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Fuses two or more rankings by reciprocal rank.

public static IReadOnlyList<SearchHit> Rrf(IEnumerable<IEnumerable<int>> rankings, int k = 60)

Parameters — rankings is one sequence of document identifiers per ranking, each ordered best first. k is the rank offset, defaulting to RankFusion.DefaultK, and must be positive.

Returns — IReadOnlyList<SearchHit> holding every document that appeared in any ranking, best fused score first.

Exceptions — ArgumentNullException when rankings, or one of them, is null; ArgumentOutOfRangeException when k is not positive.

Example — the formula, on a document that appears in both rankings.

using Lodestar.Text.Search;

IReadOnlyList<SearchHit> fused = RankFusion.Rrf([[1, 2], [3, 1]], k: 60);

// Document 1 is first in one ranking and second in the other: 1/61 + 1/62.
double score = fused[0].Score;  // => 0.03252…
int document = fused[0].Document;  // => 1
int total = fused.Count;  // => 3

Remarks — the score of a document is Σ 1 / (k + rank) over the rankings that hold it.

Rank counts from 1. Counting from zero is the common mistake and it changes every number: at k = 60 the first position would be worth 1/60 rather than 1/61.

A document absent from a ranking contributes nothing from it, which is what lets rankings of different lengths be fused without padding — a BM25 top-10 and a vector top-100 need no alignment step.

A document repeated inside one ranking is scored at its first position there. The later occurrences are ignored rather than summed, because a ranking that lists a document twice still ranks it once.

Larger k flattens the weight of rank. At k = 1 first place is worth 1/2 and second 1/3; at k = 1000 the two are within a thousandth of each other. 60 is the published default and is a compromise, not a derivation.

Ties break by score alone, and among equal scores the order is the order the documents were first seen — stable, so fusing the same rankings twice gives one answer.

Applies to — net10.0, netstandard2.0.

See also — RankFusion, SearchHit.