Maintaining Balanced Trees for Structured Distributed Streaming Systems

Abstract

In this paper, we propose and analyze a simple localized algorithm to balance a tree. The motivation comes from live distributed streaming systems in which a source diffuses a content to peers via a tree, a node forwarding the data to its children. Such systems are subject to a high churn, peers frequently joining and leaving the system. It is thus crucial to be able to repair the diffusion tree to allow an efficient data distribution. In particular, due to bandwidth limitations, an efficient diffusion tree must ensure that node degrees are bounded. Moreover, to minimize the delay of the streaming, the depth of the diffusion tree must also be controlled. We propose here a simple distributed repair algorithm in which each node carries out local operations based on its degree and on the subtree sizes of its children. In a synchronous setting, we first prove that starting from any n-node tree our process converges to a balanced tree in O(n ²) turns. We then describe a more restrictive model, adding a small extra information to each node, under which we adopt our algorithm to converge in $\Theta(\emph{n}log\emph{n})$ turns. We then exhibit by simulation that the convergence is much faster (logarithmic number of turns in average) for a random tree.

The research leading to these results has received funding from the European Project FP7 EULER, ANR CEDRE, ANR AGAPE, Associated Team AlDyNet, project ECOS-Sud Chile and région PACA.

The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-3-319-03578-9_29