In this paper and two companion papers, we produce efficient algorithms to solve the following interpolation problem: Let m \geq 1 and p > n \geq 1 . Given a finite set E \subset \mathbb{R}^n and a function f: E \rightarrow \mathbb{R} , compute an extension F of f belonging to the Sobolev space W^{m,p}(\mathbb{R}^n) with norm having the smallest possible order of magnitude; secondly, compute the order of magnitude of the norm of F . The combined running time of our algorithms is at most C N \mathrm{log} N , where N denotes the cardinality of E , and C depends only on m , n , and p .