Asymptotic theory of greedy approximations to minimal k-point random graphs

Let χ_n=(x₁,...,x_n), be an independent and identically distributed (i.i.d.) sample having multivariate distribution P. We derive almost sure (a.s.) limits for the power-weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples. The class includes minimal k-point graphs constructed by the partitioning method of Ravi, Sundaram, Marathe, Rosenkrantz, and Ravi (see Proc. 5th Annu. ACM-SIAM Symp. Discrete Algorithms, Arlington, VA, p.546-55, 1994), where the edge weight function satisfies the quasi-additive property of Redmond and Yukich (see Ann. Appl. Probab., vol.4, no.4, p.1057-73, 1994). In particular, this includes greedy approximations to the k-point minimal spanning tree (k-MST), Steiner tree (k-ST), and the traveling salesman problem (k-TSP). An expression for the influence function of the minimal-weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution. The influence function takes a form which indicates that the k-point minimal graph in d>1 dimensions has robustness properties in R^d which are analogous to those of rank-order statistics in one dimension. A direct result of our theory is that the log-weight of the k-point minimal graph is a consistent nonparametric estimate of the Renyi entropy of the distribution P. Possible applications of this work include: analysis of random communication network topologies, estimation of the mixing coefficient in ε-contaminated mixture models, outlier discrimination and rejection, clustering, and pattern recognition, robust nonparametric regression, two-sample matching, and image registration