Compressibility-Aware Quantum Algorithms on Strings

Sublinear time quantum algorithms have been established for many fundamental problems on strings. This work demonstrates that new, faster quantum algorithms can be designed when the string is highly compressible. We focus on two popular and theoretically significant compression algorithms -- the Lempel-Ziv77 algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT), and obtain the results below. We first provide a quantum algorithm running in $\tilde{O}(\sqrt{zn})$ time for finding the LZ77 factorization of an input string $T[1..n]$ with $z$ factors. Combined with multiple existing results, this yields an $\tilde{O}(\sqrt{rn})$ time quantum algorithm for finding the RL-BWT encoding with $r$ BWT runs. Note that $r = \tilde{\Theta}(z)$. We complement these results with lower bounds proving that our algorithms are optimal (up to polylog factors). Next, we study the problem of compressed indexing, where we provide a $\tilde{O}(\sqrt{rn})$ time quantum algorithm for constructing a recently designed $\tilde{O}(r)$ space structure with equivalent capabilities as the suffix tree. This data structure is then applied to numerous problems to obtain sublinear time quantum algorithms when the input is highly compressible. For example, we show that the longest common substring of two strings of total length $n$ can be computed in $\tilde{O}(\sqrt{zn})$ time, where $z$ is the number of factors in the LZ77 factorization of their concatenation. This beats the best known $\tilde{O}(n^\frac{2}{3})$ time quantum algorithm when $z$ is sufficiently small

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

PRIVACY-PRESERVING QUERY PROCESSING ON OUTSOURCED DATABASES IN CLOUD COMPUTING

Database-as-a-Service (DBaaS) is a category of cloud computing services that enables IT providers to deliver database functionality as a service. In this model, a third party service provider known as a cloud server hosts a database and provides the associated software and hardware supports. Database outsourcing reduces the workload of the data owner in answering queries by delegating the tasks to powerful third-party servers with large computational and network resources. Despite the economic and technical benefits, privacy is the primary challenge posed by this category of services. By using these services, the data owners will lose the control of their databases. Moreover, the privacy of clients may be compromised since a curious cloud operator can follow the queries of a client and infer what the client is after. The challenge is to fulfill the main privacy goals of both the data owner and the clients without undermining the ability of the cloud server to return the correct query results. This thesis considers the design of protocols that protect the privacy of the clients and the data owners in the DBaaS model. Such protocols must protect the privacy of the clients so that the data owner and the cloud server cannot infer the constants contained in the query predicate as well as the query result. Moreover, the data owner privacy should be preserved by ensuring that the sensitive information in the database is not leaked to the cloud server and nothing beyond the query result is revealed to the clients. The results of the complexity and performance analysis indicates that the proposed protocols incur reasonable communication and computation overhead on the client and the data owner, considering the added advantage of being able to perform the symmetrically-private database search

Approximating Spectral Clustering via Sampling: a Review

Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of these algorithms' success and their Achilles heel: forming a graph and computing its dominant eigenvectors can indeed be computationally prohibitive when dealing with more that a few tens of thousands of points. In this paper, we review the principal research efforts aiming to reduce this computational cost. We focus on methods that come with a theoretical control on the clustering performance and incorporate some form of sampling in their operation. Such methods abound in the machine learning, numerical linear algebra, and graph signal processing literature and, amongst others, include Nystr\"om-approximation, landmarks, coarsening, coresets, and compressive spectral clustering. We present the approximation guarantees available for each and discuss practical merits and limitations. Surprisingly, despite the breadth of the literature explored, we conclude that there is still a gap between theory and practice: the most scalable methods are only intuitively motivated or loosely controlled, whereas those that come with end-to-end guarantees rely on strong assumptions or enable a limited gain of computation time