Information Processing Letters 110 (2010) 356–359 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl On the connectivity threshold for general uniform metric spaces Gady Kozma a , Zvi Lotker b,∗ , Gideon Stupp c a Department of Mathematics, Weizmann Institute of Science, PO Box 26, Rehovot 76100, Israel b Department of Communication Systems Engineering, Ben Gurion University, Beer Sheva 84105, Israel c Cisco systems, 32 Hamelacha st, Netanya, Israel article info abstract Article history: Received 8 December 2008 Received in revised form 20 January 2010 Accepted 24 February 2010 Available online 26 February 2010 Communicated by A. Tarlecki Keywords: Random networks Sensor networks Connectivity threshold Fractal Let μ be a measure supported on a compact connected subset of an Euclidean space, which satisfies a uniform d-dimensional decay of the volume of balls of the type αδ d  μ ( B (x,δ) )  βδ d (1) where d is a fixed constant. We show that the maximal edge in the minimum spanning tree of n independent samples from μ is, with high probability ≈ ( log n n ) 1/d . While previous studies on the maximal edge of the minimum spanning tree attempted to obtain the exact asymptotic, we on the other hand are interested only on the asymptotic up to multiplication by a constant. This allows us to obtain a more general and simpler proof than previous ones.  2010 Elsevier B.V. All rights reserved. 1. Introduction The laws governing the behavior of the minimum span- ning tree (MST) on points in the unit disk and ball have been thoroughly researched and applied in the field of mo- bile computing; see [12] for a survey of the worst case. For the average case (or points taken randomly), extremely fine results have been obtained: see [4] for the central limit theorem for the total length, [2,1] for the dimension of a typical path, [3, Chapter 6] for an “objective” approach, [13] for intriguing simulation results, and [6] for efforts to prove them. See also the book [9] for the strongly related continuum percolation. The exact asymptotics of the length of the longest edge of the MST was studied in [10,11,7]. Although adequate for modeling mobile systems in man-made environments such as inside a room or a build- ing, Euclidean geometry is perhaps too restrictive for mod- eling systems in natural settings, such as woods or rugged terrains. However, there has been much less research on * Corresponding author. E-mail addresses: gady.kozma@weizmann.ac.il (G. Kozma), zvilo@cse.bgu.ac.il (Z. Lotker), gstupp@cisco.com (G. Stupp). MST behavior in other metric spaces and in particular on fractals. In [8], we studied the worst case problem for the total weighted MST length. Here, we switch to the average case and are interested in the length of the longest edge, which, by the greedy algorithm, is the same as the con- nectivity threshold, i.e., the minimal number r such that the graph in which two points are connected if and only if their metric distance is  r , is connected. In the setting of a ball in R d this is known to be, with high probability ≈  log n n  1/d where ≈ means that the ratio of the two quantities is bounded between two absolute constants. We wish to ex- tend this result to fractal sets. Clearly, to get any kind of estimate, one has to as- sume that the fractal is connected. Further, it is clear that some kind of regularity is needed. To see why regularity is needed, it might be instructive to consider the following example: in R 2 take the set F =  ∞ i=1 ( A i ∪ B i ), consist- ing of a set of “thick” vertical slabs A i =[ 1 2 i , 1 (2 i−1 ) ]×[0, 1] connected by “thin” horizontal bridges B i =[ 1 (2 i+1 ) , 1 2 i ]× [0, 3 −i ]; see Fig. 1. Take the normalized Lebesgue mea- 0020-0190/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2010.02.015