On optimal embeddings for distributed computation of arbitrary functions

We consider optimal embedding of weighted directed acyclic graph (DAG), representing the sequence of computation steps to evaluate a function of distributed data, on an arbitrary undirected weighted graph that represents a distributed system. This has applications in in-network computation, operator placement for distributed query processing, and module placement in distributed software design. We first formulate a Quadratic Integer Program to obtain the optimal embedding. Then, for the case when the DAG is a tree, we show its connection to shortest path algorithms and review two algorithms from the literature. For the case when the DAG is a tree, we describe a polynomial-time algorithm that finds the two best embeddings. Arguing that the general embedding problem is NP-hard, we present two heuristic algorithms to solve the problem when the function graph is a general DAG and analyse their performance via simulations.