Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges

The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

“…where f is the density of the absolutely continuous part of μ and c(α, d) is a constant that depends only on α and d. This result goes beyond Bland's conjecture in several respects, but, ironically, the subadditive method used in Steele [58] falls short of covering the motivating case α = d. This is where the objective method entered the picture.…”

“…Confirmed cases on this list include the limit theory for MST with power weighted edges [58] and MST vertex degrees [61]; and, with some systematic effort, the list can probably be extended to include the theory of Euclidean semi-matchings [59], optimal cost triangulations [56] and the K-median problem [35]. Moreover, there are many cases where the objective method quickly gives one the essential limit theory, yet subadditive methods appear to be awkward to apply.…”

“…Stimulated by Bland's observation, Steele [58] proved that if the random variables {X i } are independent and have a distribution μ with compact support, then for all 0 < α < d one actually has…”

Rate of convergence for the Euclidean minimum spanning tree limit law

“…First let Xi, 1 _ i < o0, be uniformly and independently distributed random variables in [0,1] 2 and let LMsT(n) be the length of the shortest tree spanning {X 1, X 2 ..... Xn}. Steele [9] proved that LMsT(n) is asymptotic to /3MSTX/n with probability one (the same being true in expectation). In fact this result is valid for any uniform i.i.d, random variables with compact support of measure one in R d, d ~ 2, provided v~ is replaced by n (d-t)/d, the constant depending only on the dimension of the space and not on the shape of the compact support.…”

“…Questions about rates of convergence for these limit laws have been raised many times in the literature (see for example [2,5,8,9]). There are in fact two issues concerning information on rates of convergence (let P be a generic symbol representing any of the problems pre-cited):…”