A dictionary data structure maintains a set of at most $n$ keys from the universe $[U]$ under key insertions and deletions, such that given a query $x \in [U]$, it returns if $x$ is in the set. Some variants also store values associated to the keys such that given a query $x$, the value associated to $x$ is returned when $x$ is in the set. This fundamental data structure problem has been studied for six decades since the introduction of hash tables in 1953. A hash table occupies $O(n\log U)$ bits of space with constant time per operation in expectation. There has been a vast literature on improving its time and space usage. The state-of-the-art dictionary by Bender, Farach-Colton, Kuszmaul, Kuszmaul and Liu [BFCK+22] has space consumption close to the information-theoretic optimum, using a total of \[ \log\binom{U}{n}+O(n\log^{(k)} n) \] bits, while supporting all operations in $O(k)$ time, for any parameter $k \leq \log^* n$. The term $O(\log^{(k)} n) = O(\underbrace{\log\cdots\log}_k n)$ is referred to as the wasted bits per key. In this paper, we prove a matching cell-probe lower bound: For $U=n^{1+\Theta(1)}$, any dictionary with $O(\log^{(k)} n)$ wasted bits per key must have expected operational time $\Omega(k)$, in the cell-probe model with word-size $w=\Theta(\log U)$. Furthermore, if a dictionary stores values of $\Theta(\log U)$ bits, we show that regardless of the query time, it must have $\Omega(k)$ expected update time. It is worth noting that this is the first cell-probe lower bound on the trade-off between space and update time for general data structures.
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One of the main tasks of actuaries and data scientists is to build good predictive models for certain phenomena such as the claim size or the number of claims in insurance. These models ideally exploit given feature information to enhance the accuracy of prediction. This user guide revisits and clarifies statistical techniques to assess the calibration or adequacy of a model on the one hand, and to compare and rank different models on the other hand. In doing so, it emphasises the importance of specifying the prediction target functional at hand a priori (e.g. the mean or a quantile) and of choosing the scoring function in model comparison in line with this target functional. Guidance for the practical choice of the scoring function is provided. Striving to bridge the gap between science and daily practice in application, it focuses mainly on the pedagogical presentation of existing results and of best practice. The results are accompanied and illustrated by two real data case studies on workers' compensation and customer churn.
Wireless communication is enabling billions of people to connect to each other and the internet, transforming every sector of the economy, and building the foundations for powerful new technologies that hold great promise to improve lives at an unprecedented rate and scale. The rapid increase in the number of devices and the associated demands for higher data rates and broader network coverage fuels the need for more robust wireless technologies. The key technology identified to address this problem is referred to as Cell-Free Massive MIMO (CF-mMIMO). CF-mMIMO is accompanied by many challenges, one of which is efficiently allocating limited resources. In this paper, we focus on a major resource allocation problem in wireless networks, namely the Pilot Assignment problem (PA). We show that PA is strongly NP-hard and that it does not admit a polynomial-time constant-factor approximation algorithm. Further, we show that PA cannot be approximated in polynomial time within $\mathcal{O}(K^2)$ (where $K$ is the number of users) when the system consists of at least three pilots. Finally, we present an approximation lower bound of $1.058$ (resp. $\epsilon|K|^2$, for $\epsilon >0$) in special cases where the system consists of exactly two (resp. three) pilots.
According to ICH Q8 guidelines, the biopharmaceutical manufacturer submits a design space (DS) definition as part of the regulatory approval application, in which case process parameter (PP) deviations within this space are not considered a change and do not trigger a regulatory post approval procedure. A DS can be described by non-linear PP ranges, i.e., the range of one PP conditioned on specific values of another. However, independent PP ranges (linear combinations) are often preferred in biopharmaceutical manufacturing due to their operation simplicity. While some statistical software supports the calculation of a DS comprised of linear combinations, such methods are generally based on discretizing the parameter space - an approach that scales poorly as the number of PPs increases. Here, we introduce a novel method for finding linear PP combinations using a numeric optimizer to calculate the largest design space within the parameter space that results in critical quality attribute (CQA) boundaries within acceptance criteria, predicted by a regression model. A precomputed approximation of tolerance intervals is used in inequality constraints to facilitate fast evaluations of this boundary using a single matrix multiplication. Correctness of the method was validated against different ground truths with known design spaces. Compared to stateof-the-art, grid-based approaches, the optimizer-based procedure is more accurate, generally yields a larger DS and enables the calculation in higher dimensions. Furthermore, a proposed weighting scheme can be used to favor certain PPs over others and therefore enabling a more dynamic approach to DS definition and exploration. The increased PP ranges of the larger DS provide greater operational flexibility for biopharmaceutical manufacturers.
The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.
We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. Since MMS allocations need not exist when $n>2$, a series of works showed the existence of approximate MMS allocations with the current best factor of $\frac34 + O(\frac{1}{n})$. However, a simple example in [DFL82, BEF21, AGST23] showed the limitations of existing approaches and proved that they cannot improve this factor to $3/4 + \Omega(1)$. In this paper, we bypass these barriers to show the existence of $(\frac{3}{4} + \frac{3}{3836})$-MMS allocations by developing new reduction rules and analysis techniques.
Non-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.
In this paper, necessary and sufficient conditions for the reversibility of a cyclic code of arbitrary length over a finite commutative chain ring have been derived. MDS reversible cyclic codes having length p^s over a finite chain ring with nilpotency index 2 have been characterized and a few examples of MDS reversible cyclic codes have been presented. Further, it is shown that the torsion codes of a reversible cyclic code over a finite chain ring are reversible. Also, an example of a non-reversible cyclic code for which all its torsion codes are reversible has been presented to show that the converse of this statement is not true. The cardinality and Hamming distance of a cyclic code over a finite commutative chain ring have also been determined.
The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.
While most theoretical run time analyses of discrete randomized search heuristics focused on finite search spaces, we consider the search space $\mathbb{Z}^n$. This is a further generalization of the search space of multi-valued decision variables $\{0,\ldots,r-1\}^n$. We consider as fitness functions the distance to the (unique) non-zero optimum $a$ (based on the $L_1$-metric) and the \ooea which mutates by applying a step-operator on each component that is determined to be varied. For changing by $\pm 1$, we show that the expected optimization time is $\Theta(n \cdot (|a|_{\infty} + \log(|a|_H)))$. In particular, the time is linear in the maximum value of the optimum $a$. Employing a different step operator which chooses a step size from a distribution so heavy-tailed that the expectation is infinite, we get an optimization time of $O(n \cdot \log^2 (|a|_1) \cdot \left(\log (\log (|a|_1))\right)^{1 + \epsilon})$. Furthermore, we show that RLS with step size adaptation achieves an optimization time of $\Theta(n \cdot \log(|a|_1))$. We conclude with an empirical analysis, comparing the above algorithms also with a variant of CMA-ES for discrete search spaces.
The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in $2^{(1-\epsilon)n}$ time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite the wide applicability of SETH, for many problems, there are no known SETH-based lower bounds, so the quest for new reductions continues. Two barriers for proving SETH-based lower bounds are known. Carmosino et al. (ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis (NSETH) stating that TAUT cannot be solved in time $2^{(1-\epsilon)n}$ even if one allows nondeterminism. They used this hypothesis to show that some natural fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM requires time $n^{1.5+\epsilon}$ under SETH, breaks NSETH and this, in turn, implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023) introduced the so-called polynomial formulations to show that for many NP-hard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds. We prove that for a range of problems from P, including $k$-SUM and triangle detection, proving superlinear lower bounds under SETH is challenging as it implies new circuit lower bounds. To this end, we show that these problems can be solved in nearly linear time with oracle calls to evaluating a polynomial of constant degree. Then, we introduce a strengthening of SETH stating that solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has constant degree polynomial evaluation oracle calls. This hypothesis is stronger and less believable than SETH, but refuting it is still challenging: we show that this implies circuit lower bounds.
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