1. Tight Lower Bounds for Data- Dependent Locality-Sensitive Hashing Alexandr Andoni (Columbia) Ilya Razenshteyn (MIT CSAIL)

2. Near Neighbor Search

3. Approximate Near Neighbor Search (ANN)

4. Locality-Sensitive Hashing (LSH) From the definition of ANN

5. From LSH to ANN

6. Bounds on LSH Distance metricReference 1/4(Andoni-Indyk 2006) (O’Donnell-Wu-Zhou 2011) 1/2(Indyk-Motwani 1998) (O’Donnell-Wu-Zhou 2011) Can one improve upon LSH? Yes! (Andoni-Indyk-Nguyen-R 2014) (Andoni-R 2015)

7. How to do better than LSH?

8. Bounds on data-dependent LSH Distance metricReference 1/4(Andoni-Indyk 2006) (O’Donnell-Wu-Zhou 2011) 1/7(Andoni-R 2015) 1/2(Indyk-Motwani 1998) (O’Donnell-Wu-Zhou 2011) 1/3(Andoni-R 2015) Optimal !

9. The main result The data-dependent space partitions for the Euclidean and Manhattan/Hamming distances from (Andoni-R 2015) are optimal* * After proper formalization

10. Hard instance

11. Fine print Voronoi diagram: a perfect partition Useless: hard to locate points Need to define what is allowed properly to rule it out

12. Formalizing the model Restricted computational complexity: data structure lower bounds Bounded number of parts: can tweak the Voronoi diagram example

13. The main result

14. Proof outline

15. Conclusions SpaceQuery time Can prove matching data-independent lower bounds for the hard instance (in an appropriate model) What about data-dependent? Question s?