The tvdenoising package provides an implementation of an efficient linear-time dynamic programming algorithm for univariate total variation denoising (also called fused lasso signal approximation), due to Johnson (2013), which computes the exact solution, for a given regularization level $\lambda$. You can also find a concise explanation of the algorithm at this link.

To install the released version from CRAN:

install.packages("tvdenoising")

To install the development version from GitHub:

# install.packages("pak")
pak::pak("glmgen/tvdenoising")