最短路径树：Dijstra算法
一、背景
全文根据《算法-第四版》，Dijkstra（迪杰斯特拉）算法，一种单源最短路径算法。我们把问题抽象为2步：1.数据结构抽象 2.实现。 分别对应第二章、第三章。
二、算法分析
2.1 数据结构
顶点+边->图。注意：Dijkstra算法的限定：
- 1.边有权重，且非负
- 2.边有向
2.1.1 加权有向边
DirectedEdge，API抽象如下：
| 方法 | 描述 |
| DirectedEdge(int v, int w, double weight) | 构造边 |
| double weight() | 边的权重 |
| int from() | 边的起点 |
| int to() | 边的终点 |
2.1.2 加权有向图
EdgeWeightedDigraph，API抽象如下：
| 方法 | 描述 |
| EdgeWeightedDigraph(In in) | 从输入流中构造图 |
| int V() | 顶点总数 |
| int E() | 边总数 |
| void addEdge(DirectedEdge e) | 将边e添加到图中 |
| Iterable<DirectedEdge> adj(int v) | 从顶点v指出的边（邻接表,一个哈希链表，key=顶点，value=顶点指出的边链表） |
| Iterable<DirectedEdge> edges() | 图中全部边 |
2.1.3 最短路径
DijkstraSP, API抽象如下：
| 方法 | 描述 |
| DijkstraSP(EdgeWeightedDigraph G, int s) | 构造最短路径树 |
| double distTo(int v) | 顶点s->v的距离，初始化无穷大 |
| boolean hasPathTo(int v) | 是否存在顶点s->v的路径 |
| Iterable<DirectedEdge> pathTo(int v) | s->v的路径，不存在为null |
元素：
最短路径树中的边（DirectedEdge[] edgeTo）：
edgeTo[v]代表树中连接v和父节点的边（最短路径最后一条边数组），每个顶点都有一条这样的边，就组成了最短路径树。
原点到达顶点的距离：由顶点索引的数组 double[] distTo：
distTo[v] 代表原点到达顶点v的最短距离。
索引最小优先级队列: IndexMinPQ<Double> pq：
int[] pq：索引二叉堆（元素=顶点v，对应keys[v]）：数组从pq[0]代表原点其它顶点从pq[1]开始插入
Key[] keys：元素有序数组（按照pq值作为下标赋值）存储到顶点的最短距离
2.2 算法核心
计算最短路径，三步骤：
- 1.每次选取最小节顶点：如果选择？使用最小堆排序，每次取堆顶元素即可。
- 2.遍历从顶点的发出的全部边
- 3.放松操作
三、具体实现
3.1 构造
3.1.1 元素迭代器
因为有遍历需要，这里定义Bag<Item>类实现了Iterable<Item>迭代器接口，Item是元素。就是个简单的某个元素的迭代器基本实现。
package study.algorithm.base; import java.util.Iterator;
import java.util.NoSuchElementException; /**
* The {@code Bag} class represents a bag (or multiset) of
* generic items. It supports insertion and iterating over the
* items in arbitrary order.
* <p>
* This implementation uses a singly linked list with a static nested class Node.
* See {@link LinkedBag} for the version from the
* textbook that uses a non-static nested class.
* See {@link ResizingArrayBag} for a version that uses a resizing array.
* The <em>add</em>, <em>isEmpty</em>, and <em>size</em> operations
* take constant time. Iteration takes time proportional to the number of items.
* <p>
* For additional documentation, see <a href="https://algs4.cs.princeton.edu/13stacks">Section 1.3</a> of
* <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*
* @author Robert Sedgewick
* @author Kevin Wayne
*
* @param <Item> the generic type of an item in this bag
*/
public class Bag<Item> implements Iterable<Item> {
/**
* 首节点
*/
private Node<Item> first;
/**
* 元素个数
*/
private int n; /**
* 链接表
* @param <Item>
*/
private static class Node<Item> {
private Item item;
private Node<Item> next;
} /**
* 初始化一个空包
*/
public Bag() {
first = null;
n = 0;
} /**
* Returns true if this bag is empty.
*
* @return {@code true} if this bag is empty;
* {@code false} otherwise
*/
public boolean isEmpty() {
return first == null;
} /**
* Returns the number of items in this bag.
*
* @return the number of items in this bag
*/
public int size() {
return n;
} /**
* Adds the item to this bag.
*
* @param item the item to add to this bag
*/
public void add(Item item) {
// 保留老的首节点
Node<Item> oldfirst = first;
// 构造一个新首节点
first = new Node<Item>();
// item为新首节点item
first.item = item;
// 新节点的next节点指向老的首节点
first.next = oldfirst;
n++;
} /**
* Returns an iterator that iterates over the items in this bag in arbitrary order.
*
* @return an iterator that iterates over the items in this bag in arbitrary order
*/
@Override
public Iterator<Item> iterator() {
return new LinkedIterator(first);
} /**
* 链接迭代器，不支持移除
*/
private class LinkedIterator implements Iterator<Item> {
private Node<Item> current; public LinkedIterator(Node<Item> first) {
current = first;
} @Override
public boolean hasNext() { return current != null; }
@Override
public void remove() { throw new UnsupportedOperationException(); } @Override
public Item next() {
if (!hasNext()) {
throw new NoSuchElementException();
}
Item item = current.item;
// 下一节点
current = current.next;
return item;
}
} /**
* Unit tests the {@code Bag} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
Bag<String> bag = new Bag<String>();
while (!StdIn.isEmpty()) {
String item = StdIn.readString();
bag.add(item);
} StdOut.println("size of bag = " + bag.size());
for (String s : bag) {
StdOut.println(s);
}
} }
3.1.2 具体构造
1. 从输入流中初始化图，输入流格式（括号内为注释，实际文件中不存在）：
8（顶点数）
15（边数）
4 5 0.35（边4->5 权重=0.35）
5 4 0.35
4 7 0.37
5 7 0.28
7 5 0.28
5 1 0.32
0 4 0.38
0 2 0.26
7 3 0.39
1 3 0.29
2 7 0.34
6 2 0.40
3 6 0.52
6 0 0.58
6 4 0.93
如下图中public EdgeWeightedDigraph(In in)构造方法，核心：
往邻接表（顶点作为数组下标）中添加带权重边。
package study.algorithm.graph; import study.algorithm.base.*; import java.util.NoSuchElementException; /***
* @Description 边权重有向图
* @author denny.zhang
* @date 2020/4/24 9:58 上午
*/
public class EdgeWeightedDigraph {
private static final String NEWLINE = System.getProperty("line.separator"); /**
* 顶点总数
*/
private final int V;
/**
* 边总数
*/
private int E;
/**
* 邻接表（每个元素Bag代表 由某个顶点为起点的边数组,按顶点顺序排列），adjacency list
*/
private Bag<DirectedEdge>[] adj; /**
* 从输入流中初始化图，输入流格式：
* 8（顶点数）
* 15（边总数）
* 4 5 0.35（每一条边 4->5 权重0.35）
* 5 4 0.35
* 4 7 0.37
* ...
*
* @param in the input stream
* @throws IllegalArgumentException if {@code in} is {@code null}
* @throws IllegalArgumentException if the endpoints of any edge are not in prescribed range
* @throws IllegalArgumentException if the number of vertices or edges is negative
*/
public EdgeWeightedDigraph(In in) {
if (in == null) {
throw new IllegalArgumentException("argument is null");
}
try {
// 1.读取顶点数
this.V = in.readInt(); // 初始化邻接表
adj = (Bag<DirectedEdge>[]) new Bag[V];
for (int v = 0; v < V; v++) {
adj[v] = new Bag<DirectedEdge>();
}
// 2.读取边数
int E = in.readInt();
for (int i = 0; i < E; i++) {
int v = in.readInt();
int w = in.readInt();
// 3.读取边的权重
double weight = in.readDouble();
// 添加权重边
addEdge(new DirectedEdge(v, w, weight));
}
}
catch (NoSuchElementException e) {
throw new IllegalArgumentException("invalid input format in EdgeWeightedDigraph constructor", e);
}
} /**
* 顶点数
*
* @return the number of vertices in this edge-weighted digraph
*/
public int V() {
return V;
} /**
* 边数
*
* @return the number of edges in this edge-weighted digraph
*/
public int E() {
return E;
} /**
* 往图中添加边
*
* @param e the edge
* @throws IllegalArgumentException unless endpoints of edge are between {@code 0}
* and {@code V-1}
*/
public void addEdge(DirectedEdge e) {
// 边的起点
int v = e.from();
// 边的终点
int w = e.to(); // 起点v的邻接表，加入一条边
adj[v].add(e);
// 边总数+1
E++;
} /**
* 返回从顶点V 指出的全部可迭代边（邻接表）
*
* @param v the vertex
* @return the directed edges incident from vertex {@code v} as an Iterable
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public Iterable<DirectedEdge> adj(int v) {
validateVertex(v);
return adj[v];
} /**
* 返回全部有向边
*
* @return all edges in this edge-weighted digraph, as an iterable
*/
public Iterable<DirectedEdge> edges() {
Bag<DirectedEdge> list = new Bag<DirectedEdge>();
// 遍历全部顶点
for (int v = 0; v < V; v++) {
// 每个顶点的邻接表（指出边）
for (DirectedEdge e : adj(v)) {
// 指出边入list
list.add(e);
}
}
return list;
} /**
* Returns a string representation of this edge-weighted digraph.
*
* @return the number of vertices <em>V</em>, followed by the number of edges <em>E</em>,
* followed by the <em>V</em> adjacency lists of edges
*/
@Override
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();
} /**
* Unit tests the {@code EdgeWeightedDigraph} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
In in = new In(args[0]);
EdgeWeightedDigraph G = new EdgeWeightedDigraph(in);
StdOut.println(G);
} }
3.2 计算最短路径
3.2.1 索引优先队列
package study.algorithm.base; import java.util.Iterator;
import java.util.NoSuchElementException; /**
* 索引最小优先级队列
*
* @param <Key>
*/
public class IndexMinPQ<Key extends Comparable<Key>> implements Iterable<Integer> {
/**
* 元素数量上限
*/
private int maxN;
/**
* 元素数量
*/
private int n;
/**
* 索引二叉堆（元素=顶点v，对应keys[v]）：pq[0]代表原点，其它顶点从pq[1]开始插入
*/
private int[] pq;
/**
* 标记索引为i的元素在二叉堆中的位置。pq的反转数组（qp[index]=i）：qp[pq[i]] = pq[qp[i]] = i
*/
private int[] qp; /**
* 元素有序数组（按照pq的索引赋值）
*/
private Key[] keys; /**
* 初始化一个空索引优先队列，索引范围:0 ~ maxN-1
*
* @param maxN the keys on this priority queue are index from {@code 0}
* {@code maxN - 1}
* @throws IllegalArgumentException if {@code maxN < 0}
*/
public IndexMinPQ(int maxN) {
if (maxN < 0) throw new IllegalArgumentException();
this.maxN = maxN;
// 初始有0个元素
n = 0;
// 初始化键数组长度为maxN + 1
keys = (Key[]) new Comparable[maxN + 1];
// 初始化"键值对"数组长度为maxN + 1
pq = new int[maxN + 1];
// 初始化"值键对"数组长度为maxN + 1
qp = new int[maxN + 1];
// 遍历给"值键对"数组赋值-1，后续只要!=-1,即包含i
for (int i = 0; i <= maxN; i++)
qp[i] = -1;
} /**
* Returns true if this priority queue is empty.
*
* @return {@code true} if this priority queue is empty;
* {@code false} otherwise
*/
public boolean isEmpty() {
return n == 0;
} /**
* Is {@code i} an index on this priority queue?
*
* @param i an index
* @return {@code true} if {@code i} is an index on this priority queue;
* {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
*/
public boolean contains(int i) {
validateIndex(i);
return qp[i] != -1;
} /**
* Returns the number of keys on this priority queue.
*
* @return the number of keys on this priority queue
*/
public int size() {
return n;
} /**
* 插入一个元素，将元素key关联索引i
*
* @param i an index
* @param key the key to associate with index {@code i}
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws IllegalArgumentException if there already is an item associated
* with index {@code i}
*/
public void insert(int i, Key key) {
validateIndex(i);
if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
// 元素个数+1
n++;
// 索引为i的二叉堆位置为n
qp[i] = n;
// 二叉堆底部插入新元素，值=i
pq[n] = i;
// 索引i对应的元素赋值
keys[i] = key;
// 二叉堆中，上浮最后一个元素（小值上浮）
swim(n);
} /**
* 返回最小元素的索引
*
* @return an index associated with a minimum key
* @throws NoSuchElementException if this priority queue is empty
*/
public int minIndex() {
if (n == 0) throw new NoSuchElementException("Priority queue underflow");
return pq[1];
} /**
* 返回最小元素（key）
*
* @return a minimum key
* @throws NoSuchElementException if this priority queue is empty
*/
public Key minKey() {
if (n == 0) throw new NoSuchElementException("Priority queue underflow");
return keys[pq[1]];
} /**
* 删除最小值key,并返回最小值
*
* @return an index associated with a minimum key
* @throws NoSuchElementException if this priority queue is empty
*/
public int delMin() {
if (n == 0) throw new NoSuchElementException("Priority queue underflow");
// pq[1]即为索引最小值
int min = pq[1];
// 交换第一个元素和最后一个元素
exch(1, n--);
// 把新换来的第一个元素下沉
sink(1);
// 校验下沉后，最后一个元素是最小值
assert min == pq[n+1];
// 恢复初始值，-1即代表该元素已删除
qp[min] = -1; // delete
// 方便垃圾回收
keys[min] = null;
// 最后一个元素（索引）赋值-1
pq[n+1] = -1; // not needed
return min;
} /**
* Returns the key associated with index {@code i}.
*
* @param i the index of the key to return
* @return the key associated with index {@code i}
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws NoSuchElementException no key is associated with index {@code i}
*/
public Key keyOf(int i) {
validateIndex(i);
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
else return keys[i];
} /**
* Change the key associated with index {@code i} to the specified value.
*
* @param i the index of the key to change
* @param key change the key associated with index {@code i} to this key
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws NoSuchElementException no key is associated with index {@code i}
*/
public void changeKey(int i, Key key) {
validateIndex(i);
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
keys[i] = key;
swim(qp[i]);
sink(qp[i]);
} /**
* Change the key associated with index {@code i} to the specified value.
*
* @param i the index of the key to change
* @param key change the key associated with index {@code i} to this key
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @deprecated Replaced by {@code changeKey(int, Key)}.
*/
@Deprecated
public void change(int i, Key key) {
changeKey(i, key);
} /**
* 减小索引i对应的值为key
* 更新：
* 1.元素数组keys[]
* 2.小顶二叉堆pq[]
*
* @param i the index of the key to decrease
* @param key decrease the key associated with index {@code i} to this key
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws IllegalArgumentException if {@code key >= keyOf(i)}
* @throws NoSuchElementException no key is associated with index {@code i}
*/
public void decreaseKey(int i, Key key) {
validateIndex(i);
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
// key 值一样，报错
if (keys[i].compareTo(key) == 0)
throw new IllegalArgumentException("Calling decreaseKey() with a key equal to the key in the priority queue");
// key比当前值大，报错
if (keys[i].compareTo(key) < 0)
throw new IllegalArgumentException("Calling decreaseKey() with a key strictly greater than the key in the priority queue");
// key比当前值小，把key赋值进去
keys[i] = key;
// 小值上浮（qp[i]=索引i在二叉堆pq[]中的位置）
swim(qp[i]);
} /**
* Increase the key associated with index {@code i} to the specified value.
*
* @param i the index of the key to increase
* @param key increase the key associated with index {@code i} to this key
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws IllegalArgumentException if {@code key <= keyOf(i)}
* @throws NoSuchElementException no key is associated with index {@code i}
*/
public void increaseKey(int i, Key key) {
validateIndex(i);
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
if (keys[i].compareTo(key) == 0)
throw new IllegalArgumentException("Calling increaseKey() with a key equal to the key in the priority queue");
if (keys[i].compareTo(key) > 0)
throw new IllegalArgumentException("Calling increaseKey() with a key strictly less than the key in the priority queue");
keys[i] = key;
sink(qp[i]);
} /**
* Remove the key associated with index {@code i}.
*
* @param i the index of the key to remove
* @throws IllegalArgumentException unless {@code 0 <= i < maxN}
* @throws NoSuchElementException no key is associated with index {@code i}
*/
public void delete(int i) {
validateIndex(i);
if (!contains(i)) throw new NoSuchElementException("index is not in the priority queue");
int index = qp[i];
exch(index, n--);
swim(index);
sink(index);
keys[i] = null;
qp[i] = -1;
} // throw an IllegalArgumentException if i is an invalid index
private void validateIndex(int i) {
if (i < 0) throw new IllegalArgumentException("index is negative: " + i);
if (i >= maxN) throw new IllegalArgumentException("index >= capacity: " + i);
} /***************************************************************************
* General helper functions.
***************************************************************************/
private boolean greater(int i, int j) {
return keys[pq[i]].compareTo(keys[pq[j]]) > 0;
} private void exch(int i, int j) {
int swap = pq[i];
pq[i] = pq[j];
pq[j] = swap;
qp[pq[i]] = i;
qp[pq[j]] = j;
} /***************************************************************************
* Heap helper functions.
***************************************************************************/
private void swim(int k) {
// 如果父节点值比当前节点值大，交换，父节点作为当前节点，轮询。即小值上浮。
while (k > 1 && greater(k/2, k)) {
exch(k, k/2);
k = k/2;
}
} private void sink(int k) {
while (2*k <= n) {
int j = 2*k;
if (j < n && greater(j, j+1)) j++;
if (!greater(k, j)) break;
exch(k, j);
k = j;
}
} /***************************************************************************
* Iterators.
***************************************************************************/ /**
* Returns an iterator that iterates over the keys on the
* priority queue in ascending order.
* The iterator doesn't implement {@code remove()} since it's optional.
*
* @return an iterator that iterates over the keys in ascending order
*/
@Override
public Iterator<Integer> iterator() { return new HeapIterator(); } private class HeapIterator implements Iterator<Integer> {
// create a new pq
private IndexMinPQ<Key> copy; // add all elements to copy of heap
// takes linear time since already in heap order so no keys move
public HeapIterator() {
copy = new IndexMinPQ<Key>(pq.length - 1);
for (int i = 1; i <= n; i++)
copy.insert(pq[i], keys[pq[i]]);
} @Override
public boolean hasNext() { return !copy.isEmpty(); }
@Override
public void remove() { throw new UnsupportedOperationException(); } @Override
public Integer next() {
if (!hasNext()) throw new NoSuchElementException();
return copy.delMin();
}
} /**
* Unit tests the {@code IndexMinPQ} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// insert a bunch of strings
String[] strings = { "it", "was", "the", "best", "of", "times", "it", "was", "the", "worst" }; IndexMinPQ<String> pq = new IndexMinPQ<String>(strings.length);
for (int i = 0; i < strings.length; i++) {
pq.insert(i, strings[i]);
} // delete and print each key
while (!pq.isEmpty()) {
int i = pq.delMin();
StdOut.println(i + " " + strings[i]);
}
StdOut.println(); // reinsert the same strings
for (int i = 0; i < strings.length; i++) {
pq.insert(i, strings[i]);
} // print each key using the iterator
for (int i : pq) {
StdOut.println(i + " " + strings[i]);
}
while (!pq.isEmpty()) {
pq.delMin();
} }
}
3.2.2 最短路径
package study.algorithm.graph; import study.algorithm.base.In;
import study.algorithm.base.IndexMinPQ;
import study.algorithm.base.Stack;
import study.algorithm.base.StdOut; /***
* @Description 边权重非负的加权有向图的单起点最短路径树
* @author denny.zhang
* @date 2020/4/23 11:29 上午
*/
public class DijkstraSP { /**
* 最短路径数组，元素：到所有顶点的最短路径
*/
private double[] distTo; /**
* 有向边数组：最短路径最后一条边数组
*/
private DirectedEdge[] edgeTo; /**
* 顶点作为下标，索引最小优先级队列
*/
private IndexMinPQ<Double> pq; /**
* 计算从原点S 到 其它所有顶点 的"最短路径" 边权重 图
*
* @param G the edge-weighted digraph 边权重图
* @param s the source vertex 原点
* @throws IllegalArgumentException if an edge weight is negative
* @throws IllegalArgumentException unless {@code 0 <= s < V}
*/
public DijkstraSP(EdgeWeightedDigraph G, int s) {
// 负权重校验
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0) {
throw new IllegalArgumentException("edge " + e + " has negative weight");
}
}
// 最短路径数组长度=顶点个数
distTo = new double[G.V()];
// 构造长度为顶点总数的最短路径边数组
edgeTo = new DirectedEdge[G.V()];
// 校验原点值
validateVertex(s);
// 初始化所有顶点的路径为无穷大
for (int v = 0; v < G.V(); v++) {
distTo[v] = Double.POSITIVE_INFINITY;
}
// 初始化到原点最小路径为0
distTo[s] = 0.0; // 构造一个长度为 顶点总数的 索引最小优先队列
pq = new IndexMinPQ<Double>(G.V());
// 把原点插入，路径为0
pq.insert(s, distTo[s]);
// 只要队列不空(从上往下，顺序遍历一遍pq[])，
while (!pq.isEmpty()) {
// 删除最小key（即pq[1]），并返回最小值（顶点）
int v = pq.delMin();
// 遍历顶点v的邻接表，每一条边
for (DirectedEdge e : G.adj(v)) {
// 放松边
relax(e);
}
} // 校验
assert check(G, s);
} /**
* 放松并更新pq
* @param e
*/
private void relax(DirectedEdge e) {
// 起点、终点
int v = e.from(), w = e.to();
// 如果原点到终点w的距离 > 原点到起点v的距离+边权重 说明原点到w松弛了
if (distTo[w] > distTo[v] + e.weight()) {
// 最新距离
distTo[w] = distTo[v] + e.weight();
// 到终点w的边赋值为新边
edgeTo[w] = e;
// 如果优先队列已经包含终点w
if (pq.contains(w)) {
// 比较下标为w的key如果>当前路径（即当前值比队列中值小），重新排序
pq.decreaseKey(w, distTo[w]);
} else {
// 不包含，插入并排序
pq.insert(w, distTo[w]);
}
}
} /**
* s->v的最短路径
* @param v the destination vertex
* @return the length of a shortest path from the source vertex {@code s} to vertex {@code v};
* {@code Double.POSITIVE_INFINITY} if no such path
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public double distTo(int v) {
validateVertex(v);
return distTo[v];
} /**
* s->v是否可达
*
* @param v the destination vertex
* @return {@code true} if there is a path from the source vertex
* {@code s} to vertex {@code v}; {@code false} otherwise
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public boolean hasPathTo(int v) {
validateVertex(v);
return distTo[v] < Double.POSITIVE_INFINITY;
} /**
* s->v的最短可迭代边（1->2->3）
*
* @param v the destination vertex
* @return a shortest path from the source vertex {@code s} to vertex {@code v}
* as an iterable of edges, and {@code null} if no such path
* @throws IllegalArgumentException unless {@code 0 <= v < V}
*/
public Iterable<DirectedEdge> pathTo(int v) {
validateVertex(v);
if (!hasPathTo(v)) {
return null;
}
// 可迭代有向边栈
Stack<DirectedEdge> path = new Stack<DirectedEdge>();
// e是顶点v的最短路径树的最后一条边，沿着边往上追溯上一个顶点 3->2->1
for (DirectedEdge e = edgeTo[v]; e != null; e = edgeTo[e.from()]) {
// 压栈
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) { // 校验边权重不为负值
for (DirectedEdge e : G.edges()) {
if (e.weight() < 0) {
System.err.println("negative edge weight detected");
return false;
}
} // 校验到顶点的路径为0且到顶点的边为空
if (distTo[s] != 0.0 || edgeTo[s] != null) {
System.err.println("distTo[s] and edgeTo[s] inconsistent");
return false;
}
// 遍历顶点
for (int v = 0; v < G.V(); v++) {
// 起点跳过
if (v == s) {
continue;
}
// 到顶点v的最后一条边为空（不可达） 且 到顶点v的最短路径不是无穷大（即有值）两者冲突
if (edgeTo[v] == null && distTo[v] != Double.POSITIVE_INFINITY) {
System.err.println("distTo[] and edgeTo[] inconsistent");
return false;
}
} // 校验所有边非松弛
for (int v = 0; v < G.V(); v++) {
// 遍历顶点v的邻接边
for (DirectedEdge e : G.adj(v)) {
int w = e.to();
// 校验松弛
if (distTo[v] + e.weight() < distTo[w]) {
System.err.println("edge " + e + " not relaxed");
return false;
}
}
} // 校验最短路径树：满足 distTo[w] == distTo[v] + e.weight()
for (int w = 0; w < G.V(); w++) {
// 跳过不可达顶点
if (edgeTo[w] == null) {
continue;
}
// 最后一条边
DirectedEdge e = edgeTo[w];
// 起点
int v = e.from();
//终点
if (w != e.to()) {
return false;
}
// 校验：最短路劲树，起点路径+权重=终点路径
if (distTo[v] + e.weight() != distTo[w]) {
System.err.println("edge " + e + " on shortest path not tight");
return false;
}
}
return true;
} // throw an IllegalArgumentException unless {@code 0 <= v < V}
private void validateVertex(int v) {
int V = distTo.length;
if (v < 0 || v >= V) {
throw new IllegalArgumentException("vertex " + v + " is not between 0 and " + (V-1));
}
} /**
* Unit tests the {@code DijkstraSP} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// 图文件名称
In in = new In(args[0]);
// 构造边权重有向图
EdgeWeightedDigraph G = new EdgeWeightedDigraph(in);
// 顶点
int s = Integer.parseInt(args[1]); // 计算最短路径
DijkstraSP sp = new DijkstraSP(G, s); // 遍历所有顶点
for (int t = 0; t < G.V(); t++) {
// 可达
if (sp.hasPathTo(t)) {
// 原点到t的路径 长度
StdOut.printf("%d to %d (%.2f) ", s, t, sp.distTo(t));
// 原点到t的路径图
for (DirectedEdge e : sp.pathTo(t)) {
StdOut.print(e + " ");
}
// 换行
StdOut.println();
}
// 不可达
else {
StdOut.printf("%d to %d no path\n", s, t);
}
}
} }
四、测试结果
4.1 测试准备
本地生存一个文件 tinyEWD.txt,内容如下：
8
15
4 5 0.35
5 4 0.35
4 7 0.37
5 7 0.28
7 5 0.28
5 1 0.32
0 4 0.38
0 2 0.26
7 3 0.39
1 3 0.29
2 7 0.34
6 2 0.40
3 6 0.52
6 0 0.58
6 4 0.93
4.2 测试
本地运行DijkstraSP，配置运行参数，以idea为例：第一个入参是文件地址，第二个参数代表原点是0，计算从原点（顶点0）到 其它所有顶点 的"最短路径" 边权重 图：
运行的最短路径，结果如下：
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