十八、图算法之最短路径
最短路径
这里基于的是加权有向图的讨论，而且是单源最短路径问题。我们称为最短路径的结果是一棵最短路径树。
给定一幅加权有向图和一个顶点s，以s为起点的一棵最短路径树是图的一幅子图，它包含s和从s可达的所有顶点。这棵有向树的根节点为s，树的每条路径（到每个点的每条路径）都是有向图中的一条最短路径。
加权有向图的数据结构
有向边类：
public class EdgeWeightedDigraph {
private static final String NEWLINE = System.getProperty("line.separator");
private final int V; // number of vertices in this digraph
private int E; // number of edges in this digraph
private Bag<DirectedEdge>[] adj; // adj[v] = adjacency list for vertex v
private int[] indegree; // indegree[v] = indegree of vertex v
public EdgeWeightedDigraph(int V) {
if (V < 0) throw new IllegalArgumentException("Number of vertices in a Digraph must be nonnegative");
this.V = V;
this.E = 0;
this.indegree = new int[V];
adj = (Bag<DirectedEdge>[]) new Bag[V];
for (int v = 0; v < V; v++)
adj[v] = new Bag<DirectedEdge>();
}
public EdgeWeightedDigraph(int V, int E) {
this(V);
if (E < 0) throw new IllegalArgumentException("Number of edges in a Digraph must be nonnegative");
for (int i = 0; i < E; i++) {
int v = StdRandom.uniform(V);
int w = StdRandom.uniform(V);
double weight = .01 * StdRandom.uniform(100);
DirectedEdge e = new DirectedEdge(v, w, weight);
addEdge(e);
}
}
public EdgeWeightedDigraph(In in) {
this(in.readInt());
int E = in.readInt();
if (E < 0) throw new IllegalArgumentException("Number of edges must be nonnegative");
for (int i = 0; i < E; i++) {
int v = in.readInt();
int w = in.readInt();
if (v < 0 || v >= V) throw new IndexOutOfBoundsException("vertex " + v + " is not between 0 and " + (V-1));
if (w < 0 || w >= V) throw new IndexOutOfBoundsException("vertex " + w + " is not between 0 and " + (V-1));
double weight = in.readDouble();
addEdge(new DirectedEdge(v, w, weight));
}
}
public EdgeWeightedDigraph(EdgeWeightedDigraph G) {
this(G.V());
this.E = G.E();
for (int v = 0; v < G.V(); v++)
this.indegree[v] = G.indegree(v);
for (int v = 0; v < G.V(); v++) {
// reverse so that adjacency list is in same order as original
Stack<DirectedEdge> reverse = new Stack<DirectedEdge>();
for (DirectedEdge e : G.adj[v]) {
reverse.push(e);
}
for (DirectedEdge e : reverse) {
adj[v].add(e);
}
}
}
public int V() {
return V;
}
public int E() {
return E;
}
// throw an IndexOutOfBoundsException unless 0 <= v < V
private void validateVertex(int v) {
if (v < 0 || v >= V)
throw new IndexOutOfBoundsException("vertex " + v + " is not between 0 and " + (V-1));
}
public void addEdge(DirectedEdge e) {
int v = e.from();
int w = e.to();
validateVertex(v);
validateVertex(w);
adj[v].add(e);
indegree[w]++;
E++;
}
public Iterable<DirectedEdge> adj(int v) {
validateVertex(v);
return adj[v];
}
public int outdegree(int v) {
validateVertex(v);
return adj[v].size();
}
public int indegree(int v) {
validateVertex(v);
return indegree[v];
}
public Iterable<DirectedEdge> edges() {
Bag<DirectedEdge> list = new Bag<DirectedEdge>();
for (int v = 0; v < V; v++) {
for (DirectedEdge e : adj(v)) {
list.add(e);
}
}
return list;
}
public String toString() {
StringBuilder s = new StringBuilder();
s.append(V + " " + E + NEWLINE);
for (int v = 0; v < V; v++) {
s.append(v + ": ");
for (DirectedEdge e : adj[v]) {
s.append(e + " ");
}
s.append(NEWLINE);
}
return s.toString();
}
}Dijstra算法（有向，无负边）
要求权重非负，算法很简单，不再赘述
复杂度： 空间与V成正比，时间与ElogV成正比
内部使用了索引优先队列
public class DijkstraSP {
private double[] distTo; // distTo[v] = distance of shortest s->v path
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on shortest s->v path
private IndexMinPQ<Double> pq; // priority queue of vertices
public DijkstraSP(EdgeWeightedDigraph G, int s) {
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0)
throw new IllegalArgumentException("edge " + e + " has negative weight");
}
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// relax vertices in order of distance from s
pq = new IndexMinPQ<Double>(G.V());
pq.insert(s, distTo[s]);
while (!pq.isEmpty()) {
int v = pq.delMin();
for (DirectedEdge e : G.adj(v))
relax(e);
}
// check optimality conditions
assert check(G, s);
}
// relax edge e and update pq if changed
private void relax(DirectedEdge e) {
int v = e.from(), w = e.to();
if (distTo[w] > distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
if (pq.contains(w)) pq.decreaseKey(w, distTo[w]);
else pq.insert(w, distTo[w]);
}
}
public double distTo(int v) {
return distTo[v];
}
public boolean hasPathTo(int v) {
return distTo[v] < Double.POSITIVE_INFINITY;
}
public Iterable<DirectedEdge> pathTo(int v) {
if (!hasPathTo(v)) return null;
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
path.push(e);
}
return path;
}
// check optimality conditions:
// (i) for all edges e: distTo[e.to()] <= distTo[e.from()] + e.weight()
// (ii) for all edge e on the SPT: distTo[e.to()] == distTo[e.from()] + e.weight()
private boolean check(EdgeWeightedDigraph G, int s) {
// check that edge weights are nonnegative
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0) {
System.err.println("negative edge weight detected");
return false;
}
}
// check that distTo[v] and edgeTo[v] are consistent
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distTo[s] and edgeTo[s] inconsistent");
return false;
}
for (int v = 0; v < G.V(); v++) {
if (v == s) continue;
if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
System.err.println("distTo[] and edgeTo[] inconsistent");
return false;
}
}
// check that all edges e = v->w satisfy distTo[w] <= distTo[v] + e.weight()
for (int v = 0; v < G.V(); v++) {
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
if (distTo[v] + e.weight() < distTo[w]) {
System.err.println("edge " + e + " not relaxed");
return false;
}
}
}
// check that all edges e = v->w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (int w = 0; w < G.V(); w++) {
if (edgeTo[w] == null) continue;
DirectedEdge e = edgeTo[w];
int v = e.from();
if (w != e.to()) return false;
if (distTo[v] + e.weight() != distTo[w]) {
System.err.println("edge " + e + " on shortest path not tight");
return false;
}
}
return true;
}
}无环加权有向图中的最短路径算法
无环，能有负边，比Dijstra算法快
按照拓扑排序（详见之前博客）更新权值
按照拓扑排序更新权值，就能在和E+V成正比的时间内解决无环加权有向图的电源最短路径问题
public class AcyclicSP {
private double[] distTo; // distTo[v] = distance of shortest s->v path
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on shortest s->v path
public AcyclicSP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.POSITIVE_INFINITY;
distTo[s] = 0.0;
// visit vertices in toplogical order
Topological topological = new Topological(G);
if (!topological.hasOrder())
throw new IllegalArgumentException("Digraph is not acyclic.");
for (int v : topological.order()) {
for (DirectedEdge e : G.adj(v))
relax(e);
}
}
// relax edge e
private void relax(DirectedEdge e) {
int v = e.from(), w = e.to();
if (distTo[w] > distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
}
}
public double distTo(int v) {
return distTo[v];
}
public boolean hasPathTo(int v) {
return distTo[v] < Double.POSITIVE_INFINITY;
}
public Iterable<DirectedEdge> pathTo(int v) {
if (!hasPathTo(v)) return null;
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
path.push(e);
}
return path;
}
}无环加权有向图中的最长路径算法
放松边的那个符号反过来就行了，初始化要改成负无穷
public class AcyclicLP {
private double[] distTo; // distTo[v] = distance of longest s->v path
private DirectedEdge[] edgeTo; // edgeTo[v] = last edge on longest s->v path
public AcyclicLP(EdgeWeightedDigraph G, int s) {
distTo = new double[G.V()];
edgeTo = new DirectedEdge[G.V()];
for (int v = 0; v < G.V(); v++)
distTo[v] = Double.NEGATIVE_INFINITY;
distTo[s] = 0.0;
// relax vertices in toplogical order
Topological topological = new Topological(G);
if (!topological.hasOrder())
throw new IllegalArgumentException("Digraph is not acyclic.");
for (int v : topological.order()) {
for (DirectedEdge e : G.adj(v))
relax(e);
}
}
// relax edge e, but update if you find a *longer* path
private void relax(DirectedEdge e) {
int v = e.from(), w = e.to();
if (distTo[w] < distTo[v] + e.weight()) {
distTo[w] = distTo[v] + e.weight();
edgeTo[w] = e;
}
}
public double distTo(int v) {
return distTo[v];
}
public boolean hasPathTo(int v) {
return distTo[v] > Double.NEGATIVE_INFINITY;
}
public Iterable<DirectedEdge> pathTo(int v) {
if (!hasPathTo(v)) return null;
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
path.push(e);
}
return path;
}
}