Assignment of Different-Sized Inputs in MapReduce

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 set, X , is required to meet each input of another set, Y , in at least one reducer. We prove that finding an optimal mapping schema for these families of problem is NP-hard, and ∗More details appear in [1]. †Supported by the project Handling Uncertainty in Data Intensive Applications, co-financed by the European Union (European Social Fund) and Greek national funds, through the Operational Program “Education and Lifelong Learning,” under the program THALES ‡Supported by the Rita Altura Trust Chair in Computer Sciences, Lynne and William Frankel Center for Computer Sciences, Israel Science Foundation (grant 428/11), the Israeli Internet Association, and the Ministry of Science and Technology, Infrastructure Research in the Field of Advanced Computing and Cyber Security. c ©2015, Copyright is with the authors. Published in the Workshop Proceedings of the EDBT/ICDT 2015 Joint Conference (March 27, 2015, Brussels, Belgium) on CEUR-WS.org (ISSN 1613-0073). Distribution of this paper is permitted under the terms of the Creative Commons license CC-by-nc-nd 4.0 present several approximation algorithms for finding a near optimal mapping schema.