The greedy Prefer-same de Bruijn sequence construction was first presented by Eldert, Gray, Gurk, and Rubinoff in 1958. As a greedy algorithm, it has one major downside: it requires an exponential amount of space to store the length \(2^{n}\) de Bruijn sequence. Though de Bruijn sequences have been heavily studied over the last 60 years, finding an efficient construction for the Prefer-same de Bruijn sequence has remained a tantalizing open problem. In this article, we unveil the underlying structure of the Prefer-same de Bruijn sequence and solve the open problem by presenting an efficient algorithm to construct it using \(O(n)\) time per bit and only \(O(n)\) space. Following a similar approach, we also present an efficient algorithm to construct the Prefer-opposite de Bruijn sequence.

When downloading an application by a smartphone user, the application asks the user to accept a set of permissions allowing it to access sensitive information on the phone. It is mostly unclear how, when, and why such data will be used. openPDS is a framework that was proposed to limit such privacy invasion. Although openPDS protects the user's raw data, sensitive personal information, such as location trace, can still be inferred by the service provider by analyzing the accumulated answers. In this paper, we aim to append openPDS with a module that prevents the service provider from reconstructing the trace of users. This module abides with the QoS requirements necessitated by the provider, and defined as the tolerance of the application to inaccurate answers. Our approach was tested on 10 users whose locations traces were recorded for 10 months. Results show that no user trace was successfully reconstructed even when high QoS levels were required.