questions about sensitivity in differential privacy

Question

Ted here (What does the term "differential" in "differential privacy" mean?) describes the difference between local and global sensitivity as

"By contrast, local and global sensitivity are only tools to build differentially private mechanisms: they measure how much changing the data of one user will change the output of the statistic you're trying to compute. Global sensitivity measures how big that change can get for every possible database, while local sensitivity measures it for a single database. Local sensitivity is mostly used to define another concept, smooth sensitivity, which in turn is used to design DP mechanisms."

for every possible database, we mean that ones record is to more than one database?

what is more, I see in the definition that local sensitivity takes account only for D'. why not the D as well?

Answer 1

$D$ is the database at hand. Differential Privacy is a property of a randomized algorithm, which is associated with a domain. So, DP holds for any neighboring databases in the domain. However, when you have a true database $D$, in some cases you don't need to add a noise whose distribution is associated with the global sensitivity to ensure privacy. You can add a smoothened version of the local sensitivity to add a smaller noise, where local sensitivity is roughly the sensitivity of the function restricted to the neighbors of $D$.