Computing longest common square subsequences

A square is a non-empty string of form YY. The longest common square subsequence (LCSqS) problem is to compute a longest square occurring as a subsequence in two given strings A and B. We show that the problem can easily be solved in O(n^6) time or O(|M|n^4) time with O(n^4) space, where n is the length of the strings and M is the set of matching points between A and B. Then, we show that the problem can also be solved in O(sigma |M|^3 + n) time and O(|M|^2 + n) space, or in O(|M|^3 log^2 n log log n + n) time with O(|M|^3 + n) space, where sigma is the number of distinct characters occurring in A and B. We also study lower bounds for the LCSqS problem for two or more strings

Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, K\"unnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$. [...]Comment: Presented at SODA'18. Full Version. 66 page

Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, K\"unnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$. [...

Evolution and Regularisation of Vacuum Brill Gravitational Waves in Spherical Polar Coordinates

In this thesis the universal collapse of vacuum Brill waves is demonstrated numerically and analytically. This thesis presents the mathematical and numerical methods necessary to regularise and evolve Brill Gravitational Waves in spherical polar coordinates. A Cauchy ADM formulation is used for the time evolution. We find strong evidence that all IVP formulations of pure vacuum Brill gravitational waves collapse to form singularities/black holes, and we do not observe critical black hole mass scaling phenomena in the IVP parameter phase space that has been characterised in non-vacuum systems. A theoretical framework to prove this result analytically is presented. We discuss the meaning of Brill metric variables, the topology of trapped surfaces for various scenarios, and verify other results in the field related to critical values of initial value parameters and black hole formation approaching spatial infinity. The instability of Minkowski (flat) space under Brill wave and more general perturbations is demonstrated. The main numerical tools employed to achieve a stable evolution code are (1) derivation of appropriate regularity conditions on the lapse function and metric function q, (2) the move to a 4th order correct discretisation scheme with appropriate boundary conditions, (3) the use of exponential metric terms, (4) an understanding of the right mix of free versus constrained evolution and (5) the development of appropriate numerical techniques for discretisation and differencing to reduce numerical error, along with a characterisation of condition numbers.Comment: PhD Thesis, 2014, University of Calgary, 368 page

Elements, Government, and Licensing: Developments in phonology

Elements, Government, and Licensing brings together new theoretical and empirical developments in phonology. It covers three principal domains of phonological representation: melody and segmental structure; tone, prosody and prosodic structure; and phonological relations, empty categories, and vowel-zero alternations. Theoretical topics covered include the formalisation of Element Theory, the hotly debated topic of structural recursion in phonology, and the empirical status of government. In addition, a wealth of new analyses and empirical evidence sheds new light on empty categories in phonology, the analysis of certain consonantal sequences, phonological and non-phonological alternation, the elemental composition of segments, and many more. Taking up long-standing empirical and theoretical issues informed by the Government Phonology and Element Theory, this book provides theoretical advances while also bringing to light new empirical evidence and analysis challenging previous generalisations. The insights offered here will be equally exciting for phonologists working on related issues inside and outside the Principles & Parameters programme, such as researchers working in Optimality Theory or classical rule-based phonology

Optimum Distribution System Architectures for Efficient Operation of Hybrid AC/DC Power Systems Involving Energy Storage and Pulsed Loads

After more than a century of the ultimate dominance of AC in distribution systems, DC distribution is being re-considered. However, the advantages of AC systems cannot be omitted. This is mainly due to the cheap and efficient means of generation provided by the synchronous AC machines and voltage stepping up/down allowed by the AC transformers. As an intermediate solution, hybrid AC/DC distribution systems or microgrids are proposed. This hybridization of distribution systems, incorporation of heterogeneous mix of energy sources, and introducing Pulsed Power Loads (PPL) together add more complications and challenges to the design problem of distribution systems. In this dissertation, a comprehensive multi-objective optimization approach is presented to determine the optimal design of the AC/DC distribution system architecture. The mathematical formulation of a multi-objective optimal power flow problem based on the sequential power flow method and the Pareto concept is developed and discussed. The outcome of this approach is to answer the following questions: 1) the optimal size and location of energy storage (ES) in the AC/DC distribution system, 2) optimal location of the PPLs, 3) optimal point of common coupling (PCC) between the AC and DC sides of the network, and 4) optimal network connectivity. These parameters are to be optimized to design a distribution architecture that supplies the PPLs, while fulfilling the safe operation constraints and the related standard limitations. The optimization problem is NP-hard, mixed integer and combinatorial with nonlinear constraints. Four objectives are involved in the problem: minimizing the voltage deviation (ΔV), minimizing frequency deviation (Δf), minimizing the active power losses in the distribution system and minimizing the energy storage weight. The last objective is considered in the context of ship power systems, where the equipment’s weight and size are restricted. The utilization of Hybrid Energy Storage Systems (HESS) in PPL applications is investigated. The design, hardware implementation and performance evaluation of an advanced – low cost Modular Energy Storage regulator (MESR) to efficiently integrate ES to the DC bus are depicted. MESR provides a set of unique features: 1) It is capable of controlling each individual unit within a series/parallel array (i.e. each single unit can be treated, controlled and monitored separately from the others), 2) It is able to charge some units within an ES array while other units continue to serve the load, 3) Balance the SoC without the need for power electronic converters, and 4) It is able to electrically disconnect a unit and allow the operator to perform the required maintenance or replacement without affecting the performance of the whole array. A low speed flywheel Energy Storage System (FESS) is designed and implemented to be used as an energy reservoir in PPL applications. The system was based on a separately excited DC machine and a bi-directional Buck-Boost converter as the driver to control the charging/discharging of the flywheel. Stable control loops were designed to charge the FESS off the pulse and discharge on the pulse. All the developments in this dissertation were experimentally verified at the Smart Grid Testbed

28th Annual Symposium on Combinatorial Pattern Matching : CPM 2017, July 4-6, 2017, Warsaw, Poland

Peer reviewe

Active Suppression ofAerofoil Flutter via Neural-Network-Based Adaptive Nonlinear Optimal Control

This thesis deals with active flutter suppression (AFS) on aerofoils via adaptive nonlinear optimal control using neural networks (NNs). Aeroelastic flutter can damage aerofoils if not properly controlled. AFS not only ensures flutter-free flight but also enables the use of aerodynamically more efficient lightweight aerofoils. However, existing optimal controllers for AFS are generally susceptible to modelling errors while other controllers less prone to uncertainties do not provide optimal control. This thesis, thus, aims to reduce the impact of the dilemma by proposing new solutions based on nonlinear optimal control online synthesis (NOCOS) according to online updated dynamics. Existing NOCOS methods, with NNs as essential elements, require a separate initial stabilising control law for the overall system, an additional stabilising tuning loop for the actor NN, or an additional stabilising term in the critic NN tuning law, to guarantee the closed-loop stability for unstable and marginally stable systems. The resulting complexity is undesired in AFS applications due to computational concerns in real-time implementation. Moreover, the existing NOCOS methods are confined to locally nonlinear systems, while aeroelastic systems under consideration are globally nonlinear. These make all the existing NOCOS algorithms inapplicable to AFS without modification and improvement. Therefore, this thesis solves the aforementioned problems through the following step-by-step approaches. Firstly, a four degrees-of-freedom (4-DOF) aeroelastic model is considered, where leading- and trailing-edge control surfaces of the aerofoil are used to actively suppress flutter. Accordingly, a virtual stiffness-damping system (VSDS) is developed to simulate physical stiffness in the aeroelastic system. The VSDS, together with a scaled-down typical aerofoil section placed in a wind tunnel, serve as an experimental 4-DOF aeroelastic test-bed for synthesis and validation of proposed AFS controllers that follow. Secondly, a Modified form of NN-based Value Function Approximation (MVFA), tuned by gradient-descent learning, is proposed for NOCOS to address the closedloop stability in a compact controller configuration suitable for real-time implementation. Its validity and efficacy are examined by the Lyapunov stability analysis and numerical studies. Thirdly, a systematic procedure based on linear matrix inequalities is further proposed for synthesising a scheduled parameter matrix to generalise the MVFA to to globally nonlinear cases, so that the new NN controller suits AFS applications. In addition, the extended Kalman filter (EKF) is proposed for the new NN controller for fast parameter convergence. An identifier NN is also derived to capture and update aeroelastic dynamics in real time to mitigate the impact of modelling errors. Wind-tunnel experiments were conducted for validation. Finally, a non-quadratic functional is introduced to generalise the performance index to tackle the problem where control inputs are constrained. The feasibility of including the non-quadratic cost function under the proposed control scheme based on the MVFA is examined via the Lyapunov stability analysis and was also experimentally evaluated through the wind-tunnel testings. The proposed NN controllers are compact in structure and shown capable of maintaining the closed-loop stability while eliminating the need for a separate initial stabilising control law for the overall system, an additional tuning loop for the actor NN, and an additional stabilising term in the critic NN tuning law. Under the new control schemes, online synthesised nonlinear control laws are optimal in the cases with and without constraints in control. Comparisons drawn with a popular linear-parameter-varying (LPV) controller in the form of the widely used linear quadratic regulator (LQR) in experiments show that the proposed NN controllers outperform the LPV-LQR algorithm and improve AFS from the optimal control perspective. Specifically, the proposed NN controllers can effectively mitigate the impact of modelling errors, successfully solving the mentioned dilemma involved in AFS. The results also confirm that the proposed NN controllers are suitable for real-time implementation.Thesis (Ph.D.) -- University of Adelaide, School of Mechanical Engineering, 201

Cosmic Rays and the Search for a Lorentz Invariance Violation

This is an introductory review about the on-going search for a signal of Lorentz Invariance Violation (LIV) in cosmic rays. We first summarise basic aspects of cosmic rays, focusing on rays of ultra high energy (UHECRs). We discuss the Greisen-Zatsepin-Kuz'min (GZK) energy cutoff for cosmic protons, which is predicted due to photopion production in the Cosmic Microwave Background (CMB). This is a process of modest energy in the proton rest frame. It can be investigated to a high precision in the laboratory, if Lorentz transformations apply even at factors $\gamma \sim O(10^{11})$. For heavier nuclei the energy attenuation is even faster due to photo-disintegration, again if this process is Lorentz invariant. Hence the viability of Lorentz symmetry up to tremendous gamma-factors - far beyond accelerator tests - is a central issue. Next we comment on conceptual aspects of Lorentz Invariance and the possibility of its spontaneous breaking. This could lead to slightly particle dependent ``Maximal Attainable Velocities''. We discuss their effect in decays, Cerenkov radiation, the GZK cutoff and neutrino oscillation in cosmic rays. We also review the search for LIV in cosmic gamma-rays. For multi TeV gamma-rays we possibly encounter another puzzle related to the transparency of the CMB, similar to the GZK cutoff. The photons emitted in a Gamma Ray Burst occur at lower energies, but their very long path provides access to information not far from the Planck scale. No LIV has been observed so far. However, even extremely tiny LIV effects could change the predictions for cosmic ray physics drastically. An Appendix is devoted to the recent hypothesis by the Pierre Auger Collaboration, which identifies nearby Active Galactic Nuclei - or objects next to them - as probable UHECR sources.Comment: 81 pages, 15 figures, some points extended and improved, references adde