time. The results establish an efficient polynomial-time approximation scheme (EPTAS) in higher dimensions and a tight, fast 1D algorithm, advancing the understanding of embedding ordinal neighbor relations into Euclidean space. The methods hinge on graph separators, quasi-realizations on small components, and a constructive concatenation across components, with potential extensions to ranked neighbor data and related proximity graphs. These findings facilitate practical embedding of proximity graphs while clarifying the computational limits of exact realizability in low and higher dimensions.

Abstract

Let G = (V, E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p1, p2, ..., pn} in R^d along with a one-to-one mapping phi: V -> P such that for any u, v in V, u is an out-neighbor of v in G if and only if phi(u) is one of the k nearest neighbors of phi(v); we call the map phi a "kNN realization" of G in R^d. The kNN realization problem, which aims to compute such a mapping in R^d, is known to be NP-hard already for d = 2 and k = 1 (Eades and Whitesides, Theoretical Computer Science, 1996), and to the best of our knowledge, has not been studied in dimension d = 1. The main results of this paper are the following: (1) For any fixed dimension d >= 2, we can efficiently compute an embedding realizing at least a (1 - epsilon) fraction of G's edges, or conclude that G is not kNN-realizable in R^d. (2) For d = 1, we can decide in O(kn) time whether G is kNN-realizable and, if so, compute a realization in O(n^{2.5} * polylog(n)) time.

Paper Structure

This paper contains 10 sections, 11 theorems, 3 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Basic definitions
  4. Approximate kNN-realization in Rd
  5. Splitting the graph by removing few edges
  6. Computing the approximate realization
  7. kNN-Realization in R1
  8. Finding the vertex ordering
  9. Deciding the realizability and computing the realization
  10. Concluding remarks and extensions

Key Result

Lemma 2

Let $G$ be a directed graph that is $k$NN-realizable in $\mathbb{R}^d$, and $G_0$ be the undirected graph corresponding to $G$. Then any subgraph $H$ of $G_0$ with $|V(H)| \geq 2$ admits a balanced cut of size $O(|V(H)|^{1-\frac{1}{d}})$, where the constant hidden in $O(\cdot)$ only depends on $k$ a

Figures (1)

  • Figure 1: A $2$-regular graph on $6$ vertices and its realization (top). The values of $p, p', q, q'$ for each vertex illustrate a natural ordering of the in and out-neighborhoods corresponding to the feasible vertex ordering $v_1, \cdots, v_6$ (bottom).

Theorems & Definitions (12)

  • Definition 1: approximate $k$NN-realization
  • Lemma 2
  • Theorem 3: chuzhoy2020deterministic
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Lemma 8
  • Lemma 10
  • Lemma 11
  • Theorem 12
  • ...and 2 more