【图论】深入理解Dijsktra算法
1. 介绍
Dijsktra算法是大牛Dijsktra于1956年提出，用来解决有向图单源最短路径问题；但是不能解决负权的有向图，若要解决负权图则需要用到Bellman-Ford算法。Dijsktra算法思想：在DFS遍历图的过程中，每一次取出离源点的最近距离的点，将该点标记为已访问，松弛与该点相邻的结点。
有向图记为\(G=(n, m)\)，其中，\(n\)为顶点数，\(m\)为边数；且\(e[a,b]\)表示从结点\(a\)到结点\(b\)的边。\(d[i]\)记录源点到结点i的距离，\(U\)为未访问的结点集合，\(V\)为已访问的结点集合。Dijsktra算法具体步骤如下：
- 从集合\(U\)中寻找离源点最近的结点\(u\)，并将结点\(u\)标记为已访问（从集合\(U\)中移到集合\(V\)中）
\[u = \mathop {\arg \min } \limits_{i \in U} d[i]
\]
- 松弛与结点\(u\)相邻的未访问结点，更新d数组
\[\mathop {d[i]} \limits_{i \in U} = \min \lbrace d[i]\ , \ d[u] + e[u,i] \rbrace
\]
- 重复上述操作\(n\)次，即访问了所有结点，集合\(U\)为空
Dijsktra算法的Java实现
/**
* Dijkstra's Algorithm for finding the shortest path
*
* @param adjMatrix adjacency matrix representation of the graph
* @param source the source vertex
* @param dest the destination vertex
* @return the cost for the shortest path
*/
public static int dijkstra(int[][] adjMatrix, int source, int dest) {
int numVertex = adjMatrix.length, minVertex = source;
// `d` marks the cost for the shortest path, `visit` marks whether has been visited or not
int[] d = new int[numVertex], visit = new int[numVertex];
Arrays.fill(d, Integer.MAX_VALUE);
d[source] = 0;
for (int cnt = 1; cnt <= numVertex; cnt++) {
int lowCost = Integer.MAX_VALUE;
// find the min-vertex which is the nearest among the unvisited vertices
for (int i = 0; i < numVertex; i++) {
if (visit[i] == 0 && d[i] < lowCost) {
lowCost = d[i];
minVertex = i;
}
}
visit[minVertex] = 1;
if (minVertex == dest) return d[dest];
// relax the minVertex's adjacency vertices
for (int i = 0; i < numVertex; i++) {
if (visit[i] == 0 && adjMatrix[minVertex][i] != Integer.MAX_VALUE) {
d[i] = Math.min(d[i], d[minVertex] + adjMatrix[minVertex][i]);
}
}
}
return d[dest];
}
复杂度分析
- 时间复杂度：重复操作（即最外层for循环）n次，找出minNode操作、松弛操作需遍历所有结点，因此复杂度为\(O(n^2)\).
- 空间复杂度：开辟两个长度为n的数组d与visit，因此复杂度为\(T(n)\).
2. 优化
从上述Java实现中，我们发现：（里层for循环）寻找距离源点最近的未访问结点\(u\)，通过遍历整个数组来实现的，缺点是重复访问已经访问过的结点，浪费了时间。首先，我们来看看堆的性质。
堆
堆是一种完全二叉树(complete binary tree)；若其高度为h，则1~h-1层都是满的。如果从左至右从上至下从1开始给结点编号，堆满足：
- 结点\(i\)的父结点编号为\(i/2\).
- 结点\(i\)的左右孩子结点编号分别为\(2*i\), \(2*i+1\).
如果结点\(i\)的关键值小于父结点的关键值，则需要进行上浮操作（move up）；如果结点\(i\)的关键值大于父结点的，则需要下沉操作（move down）。为了保持堆的整体有序性，通常下沉操作从根结点开始。
小顶堆优化Dijsktra算法
我们可以用小顶堆来代替d数组，堆顶对应于结点\(u\)；取出堆顶，然后删除，如此堆中结点都是未访问的。同时为了记录数组\(d[i]\)中索引\(i\)值，我们让每个堆结点挂两个值——顶点、源点到该顶点的距离。算法伪代码如下：
Insert(vertex 0, 0) // 插入源点
FOR i from 1 to n-1: // 初始化堆
Insert(vertex i, infinity)
FOR k from 1 to n:
(i, d) := DeleteMin()
FOR all edges ij:
IF d + edge(i,j) < j’s key
DecreaseKey(vertex j, d + edge(i,j))
- Insert(vertex i, d)指在堆中插入堆结点(i, d)。
- DeleteMin()指取出堆顶并删除，时间复杂度为\(O(\log n)\)。
- DecreaseKey(vertex j, d + edge(i,j))是松弛操作，更新结点(vertex j, d + edge(i,j))，需要进行上浮，时间复杂度为\(O(\log n)\)。
我们需要n次DeleteMin，m次DecreaseKey，优化版的算法时间复杂度为\(O((n+m)\log n)\).
代码实现
邻接表
每一个邻接表的表项包含两个部分：头结点、表结点，整个邻接表可以用一个头结点数组表示。下面给出其Java实现
public class AdjList {
private int V = 0;
private HNode[] adjList =null; //邻接表
/*表结点*/
class ArcNode {
int adjvex, weight;
ArcNode next;
public ArcNode(int adjvex, int weight) {
this.adjvex = adjvex;
this.weight = weight;
next = null;
}
}
/*头结点*/
class HNode {
int vertex;
ArcNode firstArc;
public HNode(int vertex) {
this.vertex = vertex;
firstArc = null;
}
}
/*构造函数*/
public AdjList(int V) {
this.V = V;
adjList = new HNode[V+1];
for(int i = 1; i <= V; i++) {
adjList[i] = new HNode(i);
}
}
/*添加边*/
public void addEdge(int start, int end, int weight) {
ArcNode arc = new ArcNode(end, weight);
ArcNode temp = adjList[start].firstArc;
adjList[start].firstArc = arc;
arc.next = temp;
}
public int getV() {
return V;
}
public HNode[] getAdjList() {
return adjList;
}
}
小顶堆
public class Heap {
public int size = 0 ;
public Node[] h = null; //堆结点
/*记录Node中vertex对应堆的位置*/
public int[] index = null;
/*堆结点：
* 存储结点+源点到该结点的距离
*/
public class Node {
int vertex, weight;
public Node(int vertex, int weight) {
this.vertex = vertex;
this.weight = weight;
}
}
public Heap(int maximum) {
h = new Node[maximum];
index = new int[maximum];
}
/*上浮*/
public void moveUp(int pos) {
Node temp = h[pos];
for (; pos > 1 && h[pos/2].weight > temp.weight; pos/=2) {
h[pos] = h[pos/2];
index[h[pos].vertex] = pos; //更新位置
}
h[pos] = temp;
index[h[pos].vertex] = pos;
}
/*下沉*/
public void moveDown( ) {
Node root = h[1];
int pos = 1, child = 1;
for(; pos <= size; pos = child) {
child = 2*pos;
if(child < size && h[child+1].weight < h[child].weight)
child++;
if(h[child].weight < root.weight) {
h[pos] = h[child];
index[h[pos].vertex] = pos;
} else {
break;
}
}
h[pos] = root;
index[h[pos].vertex] = pos;
}
/*插入操作*/
public void insert(int v, int w) {
h[++size] = new Node(v, w);
moveUp(size);
}
/*删除堆顶元素*/
public Node deleteMin( ) {
Node result = h[1];
h[1] = h[size--];
moveDown();
return result;
}
}
优化算法
public class ShortestPath {
private static final int inf = 0xffffff;
public static void dijkstra(AdjList al) {
int V = al.getV();
Heap heap = new Heap(V+1);
heap.insert(1, 0);
for(int i = 2; i <= V; i++) {
heap.insert(i, inf);
}
for(int k =1; k <= V; k++) {
Heap.Node min = heap.deleteMin();
if(min.vertex == V) {
System.out.println(min.weight);
break;
}
AdjList.ArcNode arc = al.getAdjList()[min.vertex].firstArc;
while(arc != null) {
if((min.weight+ arc.weight) < heap.h[heap.index[arc.adjvex]].weight) {
heap.h[heap.index[arc.adjvex]].weight = min.weight+ arc.weight;
heap.moveUp(heap.index[arc.adjvex]);
}
arc = arc.next;
}
}
}
/*main方法用于测试*/
public static void main(String[] args) {
AdjList al = new AdjList(5);
al.addEdge(1, 2, 20);
al.addEdge(2, 3, 30);
al.addEdge(3, 4, 20);
al.addEdge(4, 5, 20);
al.addEdge(1, 5, 100);
dijkstra(al);
}
}
3. 参考资料#
[1] FRWMM, ALGORITHMS - DIJKSTRA WITH HEAPS.
