Characterizing Algorithmic Efficiencies through Concentration

Abstract

Understanding inherent bottlenecks to efficient algorithm design lies at the heart of computer science. This question is significant both in the classical computing domain as well as in the emerging context of quantum computing. In this thesis, my goal is to characterize bottlenecks in designing efficient algorithms through the lens of a parameter called concentration of functions, starting with the domain of quantum information. My primary focus is probabilistically approximately correct (PAC) learning. I chose this domain since it allows us to approach the important subject of supervised learning in a rigorous and principled manner. For PAC learning, I propose a quantum algorithm to learn the class of concentrated Boolean functions with complexity $ O(\frac{M}{\epsilon^2})$ which offers an advantage over the best known classical PAC algorithms with complexity $\tilde{O}(n^2M)$, where M denotes the number of the concentration terms. I also show a lower bound $\Omega(M)$ to PAC learn this class of functions in distribution-independent settings. All this work is done in the context of the standard query model for PAC learning where the complexity measure is the number of queries, dubbed query complexity. I extend this work to include the learning model where functions are learned without any error, which is often called exact learning, and prove a query complexity lower bound of $\Omega(\frac{\epsilon \log M}{n} 2^n)$ in exact learning the class of concentrated Boolean functions. In the next part of the thesis, I focus my work on classical algorithms and explore a combinatorial counterpart of concentration called degree of symmetry. In this arena, graph isomorphism is my problem of choice. Once again, my goal is to characterize the efficiency of algorithms, in particular parallel algorithms, for graph isomorphism based on a concentration-related parameter. In particular, I propose a parallel algorithm in polynomial time using a quasi-polynomial number of processors for the Graph Isomorphism problem. My work builds on Babai’s celebrated quasi-polynomial algorithm and is work-preserving. The idea behind the parallelization explores the symmetry of the input structure for easier parallelization.