  In a district-based election, we apply a voting rule $r$ to decide the
winners in each district, and a candidate who wins in a maximum number of
districts is the winner of the election. We present efficient sampling-based
algorithms to predict the winner of such district-based election systems in
this paper. When $r$ is plurality and the margin of victory is known to be at
least $\varepsilon$ fraction of the total population, we present an algorithm
to predict the winner. The sample complexity of our algorithm is
$\mathcal{O}\left(\frac{1}{\varepsilon^4}\log
\frac{1}{\varepsilon}\log\frac{1}{\delta}\right)$. We complement this result by
proving that any algorithm, from a natural class of algorithms, for predicting
the winner in a district-based election when $r$ is plurality, must sample at
least $\Omega\left(\frac{1}{\varepsilon^4}\log\frac{1}{\delta}\right)$ votes.
We then extend this result to any voting rule $r$. Loosely speaking, we show
that we can predict the winner of a district-based election with an extra
overhead of
$\mathcal{O}\left(\frac{1}{\varepsilon^2}\log\frac{1}{\delta}\right)$ over the
sample complexity of predicting the single-district winner under $r$. We
further extend our algorithm for the case when the margin of victory is
unknown, but we have only two candidates. We then consider the median voting
rule when the set of preferences in each district is single-peaked. We show
that the winner of a district-based election can be predicted with
$\mathcal{O}\left(\frac{1}{\varepsilon^4}\log\frac{1}{\varepsilon}\log\frac{1}{\delta}\right)$
samples even when the harmonious order in different districts can be different
and even unknown. Finally, we also show some results for estimating the margin
of victory of a district-based election within both additive and multiplicative
error bounds.
