13,986 research outputs found
Upper and lower bounds for dynamic data structures on strings
We consider a range of simply stated dynamic data structure problems on
strings. An update changes one symbol in the input and a query asks us to
compute some function of the pattern of length and a substring of a longer
text. We give both conditional and unconditional lower bounds for variants of
exact matching with wildcards, inner product, and Hamming distance computation
via a sequence of reductions. As an example, we show that there does not exist
an time algorithm for a large range of these problems
unless the online Boolean matrix-vector multiplication conjecture is false. We
also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1
The Geometry of Differential Privacy: the Sparse and Approximate Cases
In this work, we study trade-offs between accuracy and privacy in the context
of linear queries over histograms. This is a rich class of queries that
includes contingency tables and range queries, and has been a focus of a long
line of work. For a set of linear queries over a database , we
seek to find the differentially private mechanism that has the minimum mean
squared error. For pure differential privacy, an approximation to
the optimal mechanism is known. Our first contribution is to give an approximation guarantee for the case of (\eps,\delta)-differential
privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise
to the answers. We prove its approximation guarantee relative to the hereditary
discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex
geometry.
We next consider this question in the case when the number of queries exceeds
the number of individuals in the database, i.e. when . It is known that better mechanisms exist in this setting. Our second
main contribution is to give an (\eps,\delta)-differentially private
mechanism which is optimal up to a \polylog(d,N) factor for any given query
set and any given upper bound on . This approximation is
achieved by coupling the Gaussian noise addition approach with a linear
regression step. We give an analogous result for the \eps-differential
privacy setting. We also improve on the mean squared error upper bound for
answering counting queries on a database of size by Blum, Ligett, and Roth,
and match the lower bound implied by the work of Dinur and Nissim up to
logarithmic factors.
The connection between hereditary discrepancy and the privacy mechanism
enables us to derive the first polylogarithmic approximation to the hereditary
discrepancy of a matrix
Linear and Range Counting under Metric-based Local Differential Privacy
Local differential privacy (LDP) enables private data sharing and analytics
without the need for a trusted data collector. Error-optimal primitives (for,
e.g., estimating means and item frequencies) under LDP have been well studied.
For analytical tasks such as range queries, however, the best known error bound
is dependent on the domain size of private data, which is potentially
prohibitive. This deficiency is inherent as LDP protects the same level of
indistinguishability between any pair of private data values for each data
downer.
In this paper, we utilize an extension of -LDP called Metric-LDP or
-LDP, where a metric defines heterogeneous privacy guarantees for
different pairs of private data values and thus provides a more flexible knob
than does to relax LDP and tune utility-privacy trade-offs. We show
that, under such privacy relaxations, for analytical workloads such as linear
counting, multi-dimensional range counting queries, and quantile queries, we
can achieve significant gains in utility. In particular, for range queries
under -LDP where the metric is the -distance function scaled by
, we design mechanisms with errors independent on the domain sizes;
instead, their errors depend on the metric , which specifies in what
granularity the private data is protected. We believe that the primitives we
design for -LDP will be useful in developing mechanisms for other analytical
tasks, and encourage the adoption of LDP in practice
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