You are given an undirected complete graph on n vertices. A complete graph is a graph where each pair of vertices is connected by an edge. You have to paint the edges of the graph into two colors, red and blue (each edge will have one color). A set of vertices S is red-connected if, for every pair of vertices (v1,v2) such that v1∈S and v2∈S, there exists a path from v1 to v2 that goes only through red edges and vertices from S. Similarly, a set of vertices S is blue-connected if, for every pair of vertices (v1,v2) such that v1∈S and v2∈S, there exists a path from v1 to v2 that goes only through blue edges and vertices from S. You have to paint the graph in such a way that: there is at least one red edge; there is at least one blue edge; for each set of vertices S such that |S|≥2, S is either red-connected or blue-connected, but not both. Calculate the number of ways to paint the graph, and print it modulo 998244353.

给定一个n个顶点的无向完全图。完全图是指每对顶点之间都有一条边相连的图。你需要将图中的边涂上红色和蓝色两种颜色（每条边只能有一种颜色）。一组顶点S是红色连通的，如果对于任意一对顶点（v1，v2），满足v1∈S且v2∈S，都存在一条从v1到v2的路径，该路径只经过红色边和S中的顶点。同样，一组顶点S是蓝色连通的，如果对于任意一对顶点（v1，v2），满足v1∈S且v2∈S，都存在一条从v1到v2的路径，该路径只经过蓝色边和S中的顶点。你需要涂色使得：至少有一条红色边，至少有一条蓝色边；对于每一组顶点S，满足|S|≥2，S要么是红色连通的，要么是蓝色连通的，但不能既是红色连通又是蓝色连通。请计算出涂色的方案数，并将其对 998244353 取模后输出。

相关问题

Compare adjacency matrix and adjacency list for graph representation. How you choose which presentation to use depending on task and graph?

Both adjacency matrix and adjacency list are used to represent graphs in computer science.

Adjacency Matrix:
- An adjacency matrix is a 2D array where the rows and columns represent vertices in a graph.
- If there is an edge between two vertices, then the corresponding cell in the matrix is marked with a 1, otherwise it is marked with a 0.
- For undirected graphs, the matrix is symmetric along the diagonal.

Adjacency List:
- An adjacency list is a collection of linked lists or arrays where each vertex has a list of its neighboring vertices.
- Each list contains all the vertices adjacent to a particular vertex.
- For directed graphs, the list is not necessarily symmetric.

Choosing between the two representations depends on the task and the graph. Here are some guidelines:

Adjacency Matrix:
- Good for dense graphs with many edges.
- Provides constant time access to edges.
- Takes up more space than adjacency lists, especially for sparse graphs.
- Slower for adding or removing nodes or edges.

Adjacency List:
- Good for sparse graphs with fewer edges.
- Takes less space than an adjacency matrix, especially for sparse graphs.
- Faster for adding or removing nodes or edges.
- Slower for accessing edges, as it requires traversing through the list.

In summary, if the graph is dense, use an adjacency matrix. If the graph is sparse, use an adjacency list. If the task involves frequently adding or removing nodes or edges, use an adjacency list. If the task involves frequently accessing edges, use an adjacency matrix.

Error in qgraph(mymatrix, layout = qgraph.layout.fruchtermanreingold) : Input is not a weights matrix or an edgelist.

这个错误通常是因为输入的数据格式不正确导致的。qgraph包要求输入的网络图数据必须是一个权重矩阵或者一份边列表。如果出现上述错误提示，那么有可能是以下原因导致的：

1.输入数据不是一个矩阵或者数据框

请确保输入的数据是一个正常的矩阵或者数据框格式，如果不是，可以尝试对数据进行转换。例如，如果输入数据是一个向量，可以使用以下代码将其转换为矩阵格式：

mymatrix <- matrix(mymatrix, nrow = length(mymatrix), ncol = length(mymatrix))

2.输入数据不是一个权重矩阵或者边列表

请确保输入的数据是一个符合qgraph包要求的权重矩阵或者边列表格式。如果不是，可以尝试对数据进行转换。例如，如果输入数据是一个邻接矩阵，可以使用以下代码将其转换为权重矩阵：

mymatrix <- as.matrix(mymatrix)
mymatrix[mymatrix != 0] <- 1

或者使用以下代码将其转换为边列表：

library(igraph)
g <- graph_from_adjacency_matrix(mymatrix, mode = "undirected")
mymatrix <- as.matrix(get.edgelist(g))

3.输入数据中存在缺失值或者非数值型数据

请确保输入的数据中不包含缺失值或者非数值型数据。如果存在，可以尝试对数据进行清洗或者填充。例如，可以使用以下代码将缺失值填充为0：

mymatrix[is.na(mymatrix)] <- 0

总之，当出现“Input is not a weights matrix or an edgelist”的错误提示时，需要认真检查输入数据的格式和内容，并尝试对数据进行转换和清洗。