Advances in topological data analysis (TDA) require efficient algorithm designs that can extract algebraic structures hidden in the data. This talk centers around the theme that special geometric/combinatorial constructions underlying the algebra can facilitate designing efficient algorithms in TDA. As examples of this premise, we present three cases: (i) how a geometric viewpoint on zigzag persistence in terms of the Mayer-Vietoris pyramid helped designing a fast algorithm for computing zigzag persistence from an input zigzag filtration, (ii) how the special structure of two-dimensional grid ($\mathbb{Z}^2$) helped designing an efficient algorithm for computing the generalized rank (rank of the limit-to-colimit map) for $2$-parameter persistence, and (iii) how combinatorial multivectors representing dynamical systems helped designing efficient algorithms for computing algebraic summaries such as Connection Matrices and Conley-Morse barcodes.