Bichromatic Closest Pair

• Bichromatic closest pair is defined as finding the minimum-distance pair between two disjoint sets ('red' and 'blue'), which is pivotal in both theoretical reductions and practical clustering applications.
• Algorithmic strategies range from classical plane sweep and divide-and-conquer methods to advanced data structures like multi-level range trees and quantum walks for approximate solutions.
• Recent work links BCP's complexity to conjectures such as SETH and OVC, providing conditional lower bounds and insights into similarity measures and high-dimensional challenges.

The bichromatic closest pair problem arises in computational geometry and fine-grained complexity as the task of identifying the minimum-distance pair between two disjoint sets of points, commonly referred to as “red” and “blue.” This setting is crucial in both theoretical investigations—such as reductions between computational problems—and practical domains including data analysis, computational biology, clustering, and information retrieval. The problem exhibits markedly different algorithmic and complexity properties compared to its monochromatic counterpart, with distinctions sharpening in high dimensions, under different metrics, and when considering conditional lower bounds grounded in central conjectures such as SETH and the Orthogonal Vectors Conjecture.

1. Formal Definition and Variants

Given two sets $R$ and $B$, each consisting of $n$ points in $\mathbb{R}^d$ or a general metric space, the bichromatic closest pair (BCP) problem seeks a pair $(r, b) \in R \times B$ that minimizes the distance $d(r, b)$. Standard distance functions include $\ell_p$ norms (e.g., Euclidean, Manhattan, Chebyshev), Hamming distance, or set similarity metrics such as Jaccard.

Extensions of BCP include:

• Approximate BCP: Reporting a pair whose distance is at most $(1+\varepsilon)$ times the true bichromatic minimum.
• Range BCP/CRCP: Locating the closest bichromatic pair within a query range (e.g., a rectangle or slab in data structures).
• BCP under similarity measures: Maximizing Jaccard or Braun-Blanquet similarity instead of minimizing a metric distance.

These variants interact with data structuring, bounded-range queries, and complexity-theoretic reductions.

2. Algorithmic Approaches

Classical Algorithms

Early approaches mirror the monochromatic closest pair methods—plane sweep, divide-and-conquer, and brute-force comparison—yielding $O(n \log n)$ time in constant dimensions and $O(n^{2-1/\Theta(d)})$ time for moderate $d$. The MPR algorithm (Rajasekaran et al., 2014) introduces multiple projected reference points and early abandonment to prune candidate pairs, extending efficiently to bichromatic settings by filtering only cross-set pairs: $d(p_i, p_j) \ge |d(r_k, p_i) - d(r_k, p_j)|$ Candidate pair $(p_i,p_j)$ is pruned if the lower bound from any reference exceeds the current best $\delta$.

Advanced Data Structures

For colored range queries, RCP coreset techniques assemble a small subset of candidate bichromatic pairs guaranteeing that, for any query $X$, the closest pair found is within $(1+\varepsilon)$ of optimal. In rectangle or slab queries, multi-level range trees and quadrant decompositions yield near-linear space and polylogarithmic query time for $(1+\varepsilon)$-approximate BCP (Xue, 2018).

A representative pseudocode for coreset construction:

Initialize coreset Π' ← ∅ While ∃ query range X with nonempty candidate set φ* ← ClosestPair(S_bich ∩ X) If (1+ε) * |φ*| improves best in Π' Add φ* to Π' Remove φ* from S_bich Return Π'

Quantum Algorithms

Quantum walks on tensor products of Johnson graphs enable $O(n^{2/3})$ time for approximate BCP in constant dimensions (Aaronson et al., 2019). These methods require history-independent data structures to preserve quantum interference, achieved by maintaining uniquely represented radix trees and skip lists over discretized cells.

3. Fine-Grained Complexity and Conditional Hardness

Recent work has established strong lower bounds for BCP in moderate and high dimensions by reductions from canonical hard problems under SETH and OVC (Williams, 2017). Particularly:

• BCP in $\ell_2$ with $d = \omega(\log \log n)^2$ cannot be solved in $O(n^{2-\epsilon})$ time, unless SETH/OVC fails.
• Gadget-based reductions, exploiting polar-pair constructions and the sphericity/contact-dimension of $K_{n,n}$ (David et al., 2016), connect the complexity of BCP to monochromatic CP. In metrics where these parameters are low (e.g., $\ell_p$ for $p>2$), improvements transfer, while Euclidean and $\ell_1$ incur dimension blow-up that protects BCP from certain reductions.

In table form:

Domain	Hardness under SETH/OVC	Efficient Algorithms Exist
BCP, $\ell_2$, $d=\omega(\log \log n)^2$	Yes	No
BCP, $\ell_\infty$, $d=O(\log n)$	No	Yes (logarithmic dimension overhead)
Range BCP, $\ell_p$, low $d$	No	Yes (approximate, data structures)

4. Graph-Based Gadget Constructions

Hardness reductions for BCP and equivalence with CP exploit dense bipartite graphs with low contact dimension (S. et al., 2018). Embedding vertex sets $A^*$ and $B^*$ of a bipartite graph into $\mathbb{R}^d$, with cross-pair distances exactly $\beta$ and intra-set pair distances $>(1+\delta)\beta$, is achieved via codes (e.g., Reed–Solomon, algebraic-geometric). For instance:

• If $a\in A^*$ and $b\in B^*$, define $\tau^*(a)$ and $\tau^*(b)$ by evaluating polynomials at finite field points; then for $(a,b)\in E^*$, $||\tau^*(a)-\tau^*(b)||_0 = \beta$.
• Improved gadgets broaden the reach of reductions; open questions remain on constructing codes with larger gaps and smaller dimensions.

5. BCP under Similarity Measures

BCP generalizes naturally to maximization under set similarities. For Jaccard similarity, MinHash-based LSH yields $\tilde{O}(n^{2-\delta})$ algorithms when the gap $j_1 \geq j_2^{1-\delta}$ (Pagh et al., 2019). Conditional hardness shows that narrowing this gap ($j_1 \leq j_2^{1-\varepsilon}$) renders subquadratic solutions impossible under OVC. The reduction employs characteristic vector squaring and sampling, and the thresholds interact as: $$J(f(a, i), f(b, i)) \approx \frac{|a \cap b|^{2^i}}{|a|^{2^i} + |b|^{2^i} - |a \cap b|^{2^i}}$$ thus amplifying the gap for the reduction.

6. Geometric and Structural Insights

In the geometric plane, the paper of non-crossing bichromatic matchings (Aloupis et al., 2012)—while not directly solving BCP—provides combinatorial structure. Ham-sandwich cuts, convex partitions, and planarity-preserving operators (GLUE, CUT) serve dual purposes: they facilitate geometric reconfiguration and inspire balanced partitioning, essential to divide-and-conquer approaches for BCP.

Moreover, range-restricted BCP queries leverage such decompositions, with anchors and sector partitions used to approximate closest pairs efficiently within subdomains.

7. Open Problems and Future Directions

Key challenges include:

• Tightening bounds on the diameter of compatible matching graphs for geometric BCP settings (Aloupis et al., 2012).
• Constructing gadgets with better gap and dimensionality properties for harder lower bounds (S. et al., 2018).
• Extending reductions to more general metrics (e.g., set similarity, $\ell_p$ with $p\neq 2$) and to $k$-vector or $k$-biclique analogues.
• Overcoming the triangle inequality barrier for approximation hardness (S. et al., 2018).
• Quantum complexity: validating or refuting QSETH, and designing history-independent data structures for quantum walks in higher dimensions (Aaronson et al., 2019).

These advances impact not only theoretical boundaries but also practical data analysis, search, and clustering across domains where efficient cross-category proximity search is required.

• “Bichromatic compatible matchings” (Aloupis et al., 2012)
• “Efficient algorithms for the closest pair problem and applications” (Rajasekaran et al., 2014)
• “On the complexity of closest pair via polar-pair of point-sets” (David et al., 2016)
• “On the difference between closest, furthest, and orthogonal pairs: nearly-linear vs barely-subquadratic complexity in computational geometry” (Williams, 2017)
• “Colored range closest-pair problem under general distance functions” (Xue, 2018)
• “On closest pair in Euclidean metric: monochromatic is as hard as bichromatic” (S. et al., 2018)
• “Hardness of bichromatic closest pair with Jaccard similarity” (Pagh et al., 2019)
• “On the quantum complexity of closest pair and related problems” (Aaronson et al., 2019)