一、定義
圖的搜索算法的目標(biāo)是：從某個指定源點(diǎn)開始，遍歷圖中其它頂點(diǎn)，并作相應(yīng)的處理。
API定義：
二、深度優(yōu)先搜索（DFS）
基本思想：
深度優(yōu)先搜索基于遞歸的思想：
- 首先以一個未被訪問過的頂點(diǎn)作為起始頂點(diǎn)；
- 沿當(dāng)前頂點(diǎn)的邊走到一個未被訪問過的頂點(diǎn)；
- 當(dāng)已經(jīng)沒有未被訪問過的頂點(diǎn)時，則回到上一個頂點(diǎn)，繼續(xù)試探訪問別的頂點(diǎn)，直到所有頂點(diǎn)都被訪問過。
源碼：
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示意圖
三、廣度優(yōu)先搜索（BFS）
基本思想：
廣度優(yōu)先搜索首先訪問與源點(diǎn)最近的頂點(diǎn)，然后一層層向外擴(kuò)展。
該算法用一個隊(duì)列保存所有已經(jīng)被標(biāo)記過但其鄰接表還未被檢查過的頂點(diǎn)：
- 將源點(diǎn)加入隊(duì)列，然后重復(fù)以下步驟直到隊(duì)列為空；
- 取出隊(duì)列中的下一個頂點(diǎn)v并標(biāo)記它；
- 將與v相鄰的所有未被標(biāo)記過的頂點(diǎn)加入隊(duì)列。
源碼：
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]);
BreadthFirstPaths bfs = new BreadthFirstPaths(G, s);
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示意圖
