The k-mismatch problem revisited

We revisit the complexity of one of the most basic problems in pattern matching. In the k-mismatch problem we must compute the Hamming distance between a pattern of length m and every m-length substring of a text of length n, as long as that Hamming distance is at most k. Where the Hamming distance is greater than k at some alignment of the pattern and text, we simply output "No". We study this problem in both the standard offline setting and also as a streaming problem. In the streaming k-mismatch problem the text arrives one symbol at a time and we must give an output before processing any future symbols. Our main results are as follows: 1) Our first result is a deterministic $O(n k^2\log{k} / m+n \text{polylog} m)$ time offline algorithm for k-mismatch on a text of length n. This is a factor of k improvement over the fastest previous result of this form from SODA 2000 by Amihood Amir et al. 2) We then give a randomised and online algorithm which runs in the same time complexity but requires only $O(k^2\text{polylog} {m})$ space in total. 3) Next we give a randomised $(1+\epsilon)$-approximation algorithm for the streaming k-mismatch problem which uses $O(k^2\text{polylog} m / \epsilon^2)$ space and runs in $O(\text{polylog} m / \epsilon^2)$ worst-case time per arriving symbol. 4) Finally we combine our new results to derive a randomised $O(k^2\text{polylog} {m})$ space algorithm for the streaming k-mismatch problem which runs in $O(\sqrt{k}\log{k} + \text{polylog} {m})$ worst-case time per arriving symbol. This improves the best previous space complexity for streaming k-mismatch from FOCS 2009 by Benny Porat and Ely Porat by a factor of k. We also improve the time complexity of this previous result by an even greater factor to match the fastest known offline algorithm (up to logarithmic factors)

Parameterized Matching in the Streaming Model

We study the problem of parameterized matching in a stream where we want to output matches between a pattern of length m and the last m symbols of the stream before the next symbol arrives. Parameterized matching is a natural generalisation of exact matching where an arbitrary one-to-one relabelling of pattern symbols is allowed. We show how this problem can be solved in constant time per arriving stream symbol and sublinear, near optimal space with high probability. Our results are surprising and important: it has been shown that almost no streaming pattern matching problems can be solved (not even randomised) in less than Theta(m) space, with exact matching as the only known problem to have a sublinear, near optimal space solution. Here we demonstrate that a similar sublinear, near optimal space solution is achievable for an even more challenging problem. The proof is considerably more complex than that for exact matching.Comment: 19 pages, 3 figure

Probing the Optical Dynamics of Quantum Emitters in Hexagonal Boron Nitride

Hexagonal boron nitride is a van der Waals material that hosts visible-wavelength quantum emitters at room temperature. However, experimental identification of the quantum emitters' electronic structure is lacking, and key details of their charge and spin properties remain unknown. Here, we probe the optical dynamics of quantum emitters in hexagonal boron nitride using photon emission correlation spectroscopy. Several quantum emitters exhibit ideal single-photon emission with noise-limited photon antibunching, $g^{(2)}(0)=0$. The photoluminescence emission lineshapes are consistent with individual vibronic transitions. However, polarization-resolved excitation and emission suggests the role of multiple optical transitions, and photon emission correlation spectroscopy reveals complicated optical dynamics associated with excitation and relaxation through multiple electronic excited states. We compare the experimental results to quantitative optical dynamics simulations, develop electronic structure models that are consistent with the observations, and discuss the results in the context of ab initio theoretical calculations.Comment: 31 pages, 16 figures, 6 table

Dynamical Characterization and Room-Temperature Control of an Optically Addressable Single Spin in Hexagonal Boron Nitride

Hexagonal boron nitride (h-BN), a wide bandgap, two-dimensional solid-state material, hosts pure single-photon emitters that have shown signatures of optically-addressable electronic spins. Here, we report on a single emitter in h-BN exhibiting optically detected magnetic resonance at room temperature, and we propose a model for its electronic structure and optical dynamics. Using photon emission correlation spectroscopy in conjunction with time-domain optical and microwave experiments, we establish key features of the emitter's electronic structure. Specifically, we propose a model that includes a spinless optical ground and excited state, a metastable spin-1/2 configuration, and an emission modulation mechanism. Using optical and spin dynamics simulations, we constrain and quantify transition rates in the model, and we design protocols that optimize the signal-to-noise ratio for spin readout. This constitutes a necessary step toward quantum control of spin states in h-BN.Comment: 14 pages, 15 figures. arXiv admin note: text overlap with arXiv:2201.0888

Role of biomechanics in the understanding of normal, injured, and healing ligaments and tendons

Ligaments and tendons are soft connective tissues which serve essential roles for biomechanical function of the musculoskeletal system by stabilizing and guiding the motion of diarthrodial joints. Nevertheless, these tissues are frequently injured due to repetition and overuse as well as quick cutting motions that involve acceleration and deceleration. These injuries often upset this balance between mobility and stability of the joint which causes damage to other soft tissues manifested as pain and other morbidity, such as osteoarthritis

Late Quaternary sea-level change and early human societies in the central and eastern Mediterranean Basin : an interdisciplinary review

This article reviews key data and debates focused on relative sea-level changes since the Last Interglacial (approximately the last 132,000 years) in the Mediterranean Basin, and their implications for past human populations. Geological and geomorphological landscape studies are critical to archaeology. Coastal regions provide a wide range of resources to the populations that inhabit them. Coastal landscapes are increasingly the focus of scholarly discussions from the earliest exploitation of littoral resources and early hominin cognition, to the inundation of the earliest permanently settled fishing villages and eventually, formative centres of urbanisation. In the Mediterranean, these would become hubs of maritime transportation that gave rise to the roots of modern seaborne trade. As such, this article represents an original review of both the geo-scientific and archaeological data that specifically relate to sea-level changes and resulting impacts on both physical and cultural landscapes from the Palaeolithic until the emergence of the Classical periods. Our review highlights that the interdisciplinary links between coastal archaeology, geomorphology and sea-level changes are important to explain environmental impacts on coastal human societies and human migration. We review geological indicators of sea level and outline how archaeological features are commonly used as proxies for measuring past sea levels, both gradual changes and catastrophic events. We argue that coastal archaeologists should, as a part of their analyses, incorporate important sea-level concepts, such as indicative meaning. The interpretation of the indicative meaning of Roman fishtanks, for example, plays a critical role in reconstructions of late Holocene Mediterranean sea levels. We identify avenues for future work, which include the consideration of glacial isostatic adjustment (GIA) in addition to coastal tectonics to explain vertical movements of coastlines, more research on Palaeolithic island colonisation, broadening of Palaeolithic studies to include materials from the entire coastal landscape and not just coastal resources, a focus on rescue of archaeological sites under threat by coastal change, and expansion of underwater archaeological explorations in combination with submarine geomorphology. This article presents a collaborative synthesis of data, some of which have been collected and analysed by the authors, as the MEDFLOOD (MEDiterranean sea-level change and projection for future FLOODing) community, and highlights key sites, data, concepts and ongoing debates