Sparse structures are widely recognized and utilized in channel estimation. Two typical mechanisms, namely proportionate updating (PU) and zero-attracting (ZA) techniques, achieve better performance, but their computational complexity are higher than non-sparse counterparts. In this paper, we propose a DCS technique based on the recursive least squares (RLS) algorithm which can simultaneously achieve improved performance and reduced computational complexity. Specifically, we develop the sparse adaptive subspace pursuit-improved RLS (SpAdSP-IRLS) algorithm by updating only the sparse structure in the IRLS to track significant coefficients. The complexity of the SpAdSP-IRLS algorithm is successfully reduced to $\mathcal{O}(L^2+2L(s+1)+10s)$, compared with the order of $\mathcal{O}(3L^2+4L)$ for the standard RLS. Here, $L$ represents the length of the channel, and $s$ represents the size of the support set. Our experiments on both synthetic and real data show the superiority of the proposed SpAdSP-IRLS, even though only $s$ elements are updated in the channel estimation.