A snap-stabilizing protocol, starting from any arbitrary initial configuration, always behaves according to its specification. In [4], we presented the first snap-stabilizing depth-first search (DFS) wave protocol for arbitrary rooted networks working under an unfair daemon. However, this protocol needs O(N ^N ) states per processors (where N is the number of processors) and needs ids on processors. In this paper, we propose an original snap-stabilizing solution for this problem with a strongly enhanced space complexity, i.e., O(Δ² × N) states where Δ is the degree of the network. Furthermore, this new protocol does not need a completely identified network: only the root needs to be identified, i.e., the network is semi-anonymous.