The Cover Suffix Tree (CST) of a string $T$ is the suffix tree of $T$ with additional explicit nodes corresponding to halves of square substrings of $T$. In the CST an explicit node corresponding to a substring $C$ of $T$ is annotated with two numbers: the number of non-overlapping consecutive occurrences of $C$ and the total number of positions in $T$ that are covered by occurrences of $C$ in $T$. Kociumaka et al. (Algorithmica, 2015) have shown how to compute the CST of a length-$n$ string in $O(n \log n)$ time. We show how to compute the CST in $O(n)$ time assuming that $T$ is over an integer alphabet. Kociumaka et al. (Algorithmica, 2015; Theor. Comput. Sci., 2018) have shown that knowing the CST of a length-$n$ string $T$, one can compute a linear-sized representation of all seeds of $T$ as well as all shortest $\alpha$-partial covers and seeds in $T$ for a given $\alpha$ in $O(n)$ time. Thus our result implies linear-time algorithms computing these notions of quasiperiodicity. The resulting algorithm computing seeds is substantially different from the previous one (Kociumaka et al., SODA 2012, ACM Trans. Algorithms, 2020). Kociumaka et al. (Algorithmica, 2015) proposed an $O(n \log n)$-time algorithm for computing a shortest $\alpha$-partial cover for each $\alpha=1,\ldots,n$; we improve this complexity to $O(n)$. Our results are based on a new characterization of consecutive overlapping occurrences of a substring $S$ of $T$ in terms of the set of runs (see Kolpakov and Kucherov, FOCS 1999) in $T$. This new insight also leads to an $O(n)$-sized index for reporting overlapping consecutive occurrences of a given pattern $P$ of length $m$ in $O(m+output)$ time, where $output$ is the number of occurrences reported. In comparison, a general index for reporting bounded-gap consecutive occurrences of Navarro and Thankachan (Theor. Comput. Sci., 2016) uses $O(n \log n)$ space.
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Algorithmica是一本國際性的期刊，它出版關于解決實際領域中出現的問題的算法的理論論文，以及對實際重要性或技術具有普遍吸引力的實驗論文。算法的發展是計算機科學的一個組成部分。計算機應用的日益復雜和范圍使得高效算法的設計必不可少。此外，該雜志還設有兩個專區：應用經驗、將理論成果應用到實際情況中的發現和問題、提供有關計算機科學選定主題的問題的短文。官網鏈接： ·
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state-of-the-art
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It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\Delta$ is NP-hard if $\Delta\geq 5\cdot 2^{k-1}+1$. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon, Sly and Zhang (RSA, 2019), the perfect sampler by Qiu, Wang and Zhang (ICALP, 2022), and the deterministic approximation scheme by Feng, Guo, Wang, Wang and Yin (FOCS, 2023).
We study the problem of cutting a length-$n$ string of positive real numbers into $k$ pieces so that every piece has sum at least $b$. The problem can also be phrased as transforming such a string into a new one by merging adjacent numbers. We discuss connections with other problems and present several algorithms in connection with the problem: an $O(n)$-time greedy algorithm, an $O(kn\log n)$-time dynamic programming algorithm, and an $O(n+ k \log n + 2^kk)$-time FPT algorithm for $n-k$ pieces.
A combinatorial problem concerning the maximum size of the (hamming) weight set of an $[n,k]_q$ linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those $[n,k]_q $ codes with the same weight set as $ \mathbb{F}_q^n $ are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed $ k,q $ the values of $ n $ for which an $ [n,k]_q $-FWS code exists are completely determined, but the determination of the minimum length $ M(H,k,q) $ of an $ [n,k]_q $-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on $ n $ for which an FWS code exists, and bounds on $ n $ for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on $ M(\mathcal{L},k,q) $ (the minimum length of Lee MWS codes), and pose the determination of $ M(\mathcal{L},k,q) $ as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.
A differentially private computation often begins with a bound on a $d$-dimensional statistic's $\ell_p$ sensitivity. The $K$-norm mechanism can yield more accurate additive noise by using a statistic-specific (and possibly non-$\ell_p$) norm. However, sampling such mechanisms requires sampling from the corresponding norm balls. These are $d$-dimensional convex polytopes, and the fastest known general algorithm for approximately sampling such polytopes takes time $\tilde O(d^{3+\omega})$, where $\omega \geq 2$ is the matrix multiplication exponent. For the simple problems of sum and ranked vote, this paper constructs samplers that run in time $\tilde O(d^2)$. More broadly, we suggest that problem-specific $K$-norm mechanisms may be an overlooked practical tool for private additive noise.
Relative entropy coding (REC) algorithms encode a sample from a target distribution $Q$ using a proposal distribution $P$ using as few bits as possible. Unlike entropy coding, REC does not assume discrete distributions or require quantisation. As such, it can be naturally integrated into communication pipelines such as learnt compression and differentially private federated learning. Unfortunately, despite their practical benefits, REC algorithms have not seen widespread application, due to their prohibitively slow runtimes or restrictive assumptions. In this paper, we make progress towards addressing these issues. We introduce Greedy Rejection Coding (GRC), which generalises the rejection based-algorithm of Harsha et al. (2007) to arbitrary probability spaces and partitioning schemes. We first show that GRC terminates almost surely and returns unbiased samples from $Q$, after which we focus on two of its variants: GRCS and GRCD. We show that for continuous $Q$ and $P$ over $\mathbb{R}$ with unimodal density ratio $dQ/dP$, the expected runtime of GRCS is upper bounded by $\beta D_{KL}[Q || P] + O(1)$ where $\beta \approx 4.82$, and its expected codelength is optimal. This makes GRCS the first REC algorithm with guaranteed optimal runtime for this class of distributions, up to the multiplicative constant $\beta$. This significantly improves upon the previous state-of-the-art method, A* coding (Flamich et al., 2022). Under the same assumptions, we experimentally observe and conjecture that the expected runtime and codelength of GRCD are upper bounded by $D_{KL}[Q || P] + O(1)$. Finally, we evaluate GRC in a variational autoencoder-based compression pipeline on MNIST, and show that a modified ELBO and an index-compression method can further improve compression efficiency.
Imagine a polygon-shaped platform $P$ and only one static spotlight outside $P$; which direction should the spotlight face to light most of $P$? This problem occurs in maximising the visibility, as well as in limiting the uncertainty in localisation problems. More formally, we define the following maximum cover problem: "Given a convex polygon $P$ and a Field Of View (FOV) with a given centre and inner angle $\phi$; find the direction (an angle of rotation $\theta$) of the FOV such that the intersection between the FOV and $P$ has the maximum area". In this paper, we provide the theoretical foundation for the analysis of the maximum cover with a rotating field of view. The main challenge is that the function of the area $A_{\phi}(\theta)$, with the angle of rotation $\theta$ and the fixed inner angle $\phi$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_{\theta}(\phi)$ (with a restricted inner angle $\phi$ and a fixed direction $\theta$). We show that $A_{\theta}(\phi)$ has an analytical solution in the special case of a two-sector intersection and later provides a constrictive solution for the original problem. Since the optimal solution is a real number, we develop an algorithm that approximates the direction of the field of view, with precision $\varepsilon$, and complexity $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.
The hitting set problem asks for a collection of sets over a universe $U$ to find a minimum subset of $U$ that intersects each of the given sets. It is NP-hard and equivalent to the problem set cover. We give a branch-and-bound algorithm to solve hitting set. Though it requires exponential time in the worst case, it can solve many practical instances from different domains in reasonable time. Our algorithm outperforms a modern ILP solver, the state-of-the-art for hitting set, by at least an order of magnitude on most instances.
We show that the permanent of an $n\times n$ matrix of $\operatorname{poly}(n)$-bit integers and the number of Hamiltonian cycles of an $n$-vertex graph can both be computed in time $2^{n-\Omega(\sqrt{n})}$, improving an earlier algorithm of Bj\"orklund, Kaski, and Williams (Algorithmica 2019) that runs in time $2^{n - \Omega\left(\sqrt{n/\log \log n}\right)}$. A key tool of our approach is to design a data structure that supports fast "$r$-order evaluation" of permanent and Hamiltonian cycles, which cooperates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022).
We present a novel objective function for cluster-based self-supervised learning (SSL) that is designed to circumvent the triad of failure modes, namely representation collapse, cluster collapse, and the problem of invariance to permutations of cluster assignments. This objective consists of three key components: (i) A generative term that penalizes representation collapse, (ii) a term that promotes invariance to data augmentations, thereby addressing the issue of label permutations and (ii) a uniformity term that penalizes cluster collapse. Additionally, our proposed objective possesses two notable advantages. Firstly, it can be interpreted from a Bayesian perspective as a lower bound on the data log-likelihood. Secondly, it enables the training of a standard backbone architecture without the need for asymmetric elements like stop gradients, momentum encoders, or specialized clustering layers. Due to its simplicity and theoretical foundation, our proposed objective is well-suited for optimization. Experiments on both toy and real world data demonstrate its effectiveness
We study the following two related problems. The first is to determine to what error an arbitrary zonoid in $\mathbb{R}^{d+1}$ can be approximated in the Hausdorff distance by a sum of $n$ line segments. The second is to determine optimal approximation rates in the uniform norm for shallow ReLU$^k$ neural networks on their variation spaces. The first of these problems has been solved for $d\neq 2,3$, but when $d=2,3$ a logarithmic gap between the best upper and lower bounds remains. We close this gap, which completes the solution in all dimensions. For the second problem, our techniques significantly improve upon existing approximation rates when $k\geq 1$, and enable uniform approximation of both the target function and its derivatives.
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