## when to use adjacency matrix vs list

In the case of the adjacency matrix, we store 1 when there is an edge between two vertices else we store infinity. The adjacency matrix is a good way to represent a weighted graph. 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. Instead of a list of lists, it is a 2D matrix that maps the connections to nodes as seen in figure 4. 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. Up to O(v2) edges if fully connected. Implementation of DFS using adjacency matrix Depth First Search (DFS) has been discussed before as well which uses adjacency list for the graph representation. Fig 3: Adjacency Matrix . • Sparse graph: very few edges. Update matrix entry to contain the weight. • The adjacency matrix is a good way to represent a weighted graph. We will treat "self-ties" as … Adjacency List vs Adjacency Matrix. Usually easier to implement and perform lookup than an adjacency list. In a weighted graph, the edges Up to v2 edges if fully connected. In an adjacency list, each vertex is followed by a list, which contains only the n adjacent vertices. an adjacency list. This O(V)-space cost leads to fast (O(1)-time) searching of edges. 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 of an empty graph may be a zero matrix. Now in this section, the adjacency matrix will be used to represent the graph. Adjacency Matrix: Use this when you need to access the edge [math]a[i][j] [/math]as an [math]O(1)[/math] lookup often. Adjacency Matrix vs. Each Node in this Linked list represents the reference to the other vertices which share an edge with the current vertex. Adjacency List: Adjacency List is the Array[] of Linked List, where array size is same as number of Vertices in the graph. Every Vertex has a Linked List. In a sparse graph most entries will be 0 and waste a bunch of space. So what we can do is just store the edges from a given vertex as an array or list. If you notice, we are storing those infinity values unnecessarily, as they have no use for us. This space-efficient way leads to slow searching (O(n)). This is usually a space vs. time tradeoff. Now suppose that we multiply this adjacency matrix times itself (i.e. • Dense graph: lots of edges. raise the matrix to the 2nd power, or square it). 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. The weights can also be stored in the Linked List Node. Dense graph: lots of edges. Fig 4. • The matrix always uses Θ(v2) memory. List? An example of an adjacency matrix. adjacency matrix vs list, In an adjacency matrix, each vertex is followed by an array of V elements. Weights could indicate distance, cost, etc. An Adjacency matrix is just another way of representing a graph when using a graph algorithm. 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. 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. What I meant was that the vertex marking considered for the construction of the matrices is the same. Sparse graph: very few edges. 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