We study distributed algorithms for some fundamental problems in data summarization. Given a communication graph G of n nodes each of which may hold a value initially, we focus on computing sum_{i=1}^N g(f_i), where f_i is the number of occurrences of value i and g is some fixed function. This includes important statistics such as the number of distinct elements, frequency moments, and the empirical entropy of the data.
In the CONGEST~ model, a simple adaptation from streaming lower bounds shows that it requires Omega~(D+ n) rounds, where D is the diameter of the graph, to compute some of these statistics exactly. However, these lower bounds do not hold for graphs that are well-connected. We give an algorithm that computes sum_{i=1}^{N} g(f_i) exactly in {tau_{G}} * 2^{O(sqrt{log n})} rounds where {tau_{G}} is the mixing time of G. This also has applications in computing the top k most frequent elements.
We demonstrate that there is a high similarity between the GOSSIP~ model and the CONGEST~ model in well-connected graphs. In particular, we show that each round of the GOSSIP~ model can be simulated almost perfectly in O~({tau_{G}}) rounds of the CONGEST~ model. To this end, we develop a new algorithm for the GOSSIP~ model that 1 +/- epsilon approximates the p-th frequency moment F_p = sum_{i=1}^N f_i^p in O~(epsilon^{-2} n^{1-k/p}) roundsfor p >= 2, when the number of distinct elements F_0 is at most O(n^{1/(k-1)}). This result can be translated back to the CONGEST~ model with a factor O~({tau_{G}}) blow-up in the number of rounds
