Distributed K-Backup-Placement and Applications to Virtual Memory in Heterogeneous Networks
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The Backup Placement problem in networks in the CONGEST distributed setting considers a network graph G = (V,E), in which the goal of each vertex v ∈ V is selecting a neighbor, such that the maximum number of vertices in V that select the same vertex is minimized [Halldorsson et al., 2015]. The problem is called 1-Backup-Placement when each vertex selects exactly one neighbor. A more general version, in which each vertex selects K neighbors, for a positive parameter K, is called K-Backup-Placement. In [SOSA APOCS, 2020] Barenboim and Oren studied approximate backup placement algorithms in sparse and dense networks and proved their correctness, specifically that each vertex is selected by at most O(1) neighbors, and achieving a desired load-balancing. However, those algorithm suffer from obliviousness to some main factors of heterogeneous wireless network. Specifically, there is no consideration of the nodes memory and storage capacities and the heterogeneity of capacity, and no reference to a case in which nodes have different energy capacity, and thus can leave (or join) the network at any time. Those parameters coupled in wireless networks, as the load on different parts of the network can differ greatly, thus requiring more messages, energy, memory and storage. In this work we suggest two complementary algorithms which address this problem. In the first one we divide the memory of each node to many small parts, but assign each vertex the memories of a large number of its neighbors. Thus more memory for more vertices is gained but with much fragmentation. In the second algorithm we execute more rounds, but do not produce fragmentation, and allow for a larger virtual memory per vertex. A key technical contribution of ours is an algorithm for K-Backup-Placement, that is used as a building block for these two procedures. Its running time is just one round.
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