Competitive Self-Stabilizing k-Clustering

• Main Authors: Datta, A. K., Larmore, L. L., Devismes, S., Heurtefeux, K., Rivierre, Y.
• Format: Conference Proceeding
• Language: English
• Subjects: Algorithm design and analysis; Approximation algorithms; Clustering algorithms; competitiveness; Complexity theory; Data structures; Encoding; k-clustering; maximal independent set; MIS tree; self-stabilization; Vegetation
• ISSN: 1063-6927; 2575-8411
• DOI: 10.1109/ICDCS.2012.72

In this paper, we propose a silent self-stabilizing asynchronous distributed algorithm for constructing a kclustering of any connected network with unique IDs. Our algorithm stabilizes in O(n) rounds, using O(log n) space per process, where n is the number of processes. In the general case, our algorithm constructs O(n/k) k-clusters. If the network is a Unit Disk Graph (UDG), then our algorithm is 7.2552k+O(1)competitive, that is, the number of k-clusters constructed by the algorithm is at most 7.2552k + O(1) times the minimum possible number of k-clusters in any k-clustering of the same network. More generally, if the network is an Approximate Disk Graph (ADG) with approximation ratio λ, then our algorithm is 7.2552λ 2 k + O(λ)-competitive. Our solution is based on the self-stabilizing construction of a data structure called the MIS Tree, a spanning tree of the network whose processes at even levels form a maximal independent set of the network. The MIS tree construction is the time bottleneck of our k-clustering algorithm, as it takes Θ(n) rounds in the worst case, while the rest of the algorithm takes O(D) rounds, where V is the diameter of the network. We would like to improve that time to be O(D), but we show that our distributed MIS tree construction is a P-complete problem.