We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for N-body problems in d dimensions, namely efficient $\mathcal{H}^2_∗$ (fully nested algorithm, i.e., $\mathcal{H}^2$ matrix algorithm) and $(\mathcal{H}^2+\mathcal{H})_∗$ (semi-nested algorithm, i.e., cross of $\mathcal{H}^2$ and $\mathcal{H}$ matrix algorithms). The efficient $\mathcal{H}^2_*$ and $(\mathcal{H}^2+\mathcal{H})_∗$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions (Khan et al., J. Comput. Phys. 2024), where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA (Bebendorf et al., Computing 2003) and NCA (Bebendorf et al., Numer. Math. 2012; Gujjula and Ambikasaran, arXiv:2203.14832 2022; Zhao et al., IEEE Trans. Microw. Theory Tech. 2019)) and develop both algorithms in a black-box (kernel-independent) fashion. The initialization time of the proposed algorithms scales quasi-linearly, i.e., complexity $\mathcal{O}(N{\rm log}^α (N)),$ $α ≥0$ and small. Using the proposed hierarchical representations, one can perform the MVP that scales at most quasi-linearly. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms (e.g.,$\mathcal{H}^2, $ $\mathcal{H},$ etc.) in 2D and 3D ($d = 2,3$). The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. The article also discusses the scalability of the algorithms and provides various benchmarks. Notably, all the algorithms are developed in a similar fashion in C++ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\mathcal{H}^2$ matrix algorithm (where the admissible clusters are the far-field clusters) with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/.