  There have been emerging lots of applications for distributed storage systems
e.g., those in wireless sensor networks or cloud storage. Since storage nodes
in wireless sensor networks have limited battery, it is valuable to find a
repair scheme with optimal transmission costs (e.g., energy). The optimal-cost
repair has been recently investigated in a centralized way. However a
centralized control mechanism may not be available or is very expensive. For
the scenarios, it is interesting to study optimal-cost repair in a
decentralized setup. We formulate the optimal-cost repair as convex
optimization problems for the network with convex transmission costs. Then we
use primal and dual decomposition approaches to decouple the problem into
subproblems to be solved locally. Thus, each surviving node, collaborating with
other nodes, can minimize its transmission cost such that the global cost is
minimized. We further study the optimality and convergence of the algorithms.
Finally, we discuss the code construction and determine the field size for
finding feasible network codes in our approaches.
