A practical approach for successive omniscience

Abstract

Abstract-The system that we study in this paper contains a set of users that observe a discrete memoryless multiple source and communicate via noise-free channels with the aim of attaining omniscience, the state that all users recover the entire multiple source. We adopt the concept of successive omniscience (SO), i.e., letting the local omniscience in some user subset be attained before the global omniscience in the entire system, and consider the problem of how to efficiently attain omniscience in a successive manner. Based on the existing results on SO, we propose a CompSetSO algorithm for determining a complimentary set, a user subset in which the local omniscience can be attained first without increasing the sum-rate, the total number of communications, for the global omniscience. We also derive a sufficient condition for a user subset to be complimentary so that running the CompSetSO algorithm only requires a lower bound, instead of the exact value, of the minimum sum-rate for attaining global omniscience. The CompSetSO algorithm returns a complimentary user subset in polynomial time. We show by example how to recursively apply the CompSetSO algorithm so that the global omniscience can be attained by multi-stages of SO.

Improving Computational Efficiency of Communication for Omniscience and Successive Omniscience

“…, 5}) > λ. In Part II of this paper [30], we show that the dendrogram indicates a bottom-up successive omniscience approach for the asymptotic model. Algorithm 3].…”

“…We call Lemma III.3(a) to (c) the essential properties since they hold the strict strong map property in the Theorem III.5 in Section III-B, the main theorem that ensures the validity of the PAR algorithm. In addition, in Part II [30], we show that the monotonicity of the sum-rate in Lemma III.3(c) also guarantees the feasibility of a multi-stage SO.…”

“…The proposed PAR algorithm also solves a successive omniscience (SO) problem, where the omniscience process takes stages: a user subset attains the local omniscience each time. In Part II [30] of this paper, we derive the achievability conditions for the multi-stage SO for both asymptotic and non-asymptotic models. By using the segmented Dilworth truncation and rate vector r α,V returned by PAR, we propose algorithms extracting the user subset and achievable rate vector for each stage of SO such that, at the final stage, the global omniscience in V is attained by the minimum sum-rate.…”

“…That is, the partition Q α,Vi and r α,Vi are segmented in α and determine the solution to the minimum sum-rate problem in a subsystem V i . This fact will be utilized in Section V to propose a distributed algorithm for solving the CO problem and in Part II of this paper [30] to solve the successive omniscience problem.…”

“…It means that the local omniscience problem in V i is solved before the global omniscience. This fact will be utilized in Part II [30] for solving the SO problem, where it is also shown that an optimal local omniscience in X ⊆ V i can be directly determined from r λ,Vi .…”