Adaptive and Dynamic Multi-Resolution Hashing for Pairwise Summations

In this paper, we propose Adam-Hash: an adaptive and dynamic multi-resolution hashing data-structure for fast pairwise summation estimation. Given a data-set X ⊂ ℝ
^d
, a binary function f : ℝ
^d
× ℝ
^d
→ ℝ, and a point y ∈ ℝ
^d
, the Pairwise Summation Estimate $PS{E_X}(y): = \frac{1}{{\left| X \right|}}\sum\nolimits_{x \in X} {f(x,y)} $. For any given data-set X, we need to design a data-structure such that given any query point y ∈ ℝ
^d
, the data-structure approximately estimates PSE
X
(y) in time that is sub-linear in |X|. Prior works on this problem have focused exclusively on the case where the data-set is static, and the queries are independent. In this paper, we design a hashing-based PSE data-structure which works for the more practical dynamic setting in which insertions, deletions, and replacements of points are allowed. Moreover, our proposed Adam-Hash is also robust to adaptive PSE queries, where an adversary can choose query q
j
∈ ℝ
^d
depending on the output from previous queries q
1
, q
2
, …, q
j–1
.