We consider the problems of distributed ranking and sorting on a Coterie, a communication structure which has proven to be a good candidate as underlying interconnection network for distributed processing. Ranking and sorting problems are harder than a consensus one, a vital and well studied problem in distributed processing, in that the later one computes for only one function (e.g. summation), while the former one actually performs n functions, as ranking is to rank the key in each of n sites. The currently best known decentralized consensus protocols on a coterie uses O(n * square-root of n) messages, and requires two rounds of message exchange. In this paper we show that both ranking and sorting can be done on a coterie with the same message complexity although the problems we investigate are much harder. We first present a two-round ranking algorithm which requires only O(n * square-root of n) messages. Then using this ranking algorithm, we obtain a sorting algorithm which also uses only O(n * square-root of n) messages, but requires two more rounds of message exchange. Our schemes are optimal in the sense that the lower bound of messages needed for achieving a consensus is at least(n * square-root of n) (a trivial lower bound shown in the paper of Lakshman's).