An O(n log n)-Time Approximation Scheme for Geometric Many-to-Many Matching
CoRR(2024)
摘要
Geometric matching is an important topic in computational geometry and has
been extensively studied over decades. In this paper, we study a
geometric-matching problem, known as geometric many-to-many matching. In this
problem, the input is a set S of n colored points in ℝ^d, which
implicitly defines a graph G = (S,E(S)) where E(S) = {(p,q): p,q ∈ S
have different colors}, and the goal is to compute a minimum-cost
subset E^* ⊆ E(S) of edges that cover all points in S. Here the
cost of E^* is the sum of the costs of all edges in E^*, where the cost of
a single edge e is the Euclidean distance (or more generally, the
L_p-distance) between the two endpoints of e. Our main result is a
(1+ε)-approximation algorithm with an optimal running time
O_ε(n log n) for geometric many-to-many matching in any fixed
dimension, which works under any L_p-norm. This is the first near-linear
approximation scheme for the problem in any d ≥ 2. Prior to this work,
only the bipartite case of geometric many-to-many matching was considered in
ℝ^1 and ℝ^2, and the best known approximation scheme in
ℝ^2 takes O_ε(n^1.5·𝗉𝗈𝗅𝗒(log n)) time.
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