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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

An alternative way to manage the list of nodes to be visited is the last-in first-out. In this way, when a node is visited, all the adjacent nodes are added into the list. Then one of them is visited, and its adjacent nodes are inserted. At this point, the algorithm selects one of them and this process iterate until there exists at least one more adjacent node. It is evident that the algorithm for each step $i+1$ visits a node at distance $i+1$, and stores the node at distance i. This traversal, called depth-first search, uses a stack, as depicted in the algorithm (Fig. 3.4).

Similarly to BFS, the time complexity for DFS is $\mathcal{O}(|\mathcal{V}|+|\mathcal{E}|)$, where $|\mathcal{V}|$ and $|\mathcal{E}|$ are, respectively, the number of nodes and edges. The space needed for the execution is related to the storing of the stack; therefore it needs additional $\mathcal{O}(|\mathcal{V}|)$ space as well as the set of already-visited vertices.
Few of the well-known applications of DFS are:

• Finding connected and strongly connected components;
• Finding the bridges of a graph;
• Finding biconnectivity in graphs.

The followings hold in connection to graph traversal at a glance:

• BFS considers all of the adjacents of a node before visiting other nodes. DFS visits all the children nodes (i.e., neighbors of the adjacent node) before visiting neighbors.
• BFS and DFS are based on a similar algorithm, and they differ in that BFS uses a queue data structure, whereas DFS uses a stack.
• BFS needs of a larger amount of memory in general, because it expands all children of a vertex and stores them in memory. DFS has much lower memory requirements than BFS, because it iteratively expands only one child until it cannot proceed anymore and backtracks after that.
• Both have the same time complexity.
• BFS is used to find the shortest path in a graph, whereas DFS is used to find a path among nodes (i.e., determining if every pair of vertices can be connected by two disjoint paths).
• The choice between both algorithms depends on the context.

## 统计代写|网络分析代写Network Analysis代考|Shortest paths in a graph

One of the most interesting problems in a graph is finding the best path connecting two nodes. Best means the shortest (in terms of geodesic distance) for unweighted graphs, or a path such that the sum of the weights of its constituent edges is minimized for weighted graphs. The shortest path problem may be defined for both directed and undirected graphs. Here, we discuss paths in undirected graphs, and the reader may easily extend the discussion for directed ones [9].

In introducing the problem, we briefly recall from Chapter 2 that a path in a graph $\mathcal{G}$ is an ordered sequence of adjacent nodes $P=\left{v_1, v_2, \ldots, v_n\right}$ such that there always exists an edge connecting $v_j, v_{j+1}$ for all $j \in 1, n-1$. The path $P$ connects nodes $v_1$ and $v_n$. The number of nodes in $P$ is called the length of the path $P$. The ordering of nodes in $P$ determines the structure of the path. Given any pair of nodes of a graph, if there exists a path between them, then they are connected, and in general, two nodes may be connected by multiple paths of different length. Given an unweighted graph $\mathcal{G}$ and two nodes, $v$ and $w$, the shortest path $\mathcal{S P}$ is the path connecting $v$ and $w$ with the minimum length. Similarly, for weighted graphs, given a real-valued weight function $f: \mathcal{E} \rightarrow \mathbb{R}$, the shortest path from $v_1$ to $v_n$ is the path $P=\left(v_1, v_2, \ldots, v_n\right)$ that minimizes the sum $\sum_{i=1}^{n-1} f\left(e_{i, i+1}\right)$ over all possible paths. When each edge in the graph has unit weight or $f: \mathcal{E} \rightarrow{1}$, this is equivalent to finding the path with fewest edges.
The shortest path problem has three main variants:

• The single-source shortest path problem, in which the algorithms aim to find all the shortest paths from a given source node $v$ to all other vertices in the graph.
• The single-destination shortest path problem, in which algorithms aim to find shortest paths from all nodes in the directed graph to a single destination node $v$.
• The all-pairs shortest path problem, in which we have to find the shortest paths between every pair of nodes in the graph.
We point out that each one of these variants have been resolved by efficient algorithms than the naive approach of running a single-pair shortest path algorithm on all relevant pairs of vertices [10].

# 网络分析代考

## 统计代写|网络分析代写Network Analysis代考|Depth-first search

DFS 的一些著名应用包括：

• 寻找连接和强连接的组件；
• 寻找图的桥梁；
• 在图中找到双连通性。

• BFS 在访问其他节点之前会考虑一个节点的所有相邻节点。DFS 在访问邻居之前访问所有子节点（即相邻节点的邻居）。
• BFS 和 DFS 基于类似的算法，不同之处在于 BFS 使用队列数据结构，而 DFS 使用堆栈。
• BFS 通常需要大量内存，因为它扩展了一个顶点的所有子节点并将它们存储在内存中。DFS 的内存需求比 BFS 低得多，因为它迭代地只扩展一个子节点，直到它不能再继续并在此之后回溯。
• 两者具有相同的时间复杂度。
• BFS 用于在图中找到最短路径，而 DFS 用于在节点之间找到路径（即确定每对顶点是否可以通过两条不相交的路径连接）。
• 两种算法之间的选择取决于上下文。

## 统计代写|网络分析代写Network Analysis代考|Shortest paths in a graph

• 单源最短路径问题，其中算法旨在找到给定源节点的所有最短路径在到图中的所有其他顶点。
• 单目标最短路径问题，其中算法旨在找到从有向图中的所有节点到单个目标节点的最短路径在.
• 全对最短路径问题，我们必须找到图中每对节点之间的最短路径。
我们指出，这些变体中的每一个都已通过有效的算法解决，而不是在所有相关顶点对上运行单对最短路径算法的简单方法[10]。

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