发表机构
Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI)（罗摩克里希纳传教会维韦卡南达教育与研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出无表索引的锥形记忆化网格，利用组合数系统实现闭式排名与反排名，显著降低内存占用并加速排序参数函数的评估，适用于 Plackett-Luce 等场景。
AI 中文摘要
许多应用需要反复评估一个关于排序分数向量的昂贵函数 f，该函数的影响随排名衰减：Plackett-Luce 选择概率、alpha-entmax 注意力阈值和排名加权聚合。Biswas 和 Regan (TCS 2015) 引入了一种锥形网格，通过预计算的节点计数表索引来记忆此类函数。我们首先明确指出，锥形网格的键集恰好是多重集组合的集合，因此其索引是经典的组合数系统：这产生了无表的闭式 O(d) 排名，消除了原始方案中 O(Bd)-O(Bd^2) 的预处理表，将其推广到固定第一坐标之外，并提供了先前缺失的 O(d) 反排名，从而实现无序并行构建和无键存储。由此产生的结构是一个仅含值的扁平数组：在 N=37.4M 条目时，它比哈希映射记忆占用内存少 5.7 倍，并且一旦两个结构都超出缓存，查询速度快 1.1-1.8 倍，并且它可以在内存映射下运行于 RAM 之外，而基于指针的替代方案无法驻留于此。我们提供了选择锥度的设计指导：最优的每层细化比率等于影响衰减比率，并且我们推导了不匹配比率的尺寸惩罚的有限 epsilon 闭式形式——在经典渐近率高估惩罚 16-43% 的情况下，其准确度在几个百分点以内。端到端地，记忆 Plackett-Luce 归一化——其精确评估是迭代超越根查找——比牛顿法快 25-55 倍，平均误差为 2.6e-3，而最初报告的约为 10 倍；我们还报告了一个负面结果，alpha-entmax 阈值，其中精确求解器的微小活动支持使其比任何表快 2.6 倍，并提炼了这暗示的范围规则。
英文摘要
Memoizing an expensive function of a sorted score vector is a data-structure problem before it is a numerical one: at a billion gridpoints, a hash map or a search tree spends most of its space on keys the grid already determines. We describe an implemented memo table that stores none. An entry's address is computed in closed form from the sorted argument itself, so $N$ values occupy $N$ slots, the argument is recoverable from the index, and the table can be memory-mapped and served from a file larger than RAM. Against a chained hash map it uses $5.7\times$ less memory at $37$M entries and $10.2\times$ less at $1.9$B, answers queries up to $2.9\times$ faster and builds up to $250\times$ faster; on 64 threads its construction needs no coordination; a sharded hash gains only $1.17\times$. Against an open-addressing table with inline keys it is $4$--$7\times$ smaller and $100\times$ faster to build but $1.5\times$ slower to query, a deficit we trace to the $O(d)$ index arithmetic. At $22$ GB on a $16$ GB desktop it serves each query in one disk access, where no key-storing container can be built; and its order-preserving addressing keeps a perturbation workload on the same pages that a hashed layout scatters. The closed form exists because the key set is the multiset combinations, whose index is the combinatorial number system. Memoizing Plackett--Luce normalization runs $25$--$55\times$ faster than Newton's method; memoizing $α$-entmax thresholds does not pay. The contrast says when this structure is worthwhile.
