## Abstract

In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in Rd is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in $(0,1)^d$. In particular, for $d=1$ and weight exponent $\alpha>1/2$, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval.

Translated title of the contribution | Limit theory for the random on-line nearest-neighbour graph |
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Original language | English |

Pages (from-to) | 125 - 156 |

Number of pages | 32 |

Journal | Random Structures and Algorithms |

Volume | 32 (2) |

DOIs | |

Publication status | Published - Mar 2008 |