  Let $f : X \to S$ be a family of smooth projective algebraic varieties over a
smooth connected quasi-projective base $S$, and let $\mathbb{V} = R^{2k} f_{*}
\mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree
$2k$ cohomology it induces. Associated to $\mathbb{V}$ one has the so-called
Hodge locus $\textrm{HL}(S) \subset S$, which is a countable union of "special"
algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb{V}$
possessing extra Hodge tensors (and so conjecturally, those fibres of $f$
possessing extra algebraic cycles). The special subvarieties belong to a larger
class of so-called weakly special subvarieties, which are subvarieties of $S$
maximal for their algebraic monodromy groups. For each positive integer $d$, we
give an algorithm to compute the set of all weakly special subvarieties $Z
\subset S$ of degree at most $d$ (with the degree taken relative to a choice of
projective compactification $S \subset \overline{S}$ and very ample line bundle
$\mathcal{L}$ on $\overline{S}$). As a corollary of our algorithm we prove
conjectures of Daw-Ren and Daw-Javanpeykar-K\"uhne on the finiteness of sets of
special and weakly special subvarieties of bounded degree.
