A Faster Closest-Point Algorithm for the $A_n^*$ Lattices

The dual root lattice $A_n^*$ is an important lattice in quantization, coding, and estimation. It can be represented as the projection of the integer lattice $\mathbb{Z}^{n+1}$ onto the $n$-dimensional hyperplane whose coordinates sum to zero. This representation makes $A_n^*$ particularly natural for quantizing simplex-constrained data, such as histograms and probability distributions. This paper studies the closest-point problem for $A_n^*$: given a query vector $y$, find the lattice point $x\in A_n^*$ minimizing $\|y-x\|^2$. The fastest previously known method is the linear-time algorithm of McKilliam, Clarkson, Smith, and Quinn (MCSQ), which employs bucket sort as a core operation. We present a faster linear-time algorithm. The key observation is that the closest-point objective depends on the rounding residuals only through two prefix aggregates: a count and a residual sum. Hence the elements inside each bucket never need to be sorted, stored, or traversed. This replaces the linked-list traversal and pointer chasing of MCSQ with a single bucketing pass over two flat arrays with counting-sort-style accumulates. A scaled objective further makes most of the computation exact integer arithmetic, and when the input coordinates are rationals with a common denominator, for example, histograms or empirical distibutions, the entire algorithm becomes exact and integer-only. Experiments on an Intel Core i9-13900H show speedups of about $1.8\times$ to $3.0\times$ over MCSQ for $n=2,\dots,100$, with larger gains at higher dimensions. The proposed algorithm is also noticeably faster than Conway and Sloan methods for other root lattices, including $A_n$, $D_n^*$, and $E_8$. An open-source implementation is available in the "fanstar" project.