---
title: Furthest Pair in Euclidean Geometry
url: https://www.emergentmind.com/topics/furthest-pair
type: topic
---

# Furthest Pair in Euclidean Geometry

Furthest Pair is the problem of finding, in a finite point set, two points at maximum mutual distance; in Euclidean space this maximum is exactly the diameter of the point set. In the literature represented here, the topic appears in several tightly related forms: the global optimization problem \(\max_{i,j}\|p_i-p_j\|_2\), per-point furthest-neighbor maps and digraphs, extremal counting problems for diameter pairs in high-dimensional Euclidean geometry, pair-valued Voronoi structures, and approximate or adversarially robust query data structures. These formulations connect extremal combinatorics, computational geometry, fine-grained complexity, and metric geometry [1709.05282] [1108.4817] [2409.14648].

## 1. Core definitions and formulations

For a set \(S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d\), the Furthest Pair problem under the Euclidean norm is to find
\[
\max_{i,j}\|p_i-p_j\|_2
\]
and a pair \((p_i,p_j)\) attaining this maximum. The value is the Euclidean diameter of \(S\). The same paper also considers bichromatic variants, but for \(\ell_2\) the bichromatic and non-bichromatic versions are essentially equivalent up to trivial reductions [1709.05282].

A local version replaces one global maximizing pair by one maximizing neighbor for each point. If \(S\subset \mathbb{R}^d\), \(|S|=n\), and \(|xy|\) denotes Euclidean distance, the furthest-distance function is
\[
D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.
\]
The furthest-neighbor digraph is
\[
\vec E_D(S):=\{(x,y)\in S^2: |xy|=D(x)\},
\]
and the corresponding extremal quantity is
\[
g_d(n):=\max\{e_D(S): S\subset\mathbb{R}^d,\ |S|=n\}.
\]
This places furthest neighbors in direct parallel with favourite-distance digraphs and with the diameter-pair counting problem [1108.4817].

In abstract metric spaces, the relevant object is often a farthest-point function rather than a single pair. A function \(g:[n]\to[n]\) without fixed points is max-realizable in a metric space if there exist distinct points \(A_1,\dots,A_n\) such that all pairwise distances are distinct and, for every \(i\), \(A_{g(i)}\) is the unique farthest point from \(A_i\). This formulation studies which farthest-neighbor patterns can occur at all, independent of algorithmic optimization [2409.14648].

Approximate formulations weaken exact maximization. In the furthest-neighbor setting, given \(P\subseteq V\) in a metric space \((V,\mathrm{dist})\), an answer \(p\in P\) to query \(v\in V\) is \(\varepsilon\)-approximate if
\[
\mathrm{dist}(v,p)\ge (1-\varepsilon)\max_{p'\in P}\mathrm{dist}(v,p').
\]
An \(\varepsilon\)-coreset is a subset \(Q\subseteq P\) that provides such an answer for every query \(v\in V\) [2603.28100].

## 2. Extremal Euclidean geometry and high-dimensional structure

In high-dimensional Euclidean extremal geometry, Furthest Pair is tightly linked to the maximum number \(M_d(n)\) of diameter pairs and to the furthest-neighbor extremal quantity \(g_d(n)\). For each fixed \(d\ge 4\), the paper on favourite distances shows
\[
g_d(n)\sim f_d(n)\sim\Bigl(1-\frac1{\lfloor d/2\rfloor}\Bigr)n^2,
\]
and, for all sufficiently large \(n\),
\[
g_d(n)=2M_d(n).
\]
Moreover, if \(S\subset\mathbb{R}^d\), \(|S|=n\), and \(e_D(S)=g_d(n)\), then \(D(x)=\operatorname{diam}(S)\) for all \(x\in S\), and \(S\) is a Lenz configuration for the distance \(\operatorname{diam}(S)\) [1108.4817].

A Lenz construction is the classical extremal configuration for unit distances and diameters in high dimensions. In even dimension \(d=2p\), it is built from \(p\) pairwise orthogonal \(2\)-dimensional subspaces carrying circles; in odd dimension \(d=2p+1\), one part is a \(2\)-sphere and the others are circles, again arranged in orthogonal subspaces with radii satisfying \(r_i^2+r_j^2=\lambda^2\). Every point in one part is at equal distance \(\lambda\) from every point in any other part, while intra-part distances do not contribute to the main term. Theorem B identifies extremal furthest-neighbor configurations with these Lenz configurations for sufficiently large \(n\), and no analogue of the \(4\)-dimensional favourite-distance exception appears for furthest neighbors [1108.4817].

For large \(n\), the exact formulas for \(g_d(n)\) follow from the exact formulas for \(M_d(n)\). Writing \(p=\lfloor d/2\rfloor\) and \(t_p(n)\) for the Turán number on \(n\) vertices with \(p\) parts, the paper gives the following large-\(n\) identities [1108.4817].

| Dimension regime | Large-\(n\) formula for \(g_d(n)\) |
|---|---|
| \(d=4\) | \(g_4(n)=2t_2(n)+n+2\) if \(n\not\equiv 3\pmod 4\); \(g_4(n)=2t_2(n)+n\) if \(n\equiv 3\pmod 4\) |
| \(d=5\) | \(g_5(n)=2t_2(n)+2n\) |
| even \(d\ge 6\), \(p=d/2\) | \(g_d(n)=2t_p(n)+2p\) |
| odd \(d\ge 7\), \(p=\lfloor d/2\rfloor\) | \(g_d(n)=2t_p(n)+2\left\lfloor\frac{n}{p}\right\rfloor+2p-2\) |

The same analysis yields asymptotic directed degrees in extremal furthest-neighbor digraphs. If the \(p\) Lenz parts are asymptotically balanced, then each vertex has out-degree and in-degree
\[
\deg^+(x)=\deg^-(x)\sim \Bigl(1-\frac1p\Bigr)n
=
\Bigl(1-\frac1{\lfloor d/2\rfloor}\Bigr)n,
\]
matching the global density \(g_d(n)\sim (1-1/p)n^2\) [1108.4817].

## 3. Exact algorithmics and conditional hardness

In computational geometry, Furthest Pair is one of the canonical examples of a barely-subquadratic problem. The best known exact algorithms for \(\ell_2\)-Furthest Pair run in time
\[
f(d)\cdot n^{2-1/\Theta(d)},
\]
and there is no known exact algorithm with running time of the nearly-linear form \(2^{O(d)}n\log^{O(1)}n\) for Euclidean Furthest Pair. By contrast, Closest Pair in \(\ell_2^d\) has nearly-linear algorithms \(2^{O(d)}n\log^{O(1)}n\), while Furthest Pair in \(\ell_1\) and \(\ell_\infty\) is much easier: \(\ell_\infty\)-Furthest Pair can be solved in \(\tilde O(dn)\) time, and \(\ell_1\)-Furthest Pair in \(\tilde O(2^d n)\) time [1709.05282].

The fine-grained lower-bound sequence is now sharp across all superconstant dimensions. An early result showed that, under SETH or OVC, finding a furthest pair in \(\omega((\log\log n)^2)\) dimensions under the \(\ell_2\) norm requires \(n^{2-o(1)}\) time, with vectors of \(O(\log n)\)-bit entries [1709.05282]. This was strengthened to dimension \(2^{O(\log^* n)}\) via a chain of reductions from OV to Hopcroft’s problem, then to exact integer Max-IP, and then to \(\ell_2\)-Furthest Pair [1802.02325]. The current endpoint is that, assuming OVH or SETH, Furthest Pair, Bichromatic Closest Pair, Maximum Inner Product, and Hopcroft’s Problem require \(n^{2-o(1)}\) time for any constructible dimension function \(D(n)=\omega(1)\); this closes the gap between the classical \(f(d)\cdot n^{2-\Theta(1/d)}\) algorithms and the hardness frontier [2606.25887].

The reduction architecture explains why Furthest Pair is grouped with Hopcroft’s problem and Max-IP. In one standard transformation, vectors from two color classes are lifted so that for cross pairs
\[
\|\tilde x-\tilde y\|_2^2 = 2W + 2(x\cdot y),
\]
making the furthest cross-pair encode the maximum inner product. This is the mechanism used to transfer exact Max-IP hardness to Euclidean Furthest Pair in very low dimensions [1802.02325].

A complementary barrier comes from circuit complexity rather than SETH. If \(\ell_2\)-Furthest Pair in \(\mathbb{R}^d\) for polylogarithmic \(d\) had a deterministic algorithm with running time
\[
n^2\mathrm{poly}(d)/\log^{\omega(1)}n,
\]
then \(NEXP\) would have no polynomial size \(THR\circ THR\) circuits. The same consequence holds for Hopcroft’s problem, bichromatic \(\ell_2\)-Closest Pair, and Integer Max-IP [1805.10698]. This suggests that even log-shaving improvements in the moderate-dimensional regime are entangled with major open problems in lower-bound complexity theory.

## 4. Approximation and adversarially robust querying

Approximate Furthest Pair in low-dimensional \(\ell_p\) geometry belongs to a broad OV-equivalence class. For constant \(p\in[1,2]\) and dimension \(d\le n^{o(1)}\), approximate \(\ell_p\)-Furthest Pair is truly-subquadratic equivalent to OV, Min-IP, Max-IP, Exact-IP, approximate bichromatic \(\ell_p\)-Closest Pair, additive approximate Max-IP, and approximate Jaccard-Index-Pair. In the refined version, a \((1+\varepsilon)\)-approximation to \(\FP_p\) is computable in \(n^{2-\varepsilon^{o(1)}}\) time if and only if analogous \(n^{2-1/c^{o(1)}}\)-type improvements hold for exact or approximate inner-product problems in dimension \(c\log n\) [1811.12017].

For query data structures, the adaptive-query model changes the problem substantially. In this model an adversary chooses \(q_i\) after observing previous answers. The first adversarially robust data structure for \(c\)-approximate furthest-neighbor queries achieves query time
\[
\tilde{O}( \min( d n^{1/c^2},\ n^{2/c^2} + d)).
\]
One variant returns a \(c\)-approximate furthest neighbor in \(\tilde O(d n^{1/c^2})\) time, and another returns a \((1+\varepsilon)c\)-approximate furthest neighbor in \(\tilde O(\min\{n^{2/c^2},n\}+d)\) time, both with preprocessing time \(\tilde O(d^2 n^{1+1/c})\) [2605.16618].

The same work also gives an adversarial attack against oblivious approximate furthest-neighbor algorithms. In particular, the data structure from Indyk’s algorithm fails to maintain its guarantees against adaptive queries [2605.16618]. This separates the oblivious and adaptive models for Furthest Neighbor, and by extension for algorithmic workflows that repeatedly probe extremal distances.

## 5. Metric realizations, simple polygons, and planar metrics

When only the farthest-neighbor function is prescribed, the combinatorics are unexpectedly rigid. A function \(g:[n]\to[n]\) without fixed points is max-realizable in some metric space if and only if the directed graph \(G_g\) has no cycles of length greater than \(2\). The same paper proves a universality theorem: any function that is max-realizable in some metric space is also max-realizable in \(\mathbb{R}^2\). Thus every realizable farthest-neighbor pattern already occurs in the Euclidean plane [2409.14648].

For geodesic distance in a simple polygon \(\mathcal P\) with \(m\) vertices and a point set \(P\) of size \(n\), there exists, for any \(\varepsilon>0\), an \(\varepsilon\)-coreset \(\mathcal C\subset P\) of size \(O(1/\varepsilon^2)\) such that for any query point \(q\in\mathcal P\), the geodesic distance from \(q\) to its furthest neighbor in \(\mathcal C\) is at least \(1-\varepsilon\) times the geodesic distance to its furthest neighbor in \(P\). The coreset can be constructed in
\[
O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)
\]
time, and then supports approximate geodesic furthest-neighbor queries with storage independent of \(|P|\) [2403.04513].

For planar metrics more generally, there always exists an \(\varepsilon\)-coreset for furthest neighbors of size bounded polynomially in \(1/\varepsilon\), constructible in polynomial time. This improves upon an exponential bound for planar and minor-free metrics and resolves the open problem for polygons with holes. The technical mechanism is an \(\varepsilon\)-comatching index: while the \(\varepsilon\)-semi-ladder index of planar metrics admits an exponential lower bound, the \(\varepsilon\)-comatching index of planar metrics is polynomial in \(1/\varepsilon\) [2603.28100].

## 6. Graph generalizations and pair-valued Voronoi structures

In graphs, the analogue of a furthest pair is a far-apart pair. In a connected unweighted graph \(G=(V,E)\), a vertex \(u\) is \(v\)-far if every neighbor \(w\) of \(u\) satisfies \(\mathrm{dist}(v,w)\le \mathrm{dist}(v,u)\); a pair \((u,v)\) is far-apart if \(u\) is \(v\)-far and \(v\) is \(u\)-far. Any diameter pair is far-apart, and the paper gives a data structure for enumerating all far-apart pairs by decreasing distance. This avoids storing the full distance matrix and, for some instances, reduces the memory consumption by at least two orders of magnitude in hyperbolicity computations [2104.12523].

A different generalization assigns distance not to one site but to an unordered pair \((p,q)\). For a finite planar set \(S\), 2-site furthest-neighbor Voronoi diagrams are defined by maximizing functions such as circumcircle radius, containing-circle radius, view angle, inradius, and several circumcenter-based quantities. The resulting furthest-neighbor diagrams are often quartic in complexity: for example, the furthest-neighbor diagrams under circumcircle radius and view angle both have \(\Omega(n^4)\) lower bounds and \(O(n^{4+\varepsilon})\) upper bounds, and the same \(O(n^{4+\varepsilon})\) upper bound holds for several other 2-site geometric distances [1105.4130].

These graph and Voronoi formulations do not replace the Euclidean diameter problem, but they show that the notion of a furthest pair has several structurally distinct incarnations: as a global diameter witness, as a per-vertex farthest-neighbor relation, as a realizability problem for metric patterns, and as a pair-valued extremal object in higher-order Voronoi geometry. Across these formulations, the recurring theme is that extremal distance is both highly structured and, in moderate or high dimension, algorithmically resistant.

Source: https://www.emergentmind.com/topics/furthest-pair