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数学代写|图论作业代写Graph Theory代考|Online colouring

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数学代写|图论作业代写Graph Theory代考|Competitive analysis

The performance of an online colouring algorithm is typically measured using worstcase analysis. More precisely, for an online algorithm $A$ and a given graph $G$, we are interested in how well $A$ does with the worst possible ordering of the vertices. In other words, let $A(G, \sigma)$ be the number of colours that $A$ uses to colour $G$ when the vertices are revealed in the order $\sigma$, and let
$$A(G)=\max A(G, \sigma)$$

数学代写|图论作业代写Graph Theory代考|Online competitive analysis

So far, we have considered the standard competitive analysis, where the performance of an online algorithm is compared with the best existing colouring, which can be obtained offline. There is however another type of analysis, called online competitive analysis, where the performance of an online algorithm is compared with the best possible performance that an online algorithm can achieve. The online chromatic number $\chi^{o}(G)$ of a graph $G$ is defined as the smallest number $k$ for which there is an online algorithm that can colour $G$ with $k$ colours, for any incoming ordering of the vertices.
The above definition implies that
$$\chi^{o}(G)=\inf {A} A(G)$$ where the infimum is taken over all online colouring algorithms $A$; for example, we have observed that $\chi^{o}\left(P{4}\right)=3$. When viewing online colouring as a two-person game, we see that the online chromatic number is exactly the number of colours that are used if both players (the online algorithm and the spoiler) play optimally.

数学代写|图论作业代写GRAPH THEORY代考|Maximum k-colourable subgraphs

The maximum k-colourable subgraph problem consists in colouring as many vertices of a given graph as possible with at most $k$ colours. The number of vertices in such a maximum $k$-colourable subgraph is denoted by $\alpha_{k}(G)$; for k=1, this is equivalent to determining a maximum stable set. Also, the chromatic number of G is the smallest integer $k$ for which the maximum $k$-colourable subgraph of G is G itself.

Here also we can consider an online version of the problem, the difference from the previous online colouring problem being that only $k$ colours are available, and we can decide (or be forced) not to colour a revealed vertex. Also, the objective does not consist in using as few colours as possible (since all $k$ colours are available), but rather to colour as many vertices as we can.

Let $n_{A}(G, \sigma, k)$ be the number of vertices coloured by $A$, when there are $k$ available colours and the vertices of $G$ are revealed in the order $\sigma$. Also, let
$$n_{A}(G, k)=\min {\sigma} n{A}(G, \sigma, k) .$$

数学代写|图论作业代写GRAPH THEORY代考|ONLINE COMPETITIVE ANALYSIS

$$\chi^{o}G= \ inf {A} AG$$ 其中下确界被所有在线着色算法取代一种; 例如，我们观察到$\chi^{o}\left(P {4}\right)=3$。当将在线填色视为两人游戏时，我们看到在线色数正好是两个玩家使用的颜色数吨H和这nl一世n和一种lG这r一世吨H米一种nd吨H和sp这一世l和r发挥最佳。

数学代写|图论作业代写GRAPH THEORY代考|MAXIMUM K-COLOURABLE SUBGRAPHS

$$n_{A}G,ķ=\min {\sigma} n {A}G,σ,ķ .$$

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。