Physical Expander in Virtual Tree Overlay

Abstract

Abstract. In this paper, we propose a new construction of constantdegree expanders motivated by their application in P2P overlay networks and in particular in the design of robust trees overlay. Our key result can be stated as follows. Consider a complete binary tree T and construct a random pairing Π between leaf nodes and internal nodes. We prove that the graph GΠ obtained from T by contracting all pairs (leaf-internal nodes) achieves a constant node expansion with high probability. The use of our result in improving the robustness of tree overlays is straightforward. That is, if each physical node participating to the overlay manages a random pair that couples one virtual internal node and one virtual leaf node then the physical-node layer exhibits a constant expansion with high probability. We encompass the difficulty of obtaining this random tree virtualization by proposing a local, selforganizing and churn resilient uniformly-random pairing algorithm with O(log 2 n) running time. Our algorithm has the merit to not modify the original tree virtual overlay (we just control the mapping between physical nodes and virtual nodes). Therefore, our scheme is general and can be applied to a large number of tree overlay implementations. We validate its performances in dynamic environments via extensive simulations.

Self-stabilizing Balancing Algorithm for Containment-Based Trees

“…To showcase the applicability of our algorithm to an existing system, trees are generated using the join protocol of [6]. Figure 6 shows the distance between the sum of initial and final tree heights and experimental termination times.…”

“…Self-stabilization [7] is a general technique to design distributed systems that can handle arbitrary transient faults. Figure 1 shows the overlay lifecycle borrowed from [6]. Rectangles refer to states of the distributed tree.…”

“…Preserving the tree nature has been addressed in previous work [4], [5], [6]. It is tightly related to the distribution model of the structure, the centralized case being far less stressing.…”

“…The works in [4], [6] have the extra property of being selfstabilizing. A self-stabilizing system, as introduced in [7], is guaranteed to converge to the intended behavior in finite time, regardless of the initial states of nodes.…”

“…Figure 1 shows the overlay lifecycle borrowed from [6]. Rectangles refer to states of the distributed tree.…”