Pseudo-Distances within a depth-determination problem

[Shaitan00]

I was given the following problem description for a depth-determination problem, however I am completly unable to figure out how this "pseudo-distance" feature works, if anyone has any clues and can provide a uber-simple example it would be much appreciated - I just can't see how it works/applies.

 

 

[Problem]

In the depth-determination problem, we maintain a forest F = {Ti} of rooted trees. We use the disjoint-set forest S=Si, where each set Si (which is itself a tree) corresponds to a tree Ti in the forest F. The tree structure within a set Si, however, does not necessarily correspond to that of Ti. In fact, the implementation of Si does not record the exact parent-child relationships but nevertheless allows us to determine any node’s depth in Ti.

 

The key idea is to maintain each node v a ”pseudo-distance” d[v], which is defined so that the sum of the pseudo-distances along the path from v to the root of its set Si equals the depth of v in Ti. That is, if the path from v to its root in v0 , v1 , . . . , vk , where v0 = v and vk is Si’s root, then the depth of v in Ti is the sum of d[vj] where j=0 to j=k.