An average case analysis of the minimum spanning tree heuristic for the range assignment problem

We present an average case analysis of the minimum spanning tree heuristic for the range assignment problem on a graph with power weighted edges. It is well-known that the worst-case approximation ratio of this heuristic is 2. Our analysis yields the following results: (1) In the one dimensional case ($d = 1$), where the weights of the edges are 1 with probability $p$ and 0 otherwise, the average-case approximation ratio is bounded from above by $2-p$. (2) When $d =1$ and the distance between neighboring vertices is drawn from a uniform $[0,1]$-distribution, the average approximation ratio is bounded from above by $2-2^{-\alpha}$ where $\alpha$ denotes the distance power radient. (3) In Euclidean 2-dimensional space, with distance power gradient $\alpha = 2$, the average performance ratio is bounded from above by $1 + \log 2$.