Improved Distance Queries and Cycle Counting by Frobenius Normal Form

Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time O-tilde(n^omega). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the following problems efficiently. * All Nodes Shortest Cycles - for every node return the length of the shortest cycle containing it. We give an O-tilde(n^omega) algorithm that improves the previous O-tilde(n^((omega + 3)/2)) algorithm for unweighted digraphs. * We show how to compute all D sets of vertices lying on cycles of length c in {1, ..., D} in randomized time O-tilde(n^omega). It improves upon an algorithm by Cygan where algorithm that computes a single set is presented. * We present a functional improvement of distance queries for directed, unweighted graphs. * All Pairs All Walks - we show almost optimal O-tilde(n^3) time algorithm for all walks counting problem. We improve upon the naive O(D n^omega) time algorithm

Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form

Algebraic techniques have had an important impact on graph algorithms so far. Porting them, e.g., the matrix inverse, into the dynamic regime improved best-known bounds for various dynamic graph problems. In this paper, we develop new algorithms for another cornerstone algebraic primitive, the Frobenius normal form (FNF). We apply our developments to dynamic and fault-tolerant exact distance oracle problems on directed graphs. For generic matrices $A$ over a finite field accompanied by an FNF, we show (1) an efficient data structure for querying submatrices of the first $k\geq 1$ powers of $A$, and (2) a near-optimal algorithm updating the FNF explicitly under rank-1 updates. By representing an unweighted digraph using a generic matrix over a sufficiently large field (obtained by random sampling) and leveraging the developed FNF toolbox, we obtain: (a) a conditionally optimal distance sensitivity oracle (DSO) in the case of single-edge or single-vertex failures, providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a multiple-failures DSO improving upon the state of the art (vd. Brand and Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved dynamic distance oracles in the case of single-edge updates, and (d) a dynamic distance oracle supporting vertex updates, i.e., changing all edges incident to a single vertex, in $\tilde{O}(n^2)$ worst-case time and distance queries in $\tilde{O}(n)$ time.Comment: To appear at FOCS 202

The hidden subgroup problem and quantum computation using group representations

The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented

Medical Image Retrieval Using Multimodal Semantic Indexing

Large collections of medical images have become a valuable source of knowledge, taking an important role in education, medical research and clinical decision making. An important unsolved issue that is actively investigated is the efficient and effective access to these repositories. This work addresses the problem of information retrieval in large collections of biomedical images, allowing to use sample images as alternative queries to the classic keywords. The proposed approach takes advantage of both modalities: text and visual information. The main drawback of the multimodal strategies is that the associated algorithms are memory and computation intensive. So, an important challenge addressed in this work is the design of scalable strategies, that can be applied efficiently and effectively in large medical image collections. The experimental evaluation shows that the proposed multimodal strategies are useful to improve the image retrieval performance, and are fully applicable to large image repositories.Maestrí

On analog quantum algorithms for the mixing of Markov chains

The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently we provided an upper bound on the quantum mixing time for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.Comment: The section concerning time-averaged mixing (Sec VIII) has been updated: Now contains numerical plots and an intuitive discussion on the random matrix theory results used to derive the results of arXiv:2001.0630

Registration and Recognition in 3D

The simplest Computer Vision algorithm can tell you what color it sees when you point it at an object, but asking that computer what it is looking at is a much harder problem. Camera and LiDAR (Light Detection And Ranging) sensors generally provide streams pixel of values and sophisticated algorithms must be engineered to recognize objects or the environment. There has been significant effort expended by the computer vision community on recognizing objects in color images; however, LiDAR sensors, which sense depth values for pixels instead of color, have been studied less. Recently we have seen a renewed interest in depth data with the democratization provided by consumer depth cameras. Detecting objects in depth data is more challenging in some ways because of the lack of texture and increased complexity of processing unordered point sets. We present three systems that contribute to solving the object recognition problem from the LiDAR perspective. They are: calibration, registration, and object recognition systems. We propose a novel calibration system that works with both line and raster based LiDAR sensors, and calibrates them with respect to image cameras. Our system can be extended to calibrate LiDAR sensors that do not give intensity information. We demonstrate a novel system that produces registrations between different LiDAR scans by transforming the input point cloud into a Constellation Extended Gaussian Image (CEGI) and then uses this CEGI to estimate the rotational alignment of the scans independently. Finally we present a method for object recognition which uses local (Spin Images) and global (CEGI) information to recognize cars in a large urban dataset. We present real world results from these three systems. Compelling experiments show that object recognition systems can gain much information using only 3D geometry. There are many object recognition and navigation algorithms that work on images; the work we propose in this thesis is more complimentary to those image based methods than competitive. This is an important step along the way to more intelligent robots