Assignment Problems of Different-Sized Inputs in MapReduce

TL;DR

This paper investigates input assignment problems in MapReduce where input sizes vary, proving NP-hardness and proposing a bin-packing-based approximation algorithm to optimize communication costs.

Contribution

It introduces the first consideration of input sizes in MapReduce mapping schemas and provides an approximation algorithm for related NP-hard problems.

Findings

Proves NP-hardness of input meeting problems in MapReduce.

Develops a bin-packing-based approximation algorithm.

Offers near-optimal solutions for input assignment problems.

Abstract

A MapReduce algorithm can be described by a mapping schema, which assigns inputs to a set of reducers, such that for each required output there exists a reducer that receives all the inputs that participate in the computation of this output. Reducers have a capacity, which limits the sets of inputs that they can be assigned. However, individual inputs may vary in terms of size. We consider, for the first time, mapping schemas where input sizes are part of the considerations and restrictions. One of the significant parameters to optimize in any MapReduce job is communication cost between the map and reduce phases. The communication cost can be optimized by minimizing the number of copies of inputs sent to the reducers. The communication cost is closely related to the number of reducers of constrained capacity that are used to accommodate appropriately the inputs, so that the requirement of how the inputs must meet in a reducer is satisfied. In this work, we consider a family of problems where it is required that each input meets with each other input in at least one reducer. We also consider a slightly different family of problems in which, each input of a list, X, is required to meet each input of another list, Y, in at least one reducer. We prove that finding an optimal mapping schema for these families of problems is NP-hard, and present a bin-packing-based approximation algorithm for finding a near optimal mapping schema.