Function Load Balancing over Networks
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Using networks as a means of computing can reduce the communication flow or the total number of bits transmitted over the networks. In this paper, we propose to distribute the computation load in stationary networks, and formulate a flow-based delay minimization problem that jointly captures the aspects of communications and computation. We exploit the distributed compression scheme of Slepian-Wolf that is applicable under any protocol information where nodes can do compression independently using different codebooks. We introduce the notion of entropic surjectivity as a measure to determine how sparse the function is and to understand the limits of functional compression for computation. We leverage Little's Law for stationary systems to provide a connection between surjectivity and the computation processing factor that reflects the proportion of flow that requires communications. This connection gives us an understanding of how much a node (in isolation) should compute to communicate the desired function within the network without putting any assumptions on the topology. Our results suggest that to effectively compute different function classes that have different entropic surjectivities, the networks can be re-structured with the transition probabilities being tailored for functions, i.e., task-based link reservations, which can enable mixing versus separately processing of a diverse function classes. They also imply that most of the available resources are reserved for computing low complexity functions versus fewer resources for processing of high complexity ones. We numerically evaluate our technique for search, MapReduce, and classification functions, and infer how sensitive the processing factor for each computation task to the entropic surjectivity is.
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