TY - JOUR
T1 - On the connectivity threshold for general uniform metric spaces
AU - Kozma, Gady
AU - Lotker, Zvi
AU - Stupp, Gideon
PY - 2010/4/30
Y1 - 2010/4/30
N2 - 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(1)α δd ≤ μ (B (x, δ)) ≤ β δd 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 ≈ (frac(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.
AB - 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(1)α δd ≤ μ (B (x, δ)) ≤ β δd 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 ≈ (frac(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.
KW - Connectivity threshold
KW - Fractal
KW - Random networks
KW - Sensor networks
UR - http://www.scopus.com/inward/record.url?scp=77949917190&partnerID=8YFLogxK
U2 - 10.1016/j.ipl.2010.02.015
DO - 10.1016/j.ipl.2010.02.015
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AN - SCOPUS:77949917190
SN - 0020-0190
VL - 110
SP - 356
EP - 359
JO - Information Processing Letters
JF - Information Processing Letters
IS - 10
ER -