Two strings $x$ and $y$ over $\Sigma \cup \Pi$ of equal length are said to \emph{parameterized match} (\emph{p-match}) if there is a renaming bijection $f:\Sigma \cup \Pi \rightarrow \Sigma \cup \Pi$ that is identity on $\Sigma$ and transforms $x$ to $y$ (or vice versa). The \emph{p-matching} problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose \emph{parameterized suffix automata} (\emph{p-suffix automata}) and \emph{parameterized directed acyclic word graphs} (\emph{PDAWGs}) which are the p-matching versions of suffix automata and DAWGs. While suffix automata and DAWGs are equivalent for standard strings, we show that p-suffix automata can have $\Theta(n^2)$ nodes and edges but PDAWGs have only $O(n)$ nodes and edges, where $n$ is the length of an input string. We also give an $O(n |\Pi| \log (|\Pi| + |\Sigma|))$-time $O(n)$-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the \emph{parameterized suffix tree} for the reversed string can also be built in the same time and space, in a right-to-left online manner. This duality also leads us to two further efficient algorithms for p-matching: Given the parameterized suffix tree for the reversal of the input string $T$, one can build the PDAWG of $T$ in $O(n)$ time in an offline manner; One can perform \emph{bidirectional} p-matching in $O(m \log (|\Pi|+|\Sigma|) + \mathit{occ})$ time using $O(n)$ space, where $m$ denotes the pattern length and $\mathit{occ}$ is the number of pattern occurrences in the text $T$.
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For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph $G$ with edge costs, a set $R$ of terminal vertices, and an integer demand $d_{s,t}$ for every terminal pair $s,t\in R$. The task is to compute a subgraph $H$ of $G$ of minimum cost, such that there are at least $d_{s,t}$ disjoint paths between $s$ and $t$ in $H$. If the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size $\ell$, the sum of demands $D$, the number of terminals $k$, and the maximum demand $d_\max$. Using simple, elegant arguments, we prove the following results. - We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter $\ell$: both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard. - We identify some special cases of VC-SNDP that are FPT: * when $d_\max\leq 3$ for parameter $\ell$, * on locally bounded treewidth graphs (e.g., planar graphs) for parameter $\ell$, and * on graphs of treewidth $tw$ for parameter $tw+D$. - The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with $d_\max=1$ on directed graphs, and is FPT parameterized by $k$ [Dreyfus & Wagner 1971]. We show that in contrast, the 2-DST problem, where $d_\max=2$, is W[1]-hard, even when parameterized by $\ell$.
Dijkstra's algorithm is one of the most popular classic path planning algorithms, achieving optimal solutions across a wide range of challenging tasks. However, it only calculates the shortest distance from one vertex to another, which is hard to directly apply to the Dynamic Multi-Sources to Single-Destination (DMS-SD) problem. This paper proposes a modified Dijkstra algorithm to address the DMS-SD problem, where the destination can be dynamically changed. Our method deploys the concept of Adjacent Matrix from Floyd's algorithm and achieves the goal with mathematical calculations. We formally show that all-pairs shortest distance information in Floyd's algorithm is not required in our algorithm. Extensive experiments verify the scalability and optimality of the proposed method.
We propose an efficient way of solving optimal control problems for rigid-body systems on the basis of inverse dynamics and the multiple-shooting method. We treat all variables, including the state, acceleration, and control input torques, as optimization variables and treat the inverse dynamics as an equality constraint. We eliminate the update of the control input torques from the linear equation of Newton's method by applying condensing for inverse dynamics. The size of the resultant linear equation is the same as that of the multiple-shooting method based on forward dynamics except for the variables related to the passive joints and contacts. Compared with the conventional methods based on forward dynamics, the proposed method reduces the computational cost of the dynamics and their sensitivities by utilizing the recursive Newton-Euler algorithm (RNEA) and its partial derivatives. In addition, it increases the sparsity of the Hessian of the Karush-Kuhn-Tucker conditions, which reduces the computational cost, e.g., of Riccati recursion. Numerical experiments show that the proposed method outperforms state-of-the-art implementations of differential dynamic programming based on forward dynamics in terms of computational time and numerical robustness.
We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect to a reference configuration. Despite its apparent simplicity, this problem exhibits a rich phenomenology. We show that different situations arise depending on the random ensemble of linear systems. When each variable is involved in at most two linear constraints, we show that the problem can be partially solved analytically, in particular we show that upon convergence, the zero-temperature limit of the cavity equations returns the optimal solution. We then study the geometrical properties of more general random ensembles. In particular we observe a range in the density of constraints at which the systems enters a glassy phase where the cost function has many minima. Interestingly, the algorithmic performances are only sensitive to another phase transition affecting the structure of configurations allowed by the linear constraints. We also extend our results to variables belonging to $\text{GF}(q)$, the Galois Field of order $q$. We show that increasing the value of $q$ allows to achieve a better optimum, which is confirmed by the Replica Symmetric cavity method predictions.
Parameter-efficient fine-tuning methods (PEFTs) offer the promise of adapting large pre-trained models while only tuning a small number of parameters. They have been shown to be competitive with full model fine-tuning for many downstream tasks. However, prior work indicates that PEFTs may not work as well for machine translation (MT), and there is no comprehensive study showing when PEFTs work for MT. We conduct a comprehensive empirical study of PEFTs for MT, considering (1) various parameter budgets, (2) a diverse set of language-pairs, and (3) different pre-trained models. We find that 'adapters', in which small feed-forward networks are added after every layer, are indeed on par with full model fine-tuning when the parameter budget corresponds to 10% of total model parameters. Nevertheless, as the number of tuned parameters decreases, the performance of PEFTs decreases. The magnitude of this decrease depends on the language pair, with PEFTs particularly struggling for distantly related language-pairs. We find that using PEFTs with a larger pre-trained model outperforms full fine-tuning with a smaller model, and for smaller training data sizes, PEFTs outperform full fine-tuning for the same pre-trained model.
We propose estimators based on kernel ridge regression for nonparametric causal functions such as dose, heterogeneous, and incremental response curves. Treatment and covariates may be discrete or continuous in general spaces. Due to a decomposition property specific to the RKHS, our estimators have simple closed form solutions. We prove uniform consistency with finite sample rates via original analysis of generalized kernel ridge regression. We extend our main results to counterfactual distributions and to causal functions identified by front and back door criteria. We achieve state-of-the-art performance in nonlinear simulations with many covariates, and conduct a policy evaluation of the US Job Corps training program for disadvantaged youths.
In a sentence, certain words are critical for its semantic. Among them, named entities (NEs) are notoriously challenging for neural models. Despite their importance, their accurate handling has been neglected in speech-to-text (S2T) translation research, and recent work has shown that S2T models perform poorly for locations and notably person names, whose spelling is challenging unless known in advance. In this work, we explore how to leverage dictionaries of NEs known to likely appear in a given context to improve S2T model outputs. Our experiments show that we can reliably detect NEs likely present in an utterance starting from S2T encoder outputs. Indeed, we demonstrate that the current detection quality is sufficient to improve NE accuracy in the translation with a 31% reduction in person name errors.
V. Levenshtein first proposed the sequence reconstruction problem in 2001. This problem studies the model where the same sequence from some set is transmitted over multiple channels, and the decoder receives the different outputs. Assume that the transmitted sequence is at distance $d$ from some code and there are at most $r$ errors in every channel. Then the sequence reconstruction problem is to find the minimum number of channels required to recover exactly the transmitted sequence that has to be greater than the maximum intersection between two metric balls of radius $r$, where the distance between their centers is at least $d$. In this paper, we study the sequence reconstruction problem of permutations under the Hamming distance. In this model, we define a Cayley graph and find the exact value of the largest intersection of two metric balls in this graph under the Hamming distance for $r=4$ with $d\geqslant 5$, and for $d=2r$.
Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.
In order to overcome the expressive limitations of graph neural networks (GNNs), we propose the first method that exploits vector flows over graphs to develop globally consistent directional and asymmetric aggregation functions. We show that our directional graph networks (DGNs) generalize convolutional neural networks (CNNs) when applied on a grid. Whereas recent theoretical works focus on understanding local neighbourhoods, local structures and local isomorphism with no global information flow, our novel theoretical framework allows directional convolutional kernels in any graph. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then we propose the use of the Laplacian eigenvectors as such vector field, and we show that the method generalizes CNNs on an n-dimensional grid, and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. Finally, we bring the power of CNN data augmentation to graphs by providing a means of doing reflection, rotation and distortion on the underlying directional field. We evaluate our method on different standard benchmarks and see a relative error reduction of 8\% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset. An important outcome of this work is that it enables to translate any physical or biological problems with intrinsic directional axes into a graph network formalism with an embedded directional field.
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