Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms

TL;DR

This paper analyzes the complexity of optimally embedding functions for in-network computation, proving NP-completeness for delay and cost minimization, and providing polynomial algorithms for specific graph structures.

Contribution

It establishes the NP-completeness of optimal embedding problems and offers polynomial algorithms for trees, layered graphs, and bounded treewidth graphs.

Findings

Minimum-delay and minimum-cost embeddings are NP-complete.

Polynomial algorithms exist for trees, layered graphs, and bounded treewidth graphs.

Cost minimization is MAX SNP-hard.

Abstract

We consider optimal distributed computation of a given function of distributed data. The input (data) nodes and the sink node that receives the function form a connected network that is described by an undirected weighted network graph. The algorithm to compute the given function is described by a weighted directed acyclic graph and is called the computation graph. An embedding defines the computation communication sequence that obtains the function at the sink. Two kinds of optimal embeddings are sought, the embedding that---(1)~minimizes delay in obtaining function at sink, and (2)~minimizes cost of one instance of computation of function. This abstraction is motivated by three applications---in-network computation over sensor networks, operator placement in distributed databases, and module placement in distributed computing. We first show that obtaining minimum-delay and minimum-cost embeddings are both NP-complete problems and that cost minimization is actually MAX SNP-hard. Next, we consider specific forms of the computation graph for which polynomial time solutions are possible. When the computation graph is a tree, a polynomial time algorithm to obtain the minimum delay embedding is described. Next, for the case when the function is described by a layered graph we describe an algorithm that obtains the minimum cost embedding in polynomial time. This algorithm can also be used to obtain an approximation for delay minimization. We then consider bounded treewidth computation graphs and give an algorithm to obtain the minimum cost embedding in polynomial time.