Optimal Triangulation on the High Bandwidth Memory Model

The High Bandwidth Memory (HBM) model is a theoretical computing model consisting of a logic circuit with a large external memory. Each address of the external memory can store $p$ elements which can be read or written at the same time. Access to $p$ elements stored at a given address in the external memory has a latency of $l$ clock cycles. However, access to any $k$ consecutive addresses can be done only in $(k+l-1)$ clock cycles in a pipeline fashion by burst mode. A hardware algorithm is implemented in a logic circuit of the HBM to solve a particular problem. In this paper, we present an optimal implementation of the $O(n^{3})$ -time dynamic programming algorithm for solving the optimal polygon triangulation (OPT) problem which is a problem to find a triangulation with minimum total weight of an input convex n-gon with weighted cords. We assume that the input weight matrix of a convex n-gon is stored in the external memory of the HBM model. Our hardware algorithm implemented in the logic circuit of size $O(s^{2})$ operates on it and computes the optimal polygon triangulation of the input polygon in $O(\frac{n^{3}}{sp}+\frac{n^{3}}{s^{2}}+\frac{n^{3}}{s^{3}}l)$ time. We also provide a theoretical proof showing that any hardware algorithm in a logic circuit of size $O(s^{2})$ takes at least $\Omega(\frac{n^{3}}{sp}+\frac{n^{3}}{s^{2}})$ time to solve the OPT problem. Thus, our implementation is optimal whenever $s^{2}\geq lp$ or $s\geq l$, and this optimality condition is always satisfied from a practical point of view.