A Paradigm for Channel Assignment and Data Migration in Distributed Systems

In this manuscript, we consider the problems of channel assignment in wireless networks and data migration in heterogeneous storage systems. We show that a soft edge coloring approach to both problems gives rigorous approximation guarantees. In the channel assignment problem arising in wireless networks, we are given a graph G = (V,E), and the number of wireless cards Cv for each vertex v. The constraint Cv limits the number of channels that edges incident to v can use. We also have the total number of channels CG available in the network. For a pair of edges incident to a vertex, they are said to be conflicting if the channels assigned to them are the same. Our goal is to assign channels (color edges) so that the number of conflicts is minimized. In this manuscript we first study the problem for a homogeneous network where Cv = k and CG ≥Cv for all nodes v. The problem is NP-hard by a reduction from EDGE COLORING and we present two combinatorial algorithms for this case. The first algorithm is based on a distributed greedy method and gives a solution with at most 2(1− k )|E| more conflicts than the optimal solution, which implies a (2− k )-approximation. We also present a soft edge coloring algorithm that yields at most 2|V |more conflicts than the optimal solution. The approximation ratio is 1+ |V | |E| , which gives a (1+ o(1))-factor for dense graphs. The algorithm generalizes Vizing’s algorithm in the sense that it gives the same result as Vizing’s algorithm when k = ∆+ 1. Moreover, we show that this approximation result is best possible unless P = NP. For the case where Cv = 1 or k, we show that the problem is NP-hard even when Cv = 1 or 2, and CG = 2, and present two approximation algorithms. The first algorithm is completely combinatorial and has an approximation ratio of 2− k . We also develop an SDP-based algorithm, producing a solution with an approximation ratio of 1.122 for k = 2, and 2−Θ( lnk k ) in general. In this manuscript, we also consider the data migration problem in heterogeneous storage systems. Large-scale storage systems are crucial components in data-intensive applications such as search engine clusters, video-on-demand servers, sensor networks, and grid computing. A storage server typically consists of a set of storage devices. In such systems, data layouts may need to be reconfigured over time for load balancing or in the event of system failure/upgrades. It is critical to migrate data to their target locations as quickly as possible to obtain the best performance of the system. Most of the previous results on data migration assume that each storage node can perform only one data transfer at a time. A storage node, however, typically can handle multiple transfers simultaneously and this can reduce the total migration time significantly. Moreover, storage devices tend to have heterogeneous capabilities as devices may be added over time due to storage demand increase. We consider the heterogeneous data migration problem where we assume that each storage node has different transfer constraint cv, representing how many simultaneous transfers the node can handle. We develop algorithms to minimize the data migration time. We show that it is possible to find an optimal migration schedule when all cv’s are even. Furthermore, though the problem is NP-hard in general, we give an efficient soft edge coloring algorithm that offers a rigorous (1+ o(1))approximation guarantee.