Memoization Without Keys: Compact, Out-of-Core Tables for Functions of Sorted Arguments

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.