Summary: We propose a new fast algorithm to compute the squared Euclidean distance transform (E$^2$DT) on every two-dimensional (2-D) irregular isothetic grids (regular square grids, quadtree-based grids, etc.). Our new fast algorithm is an extension of the E$^2$DT method proposed by {\it H. Breu, J. Gil, D. Kirkpatrick} and {\it M. Werman} [IEEE Trans. Pattern Anal. Mech. Intell. 17, No.~5, 529‒533 (1995)]. It is based on the implicit order of the cells in the grid, and builds a partial Voronoi diagram of the centers of background cells thanks to a data structure of lists. We compare the execution time of our method with the ones of others approaches we developed in previous works. In those experiments, we consider various kinds of classical 2-D grids in imagery to show the interest of our methodology, and to point out its robustness. We also show that our method may be interesting regarding an application where we extract an adaptive medial axis construction based on a quadtree decomposition scheme.