The question is a bit unclear, so I'm not sure this is what you seek.
But here is a connection to computational geoemtry.

There is considerable literature on spanners and doubling metric spaces.
Here is a sampling of references from 2006, 2008, 2019, 2022. And from their references you can locate many more.

> Damian, Mirela, Saurav Pandit, and Sriram Pemmaraju. "Distributed spanner construction in doubling metric spaces." In *International Conference on Principles of Distributed Systems*, pp. 157-171. Springer, Berlin, Heidelberg, 2006.

> Gottlieb, Lee-Ad, and Liam Roditty. "An optimal dynamic spanner for doubling metric spaces." In *European Symposium on Algorithms*, pp. 478-489. Springer, Berlin, Heidelberg, 2008.

> Borradaile, Glencora, Hung Le, and Christian Wulff-Nilsen. "Greedy spanners are optimal in doubling metrics." In *Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms*, pp. 2371-2379. Society for Industrial and Applied Mathematics, 2019.

> Bartal, Yair, Ora Nova Fandina, and Ofer Neiman. "Covering metric spaces by few trees." *Journal of Computer and System Sciences* 130 (2022): 26-42.