Indyk and Motwani (1998)

If there exists some $\mathbf{q}$ with $||\mathbf{q}-\mathbf{y}||_0 \leq R$, return a vector $\tilde{\mathbf{q}}$ with $||\tilde{\mathbf{q}}-\mathbf{y}||_0 \leq C \cdot R$ in:

• Time: $O(n^{1/C})$
• Space: $O(n^{1+1/C})$

where $||\mathbf{q}-\mathbf{y}||_0$ is the “hamming distance” number of elements that different between $\mathbf{q}$ and $\mathbf{y}$.

Used in near neighbor search problem.

#incomplete

Given a $(r_1,r_2,p_1,p_2)$-locality sensitive hash family, $(r_1,r_2)$-PLEB (point location in equal balls) can be solved with constant probability using:

• Space: $O(dn + n^{1+\gamma})$
• Query time: $O(n^\gamma)$ hash function evaluations and metric computations, $d(\cdot,\cdot)$, where $\gamma = \frac{\log(1/p_1)}{\log(1/p_2)}$.

PLEB (point location in equal balls)
For given radii r1, r2 with r1 ≤ r2, if there is at least one point p ∈ X with d(q, p) ≤ r1, return any p with d(q, p) < r2. On the other hand, if there is no point p ∈ X with d(q, p) < r2, output FAIL.

References:

1. P. Indyk and R. Motwani, “Approximate nearest neighbors: towards removing the curse of dimensionality,” in Proceedings of the thirtieth annual ACM symposium on Theory of computing  - STOC ’98, Dallas, Texas, United States: ACM Press, 1998, pp. 604–613. doi: 10.1145/276698.276876.
2. https://www.cs.princeton.edu/courses/archive/fall18/cos521/Lectures/lec12.pdf