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Near―perfect non-crossing harmonic matchings in randomly labeled points on a circle

Abstract : Consider a set $S$ of points in the plane in convex position, where each point has an integer label from $\{0,1,\ldots,n-1\}$. This naturally induces a labeling of the edges: each edge $(i,j)$ is assigned label $i+j$, modulo $n$. We propose the algorithms for finding large non―crossing $\textit{harmonic}$ matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly―perfect matching, a matching of size $n/2 - O(n^{1/3}\ln n)$.
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https://hal.inria.fr/hal-01184038
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József Balogh, Boris Pittel, Gelasio Salazar. Near―perfect non-crossing harmonic matchings in randomly labeled points on a circle. 2005 International Conference on Analysis of Algorithms, 2005, Barcelona, Spain. pp.17-26. ⟨hal-01184038⟩

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