In Java: // Exercise 4.1.16 package algs41; import stdlib.*; // This is problem
ID: 3920864 • Letter: I
Question
In Java:
// Exercise 4.1.16
package algs41;
import stdlib.*;
// This is problem 4.1.16 from the textbook
//
// You need only change the constructor.
// You can also change the main method.
// But you should not change eccentricity(), diameter(), radius(), center() or isCenter()
// You can (and should!) add more private methods (such as bfs or dfs)
//
// TODO: complete the following, using only Graph and GraphGenerator.
// You may copy code from other classes in algs41, but you should not use the classes themselves.
// In particular, you may not use BreadthFirstPaths although you may copy code from there.
//
// Definitions:
// - The "distance" from vertex v to vertex w is the length of the shortest path from v to w.
// - The "eccentricity" of a vertex v is distance to the farthest vertex from v.
// - The "diameter" of a graph is the maximum eccentricity of any vertex.
// - The "radius" of a graph is the smallest eccentricity of any vertex.
// - A "center" is a vertex whose eccentricity is the radius. The center is not necessarily unique.
public class MyGraphProperties {
int[] eccentricity;
int diameter;
int radius;
int center;
// Constructor can be linear in G.V() * G.E()
public MyGraphProperties(Graph G) {
this.eccentricity = new int[G.V()];
int diameter = Integer.MIN_VALUE;
int radius = Integer.MAX_VALUE;
int center = -1;
// If G.V()==0, then these are the correct values.
// If G is not connected, you should throw a new IllegalArgumentException()
// I suggest that you first get it to work for a connected graph
// This will require that you traverse the graph starting from some node (say 0)
// You can then adjust your code so that it throws an exception in the case that all nodes are not visited
// TODO
// compute the eccentricity, diameter, radius and center
// The center of the graph is not unique, set center to SOME center --- it does not matter which one
this.diameter = diameter;
this.radius = radius;
this.center = center;
}
// Do not change the following constant time methods
public int eccentricity(int v) { return eccentricity[v]; }
public int diameter() { return diameter; }
public int radius() { return radius; }
public int center() { return center; }
public boolean isCenter(int v) { return eccentricity[v] == radius; }
public static void main(String[] args) {
//Graph G = GraphGenerator.fromIn (new In("data/tinyG.txt")); // this is non-connected -- should throw an exception
//Graph G = GraphGenerator.connected (10, 20, 2); // Random non-connected graph -- should throw an exception
Graph G = GraphGenerator.fromIn (new In("data/tinyCG.txt")); // diameter=2, radius=2, every node is a center
//Graph G = GraphGenerator.binaryTree (10); // A complete binary tree
//Graph G = GraphGenerator.path (10); // A path -- diameter=V-1
//Graph G = GraphGenerator.connected (20, 400); // Random connected graph
StdOut.println(G);
G.toGraphviz ("g.png");
MyGraphProperties gp = new MyGraphProperties(G);
for (int v = 0; v < G.V(); v++)
StdOut.format ("eccentricity of %d: %d ", v, gp.eccentricity (v));
StdOut.format ("diameter=%d, radius=%d, center=%d ", gp.diameter(), gp.radius (), gp.center ());
}
}
Explanation / Answer
public class GraphGenerator {
private static final class Edge implements Comparable<Edge> {
private int v;
private int w;
private Edge(int v, int w) {
if (v < w) {
this.v = v;
this.w = w;
}
else {
this.v = w;
this.w = v;
}
}
public int compareTo(Edge that) {
if (this.v < that.v) return -1;
if (this.v > that.v) return +1;
if (this.w < that.w) return -1;
if (this.w > that.w) return +1;
return 0;
}
}
// this class cannot be instantiated
private GraphGenerator() { }
/**
* Returns a random simple graph containing {@code V} vertices and {@code E} edges.
* @param V the number of vertices
* @param E the number of vertices
* @return a random simple graph on {@code V} vertices, containing a total
* of {@code E} edges
* @throws IllegalArgumentException if no such simple graph exists
*/
public static Graph simple(int V, int E) {
if (E > (long) V*(V-1)/2) throw new IllegalArgumentException("Too many edges");
if (E < 0) throw new IllegalArgumentException("Too few edges");
Graph G = new Graph(V);
SET<Edge> set = new SET<Edge>();
while (G.E() < E) {
int v = StdRandom.uniform(V);
int w = StdRandom.uniform(V);
Edge e = new Edge(v, w);
if ((v != w) && !set.contains(e)) {
set.add(e);
G.addEdge(v, w);
}
}
return G;
}
/**
* Returns a random simple graph on {@code V} vertices, with an
* edge between any two vertices with probability {@code p}. This is sometimes
* referred to as the Erdos-Renyi random graph model.
* @param V the number of vertices
* @param p the probability of choosing an edge
* @return a random simple graph on {@code V} vertices, with an edge between
* any two vertices with probability {@code p}
* @throws IllegalArgumentException if probability is not between 0 and 1
*/
public static Graph simple(int V, double p) {
if (p < 0.0 || p > 1.0)
throw new IllegalArgumentException("Probability must be between 0 and 1");
Graph G = new Graph(V);
for (int v = 0; v < V; v++)
for (int w = v+1; w < V; w++)
if (StdRandom.bernoulli(p))
G.addEdge(v, w);
return G;
}
/**
* Returns the complete graph on {@code V} vertices.
* @param V the number of vertices
* @return the complete graph on {@code V} vertices
*/
public static Graph complete(int V) {
return simple(V, 1.0);
}
/**
* Returns a complete bipartite graph on {@code V1} and {@code V2} vertices.
* @param V1 the number of vertices in one partition
* @param V2 the number of vertices in the other partition
* @return a complete bipartite graph on {@code V1} and {@code V2} vertices
* @throws IllegalArgumentException if probability is not between 0 and 1
*/
public static Graph completeBipartite(int V1, int V2) {
return bipartite(V1, V2, V1*V2);
}
/**
* Returns a random simple bipartite graph on {@code V1} and {@code V2} vertices
* with {@code E} edges.
* @param V1 the number of vertices in one partition
* @param V2 the number of vertices in the other partition
* @param E the number of edges
* @return a random simple bipartite graph on {@code V1} and {@code V2} vertices,
* containing a total of {@code E} edges
* @throws IllegalArgumentException if no such simple bipartite graph exists
*/
public static Graph bipartite(int V1, int V2, int E) {
if (E > (long) V1*V2) throw new IllegalArgumentException("Too many edges");
if (E < 0) throw new IllegalArgumentException("Too few edges");
Graph G = new Graph(V1 + V2);
int[] vertices = new int[V1 + V2];
for (int i = 0; i < V1 + V2; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
SET<Edge> set = new SET<Edge>();
while (G.E() < E) {
int i = StdRandom.uniform(V1);
int j = V1 + StdRandom.uniform(V2);
Edge e = new Edge(vertices[i], vertices[j]);
if (!set.contains(e)) {
set.add(e);
G.addEdge(vertices[i], vertices[j]);
}
}
return G;
}
/**
* Returns a random simple bipartite graph on {@code V1} and {@code V2} vertices,
* containing each possible edge with probability {@code p}.
* @param V1 the number of vertices in one partition
* @param V2 the number of vertices in the other partition
* @param p the probability that the graph contains an edge with one endpoint in either side
* @return a random simple bipartite graph on {@code V1} and {@code V2} vertices,
* containing each possible edge with probability {@code p}
* @throws IllegalArgumentException if probability is not between 0 and 1
*/
public static Graph bipartite(int V1, int V2, double p) {
if (p < 0.0 || p > 1.0)
throw new IllegalArgumentException("Probability must be between 0 and 1");
int[] vertices = new int[V1 + V2];
for (int i = 0; i < V1 + V2; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
Graph G = new Graph(V1 + V2);
for (int i = 0; i < V1; i++)
for (int j = 0; j < V2; j++)
if (StdRandom.bernoulli(p))
G.addEdge(vertices[i], vertices[V1+j]);
return G;
}
/**
* Returns a path graph on {@code V} vertices.
* @param V the number of vertices in the path
* @return a path graph on {@code V} vertices
*/
public static Graph path(int V) {
Graph G = new Graph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
for (int i = 0; i < V-1; i++) {
G.addEdge(vertices[i], vertices[i+1]);
}
return G;
}
/**
* Returns a complete binary tree graph on {@code V} vertices.
* @param V the number of vertices in the binary tree
* @return a complete binary tree graph on {@code V} vertices
*/
public static Graph binaryTree(int V) {
Graph G = new Graph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
for (int i = 1; i < V; i++) {
G.addEdge(vertices[i], vertices[(i-1)/2]);
}
return G;
}
/**
* Returns a cycle graph on {@code V} vertices.
* @param V the number of vertices in the cycle
* @return a cycle graph on {@code V} vertices
*/
public static Graph cycle(int V) {
Graph G = new Graph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
for (int i = 0; i < V-1; i++) {
G.addEdge(vertices[i], vertices[i+1]);
}
G.addEdge(vertices[V-1], vertices[0]);
return G;
}
/**
* Returns an Eulerian cycle graph on {@code V} vertices.
*
* @param V the number of vertices in the cycle
* @param E the number of edges in the cycle
* @return a graph that is an Eulerian cycle on {@code V} vertices
* and {@code E} edges
* @throws IllegalArgumentException if either {@code V <= 0} or {@code E <= 0}
*/
public static Graph eulerianCycle(int V, int E) {
if (E <= 0)
throw new IllegalArgumentException("An Eulerian cycle must have at least one edge");
if (V <= 0)
throw new IllegalArgumentException("An Eulerian cycle must have at least one vertex");
Graph G = new Graph(V);
int[] vertices = new int[E];
for (int i = 0; i < E; i++)
vertices[i] = StdRandom.uniform(V);
for (int i = 0; i < E-1; i++) {
G.addEdge(vertices[i], vertices[i+1]);
}
G.addEdge(vertices[E-1], vertices[0]);
return G;
}
/**
* Returns an Eulerian path graph on {@code V} vertices.
*
* @param V the number of vertices in the path
* @param E the number of edges in the path
* @return a graph that is an Eulerian path on {@code V} vertices
* and {@code E} edges
* @throws IllegalArgumentException if either {@code V <= 0} or {@code E < 0}
*/
public static Graph eulerianPath(int V, int E) {
if (E < 0)
throw new IllegalArgumentException("negative number of edges");
if (V <= 0)
throw new IllegalArgumentException("An Eulerian path must have at least one vertex");
Graph G = new Graph(V);
int[] vertices = new int[E+1];
for (int i = 0; i < E+1; i++)
vertices[i] = StdRandom.uniform(V);
for (int i = 0; i < E; i++) {
G.addEdge(vertices[i], vertices[i+1]);
}
return G;
}
/**
* Returns a wheel graph on {@code V} vertices.
* @param V the number of vertices in the wheel
* @return a wheel graph on {@code V} vertices: a single vertex connected to
* every vertex in a cycle on {@code V-1} vertices
*/
public static Graph wheel(int V) {
if (V <= 1) throw new IllegalArgumentException("Number of vertices must be at least 2");
Graph G = new Graph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
// simple cycle on V-1 vertices
for (int i = 1; i < V-1; i++) {
G.addEdge(vertices[i], vertices[i+1]);
}
G.addEdge(vertices[V-1], vertices[1]);
// connect vertices[0] to every vertex on cycle
for (int i = 1; i < V; i++) {
G.addEdge(vertices[0], vertices[i]);
}
return G;
}
/**
* Returns a star graph on {@code V} vertices.
* @param V the number of vertices in the star
* @return a star graph on {@code V} vertices: a single vertex connected to
* every other vertex
*/
public static Graph star(int V) {
if (V <= 0) throw new IllegalArgumentException("Number of vertices must be at least 1");
Graph G = new Graph(V);
int[] vertices = new int[V];
for (int i = 0; i < V; i++)
vertices[i] = i;
StdRandom.shuffle(vertices);
// connect vertices[0] to every other vertex
for (int i = 1; i < V; i++) {
G.addEdge(vertices[0], vertices[i]);
}
return G;
}
/**
* Returns a uniformly random {@code k}-regular graph on {@code V} vertices
* (not necessarily simple). The graph is simple with probability only about e^(-k^2/4),
* which is tiny when k = 14.
*
* @param V the number of vertices in the graph
* @param k degree of each vertex
* @return a uniformly random {@code k}-regular graph on {@code V} vertices.
*/
public static Graph regular(int V, int k) {
if (V*k % 2 != 0) throw new IllegalArgumentException("Number of vertices * k must be even");
Graph G = new Graph(V);
// create k copies of each vertex
int[] vertices = new int[V*k];
for (int v = 0; v < V; v++) {
for (int j = 0; j < k; j++) {
vertices[v + V*j] = v;
}
}
// pick a random perfect matching
StdRandom.shuffle(vertices);
for (int i = 0; i < V*k/2; i++) {
G.addEdge(vertices[2*i], vertices[2*i + 1]);
}
return G;
}
// http://www.proofwiki.org/wiki/Labeled_Tree_from_Prüfer_Sequence
// http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.6484&rep=rep1&type=pdf
/**
* Returns a uniformly random tree on {@code V} vertices.
* This algorithm uses a Prufer sequence and takes time proportional to <em>V log V</em>.
* @param V the number of vertices in the tree
* @return a uniformly random tree on {@code V} vertices
*/
public static Graph tree(int V) {
Graph G = new Graph(V);
// special case
if (V == 1) return G;
// Cayley's theorem: there are V^(V-2) labeled trees on V vertices
// Prufer sequence: sequence of V-2 values between 0 and V-1
// Prufer's proof of Cayley's theorem: Prufer sequences are in 1-1
// with labeled trees on V vertices
int[] prufer = new int[V-2];
for (int i = 0; i < V-2; i++)
prufer[i] = StdRandom.uniform(V);
// degree of vertex v = 1 + number of times it appers in Prufer sequence
int[] degree = new int[V];
for (int v = 0; v < V; v++)
degree[v] = 1;
for (int i = 0; i < V-2; i++)
degree[prufer[i]]++;
// pq contains all vertices of degree 1
MinPQ<Integer> pq = new MinPQ<Integer>();
for (int v = 0; v < V; v++)
if (degree[v] == 1) pq.insert(v);
// repeatedly delMin() degree 1 vertex that has the minimum index
for (int i = 0; i < V-2; i++) {
int v = pq.delMin();
G.addEdge(v, prufer[i]);
degree[v]--;
degree[prufer[i]]--;
if (degree[prufer[i]] == 1) pq.insert(prufer[i]);
}
G.addEdge(pq.delMin(), pq.delMin());
return G;
}
/**
* Unit tests the {@code GraphGenerator} library.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
int V = Integer.parseInt(args[0]);
int E = Integer.parseInt(args[1]);
int V1 = V/2;
int V2 = V - V1;
StdOut.println("complete graph");
StdOut.println(complete(V));
StdOut.println();
StdOut.println("simple");
StdOut.println(simple(V, E));
StdOut.println();
StdOut.println("Erdos-Renyi");
double p = (double) E / (V*(V-1)/2.0);
StdOut.println(simple(V, p));
StdOut.println();
StdOut.println("complete bipartite");
StdOut.println(completeBipartite(V1, V2));
StdOut.println();
StdOut.println("bipartite");
StdOut.println(bipartite(V1, V2, E));
StdOut.println();
StdOut.println("Erdos Renyi bipartite");
double q = (double) E / (V1*V2);
StdOut.println(bipartite(V1, V2, q));
StdOut.println();
StdOut.println("path");
StdOut.println(path(V));
StdOut.println();
StdOut.println("cycle");
StdOut.println(cycle(V));
StdOut.println();
StdOut.println("binary tree");
StdOut.println(binaryTree(V));
StdOut.println();
StdOut.println("tree");
StdOut.println(tree(V));
StdOut.println();
StdOut.println("4-regular");
StdOut.println(regular(V, 4));
StdOut.println();
StdOut.println("star");
StdOut.println(star(V));
StdOut.println();
StdOut.println("wheel");
StdOut.println(wheel(V));
StdOut.println();
}
}
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