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# 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|CSCIE121 Fixed-parameter algorithms

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## 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|The MIN-CC problem is fixed-parameter tractable

We only sketch the fixed-parameter tractability result. Let $G$ be a graph and $k$ be a positive integer. Recall that a family $\mathcal{F}$ of functions from $\mathbf{V}(G)$ to ${1,2, \ldots, k}$ is perfect if for any subset $V \subseteq \mathbf{V}(G)$ of $k$ vertices there is a function $f \in \mathcal{F}$ which is injective on $V[1]$. Let $(G, \mathcal{M})$ be an instance of the MıN-CC problem, where $\mathcal{M}$ is a motif of size $k$. Then there is an occurrence of $\mathcal{M}$ in $G$, say $V \subseteq \mathbf{V}(G)$, that results in a minimum number of connected components. Furthermore, suppose we are provided with a perfect family $\mathcal{F}$ of functions from $\mathbf{V}(G)$ to ${1,2, \ldots, k}$. Since $\mathcal{F}$ is perfect, we are guaranteed that at least one function in $\mathcal{F}$ assigns $V$ with $k$ distinct labels. Let $f \in \mathcal{F}$ be such a function. We now turn to defining a dynamic programming table $T$ indexed by vertices of $G$ and subsets of ${1,2, \ldots, k}$. For any $v \in \mathbf{V}(G)$ and any $L \subseteq{1,2, \ldots, k}$, we define $T_{L}[v]$ to be the family of all motifs $\mathcal{M}^{\prime} \subseteq \mathcal{M},\left|\mathcal{M}^{\prime}\right|=|L|$, for which there exists an exact occurrence of $\mathcal{M}^{\prime}$ in $G$, say $V$, such that $v \in V$ and the set of (unique) labels that $f$ assigns to $V$ is exactly $L$. We need the following lemma [8].

## 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|A faster fixed-parameter algorithm for trees

We proved in Section 3 that the MıN-CC problem is APX-hard even if the target graph is a path. To complement Proposition 4.1, we give here a dynamic programming algorithm for trees that does not rely on the color-coding technique (approaches based on the color-coding technique usually suffer from bad running time performances).

Let $(G, \mathcal{M})$ be an instance of the MIN-CC problem for trees where both $G$ and $\mathcal{M}$ are built upon a set of colors $\mathcal{C}$. Let $k=|\mathcal{M}|$ and $q=|\mathcal{C}|$. Furthermore, for ease of exposition, write $\mathbf{V}(G)={1,2, \ldots, n}$ and assume $G$ is rooted at some arbitrary vertex $r(G)$.

Our dynamic programming algorithm is basically an exhaustive search procedure. The basic idea is to store – in a bottom-up fashion – for each vertex $i$ of $G$ and each submotif $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ that occurs in $T(i)$, i.e., the subtree rooted at $i$, the minimum number of connected components that results in an occurrence of $\mathcal{M}^{\prime}$ in $T(i)$. More precisely, for each vertex $i$ of $G$, we compute two dynamic programming tables $X[i]$ and $Y[i]$. The dynamic programming table $X[i]$ stores all pairs $\left(\mathcal{M}^{\prime}, c\right)$, where $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ is a submotif and $c$ is a positive integer, such that (1) there exists an occurrence of $\mathcal{M}^{\prime}$ in $T(i)$ that matches vertex $i,(2)$ the minimum number of connected components of an occurrence of $\mathcal{M}^{\prime}$ in $T(i)$ that matches vertex $i$ is $c$. The dynamic programming table $Y[i]$ stores all pairs $\left(\mathcal{M}^{\prime}, c\right)$, where $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ is a submotif and $c$ is a positive integer, such that (1′) there exists an occurrence of $\mathcal{M}^{\prime}$ in $T(i)$ that does not match vertex $i,\left(2^{\prime}\right)$ the minimum number of connected components of an occurrence of $\mathcal{M}^{\prime}$ in $T(i)$ that does not match vertex $i$ is $c$.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。