一种针对$A_n^*$格的更快最近点算法
A Faster Closest-Point Algorithm for the $A_n^*$ Lattices
AI总结：
研究$A_n^*$格的最近点问题，提出比McKilliam等人算法更快的线性时间算法，通过特定观察简化操作，多数计算为精确整数运算，实验显示在多维度下相比原算法有显著加速，且比其他根格算法更快。
AI中文摘要：
对偶根格$A_n^*$在量化、编码和估计中是重要格。它可表示为整数格$\mathbb{Z}^{n + 1}$在坐标和为零的$n$维超平面上的投影，适合量化单纯形约束数据。本文研究$A_n^*$的最近点问题：给定查询向量$y$，在$A_n^*$中找到使$\|y - x\|^2$最小的格点$x$。此前最快方法是McKilliam等人的线性时间算法（MCSQ），采用桶排序。本文提出更快线性时间算法，关键在于最近点目标仅通过两个前缀聚合依赖于舍入残差，无需对桶内元素排序、存储或遍历，用计数排序风格累加的两个扁平数组单次桶排序取代MCSQ的链表遍历和指针追踪。缩放目标使多数计算为精确整数运算，输入坐标为有公分母的有理数时算法完全为精确整数运算。在Intel Core i9 - 13900H上实验表明，对于$n = 2,\dots,100$，比MCSQ快约$1.8\times$到$3.0\times$，维度越高加速越大。该算法比Conway和Sloan针对其他根格（包括$A_n$、$D_n^*$和$E_8$）的方法也明显更快，“fanstar”项目有开源实现。
英文摘要：
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.
