A Table-Free Index for Tapered Memoization Grids: Compact Out-of-Core Evaluation of Functions of Sorted Arguments

Many applications must repeatedly evaluate an expensive function f of a sorted score vector whose influence decays with rank: Plackett-Luce choice probabilities, alpha-entmax attention thresholds, and rank-weighted aggregates. Biswas and Regan (TCS 2015) introduced a tapered grid that memoizes such functions, indexed through precomputed node-count tables. We first make explicit that the tapered grid's key set is exactly the set of multiset combinations, so its index is the classical combinatorial number system: this yields a table-free closed-form O(d) rank that eliminates the O(Bd)-O(Bd^2) preprocessing tables of the original scheme, generalizes it beyond a pinned first coordinate, and supplies the previously missing O(d) unranking, which enables order-free parallel construction and key-free storage. The resulting structure is a values-only flat array: at N=37.4M entries it occupies 5.7x less memory than a hash-map memo and answers queries 1.1-1.8x faster once both structures exceed cache, and it remains operable memory-mapped beyond RAM, where pointer-based alternatives cannot reside. We give design guidance for choosing the taper: the optimal per-level refinement ratio equals the influence-decay ratio, and we derive a finite-epsilon closed form for the size penalty of a mismatched ratio -- accurate to a few percent where the classical asymptotic rate overstates the penalty by 16-43%. End-to-end, memoizing the Plackett-Luce normalization -- whose exact evaluation is an iterative transcendental root-find -- is 25-55x faster than Newton's method at 2.6e-3 mean error, against the approximately 10x reported originally; we also report a negative result, alpha-entmax thresholds, where the exact solver's tiny active support makes it 2.6x faster than any table, and distil the scoping rule this implies.