Optimal Distributed Algorithms for Sorting and Ranking

We study the problems of sorting and ranking n processors that have initial values (not necessarily distinct) in a distributed system. Sorting means that the initial values have to move around in the network and be assigned to the processors according to their distinct identities, while ranking means that the numbers 1, 2,..., n have to be assigned to the processors according to their initial values; ties between initial values can be broken in any chosen way. Assuming a tree network, and assuming that a message can contain an initial value, an identity, or a rank, we present an algorithm for the ranking problem that uses, in the worst case, at most n²/2+O(n) such messages. The algorithm is then extended to perform sorting, using in the worst case at most 3n²/4+O(n) messages. Both algorithms are using a total of O(n) space. The algorithms are extended to general networks. The expected behavior of these algorithms for three classes of trees is discussed. Assuming that the initial values, identities, and ranks can be compared only within themselves, lower bounds of n²/2 and 3n²/4 messages are proved for a worst case execution of any algorithm to solve the ranking and sorting problems, respectively.