Furthest Pair in Euclidean Geometry

• Furthest Pair is the problem of determining the two points in a finite set with the maximum Euclidean distance, equivalent to its diameter.
• The topic analyzes per-point furthest-neighbor maps, extremal counting in high dimensions, and challenges in exact algorithmics under various norms.
• It further explores approximate solutions, adversarial query models, and generalizations such as graph formulations and pair-valued Voronoi structures.

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 (Williams, 2017, Swanepoel, 2011, Ranđelović, 2024).

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 (Williams, 2017).

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

$S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$0

and the corresponding extremal quantity is

$S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$1

This places furthest neighbors in direct parallel with favourite-distance digraphs and with the diameter-pair counting problem (Swanepoel, 2011).

In abstract metric spaces, the relevant object is often a farthest-point function rather than a single pair. A function $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$2 without fixed points is max-realizable in a metric space if there exist distinct points $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$3 such that all pairwise distances are distinct and, for every $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$4, $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$5 is the unique farthest point from $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$6. This formulation studies which farthest-neighbor patterns can occur at all, independent of algorithmic optimization (Ranđelović, 2024).

Approximate formulations weaken exact maximization. In the furthest-neighbor setting, given $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$7 in a metric space $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$8, an answer $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$9 to query $\max_{i,j}\|p_i-p_j\|_2$0 is $\max_{i,j}\|p_i-p_j\|_2$1-approximate if

$\max_{i,j}\|p_i-p_j\|_2$2

An $\max_{i,j}\|p_i-p_j\|_2$3-coreset is a subset $\max_{i,j}\|p_i-p_j\|_2$4 that provides such an answer for every query $\max_{i,j}\|p_i-p_j\|_2$5 (Kluk et al., 30 Mar 2026).

2. Extremal Euclidean geometry and high-dimensional structure

In high-dimensional Euclidean extremal geometry, Furthest Pair is tightly linked to the maximum number $\max_{i,j}\|p_i-p_j\|_2$6 of diameter pairs and to the furthest-neighbor extremal quantity $\max_{i,j}\|p_i-p_j\|_2$7. For each fixed $\max_{i,j}\|p_i-p_j\|_2$8, the paper on favourite distances shows

$\max_{i,j}\|p_i-p_j\|_2$9

and, for all sufficiently large $(p_i,p_j)$0,

$(p_i,p_j)$1

Moreover, if $(p_i,p_j)$2, $(p_i,p_j)$3, and $(p_i,p_j)$4, then $(p_i,p_j)$5 for all $(p_i,p_j)$6, and $(p_i,p_j)$7 is a Lenz configuration for the distance $(p_i,p_j)$8 (Swanepoel, 2011).

A Lenz construction is the classical extremal configuration for unit distances and diameters in high dimensions. In even dimension $(p_i,p_j)$9, it is built from $S$0 pairwise orthogonal $S$1-dimensional subspaces carrying circles; in odd dimension $S$2, one part is a $S$3-sphere and the others are circles, again arranged in orthogonal subspaces with radii satisfying $S$4. Every point in one part is at equal distance $S$5 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 $S$6, and no analogue of the $S$7-dimensional favourite-distance exception appears for furthest neighbors (Swanepoel, 2011).

For large $S$8, the exact formulas for $S$9 follow from the exact formulas for $\ell_2$0. Writing $\ell_2$1 and $\ell_2$2 for the Turán number on $\ell_2$3 vertices with $\ell_2$4 parts, the paper gives the following large-$\ell_2$5 identities (Swanepoel, 2011).

Dimension regime	Large-$\ell_2$6 formula for $\ell_2$7
$\ell_2$8	$\ell_2$9 if $S\subset \mathbb{R}^d$0; $S\subset \mathbb{R}^d$1 if $S\subset \mathbb{R}^d$2
$S\subset \mathbb{R}^d$3	$S\subset \mathbb{R}^d$4
even $S\subset \mathbb{R}^d$5, $S\subset \mathbb{R}^d$6	$S\subset \mathbb{R}^d$7
odd $S\subset \mathbb{R}^d$8, $S\subset \mathbb{R}^d$9	$|S|=n$0

The same analysis yields asymptotic directed degrees in extremal furthest-neighbor digraphs. If the $|S|=n$1 Lenz parts are asymptotically balanced, then each vertex has out-degree and in-degree

$|S|=n$2

matching the global density $|S|=n$3 (Swanepoel, 2011).

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 $|S|=n$4-Furthest Pair run in time

$|S|=n$5

and there is no known exact algorithm with running time of the nearly-linear form $|S|=n$6 for Euclidean Furthest Pair. By contrast, Closest Pair in $|S|=n$7 has nearly-linear algorithms $|S|=n$8, while Furthest Pair in $|S|=n$9 and $|xy|$0 is much easier: $|xy|$1-Furthest Pair can be solved in $|xy|$2 time, and $|xy|$3-Furthest Pair in $|xy|$4 time (Williams, 2017).

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 $|xy|$5 dimensions under the $|xy|$6 norm requires $|xy|$7 time, with vectors of $|xy|$8-bit entries (Williams, 2017). This was strengthened to dimension $|xy|$9 via a chain of reductions from OV to Hopcroft’s problem, then to exact integer Max-IP, and then to $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$0-Furthest Pair (Chen, 2018). The current endpoint is that, assuming OVH or SETH, Furthest Pair, Bichromatic Closest Pair, Maximum Inner Product, and Hopcroft’s Problem require $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$1 time for any constructible dimension function $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$2; this closes the gap between the classical $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$3 algorithms and the hardness frontier (Saha et al., 24 Jun 2026).

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

$$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$4

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 (Chen, 2018).

A complementary barrier comes from circuit complexity rather than SETH. If $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$5-Furthest Pair in $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$6 for polylogarithmic $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$7 had a deterministic algorithm with running time

$$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$8

then $$D=D_S:S\to (0,\infty),\qquad D(x):=\max_{y\in S}|xy|.$$9 would have no polynomial size $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$00 circuits. The same consequence holds for Hopcroft’s problem, bichromatic $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$01-Closest Pair, and Integer Max-IP (Chen, 2018). 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 $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$02 geometry belongs to a broad OV-equivalence class. For constant $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$03 and dimension $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$04, approximate $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$05-Furthest Pair is truly-subquadratic equivalent to OV, Min-IP, Max-IP, Exact-IP, approximate bichromatic $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$06-Closest Pair, additive approximate Max-IP, and approximate Jaccard-Index-Pair. In the refined version, a $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$07-approximation to $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$08 is computable in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$09 time if and only if analogous $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$10-type improvements hold for exact or approximate inner-product problems in dimension $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$11 (Chen et al., 2018).

For query data structures, the adaptive-query model changes the problem substantially. In this model an adversary chooses $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$12 after observing previous answers. The first adversarially robust data structure for $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$13-approximate furthest-neighbor queries achieves query time

$S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$14

One variant returns a $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$15-approximate furthest neighbor in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$16 time, and another returns a $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$17-approximate furthest neighbor in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$18 time, both with preprocessing time $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$19 (Banihashem et al., 15 May 2026).

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 (Banihashem et al., 15 May 2026). 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 $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$20 without fixed points is max-realizable in some metric space if and only if the directed graph $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$21 has no cycles of length greater than $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$22. The same paper proves a universality theorem: any function that is max-realizable in some metric space is also max-realizable in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$23. Thus every realizable farthest-neighbor pattern already occurs in the Euclidean plane (Ranđelović, 2024).

For geodesic distance in a simple polygon $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$24 with $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$25 vertices and a point set $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$26 of size $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$27, there exists, for any $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$28, an $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$29-coreset $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$30 of size $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$31 such that for any query point $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$32, the geodesic distance from $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$33 to its furthest neighbor in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$34 is at least $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$35 times the geodesic distance to its furthest neighbor in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$36. The coreset can be constructed in

$S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$37

time, and then supports approximate geodesic furthest-neighbor queries with storage independent of $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$38 (Berg et al., 2024).

For planar metrics more generally, there always exists an $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$39-coreset for furthest neighbors of size bounded polynomially in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$40, 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 $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$41-comatching index: while the $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$42-semi-ladder index of planar metrics admits an exponential lower bound, the $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$43-comatching index of planar metrics is polynomial in $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$44 (Kluk et al., 30 Mar 2026).

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 $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$45, a vertex $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$46 is $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$47-far if every neighbor $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$48 of $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$49 satisfies $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$50; a pair $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$51 is far-apart if $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$52 is $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$53-far and $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$54 is $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$55-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 (Coudert et al., 2021).

A different generalization assigns distance not to one site but to an unordered pair $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$56. For a finite planar set $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$57, 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 $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$58 lower bounds and $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$59 upper bounds, and the same $S=\{p_1,\dots,p_n\}\subset \mathbb{R}^d$60 upper bound holds for several other 2-site geometric distances (Barequet et al., 2011).

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.