A Bloom filter is a space-efficient probabilistic data structure for approximate set membership queries. It uses a bit array of m bits and k independent hash functions: inserting an element sets k bit positions; querying checks whether all k positions are set. False negatives are impossible (inserted elements always test positive), but false positives occur when unrelated elements happen to set the same bit positions. The false positive rate for n elements is approximately (1 - e^(-kn/m))^k, minimized when k = (m/n) ln 2. With optimal parameters, a Bloom filter uses about 1.44 log_2(1/epsilon) bits per element for false positive rate epsilon — far less than storing the elements themselves. Variants include counting Bloom filters (supporting deletion), cuckoo filters (better space for low epsilon), and Bloom filter cascades.
You have seen the basic Bloom filter idea in data structures: a bit array with hash functions that supports fast approximate membership queries. At the expert level, the focus shifts to understanding the precise mathematical tradeoffs, the information-theoretic limits, and the design space of variants that extend the basic structure.
The false positive analysis is clean. After inserting n elements with k hash functions into m bits, each specific bit remains 0 with probability (1 - 1/m)^(kn) ≈ e^(-kn/m). A false positive occurs when all k bits for a non-member happen to be 1, with probability (1 - e^(-kn/m))^k. Minimizing over k (taking the derivative and setting it to zero) gives the optimal k = (m/n) ln 2, at which point exactly half the bits are set. This yields a false positive rate of (1/2)^k = (1/2)^((m/n) ln 2) = 2^(-m ln 2 / n), or equivalently, achieving rate epsilon requires m = n * log_2(1/epsilon) / ln 2 ≈ 1.44 * n * log_2(1/epsilon) bits.
The 1.44 factor above the information-theoretic minimum of log_2(1/epsilon) bits per element is the "price of simplicity." Standard Bloom filters are not optimal data structures for approximate membership — but they are close, and their simplicity (bit-parallel operations, no pointer overhead, cache-friendly) makes them practical favorites. When the 44% overhead matters, alternatives exist: compressed Bloom filters reduce space by allowing the bit array to be entropy-coded; Golomb-coded sets achieve near-optimal space; cuckoo filters match or beat Bloom filter space while supporting deletions.
The variant landscape is rich. Counting Bloom filters replace bits with counters to support deletion (at ~4x space cost). Scalable Bloom filters grow dynamically as elements arrive. Bloomier filters store associated values, not just membership. Spectral Bloom filters count multiplicities. In distributed systems, Bloom filters are used for set reconciliation — two parties can efficiently determine which elements they share by exchanging Bloom filters. The common thread is the fundamental tradeoff: a small amount of space buys approximate answers to set queries, with a tunable, well-understood error rate.