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Instead of a list of lists, it is a 2D matrix that maps the connections to nodes as seen in figure 4. In the adjacency matrix of an undirected graph, the value is considered to be 1 if there is an edge between two vertices, else it is 0. • Sparse graph: very few edges. Update matrix entry to contain the weight. Adjacency Matrix vs. List? Adjacency List vs Adjacency Matrix. We will treat "self-ties" as … raise the matrix to the 2nd power, or square it). In an adjacency list, each vertex is followed by a list, which contains only the n adjacent vertices. In the case of the adjacency matrix, we store 1 when there is an edge between two vertices else we store infinity. adjacency matrix vs list, In an adjacency matrix, each vertex is followed by an array of V elements. Fig 3: Adjacency Matrix . So what we can do is just store the edges from a given vertex as an array or list. In a weighted graph, the edges have weights associated with them. The Right Representation: List vs. Matrix There are two classic programmatic representations of a graph: adjacency lists and adjacency matrices. Fig 4. Sparse graph: very few edges. Adjacency Matrix: Use this when you need to access the edge [math]a[i][j] [/math]as an [math]O(1)[/math] lookup often. • Dense graph: lots of edges. an adjacency list. An Adjacency matrix is just another way of representing a graph when using a graph algorithm. In a sparse graph most entries will be 0 and waste a bunch of space. Now in this section, the adjacency matrix will be used to represent the graph. Adjacency List: Adjacency List is the Array[] of Linked List, where array size is same as number of Vertices in the graph. This is usually a space vs. time tradeoff. Now suppose that we multiply this adjacency matrix times itself (i.e. Up to O(v2) edges if fully connected. The weights can also be stored in the Linked List Node. Usually easier to implement and perform lookup than an adjacency list. This O(V)-space cost leads to fast (O(1)-time) searching of edges. An example of an adjacency matrix. Up to v2 edges if fully connected. The adjacency matrix, also called the connection matrix, is a matrix containing rows and columns which is used to represent a simple labelled graph, with 0 or 1 in the position of (V i , V j) according to the condition whether V i and V j are adjacent or not. The adjacency matrix is exactly what its name suggests -- it tells us which actors are adjacent, or have a direct path from one to the other. The adjacency matrix is a good way to represent a weighted graph. Dense graph: lots of edges. The adjacency matrix of an empty graph may be a zero matrix. To construct the incidence matrix we need to mark the vertices and edges, that is, $(x_1, x_1,\ldots, x_n)$ and $(u_1, u_2,\ldots, u_m)$ respectively. Implementation of DFS using adjacency matrix Depth First Search (DFS) has been discussed before as well which uses adjacency list for the graph representation. First of all you've understand that we use mostly adjacency list for simple algorithms, but remember adjacency matrix is also equally (or more) important. This space-efficient way leads to slow searching (O(n)). In a weighted graph, the edges If you notice, we are storing those infinity values unnecessarily, as they have no use for us. What I meant was that the vertex marking considered for the construction of the matrices is the same. Every Vertex has a Linked List. • The adjacency matrix is a good way to represent a weighted graph. Each Node in this Linked list represents the reference to the other vertices which share an edge with the current vertex. Weights could indicate distance, cost, etc. • The matrix always uses Θ(v2) memory. 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