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数学代写|图论代考GRAPH THEORY代写|MATH2069 Shortest Paths

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数学代写|图论代写GRAPH THEORY代考|Shortest Paths

The shortest way to travel between two locations is perhaps one of the oldest questions. As any mathematics student knows, the answer to this question is a line. But this relies on an $x y$-plane with no barriers to traveling in a straight line. What happens when you must restrict yourself to an existing structure, such as roadways or rail lines? This problem can be described in graph theoretic terms as the search for a shortest path on a weighted graph. Recall that a path is a sequence of vertices in which there is an edge between consecutive vertices and no vertex is repeated. As with the algorithms for the Traveling Salesman Problem, the weight associated to an edge may represent more than just distance (e.g., cost or time) and the shortest path really indicates the path of least total weight.

As with the previous two topics in this chapter, our study of shortest paths can be traced to a specific moment of time. In 1956 Edsger W. Dijkstra proposed the algorithm we are about to study not out of necessity for finding a shortest route, but rather as a demonstration of the power of a new “automatic computer” at the Mathematical Centre in Amsterdam. The goal was to have a question easily understood by a general audience while also allowing for audience participation in determining the inputs of the algorithm. In Dijkstra’s own words “the demonstration was a great success” [23]. Perhaps more surprising is how important this algorithm would become to modern societyalmost every GIS (Geographic Information System, or mapping software) uses a modification of Dijkstra’s Algorithm to provide directions. In addition, Dijkstra’s Algorithm provides the backbone of many routing systems and some studies in epidemiology.

Note, we will only investigate how to find a shortest path since determining if a shortest path exists is quickly answered by simply knowing if the graph is connected. The following section will consider implications of shortest paths.

数学代写|图论代写GRAPH THEORY代考|Dijkstra’s Algorithm

Numerous versions of Dijkstra’s Algorithm exist, though two basic descriptions adhere to Dijkstra’s original design (see [22]). In one, a shortest path from your chosen starting and ending vertex is found. Though useful in its own right, we will study the more general version that finds the shortest path from a specific vertex to all other vertices in the graph (since if we only cared for the shortest path from $a$ to $b$, we could halt the algorithm once $b$ is reached).
Dijkstra’s Algorithm is a bit more complex than the algorithms we have studied so far. Each vertex is given a two-part label $L(v)=(x,(w(v))$. The first portion of the label is the name of the vertex used to travel to $v$. The second part is the weight of the path that was used to get to $v$ from the designated starting vertex. At each stage of the algorithm, we will consider a set of free vertices, denoted by an $F$ below. Free vertices are the neighbors of previously visited vertices that are themselves not yet visited.

Dijkstra’s Algorithm
Input: Weighted connected simple graph $G=(V, E, w)$ and designated Start vertex.
Steps:

1. For each vertex $x$ of $G$, assign a label $L(x)$ so that $L(x)=(-, 0)$ if $x=$ Start and $L(x)=(-, \infty)$ otherwise. Highlight Start.
2. Let $u=$ Start and define $F$ to be the neighbors of $u$. Update the labels for each vertex $v$ in $F$ as follows:
if $w(u)+w(u v)<w(v)$, then redefine $L(v)=(u, w(u)+w(u v))$ otherwise do not change $L(v)$
3. Highlight the vertex with lowest weight as well as the edge $u v$ used to update the label. Redefine $u=v$.
4. Repeat Steps (2) and (3) until each vertex has been reached. In all future iterations, $F$ consists of the un-highlighted neighbors of all previously highlighted vertices and the labels are updated only for those vertices that are adjacent to the last vertex that was highlighted.
5. The shortest path from Start to any other vertex is found by tracing back using the first component of the labels. The total weight of the path is the weight given in the second component of the ending vertex.

数学代写|图论代写GRAPH THEORY代考|Shortest Paths

• 直线飞行没有障碍的飞机。当您必须将自己限制在现有结构（例如公路或铁路线）时会发生什么？这个问题可以用图论术语描述为在加权图上搜索最短路径。回想一下，路径是一系列顶点，其中连续顶点之间有一条边，并且没有重复的顶点。与旅行商问题的算法一样，与边关联的权重可能不仅仅代表距离（例如，成本或时间），而且最短路径实际上表示总权重最小的路径。

数学代写|图论代写GRAPH THEORY代考|Dijkstra’s Algorithm

Dijkstra 算法比我们目前研究的算法要复杂一些。每个顶点都有一个两部分标签 $L(v)=(x,(w(v))$. 标签的第 一部分是用于前往的顶点的名称 $v$. 第二部分是用于到达的路径的权重 $v$ 从指定的起始顶点。在算法的每个阶段， 我们都会考虑一组自由顶点，用 $F$ 以下。自由顶点是先前访问过的顶点的邻居，这些顶点本身还没有被访问 过。
Dijkstra 算法

1. 对于每个顶点 $x$ 的 $G$, 分配标签 $L(x)$ 以便 $L(x)=(-, 0)$ 如果 $x=$ 开始和 $L(x)=(-, \infty)$ 否则。突出显 示开始。
2. 让 $u=$ 开始和定义 $F$ 成为的邻居 $u$. 更新每个顶点的标签 $v$ 在 $F$ 如下:
如果 $w(u)+w(u v)<w(v)$ ，然后重新定义 $L(v)=(u, w(u)+w(u v))$ 否则不要改变 $L(v)$
3. 突出显示权重最低的顶点和边 $u v$ 用于更新标签。重新定义 $u=v$.
4. 重复步骙 (2) 和 (3)，直到到达每个顶点。在所有末来的迭代中， $F$ 由所有先前突出显示的顶点的末突 出显示的邻居组成，并且仅针对与最后突出显示的顶点相邻的那些顶点更新标签。
5. 从 Start 到任何其他顶点的最短路径是通过使用标签的第一个组件追溯找到的。路径的总权重是在结束 顶点的第二个分量中给出的权重。

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。