  In this paper, we tackle the problem of answering multi-dimensional range
queries under local differential privacy. There are three key technical
challenges: capturing the correlations among attributes, avoiding the curse of
dimensionality, and dealing with the large domains of attributes. None of the
existing approaches satisfactorily deals with all three challenges. Overcoming
these three challenges, we first propose an approach called Two-Dimensional
Grids (TDG). Its main idea is to carefully use binning to partition the
two-dimensional (2-D) domains of all attribute pairs into 2-D grids that can
answer all 2-D range queries and then estimate the answer of a higher
dimensional range query from the answers of the associated 2-D range queries.
However, in order to reduce errors due to noises, coarse granularities are
needed for each attribute in 2-D grids, losing fine-grained distribution
information for individual attributes. To correct this deficiency, we further
propose Hybrid-Dimensional Grids (HDG), which also introduces 1-D grids to
capture finer-grained information on distribution of each individual attribute
and combines information from 1-D and 2-D grids to answer range queries. To
make HDG consistently effective, we provide a guideline for properly choosing
granularities of grids based on an analysis of how different sources of errors
are impacted by these choices. Extensive experiments conducted on real and
synthetic datasets show that HDG can give a significant improvement over the
existing approaches.
