A self-loop is an edge that connects a vertex to itself.
Two edges are parallel if they connect the same ordered pair of vertices.
The outdegree of a vertex is the number of edges pointing from it.
The indegree of a vertex is the number of edges pointing to it.
A subgraph is a subset of a digraph's edges (and associated vertices) that constitutes a digraph.
A directed path in a digraph is a sequence of vertices in which there is a (directed) edge pointing from each vertex in the sequence to its successor in the sequence, with no repeated edges.
A directed path is simple if it has no repeated vertices.
A directed cycle is a directed path (with at least one edge) whose first and last vertices are the same.
A directed cycle is simple if it has no repeated vertices (other than the requisite repetition of the first and last vertices).
The length of a path or a cycle is its number of edges.
We say that a vertex w is reachable from a vertex v if there exists a directed path from v to w.
We say that two vertices v and w are strongly connected if they are mutually reachable: there is a directed path from v to w and a directed path from w to v.
A digraph is strongly connected if there is a directed path from every vertex to every other vertex.
A digraph that is not strongly connected consists of a set of strongly connected components, which are maximal strongly connected subgraphs.
A directed acyclic graph (or DAG) is a digraph with no directed cycles.
DiGraph: Set of vertices connect pairwise by directed edges
Adjacency List Representation
Java Implementation
public class diGraph {
private final int V;
private List<Integer>[] adj;
public diGraph(int v) {
this.V = v;
adj = new List[v];
for (int i = 0; i < V; i++) {
adj[i] = new ArrayList<>();
}
}
public void addEdge(int v, int w) {
adj[v].add(w);
}
public Iterable<Integer> adj(int v) {
return adj[v];
}
public static void main(String[] args) {}
}
Space Complexity :E+V
Add Edge: 1
Edge between vand w is outdegree(v)
Iterate over vertices adjacent to v - outdegree(v)
Depth First Search
Java implementation is same as undirected graph
Breadth First Search
Java implementation is same as undirected graph
Depth First Order
Depth-first search search visits each vertex exactly once. Three vertex orderings are of interest in typical applications:
Preorder: Put the vertex on a queue before the recursive calls.
Postorder: Put the vertex on a queue after the recursive calls.
Reverse postorder: Put the vertex on a stack after the recursive calls.
Java Implementation
/******************************************************************************
* Compilation: javac DepthFirstOrder.java
* Execution: java DepthFirstOrder digraph.txt
* Dependencies: Digraph.java Queue.java Stack.java StdOut.java
* EdgeWeightedDigraph.java DirectedEdge.java
* Data files: https://algs4.cs.princeton.edu/42digraph/tinyDAG.txt
* https://algs4.cs.princeton.edu/42digraph/tinyDG.txt
*
* Compute preorder and postorder for a digraph or edge-weighted digraph.
* Runs in O(E + V) time.
*
* % java DepthFirstOrder tinyDAG.txt
* v pre post
* --------------
* 0 0 8
* 1 3 2
* 2 9 10
* 3 10 9
* 4 2 0
* 5 1 1
* 6 4 7
* 7 11 11
* 8 12 12
* 9 5 6
* 10 8 5
* 11 6 4
* 12 7 3
* Preorder: 0 5 4 1 6 9 11 12 10 2 3 7 8
* Postorder: 4 5 1 12 11 10 9 6 0 3 2 7 8
* Reverse postorder: 8 7 2 3 0 6 9 10 11 12 1 5 4
*
******************************************************************************/
/**
* The {@code DepthFirstOrder} class represents a data type for
* determining depth-first search ordering of the vertices in a digraph
* or edge-weighted digraph, including preorder, postorder, and reverse postorder.
* <p>
* This implementation uses depth-first search.
* Each constructor takes Θ(<em>V</em> + <em>E</em>) time,
* where <em>V</em> is the number of vertices and <em>E</em> is the
* number of edges.
* Each instance method takes Θ(1) time.
* It uses Θ(<em>V</em>) extra space (not including the digraph).
* <p>
* For additional documentation,
* see <a href="https://algs4.cs.princeton.edu/42digraph">Section 4.2</a> of
* <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*/
public class DepthFirstOrder {
private boolean[] marked; // marked[v] = has v been marked in dfs?
private int[] pre; // pre[v] = preorder number of v
private int[] post; // post[v] = postorder number of v
private Queue<Integer> preorder; // vertices in preorder
private Queue<Integer> postorder; // vertices in postorder
private int preCounter; // counter or preorder numbering
private int postCounter; // counter for postorder numbering
/**
* Determines a depth-first order for the digraph {@code G}.
* @param G the digraph
*/
public DepthFirstOrder(Digraph G) {
pre = new int[G.V()];
post = new int[G.V()];
postorder = new Queue<Integer>();
preorder = new Queue<Integer>();
marked = new boolean[G.V()];
for (int v = 0; v < G.V(); v++)
if (!marked[v]) dfs(G, v);
assert check();
}
/**
* Determines a depth-first order for the edge-weighted digraph {@code G}.
* @param G the edge-weighted digraph
*/
public DepthFirstOrder(EdgeWeightedDigraph G) {
pre = new int[G.V()];
post = new int[G.V()];
postorder = new Queue<Integer>();
preorder = new Queue<Integer>();
marked = new boolean[G.V()];
for (int v = 0; v < G.V(); v++)
if (!marked[v]) dfs(G, v);
}
// run DFS in digraph G from vertex v and compute preorder/postorder
private void dfs(Digraph G, int v) {
marked[v] = true;
pre[v] = preCounter++;
preorder.enqueue(v);
for (int w : G.adj(v)) {
if (!marked[w]) {
dfs(G, w);
}
}
postorder.enqueue(v);
post[v] = postCounter++;
}
// run DFS in edge-weighted digraph G from vertex v and compute preorder/postorder
private void dfs(EdgeWeightedDigraph G, int v) {
marked[v] = true;
pre[v] = preCounter++;
preorder.enqueue(v);
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (!marked[w]) {
dfs(G, w);
}
}
postorder.enqueue(v);
post[v] = postCounter++;
}
/**
* Returns the preorder number of vertex {@code v}.
* @param v the vertex
* @return the preorder number of vertex {@code v}
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public int pre(int v) {
validateVertex(v);
return pre[v];
}
/**
* Returns the postorder number of vertex {@code v}.
* @param v the vertex
* @return the postorder number of vertex {@code v}
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public int post(int v) {
validateVertex(v);
return post[v];
}
/**
* Returns the vertices in postorder.
* @return the vertices in postorder, as an iterable of vertices
*/
public Iterable<Integer> post() {
return postorder;
}
/**
* Returns the vertices in preorder.
* @return the vertices in preorder, as an iterable of vertices
*/
public Iterable<Integer> pre() {
return preorder;
}
/**
* Returns the vertices in reverse postorder.
* @return the vertices in reverse postorder, as an iterable of vertices
*/
public Iterable<Integer> reversePost() {
Stack<Integer> reverse = new Stack<Integer>();
for (int v : postorder)
reverse.push(v);
return reverse;
}
// check that pre() and post() are consistent with pre(v) and post(v)
private boolean check() {
// check that post(v) is consistent with post()
int r = 0;
for (int v : post()) {
if (post(v) != r) {
StdOut.println("post(v) and post() inconsistent");
return false;
}
r++;
}
// check that pre(v) is consistent with pre()
r = 0;
for (int v : pre()) {
if (pre(v) != r) {
StdOut.println("pre(v) and pre() inconsistent");
return false;
}
r++;
}
return true;
}
// throw an IllegalArgumentException unless {@code 0 <= v < V}
private void validateVertex(int v) {
int V = marked.length;
if (v < 0 || v >= V)
throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
}
/**
* Unit tests the {@code DepthFirstOrder} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
Digraph G = new Digraph(in);
DepthFirstOrder dfs = new DepthFirstOrder(G);
StdOut.println(" v pre post");
StdOut.println("--------------");
for (int v = 0; v < G.V(); v++) {
StdOut.printf("%4d %4d %4d\n", v, dfs.pre(v), dfs.post(v));
}
StdOut.print("Preorder: ");
for (int v : dfs.pre()) {
StdOut.print(v + " ");
}
StdOut.println();
StdOut.print("Postorder: ");
for (int v : dfs.post()) {
StdOut.print(v + " ");
}
StdOut.println();
StdOut.print("Reverse postorder: ");
for (int v : dfs.reversePost()) {
StdOut.print(v + " ");
}
StdOut.println();
}
}
