Recall@k (Retrieval)

Background

For the retrieval stage of any funnel (RAG, recsys), the right metric is recall, not precision: did the candidate set contain the gold items at all? Precision is the ranker's job at the next stage. Recall@k is the fraction of gold items that appear in the top-k retrieved.

Problem statement

Implement recall_at_k(retrieved, gold, k) where retrieved is a ranked list of document IDs (most relevant first) and gold is the set of relevant IDs.

$$\text{recall@}k \;=\; \frac{|\,\text{retrieved}[:k] \cap \text{gold}\,|}{|\,\text{gold}\,|}$$

Input

• retrieved - sequence of IDs, ranked best-first.
• gold - iterable of IDs (treated as a set).
• k - int >= 1, cutoff.

Output

• float in $[0, 1]$. Returns $0$ if gold is empty.

Examples

Input:  retrieved=["a","b","c","d","e"], gold={"a","d","z"}, k=3
Output: 1/3 = 0.333...

Explanation: top-3 contains {a, b, c}; intersect with gold {a,d,z} = {a}; recall = 1/2 ... wait |gold|=3. Actually: top-3 ∩ gold = {a}, |gold|=3, so recall = 1/3.

Constraints

• k >= 1. Raise ValueError otherwise.
• If k > len(retrieved), use the full list (no padding).
• Return Python float.

Notes

• vs precision@k. Precision@k = hit count / k. Recall@k = hit count / |gold|. Retrieval optimises recall; ranking optimises precision (and ordering).