Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a textbook algorithm solves LCS in time $O(n^2)$, but although much effort has been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work indeed shows that such an algorithm would refute the Strong Exponential Time Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann, K\"unnemann FOCS'15]. Despite the quadratic-time barrier, for over 40 years an enduring scientific interest continued to produce fast algorithms for LCS and its variations. Particular attention was put into identifying and exploiting input parameters that yield strongly subquadratic time algorithms for special cases of interest, e.g., differential file comparison. This line of research was successfully pursued until 1990, at which time significant improvements came to a halt. In this paper, using the lens of fine-grained complexity, our goal is to (1) justify the lack of further improvements and (2) determine whether some special cases of LCS admit faster algorithms than currently known. To this end, we provide a systematic study of the multivariate complexity of LCS, taking into account all parameters previously discussed in the literature: the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string $m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as the numbers of matching pairs $M$ and dominant pairs $d$. For any class of instances defined by fixing each parameter individually to a polynomial in terms of the input size, we prove a SETH-based lower bound matching one of three known algorithms. Specifically, we determine the optimal running time for LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$. [...

Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-And-Solve

Can we analyze data without decompressing it? As our data keeps growing, understanding the time complexity of problems on compressed inputs, rather than in convenient uncompressed forms, becomes more and more relevant. Suppose we are given a compression of size $n$ of data that originally has size $N$, and we want to solve a problem with time complexity $T(\cdot)$. The naive strategy of "decompress-and-solve" gives time $T(N)$, whereas "the gold standard" is time $T(n)$: to analyze the compression as efficiently as if the original data was small. We restrict our attention to data in the form of a string (text, files, genomes, etc.) and study the most ubiquitous tasks. While the challenge might seem to depend heavily on the specific compression scheme, most methods of practical relevance (Lempel-Ziv-family, dictionary methods, and others) can be unified under the elegant notion of Grammar Compressions. A vast literature, across many disciplines, established this as an influential notion for Algorithm design. We introduce a framework for proving (conditional) lower bounds in this field, allowing us to assess whether decompress-and-solve can be improved, and by how much. Our main results are: - The $O(nN\sqrt{\log{N/n}})$ bound for LCS and the $O(\min\{N \log N, nM\})$ bound for Pattern Matching with Wildcards are optimal up to $N^{o(1)}$ factors, under the Strong Exponential Time Hypothesis. (Here, $M$ denotes the uncompressed length of the compressed pattern.) - Decompress-and-solve is essentially optimal for Context-Free Grammar Parsing and RNA Folding, under the $k$-Clique conjecture. - We give an algorithm showing that decompress-and-solve is not optimal for Disjointness

Fine-Grained Completeness for Optimization in P

We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime. Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the $k$-XOR problem. Specifically, we define MaxSP as the class of problems definable as $\max_{x_1,\dots,x_k} \#\{ (y_1,\dots,y_\ell) : \phi(x_1,\dots,x_k, y_1,\dots,y_\ell) \}$, where $\phi$ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On $m$-sized structures, we can solve each such problem in time $O(m^{k+\ell-1})$. Our results are: - We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under *deterministic* fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of $O(m^{k+\ell-1})$ for *all* problems in MaxSP/MinSP by a polynomial factor. - This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic $c$-approximation would give a $(c+\varepsilon)$-approximation for all MaxSP/MinSP problems in time $O(m^{k+\ell-1-\delta})$, where $\varepsilon > 0$ can be chosen arbitrarily small. Combining our completeness with~(Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is *equivalent* to giving a $O(1)$-approximation for all MinSP problems in faster-than-$O(m^{k+\ell-1})$ time

Faixas de classificação do coeficiente de variação para avaliação da precisão em experimentos com Brachiaria spp.

O coeficiente de variação (CV%) é uma medida de dispersão relativa utilizada, tradicionalmente, para a validação de experimentos. Por meio do método proposto por Garcia (1989) buscou-se novas faixas de classificação de CV% para a validação de experimentos das mais diversas espécies de interesse e variáveis-resposta nas ciências agrárias, porém este dependia da distribuição normal dos CV%s e limitando assim a disponibilidade de classificar uma dada variável-reposta. Posteriormente com o método de Costa et al. (2002) foi possível obter faixas de classificação de CV% de distribuições desconhecidas. Devido a estas pesquisas o valor de 20%, tradicionalmente, utilizado para a validação de experimentos tem sido substituído por outros valores de referência no CV%. As únicas faixas de classificação de CV% para gramíneas forrageiras existentes, anterior a este trabalho, são as de Clemente e Muniz (2002) baseadas em publicações entre os anos 1950 e 1990. De acordo com o exposto o objetivo foi propor novas faixas de classificação do CV% por variável-resposta para Brachiaria spp. Assim, tabulou-se os CV%s por variável resposta, unidade experimental, e espécie forrageira em planilhas. Selecionou-se as variáveis de maior frequência em artigos científicos indexados no Scielo, Directory Open Access Journals, Google Academics e Associação Brasileira de Zootecnia entre os anos 2000 a 2014. Testou-se a normalidade dos coeficientes de variação com os testes de Kolmogorov-Smirnov modificado por Lilliefors e de Shapiro-Wilk, ambos a 5% de significância. Utilizou-se os métodos de obtenção de faixas de classificação de Garcia (1989) para CV%s de distribuição normal, e o de Costa et al. (2002) para todos os outros com distribuição desconhecida. Para estes CVs% buscou-se validar as faixas obtidas pelo método de Costa el al. (2002) pelo teste de aderência do qui-quadrado corrigido pelo fator Yates e quando este não foi possível utilizou-se o teste exato de Fisher, ambos a 5% de significância. Conclui-se que as faixas de classificação do CV% para as variáveis teor de proteína bruta, teor de fibra em detergente neutro, teor de fibra em detergente ácido, consumo de matéria seca do pasto, consumo de matéria seca total, relação folha/colmo, consumo de matéria orgânica, consumo de matéria seca do pasto, consumo de matéria seca total, consumo de proteína bruta, consumo de fibra em detergente neutro, consumo de carboidratos não fibrosos, consumo de nutrientes digestíveis totais, tempo de pastejo, tempo de ruminação, e tempo de ócio obtidas neste trabalho podem ser recomendadas para o gênero Brachiaria spp

Remarks on hard Lefschetz conjectures on Chow groups

We propose two conjectures of Hard Lefschetz type on Chow groups and prove them for some special cases. For abelian varieties, we shall show they are equivalent to well-known conjectures of Beauville and Murre.Comment: to appear in Sciences in China, Ser. A Mathematic