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We study fundamental theoretical aspects of probabilistic roadmaps (PRM) inthe finite time (non-asymptotic) regime. In particular, we investigate howcompleteness and optimality guarantees of the approach are influenced by theunderlying deterministic sampling distribution ${\mathcal{X}}$ and connectionradius ${r>0}$. We develop the notion of ${(\delta,\epsilon)}$-completeness ofthe parameters ${\mathcal{X}, r}$, which indicates that for everymotion-planning problem of clearance at least ${\delta>0}$, PRM using${\mathcal{X}, r}$ returns a solution no longer than ${1+\epsilon}$ times theshortest ${\delta}$-clear path. Leveraging the concept of ${\epsilon}$-nets, wecharacterize in terms of lower and upper bounds the number of samples needed toguarantee ${(\delta,\epsilon)}$-completeness. This is in contrast with previouswork which mostly considered the asymptotic regime in which the number ofsamples tends to infinity. In practice, we propose a sampling distributioninspired by ${\epsilon}$-nets that achieves nearly the same coverage as gridswhile using significantly fewer samples.

Sample Complexity of Probabilistic Roadmaps via $\epsilon$-nets