The bit complexity of distributed sorting

Abstract

We study the bit complexity of the sorting problem for asynchronous distributed systems. We show that for every network with a tree topology T, every sorting algorithm must send at least Ω(Δ _T log L/N) bits in the worst case, where {0, 1, ..., L} is the set of possible initial values, and Δ _T is the sum of distances from all the vertices to a median of the tree. In addition, we present an algorithm that sends at most O(Δ _T log L N/Δ _T) bits for such trees; These bounds are tight if either L=Ω(N ^1+ε) or Δ _T =Ω(N ²). We also present results regarding average distributions. These results suggest that sorting is an inherently non-distributive problem, since it requires an amount of information transfer, that is equal to the concentration of all the data in a single processor, which then distributes the final results to the whole network. The importance of bit complexity — as opposed to message complexity — stems also from the fact that in the lower bound discussion, no assumptions are made as to the nature of the algorithm.