Matrix Graphs Algorithm
The Matrix Graphs Algorithm is an essential technique used in computer science and mathematics, specifically in graph theory and linear algebra, for representing and analyzing graphs. A graph is a collection of nodes (vertices) connected by edges (arcs), and the Matrix Graphs Algorithm uses matrices to represent the relationship between these nodes and edges. Two common matrix representations of graphs are the adjacency matrix and the incidence matrix. The adjacency matrix is a square matrix, with its dimensions equal to the number of vertices in the graph, and the matrix elements indicating if there is an edge between two vertices. The incidence matrix, on the other hand, is a rectangular matrix with dimensions equal to the number of vertices and edges in the graph, and the matrix elements representing the connection between a vertex and an edge.
The Matrix Graphs Algorithm plays a crucial role in various applications, such as network analysis, graph traversal, pathfinding, and optimization problems. By representing graphs as matrices, one can efficiently perform complex operations and algorithms on graphs, such as calculating shortest paths, detecting cycles, and identifying strongly connected components. Furthermore, matrix operations, such as matrix multiplication and inversion, can be utilized to analyze the graph's properties and derive valuable insights into the underlying structure of the graph. For instance, matrix exponentiation can be used to find the number of paths of a certain length between two vertices, while the determinant of the Laplacian matrix can be employed to compute the number of spanning trees in the graph. As a powerful tool in graph analysis, the Matrix Graphs Algorithm has found applications in various domains, including computer networks, social network analysis, transportation systems, and more.
package DataStructures.Graphs;
public class MatrixGraphs {
public static void main(String args[]) {
AdjacencyMatrixGraph graph = new AdjacencyMatrixGraph(10);
graph.addEdge(1, 2);
graph.addEdge(1, 5);
graph.addEdge(2, 5);
graph.addEdge(1, 2);
graph.addEdge(2, 3);
graph.addEdge(3, 4);
graph.addEdge(4, 1);
graph.addEdge(2, 3);
System.out.println(graph);
}
}
class AdjacencyMatrixGraph {
private int _numberOfVertices;
private int _numberOfEdges;
private int[][] _adjacency;
static final int EDGE_EXIST = 1;
static final int EDGE_NONE = 0;
public AdjacencyMatrixGraph(int givenNumberOfVertices) {
this.setNumberOfVertices(givenNumberOfVertices);
this.setNumberOfEdges(0);
this.setAdjacency(new int[givenNumberOfVertices][givenNumberOfVertices]);
for (int i = 0; i < givenNumberOfVertices; i++) {
for (int j = 0; j < givenNumberOfVertices; j++) {
this.adjacency()[i][j] = AdjacencyMatrixGraph.EDGE_NONE;
}
}
}
private void setNumberOfVertices(int newNumberOfVertices) {
this._numberOfVertices = newNumberOfVertices;
}
public int numberOfVertices() {
return this._numberOfVertices;
}
private void setNumberOfEdges(int newNumberOfEdges) {
this._numberOfEdges = newNumberOfEdges;
}
public int numberOfEdges() {
return this._numberOfEdges;
}
private void setAdjacency(int[][] newAdjacency) {
this._adjacency = newAdjacency;
}
private int[][] adjacency() {
return this._adjacency;
}
private boolean adjacencyOfEdgeDoesExist(int from, int to) {
return (this.adjacency()[from][to] != AdjacencyMatrixGraph.EDGE_NONE);
}
public boolean vertexDoesExist(int aVertex) {
if (aVertex >= 0 && aVertex < this.numberOfVertices()) {
return true;
} else {
return false;
}
}
public boolean edgeDoesExist(int from, int to) {
if (this.vertexDoesExist(from) && this.vertexDoesExist(to)) {
return (this.adjacencyOfEdgeDoesExist(from, to));
}
return false;
}
/**
* This method adds an edge to the graph between two specified
* vertices
*
* @param from the data of the vertex the edge is from
* @param to the data of the vertex the edge is going to
* @return returns true if the edge did not exist, return false if it already did
*/
public boolean addEdge(int from, int to) {
if (this.vertexDoesExist(from) && this.vertexDoesExist(to)) {
if (!this.adjacencyOfEdgeDoesExist(from, to)) {
this.adjacency()[from][to] = AdjacencyMatrixGraph.EDGE_EXIST;
this.adjacency()[to][from] = AdjacencyMatrixGraph.EDGE_EXIST;
this.setNumberOfEdges(this.numberOfEdges() + 1);
return true;
}
}
return false;
}
/**
* this method removes an edge from the graph between two specified
* vertices
*
* @param from the data of the vertex the edge is from
* @param to the data of the vertex the edge is going to
* @return returns false if the edge doesn't exist, returns true if the edge exists and is removed
*/
public boolean removeEdge(int from, int to) {
if (!this.vertexDoesExist(from) || !this.vertexDoesExist(to)) {
if (this.adjacencyOfEdgeDoesExist(from, to)) {
this.adjacency()[from][to] = AdjacencyMatrixGraph.EDGE_NONE;
this.adjacency()[to][from] = AdjacencyMatrixGraph.EDGE_NONE;
this.setNumberOfEdges(this.numberOfEdges() - 1);
return true;
}
}
return false;
}
/**
* this gives a list of vertices in the graph and their adjacencies
*
* @return returns a string describing this graph
*/
public String toString() {
String s = " ";
for (int i = 0; i < this.numberOfVertices(); i++) {
s = s + String.valueOf(i) + " ";
}
s = s + " \n";
for (int i = 0; i < this.numberOfVertices(); i++) {
s = s + String.valueOf(i) + " : ";
for (int j = 0; j < this.numberOfVertices(); j++) {
s = s + String.valueOf(this._adjacency[i][j]) + " ";
}
s = s + "\n";
}
return s;
}
}
