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 $n/2 - O(n^{1/3}\ln n)$.

Keywords

• harmonic graph
• noncrossing
• harmonious labeling
• graceful
• convex position
• matching
• average case behavior
• algorithm
• [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-cg] computer science [cs]/computational geometry [cs.cg]
• [info.info-hc] computer science [cs]/human-computer interaction [cs.hc]