A constructive proof of the Assouad embedding theorem with bounds on the dimension

We give a constructive proof of a theorem of Naor and Neiman, (to appear, Revista Matematica Iberoamercana), which asserts that if $(E,d)$ is a doubling metric space, there is an integer $N>0$, that depends only on the metric doubling constant, such that for each exponent $\alpha \in (1\mathrm{/}2,1)$, we can find a bilipschitz mapping $F=(E,d^{\alpha })\to {\mathbb{R}}^N$.