# 图的搜索（遍历）

API定义：

### 二、深度优先搜索（DFS）

1. 首先以一个未被访问过的顶点作为起始顶点；
2. 沿当前顶点的边走到一个未被访问过的顶点；
3. 当已经没有未被访问过的顶点时，则回到上一个顶点，继续试探访问别的顶点，直到所有顶点都被访问过。

``````public class DepthFirstSearch {
private boolean[] marked;    // marked[v] = is there an s-v path?
private int count;           // number of vertices connected to s

/**
* Computes the vertices in graph {@code G} that are
* connected to the source vertex {@code s}.
* @param G the graph
* @param s the source vertex
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public DepthFirstSearch(Graph G, int s) {
marked = new boolean[G.V()];
validateVertex(s);
dfs(G, s);
}

// depth first search from v
private void dfs(Graph G, int v) {
count++;
marked[v] = true;
for (int w : G.adj(v)) {
if (!marked[w]) {
dfs(G, w);
}
}
}

/**
* Is there a path between the source vertex {@code s} and vertex {@code v}?
* @param v the vertex
* @return {@code true} if there is a path, {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public boolean marked(int v) {
validateVertex(v);
return marked[v];
}

/**
* Returns the number of vertices connected to the source vertex {@code s}.
* @return the number of vertices connected to the source vertex {@code s}
*/
public int count() {
return count;
}

// 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 DepthFirstSearch} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
Graph G = new Graph(in);
int s = Integer.parseInt(args[1]);
DepthFirstSearch search = new DepthFirstSearch(G, s);
for (int v = 0; v < G.V(); v++) {
if (search.marked(v))
StdOut.print(v + " ");
}

StdOut.println();
if (search.count() != G.V()) StdOut.println("NOT connected");
else                         StdOut.println("connected");
}
}
``````
2-1 DFS示意图

### 三、广度优先搜索（BFS）

1. 将源点加入队列，然后重复以下步骤直到队列为空；
2. 取出队列中的下一个顶点v并标记它；
3. 将与v相邻的所有未被标记过的顶点加入队列。

``````public class BreadthFirstPaths {
private static final int INFINITY = Integer.MAX_VALUE;
private boolean[] marked;  // marked[v] = is there an s-v path
private int[] edgeTo;      // edgeTo[v] = previous edge on shortest s-v path
private int[] distTo;      // distTo[v] = number of edges shortest s-v path

/**
* Computes the shortest path between the source vertex {@code s}
* and every other vertex in the graph {@code G}.
* @param G the graph
* @param s the source vertex
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public BreadthFirstPaths(Graph G, int s) {
marked = new boolean[G.V()];
distTo = new int[G.V()];
edgeTo = new int[G.V()];
validateVertex(s);
bfs(G, s);

assert check(G, s);
}

/**
* Computes the shortest path between any one of the source vertices in {@code sources}
* and every other vertex in graph {@code G}.
* @param G the graph
* @param sources the source vertices
* @throws IllegalArgumentException unless {@code 0 <= s < V} for each vertex
*         {@code s} in {@code sources}
*/
public BreadthFirstPaths(Graph G, Iterable<Integer> sources) {
marked = new boolean[G.V()];
distTo = new int[G.V()];
edgeTo = new int[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = INFINITY;
validateVertices(sources);
bfs(G, sources);
}

// breadth-first search from a single source
private void bfs(Graph G, int s) {
Queue<Integer> q = new Queue<Integer>();
for (int v = 0; v < G.V(); v++)
distTo[v] = INFINITY;
distTo[s] = 0;
marked[s] = true;
q.enqueue(s);

while (!q.isEmpty()) {
int v = q.dequeue();
for (int w : G.adj(v)) {
if (!marked[w]) {
edgeTo[w] = v;
distTo[w] = distTo[v] + 1;
marked[w] = true;
q.enqueue(w);
}
}
}
}

// breadth-first search from multiple sources
private void bfs(Graph G, Iterable<Integer> sources) {
Queue<Integer> q = new Queue<Integer>();
for (int s : sources) {
marked[s] = true;
distTo[s] = 0;
q.enqueue(s);
}
while (!q.isEmpty()) {
int v = q.dequeue();
for (int w : G.adj(v)) {
if (!marked[w]) {
edgeTo[w] = v;
distTo[w] = distTo[v] + 1;
marked[w] = true;
q.enqueue(w);
}
}
}
}

/**
* Is there a path between the source vertex {@code s} (or sources) and vertex {@code v}?
* @param v the vertex
* @return {@code true} if there is a path, and {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public boolean hasPathTo(int v) {
validateVertex(v);
return marked[v];
}

/**
* Returns the number of edges in a shortest path between the source vertex {@code s}
* (or sources) and vertex {@code v}?
* @param v the vertex
* @return the number of edges in a shortest path
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public int distTo(int v) {
validateVertex(v);
return distTo[v];
}

/**
* Returns a shortest path between the source vertex {@code s} (or sources)
* and {@code v}, or {@code null} if no such path.
* @param  v the vertex
* @return the sequence of vertices on a shortest path, as an Iterable
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public Iterable<Integer> pathTo(int v) {
validateVertex(v);
if (!hasPathTo(v)) return null;
Stack<Integer> path = new Stack<Integer>();
int x;
for (x = v; distTo[x] != 0; x = edgeTo[x])
path.push(x);
path.push(x);
return path;
}

// check optimality conditions for single source
private boolean check(Graph G, int s) {
// check that the distance of s = 0
if (distTo[s] != 0) {
StdOut.println("distance of source " + s + " to itself = " + distTo[s]);
return false;
}
// check that for each edge v-w dist[w] <= dist[v] + 1
// provided v is reachable from s
for (int v = 0; v < G.V(); v++) {
for (int w : G.adj(v)) {
if (hasPathTo(v) != hasPathTo(w)) {
StdOut.println("edge " + v + "-" + w);
StdOut.println("hasPathTo(" + v + ") = " + hasPathTo(v));
StdOut.println("hasPathTo(" + w + ") = " + hasPathTo(w));
return false;
}
if (hasPathTo(v) && (distTo[w] > distTo[v] + 1)) {
StdOut.println("edge " + v + "-" + w);
StdOut.println("distTo[" + v + "] = " + distTo[v]);
StdOut.println("distTo[" + w + "] = " + distTo[w]);
return false;
}
}
}

// check that v = edgeTo[w] satisfies distTo[w] = distTo[v] + 1
// provided v is reachable from s
for (int w = 0; w < G.V(); w++) {
if (!hasPathTo(w) || w == s) continue;
int v = edgeTo[w];
if (distTo[w] != distTo[v] + 1) {
StdOut.println("shortest path edge " + v + "-" + w);
StdOut.println("distTo[" + v + "] = " + distTo[v]);
StdOut.println("distTo[" + w + "] = " + distTo[w]);
return false;
}
}
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));
}

// throw an IllegalArgumentException unless {@code 0 <= v < V}
private void validateVertices(Iterable<Integer> vertices) {
if (vertices == null) {
throw new IllegalArgumentException("argument is null");
}
int V = marked.length;
for (int v : vertices) {
if (v < 0 || v >= V) {
throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
}
}
}

/**
* Unit tests the {@code BreadthFirstPaths} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
Graph G = new Graph(in);
// StdOut.println(G);

int s = Integer.parseInt(args[1]);
for (int v = 0; v < G.V(); v++) {
if (bfs.hasPathTo(v)) {
StdOut.printf("%d to %d (%d):  ", s, v, bfs.distTo(v));
for (int x : bfs.pathTo(v)) {
if (x == s) StdOut.print(x);
else        StdOut.print("-" + x);
}
StdOut.println();
}else {
StdOut.printf("%d to %d (-):  not connected\n", s, v);
}
}
}
}
``````
3-1 BFS示意图