  Quantiles, such as the median or percentiles, provide concise and useful
information about the distribution of a collection of items, drawn from a
totally ordered universe. We study data structures, called quantile summaries,
which keep track of all quantiles, up to an error of at most $\varepsilon$.
That is, an $\varepsilon$-approximate quantile summary first processes a stream
of items and then, given any quantile query $0\le \phi\le 1$, returns an item
from the stream, which is a $\phi'$-quantile for some $\phi' = \phi \pm
\varepsilon$. We focus on comparison-based quantile summaries that can only
compare two items and are otherwise completely oblivious of the universe.
  The best such deterministic quantile summary to date, due to Greenwald and
Khanna (SIGMOD '01), stores at most $O(\frac{1}{\varepsilon}\cdot \log
\varepsilon N)$ items, where $N$ is the number of items in the stream. We prove
that this space bound is optimal by showing a matching lower bound. Our result
thus rules out the possibility of constructing a deterministic comparison-based
quantile summary in space $f(\varepsilon)\cdot o(\log N)$, for any function $f$
that does not depend on $N$. As a corollary, we improve the lower bound for
biased quantiles, which provide a stronger, relative-error guarantee of $(1\pm
\varepsilon)\cdot \phi$, and for other related computational tasks.
