  A connected component labeling algorithm is developed for implicitly-defined
domains specified by multivariate polynomials. The algorithm operates by
recursively subdividing the constraint domain into hyperrectangular subcells
until the topology thereon is sufficiently simple; in particular, we devise a
topology test using properties of Bernstein polynomials. In many cases the
algorithm produces a certificate guaranteeing its correctness, i.e., two points
yield the same label if and only if they are path-connected. To robustly handle
various kinds of edge cases, the algorithm may assign identical labels to
distinct components, but only when they are exactly or nearly touching,
relative to a user-controlled length scale. A variety of numerical experiments
assess the effectiveness of the overall approach, including statistical
analyses on randomly generated multi-component geometry in 2D and 3D, as well
as specific examples involving cusps, self-intersections, junctions, and other
kinds of singularities.
