What kind of graph/lattice problem is this?

Question

I have a problem that can be cast as:

Mark the (a?) shortest sequence of points in an $n$-dimensional positive integer lattice to reach a given target point $(t_1, t_2,...,t_n)$ from the origin (0,0,...,0), where the rules for marking a point are:

1. Pick a pivot $(p_1,...,p_n)$. In order to mark a nearest neighbour of the pivot along an axis (e.g. from the pivot (1,2,0) mark (2,2,0)), I need to have already marked at least $n$ of the $2n$ nearest neighbours (e.g. (0,2,0), (1,2,1) and (1,2,-1)).
2. We are allowed to consider the "-1 neighbours" as marked.
3. The origin (0,0,...) is marked.

In the example above we are could mark (2,2,0), (1,3,0) and (1,1,0).

We know we can slowly crawl everywhere in the lattice, but how do we go about finding the shortest sequence of marked points between the origin and any given $(t_1,...,t_n)$? Is this a sort of graph colouring problem? Is there an algorithm for this stuff?