Modern vector databases combine sparse (BM25) and dense vectors at the storage layer via two distinct patterns. Identify them.

**Hybrid search** combines sparse (BM25 / term-frequency) signals with dense (embedding) signals to retrieve documents. The reason: BM25 wins on exact-keyword queries (where the user typed a specific term that must appear), and dense embeddings win on semantic queries (where the user paraphrases the underlying intent). Combining both is more robust than either alone. Vector databases implement this combination via two distinct architectural patterns. **Pattern 1: two separate indexes, score fusion at query time.** - One inverted index for sparse / BM25 (built like a classical search engine: term -> postings list). - One ANN index (HNSW, IVF) for dense vectors. - At query time: run both, get two ranked lists, fuse them. **Reciprocal Rank Fusion (RRF)** is the canonical fusion function: score(doc) = sum over each list of 1 / (k + rank), with k typically 60. - Vendors: Pinecone, Weaviate, Qdrant. All three ship hybrid search this way. **Pattern 2: single hybrid index, joint scoring at index traversal.** - One index that knows about both sparse and dense features per document. - At query time: traverse the index once, computing a combined score per document. - Vendors: Vespa is the canonical example (deep IR heritage from Yahoo!). Some Milvus configurations also support this. **Tradeoffs.** - *Pattern 1 (two indexes + RRF).* Simpler architecturally; you can scale and tune each index independently. Operationally: two indexes to keep in sync (writes go to both). Fusion is a constant-time operation; the choice of fusion function (RRF, weighted sum, learned to rank) is a tuning lever. - *Pattern 2 (single hybrid).* More complex index but no fusion step. Per-document scoring can be more nuanced (the index can balance sparse and dense per-document rather than uniformly). Operationally heavier. **Why the other options are wrong.** - Option B claims all vector DBs use a shared graph index; this is false (BM25 is fundamentally an inverted-list method, not a graph method). - Option C says sparse is delegated to the embedding model; sparse retrieval is a separate retrieval-time operation, not an embedding-time one. - Option D claims sparse vectors are PCA-compressed to dense; sparse representations are stored as-is in inverted lists, not converted.

**Two architectural patterns for hybrid sparse plus dense retrieval in vector databases.** **Pattern 1: two separate indexes, fuse at query time (score fusion).** The vector DB maintains two indexes per logical collection: - *Inverted index for sparse vectors.* Each term in the vocabulary maps to a postings list of (document_id, term_weight) pairs. BM25 scoring computes a per-document relevance based on term frequencies and inverse document frequencies. This is the same machinery that powers Elasticsearch, Lucene, and classical IR systems. - *ANN index for dense vectors.* HNSW or IVF over the dense embedding space. Standard vector search. At query time: 1. Run BM25 search, get ranked list of top-N sparse results. 2. Run ANN search on the query embedding, get ranked list of top-N dense results. 3. Fuse the two lists into a single ranked output. The canonical fusion function is **Reciprocal Rank Fusion (RRF)**: ``` score(doc) = sum_over_lists ( 1 / (k + rank_in_that_list) ) ``` with k = 60 by default. Documents that rank high in either list get high scores; documents that rank high in both get the highest scores. RRF is rank-based (not score-based), which means it does not require the sparse and dense scores to be on the same scale, an important property because BM25 scores are unbounded and dense scores are typically in [-1, 1] or [0, 1]. Alternatives to RRF: weighted sum (requires score normalization), Borda count, learned to rank with a trained reranker. RRF is the production default because it is parameter-light and robust. *Vendors:* Pinecone, Weaviate, Qdrant. All ship hybrid search via this two-index + RRF pattern. Pinecone calls it 'sparse-dense hybrid'; Weaviate calls it 'hybrid search'; Qdrant exposes sparse vectors and lets you compose them with dense via a fusion step. **Pattern 2: single hybrid index, joint scoring at traversal (index-time fusion).** A single index structure scores both sparse and dense per document during a single