Algorithmic Foundations of Inexact Computing

Inexact computing also referred to as approximate computing is a style of designing algorithms and computing systems wherein the accuracy of correctness of algorithms executing on them is deliberately traded for significant resource savings. Significant progress has been reported in this regard both in terms of hardware as well as software or custom algorithms that exploited this approach resulting in some loss in solution quality (accuracy) while garnering disproportionately high savings. However, these approaches tended to be ad-hoc and were tied to specific algorithms and technologies. Consequently, a principled approach to designing and analyzing algorithms was lacking. In this paper, we provide a novel model which allows us to characterize the behavior of algorithms designed to be inexact, as well as characterize opportunities and benefits that this approach offers. Our methods therefore are amenable to standard asymptotic analysis and provides a clean unified abstraction through which an algorithm's design and analysis can be conducted. With this as a backdrop, we show that inexactness can be significantly beneficial for some fundamental problems in that the quality of a solution can be exponentially better if one exploits inexactness when compared to approaches that are agnostic and are unable to exploit this approach. We show that such gains are possible in the context of evaluating Boolean functions rooted in the theory of Boolean functions and their spectra, PAC learning, and sorting. Formally, this is accomplished by introducing the twin concepts of inexactness aware and inexactness oblivious approaches to designing algorithms and the exponential gains are shown in the context of taking the ratio of the quality of the solution using the "aware" approach to the "oblivious" approach.