【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))

1. Insert(vertex i, d)指在堆中插入堆结点(i, d)。
2. DeleteMin()指取出堆顶并删除，时间复杂度为\(O(\log n)\)。
3. 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.