The integrable system governing cubic Hodge integrals satisfying local Calabi-Yau condition gives a deformation of KdV hierarchy. This new hierarchy, called special cubic Hodge hierarchy, is equivalent to a kind of reduction of 2D Toda hierarchy. In this talk, we will introduce how to establish the relation between this reduction and special cubic Hodge hierarchy based on Dubrovin-Zhang’s methods. This work is joint with Si-Qi Liu, Youjin Zhang and Di Yang.