1. Optimal Data-Dependent Hashing for Approximate Near Neighbors
Ilya Razenshteyn (CSAIL MIT)
Alexandr Andoni (Simons Institute)

2. Approximate Near Neighbors (ANN)
Dataset: n points in d dimensions
Query: a point within 1 from a data point
Want: a data point within 2 from the query
Regime: d ≈ log n

3. Locality-Sensitive Hashing (LSH)
Hash family on Rd: points at distance ≤ 1 collide more often (with probability ≥ p1) than points at distance ≥ 2 (with probability ≤ p2)
(Indyk, Motwani 1998): LSH can be used to solve ANN with space O(n1+ρ) and query time O(nρ), where ρ depends on the gap between p1 and p2
Question: what is the best ρ?

4. Bounds on LSH
(Indyk, Motwani 1998): ρ = 1/2 (defined LSH and first constructions)
(Andoni, Indyk 2006): ρ = 1/4 (better LSH construction)
(O’Donnell, Wu, Zhou 2011): ρ = 1/4 is optimal!
Can one improve upon LSH for ANN?

5. Beyond LSH
(Andoni, Indyk, Nguyen, R 2014), (Andoni, R 2015): ρ = 1/7 for ANN (the best LSH gives only ρ = 1/4)
The main idea
LSH is data-oblivious: good collision probabilities for every pair from Rd
For ANN one of the points may be assumed to lie in the dataset
Data-dependent hashing: a hash family depends on the points
Achieve improvement for every dataset
(Andoni, R 2015): ρ = 1/7 is optimal for data-dependent hashing
See the arXiv preprint (intricate proof, but very simple algorithm!)

6. Open problems
Dynamic case
Instance-optimal data-dependent hashing