Limit theory for the random on-line nearest-neighbor graph

  • Authors:
  • Mathew D. Penrose;Andrew R. Wade

  • Affiliations:
  • Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, England;Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, England

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2008

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Abstract

In the on-line nearest-neighbor graph (ONG), each point afterthe first in a sequence of points in Rd is joinedby an edge to its nearest neighbor amongst those points thatprecede it in the sequence. We study the large-sample asymptoticbehavior of the total power-weighted length of the ONG on uniformrandom points in (0,1)d. In particular, ford = 1 and weight exponent ± 1-2, thelimiting distribution of the centered total weight is characterizedby a distributional fixed-point equation. As an ancillary result,we give exact expressions for the expectation and variance of thestandard nearest-neighbor (directed) graph on uniform random pointsin the unit interval. © 2007 Wiley Periodicals, Inc. RandomStruct. Alg., 2008