Sliding suffix trees (Fiala & Greene, 1989) for an input text $T$ over an alphabet of size $\sigma$ and a sliding window $W$ of $T$ can be maintained in $O(|T| \log \sigma)$ time and $O(|W|)$ space. The two previous approaches that achieve this can be categorized into the credit-based approach of Fiala and Greene (1989) and Larsson (1996, 1999), or the batch-based approach proposed by Senft (2005). Brodnik and Jekovec (2018) showed that the sliding suffix tree can be supplemented with leaf pointers in order to find all occurrences of an online query pattern in the current window, and that leaf pointers can be maintained by credit-based arguments as well. The main difficulty in the credit-based approach is in the maintenance of index-pairs that represent each edge. In this paper, we show that valid edge index-pairs can be derived in constant time from leaf pointers, thus reducing the maintenance of edge index-pairs to the maintenance of leaf pointers. We further propose a new simple method which maintains leaf pointers without using credit-based arguments. Our algorithm and proof of correctness are much simpler compared to the credit-based approach, whose analyses were initially flawed (Senft 2005).
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We study local filters for the Lipschitz property of real-valued functions $f: V \to [0,r]$, where the Lipschitz property is defined with respect to an arbitrary undirected graph $G=(V,E)$. We give nearly optimal local Lipschitz filters both with respect to $\ell_1$ distance and $\ell_0$ distance. Previous work only considered unbounded-range functions over $[n]^d$. Jha and Raskhodnikova (SICOMP `13) gave an algorithm for such functions with lookup complexity exponential in $d$, which Awasthi et al.\ (ACM Trans. Comput. Theory) showed was necessary in this setting. By considering the natural class of functions whose range is bounded in $[0,r]$, we circumvent this lower bound and achieve running time $(d^r\log n)^{O(\log r)}$ for the $\ell_1$-respecting filter and $d^{O(r)}\text{polylog }n$ for the $\ell_0$-respecting filter for functions over $[n]^d$. Furthermore, we show that our algorithms are nearly optimal in terms of the dependence on $r$ for the domain $\{0,1\}^d$, an important special case of the domain $[n]^d$. In addition, our lower bound resolves an open question of Awasthi et al., removing one of the conditions necessary for their lower bound for general range. We prove our lower bound via a reduction from distribution-free Lipschitz testing. Finally, we provide two applications of our local filters. First, they can be used in conjunction with the Laplace mechanism for differential privacy to provide filter mechanisms for privately releasing outputs of black box functions even in the presence of malicious clients. Second, we use them to obtain the first tolerant testers for the Lipschitz property.
A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix $A$ of dimension $n$. The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by $O(n^3 (\log n + \log ||A||)^2(\log n)^2)$ bit operations, where $||A||= \max_{ij} |A_{ij}|$ denotes the largest entry of $A$ in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time $(n^3 \log ||A||)^{1+o(1)}$ bit operations, where the exponent $"+o(1)"$ captures additional factors $c_1 (\log n)^{c_2} (\log \log ||A||)^{c_3}$ for positive real constants $c_1,c_2,c_3$.
A storage code is an assignment of symbols to the vertices of a connected graph $G(V,E)$ with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in $G$. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on $\mathbb Z$ and ${\mathbb Z}^2$ (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.
Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies.
Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus.
Let $t \in \{2,8,10,12,14,16,18\}$ and $n=31s+t\geq 14$, $d_{a}(n,5)$ and $d_{l}(n,5)$ be distances of binary $[n,5]$ optimal linear codes and optimal linear complementary dual (LCD) codes, respectively. We show that an $[n,5,d_{a}(n,5)]$ optimal linear code is not an LCD code, there is an $[n,5,d_{l}(n,5)]=[n,5,d_{a}(n,5)-1]$ optimal LCD code if $t\neq 16$, and an optimal $[n,5,d_{l}(n,5)]$ optimal LCD code has $d_{l}(n,5)=16s+6=d_{a}(n,5)-2$ for $t=16$. Combined with known results on optimal LCD code, $d_{l}(n,5)$ of all $[n,5]$ LCD codes are completely determined.
We study $L_2$-approximation problems in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameters $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the strongly polynomial tractability (SPT), polynomial tractability (PT), and $(t_1,t_2)$-weak tractability ($(t_1,t_2)$-WT) for all $t_1>1$ and $t_2>0$ under the absolute or normalized error criterion. In particular, we obtain the matching necessary and sufficient condition for SPT or PT in terms of the parameters.
A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.
Partite, $3$-uniform hypergraphs are $3$-uniform hypergraphs in which each hyperedge contains exactly one point from each of the $3$ disjoint vertex classes. We consider the degree sequence problem of partite, $3$-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is $k$ or $k-1$ for some fixed $k$, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, $3$-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for $\chi^2$ testing of contingency tables. We have shown that this hypergraph-based $\chi^2$ test is more sensitive than the standard $\chi^2$ test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.
We present new Dirichlet-Neumann and Neumann-Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, three variants can be found for the Dirichlet-Neumann and Neumann-Dirichlet algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
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