Sketch of the hierarchical time-step subdivision. A generic individual time step, Δt_i is a power of two multiples of the minimum time step Δt_min, and a power of two submultiple of the maximum one, Δt_max. The integration is performed by means of elementary iterations from t^[n] to t^[n + 1] = t^[n] + Δt_min, while the particle is updated from t^[PREV] to t^[NEXT] = t^[PREV] + Δt_i. The hydrodynamics and force routines are activated only for particles that are synchronized at t^[n] = t^[PREV] or at t^[n + 1] = t^[NEXT].