The SMAWK algorithm efficiently computes the row minima of an m × n totally monotone matrix in O(m+n) time. In certain dynamic programming problems, such matrices arise implicitly, allowing SMAWK to serve as an optimization technique for such problems. This thesis explores how and when SMAWK can be applied within dynamic programming contexts. We first present the algorithm itself. We then apply SMAWK to two distinct problems: the Web Proxy Placement Problem and a variant of the Line Breaking Problem. In addition to the theoretical analysis, we implement the algorithm and analyze its practical performance in the context of the Web Proxy Placement Problem, confirming its efficiency compared to the naive approach.