Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH

Abstract: Several fundamental problems in computational geometry admit algorithms with running time $f(d) \cdot n^{2-Θ(1/d)}$ for $n$ points in $d$ dimensions, making them among the most prominent examples of barely subquadratic computation. Notable members of this class include Furthest Pair, Bichromatic Closest Pair, (Bichromatic) Maximum Innter Product, and Hopcroft's Problem. Chen [Theory Comput. 2020] proved that, assuming the Strong Exponential Time Hypothesis (SETH), these problems require $n^{2-o(1)}$ time when the dimension satisfies $d=2^{Θ(\log^* n)}$. We extend this lower bound to all efficiently constructible dimensions $d=ω(1)$. Thus, assuming SETH, the dependence of the best known algorithms on the dimension is essentially unavoidable. The proof utilizes techniques in OpenAI's recent disproof of the Erdos unit distance conjecture. The proof was initially discovered by ChatGPT 5.5 Pro. The authors have validated and substantially edited the proof to improve the presentation.

• The paper establishes that under SETH, any algorithm for the Furthest Pair in superconstant dimensions requires quadratic time.
• It employs a reduction from Boolean Orthogonal Vectors using advanced algebraic number theory and dimension-efficient encodings.
• The findings imply that improving on the n²-time bound in high dimensions is unlikely without refuting SETH.

Quadratic-Time Lower Bound for Furthest Pair in Superconstant Dimension under SETH

Problem Context and Previous Results

The Furthest Pair problem, which seeks the pair of points at maximum Euclidean distance among a set of $n$ points in $\mathbb{R}^d$, is canonically hard in computational geometry, with broad implications for related tasks such as Bichromatic Closest Pair, Maximum Inner Product (Max-IP), and Hopcroft's Problem. While Closest Pair admits truly subquadratic algorithms for moderate $d$, all algorithms for Furthest Pair (and its aforementioned relatives) degenerate to near-quadratic time once the dimension is superconstant. Chen (2020) [Che20] established that, under SETH, Furthest Pair requires $n^{2-o(1)}$ time for $d = 2^{\Theta(\log^* n)}$ dimensions, but an exponential gap remained above $d = \omega(1)$. The present work rigorously closes this gap, proving that the quadratic-time barrier is intrinsic for any efficiently constructible dimension function $d = \omega(1)$.

Main Contributions

The paper "Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH" (2606.25887) establishes that assuming the Strong Exponential Time Hypothesis (SETH), any algorithm for Furthest Pair, Bichromatic Closest Pair, Max-IP, or Hopcroft's Problem in dimension $d = \omega(1)$ cannot run in truly subquadratic time. This demonstrates the essential tightness of all known upper bounds and extends previous conditional lower bounds in the fine-grained complexity framework. The reduction framework is nontrivial, relying on deep embeddings between Boolean Orthogonal Vectors (OV) and their arithmetic (integer-valued) analogues. The construction leverages advanced machinery from algebraic number theory, building on the technical advances in OpenAI’s disproof of the classical Erdős unit distance conjecture [Ope26].

Technical Approach

The key ingredient is a reduction from the Boolean OV problem (the canonical SETH-hard problem in fine-grained complexity) to the integer variant (Z-OV) in superconstant dimensions, enabling further reductions to Furthest Pair and related geometric problems. The reduction operates blockwise, encoding bits into algebraic integers in carefully chosen number fields so that dot product orthogonality is preserved under the transformation. The construction relies heavily on:

1. Dimension-Efficient Encodings: By circumventing the classical density-of-primes bottleneck via the use of number fields in which primes split completely into many prime ideals, the work achieves block encodings that were previously unattainable. This step improves over prior barriers, which were fundamentally limited by prime densities in the integers and only allowed lower bounds at doubly or exponentially iterated logarithmic dimension.
2. Algebraic Number Theory: The reduction uses the existence of number fields and ideals as provided by recent advances by OpenAI [Ope26] in the unit distance problem, allowing the construction of norm-one codebooks and block encodings. The proof explicitly constructs field extensions with highly controlled splitting of primes, enabling an efficient Chinese Remainder Theorem mechanism over algebraic integers.
3. Uniform and Nonuniform Local Reductions: The proof carefully constructs local blockwise gadgets to embed Boolean vectors into arithmetic ones, ensuring value sets are separated efficiently. The reduction is constructed to be uniform, i.e., all gadgets are efficiently computable.
4. Compositional Reductions to Geometric Problems: Once the lower bound for Z-OV is established, reductions to Z-Max-IP, Furthest Pair, and Bichromatic Closest Pair follow by standard algebraic reductions (detailed in the paper’s appendices).

The correctness of the reduction is rigorously justified, and the core lemma has been fully formalized in Lean 4, with dependencies on modern mathematical libraries and formalizations [ABB+25, The26].

Numerical and Structural Claims

• For every efficiently constructible dimension function $d(n) = \omega(1)$ and every $\varepsilon>0$, there is no $\mathbb{R}^d$0 algorithm for Furthest Pair, Bichromatic Closest Pair, Z-Max-IP, or Z-OV in dimension $\mathbb{R}^d$1, assuming SETH.
• The construction holds for all input sizes by exhaustive search for block reductions at very small parameters.
• The lower bound relies only on SETH (or the related Orthogonal Vectors Hypothesis, OVH), and the reductions preserve efficient encodings with $\mathbb{R}^d$2-bit coordinates throughout.

Implications

This result demonstrates that for all practical purposes, any improvement on the Furthest Pair (as well as Bichromatic Closest Pair, Max-IP, and Hopcroft’s Problem) running time beyond $\mathbb{R}^d$3 is impossible unless SETH fails, even for moderate-to-large $\mathbb{R}^d$4 as soon as $\mathbb{R}^d$5. The quadratic barrier is therefore not a result of lack of algorithmic ingenuity or due to technical limitations in reductions, but is a fine-grained manifestation of fundamental structure in the Boolean and integer domains. Further, the hardness result extends directly to several linear algebraic subproblems, dynamic neural computation tasks, and attention computation [GHS+26], as many of these reduce to Max-IP or Furthest Pair.

The proof technique, which explicitly incorporates modern algebraic number theoretic tools in fine-grained complexity reductions, marks a significant step in the interplay between number theory and algorithmic lower bounds. The AI-assisted component—wherein the initial proof structure was proposed by ChatGPT 5.5 Pro—shows the growing role of automated theorem provers and LLMs in mathematical discovery, though validation and refinement remain human-led.

Theoretical and Practical Outlook

The results suggest that overcoming the quadratic complexity for these problems in superconstant dimension will require fundamentally new algorithmic paradigms or a refutation of SETH. In algorithmic geometry and high-dimensional data analysis, practitioners must therefore regard $\mathbb{R}^d$6-time as optimal for exact Furthest Pair-type problems as soon as the dimension is not bounded.

The technical machinery developed in this work also invites further exploration into the use of algebraic and number-theoretic tools for other fine-grained complexity lower bounds, especially those for problems at the boundary of combinatorics and geometry. The reduction framework, relying on splitting primes in number fields and norm-one codebooks, could be seen as a template for similar conditional lower bounds in other arithmetic or geometric computational contexts.

Conclusion

This paper (2606.25887) decisively establishes that the Furthest Pair, Bichromatic Closest Pair, Max-IP, and Hopcroft’s Problem require $\mathbb{R}^d$7 time for any efficiently constructible superconstant dimension under SETH. The proof, combining intensive number theory with fine-grained reductions and formal verification, sets a new standard for conditional lower bounds in computational geometry. These results close a longstanding gap in fine-grained hardness for key high-dimensional geometric problems and delineate the limitations of algorithmic improvements in high-dimensional spaces.