  We study the problem of fairly allocating $m$ indivisible goods to $n$
agents, where agents may have different preferences over the goods. In the
traditional setting, agents' valuations are provided as inputs to the
algorithm. In this paper, we study a new comparison-based query model where the
algorithm presents two bundles of goods to an agent and the agent responds by
telling the algorithm which bundle she prefers. We investigate the query
complexity for computing allocations with several fairness notions including
proportionality up to one good (PROP1), envy-freeness up to one good (EF1), and
maximin share (MMS). Our main result is an algorithm that computes an
allocation satisfying both PROP1 and $\frac12$-MMS within $O(\log m)$ queries
with a constant number of $n$ agents. For identical and additive valuation, we
present an algorithm for computing an EF1 allocation within $O(\log m)$ queries
with a constant number of $n$ agents. To complement the positive results, we
show that the lower bound of the query complexity for any of the three fairness
notions is $\Omega(\log m)$ even with two agents.
