  Recently, the problem of allocating one resource per agent with initial
endowments (house markets) has seen a renewed interest: indeed, while in the
domain of strict preferences the Top Trading Cycle algorithm is known to be the
only procedure guaranteeing Pareto-optimality, individual rationality, and
strategy proofness. However, the situation differs in the single-peaked domain.
Indeed, Bade presented the Crawler, an alternative procedure enjoying the same
properties, with the additional advantage of being implementable in obviously
dominant strategies. In this paper we further investigate the Crawler and
propose the Diver, a variant which checks optimally whether an allocation is
Pareto-optimal for single-peaked preferences, thus improving over known
techniques used for checking Pareto-optimality in more general domains. We also
prove that the Diver is asymptotically optimal in terms of communication
complexity.
