Heap Dijkstra算法。作为最小优先级队列的最小堆
我正在阅读《CLRS》第三版（第662页）中的Dijkstra算法。这是书中我不明白的一部分： 如果图足够稀疏-特别是，Heap Dijkstra算法。作为最小优先级队列的最小堆,heap,priority-queue,dijkstra,min-heap,Heap,Priority Queue,Dijkstra,Min Heap,我正在阅读《CLRS》第三版（第662页）中的Dijkstra算法。这是书中我不明白的一部分： 如果图足够稀疏-特别是，E=o（V^2/lg V）-我们可以通过使用二进制最小堆实现最小优先级队列来改进算法 为什么图形应该是稀疏的 以下是另一部分： 每次减少按键操作都需要时间O（log V），并且仍有 最多只能对此类操作进行重新评估 假设我的图表如下所示： 我想计算从1到6的最短路径，并使用最小堆方法。首先，我将所有节点添加到最小优先级队列中。构建min堆后，min节点就是源节点（因为它与自身
E=o（V^2/lg V）以下是另一部分： 每次减少按键操作都需要时间
O（log V）decreaseKeyprivate static class Node implements Comparable<Node> {
final int key;
int distance = Integer.MAX_VALUE;
Node prev = null;
public Node(int key) {
this.key = key;
}
@Override
public int compareTo(Node o) {
if (distance < o.distance) {
return -1;
} else if (distance > o.distance) {
return 1;
} else {
return 0;
}
}
@Override
public String toString() {
return "key=" + key + " distance=" + distance;
}
@Override
public int hashCode() {
return key;
}
@Override
public boolean equals(Object obj) {
if (this == obj) {
return true;
}
if (!(obj instanceof Node)) {
return false;
}
Node other = (Node) obj;
return key == other.key;
}
}
public static class MinPriorityQueue {
private Node[] array;
private int heapSize;
public MinPriorityQueue(Node[] array) {
this.array = array;
this.heapSize = this.array.length;
}
public Node extractMin() {
Node temp = array[0];
swap(0, heapSize - 1, array);
heapSize--;
sink(0);
return temp;
}
public boolean isEmpty() {
return heapSize == 0;
}
public void buildMinHeap() {
for (int i = heapSize / 2 - 1; i >= 0; i--) {
sink(i);
}
}
public void decreaseKey(int index, Node key) {
if (key.compareTo(array[index]) >= 0) {
throw new IllegalArgumentException("the new key must be greater than the current key");
}
array[index] = key;
while (index > 0 && array[index].compareTo(array[parentIndex(index)]) < 0) {
swap(index, parentIndex(index), array);
index = parentIndex(index);
}
}
private int parentIndex(int index) {
return (index - 1) / 2;
}
private int left(int index) {
return 2 * index + 1;
}
private int right(int index) {
return 2 * index + 2;
}
private void sink(int index) {
int smallestIndex = index;
int left = left(index);
int right = right(index);
if (left < heapSize && array[left].compareTo(array[smallestIndex]) < 0) {
smallestIndex = left;
}
if (right < heapSize && array[right].compareTo(array[smallestIndex]) < 0) {
smallestIndex = right;
}
if (index != smallestIndex) {
swap(smallestIndex, index, array);
sink(smallestIndex);
}
}
public Node min() {
return array[0];
}
private void swap(int i, int j, Node[] array) {
Node temp = array[i];
array[i] = array[j];
array[j] = temp;
}
}
私有静态类节点实现可比较{
最终整数键；
int距离=整数最大值；
Node prev=null；
公共节点（int键）{
this.key=key；
}
@凌驾
公共整数比较（节点o）{
如果（距离小于o.distance）{
返回-1；
}否则，如果（距离>o.distance）{
返回1；
}否则{
返回0；
}
}
@凌驾
公共字符串toString（）{
返回“键=”+键+”距离=+距离；
}
@凌驾
公共int hashCode（）{
返回键；
}
@凌驾
公共布尔等于（对象obj）{
if（this==obj）{
返回true；
}
如果（！（节点的obj实例））{
返回false；
}
节点其他=（节点）obj；
return key==other.key；
}
}
private static class Node implements Comparable<Node> {
final int key;
int distance = Integer.MAX_VALUE;
Node prev = null;
public Node(int key) {
this.key = key;
}
@Override
public int compareTo(Node o) {
if (distance < o.distance) {
return -1;
} else if (distance > o.distance) {
return 1;
} else {
return 0;
}
}
@Override
public String toString() {
return "key=" + key + " distance=" + distance;
}
@Override
public int hashCode() {
return key;
}
@Override
public boolean equals(Object obj) {
if (this == obj) {
return true;
}
if (!(obj instanceof Node)) {
return false;
}
Node other = (Node) obj;
return key == other.key;
}
}
public static class MinPriorityQueue {
private Node[] array;
private int heapSize;
public MinPriorityQueue(Node[] array) {
this.array = array;
this.heapSize = this.array.length;
}
public Node extractMin() {
Node temp = array[0];
swap(0, heapSize - 1, array);
heapSize--;
sink(0);
return temp;
}
public boolean isEmpty() {
return heapSize == 0;
}
public void buildMinHeap() {
for (int i = heapSize / 2 - 1; i >= 0; i--) {
sink(i);
}
}
public void decreaseKey(int index, Node key) {
if (key.compareTo(array[index]) >= 0) {
throw new IllegalArgumentException("the new key must be greater than the current key");
}
array[index] = key;
while (index > 0 && array[index].compareTo(array[parentIndex(index)]) < 0) {
swap(index, parentIndex(index), array);
index = parentIndex(index);
}
}
private int parentIndex(int index) {
return (index - 1) / 2;
}
private int left(int index) {
return 2 * index + 1;
}
private int right(int index) {
return 2 * index + 2;
}
private void sink(int index) {
int smallestIndex = index;
int left = left(index);
int right = right(index);
if (left < heapSize && array[left].compareTo(array[smallestIndex]) < 0) {
smallestIndex = left;
}
if (right < heapSize && array[right].compareTo(array[smallestIndex]) < 0) {
smallestIndex = right;
}
if (index != smallestIndex) {
swap(smallestIndex, index, array);
sink(smallestIndex);
}
}
public Node min() {
return array[0];
}
private void swap(int i, int j, Node[] array) {
Node temp = array[i];
array[i] = array[j];
array[j] = temp;
}
}
公共静态类MinPriorityQueue{
私有节点[]数组；
私人治疗；
公共最小优先级队列（节点[]数组）{
this.array=数组；
this.heapSize=this.array.length；
}
公共节点extractMin（）{
节点温度=数组[0]；
交换（0，治疗-1，阵列）；
医治--；
水槽（0）；
返回温度；
}
公共布尔值为空（）{
返回heapSize==0；
}
公共void buildMinHeap（）{
对于（inti=heapSize/2-1；i>=0；i--）{
水槽（一）；
}
}
public void decreaseKey（int索引，节点键）{
if（key.compareTo（数组[索引]）>=0）{
抛出新的IllegalArgumentException（“新密钥必须大于当前密钥”）；
}
数组[索引]=键；
while（index>0&&array[index]。比较（array[parentIndex（index）]）<0）{
交换（索引、父索引（索引）、数组）；
索引=父索引（索引）；
}
}
私有整数父索引（整数索引）{
收益率（指数-1）/2；
}
私有整数左（整数索引）{
返回2*index+1；
}
私有整数权限（整数索引）{
返回2*index+2；
}
专用void接收器（int索引）{
int smallestIndex=索引；
int left=左（索引）；
int right=right（索引）；
if（leftpublic class Vertex{
final private String id;
final private String name;
public Vertex(String id, String name) {
this.id = id;
this.name = name;
}
public String getId() {
return id;
}
public String getName() {
return name;
}
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + ((id == null) ? 0 : id.hashCode());
return result;
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
Vertex other = (Vertex) obj;
if (id == null) {
if (other.id != null)
return false;
} else if (!id.equals(other.id))
return false;
return true;
}
@Override
public String toString() {
return name;
}
}
public class Edge {
private final String id;
private final Vertex source;
private final Vertex destination;
private final int weight;
public Edge(String id, Vertex source, Vertex destination, int weight) {
this.id = id;
this.source = source;
this.destination = destination;
this.weight = weight;
}
public String getId() {
return id;
}
public Vertex getDestination() {
return destination;
}
public Vertex getSource() {
return source;
}
public int getWeight() {
return weight;
}
@Override
public String toString() {
return source + " " + destination;
}
}
public class Vertex{
final private String id;
final private String name;
public Vertex(String id, String name) {
this.id = id;
this.name = name;
}
public String getId() {
return id;
}
public String getName() {
return name;
}
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + ((id == null) ? 0 : id.hashCode());
return result;
}
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
Vertex other = (Vertex) obj;
if (id == null) {
if (other.id != null)
return false;
} else if (!id.equals(other.id))
return false;
return true;
}
@Override
public String toString() {
return name;
}
}
public class Edge {
private final String id;
private final Vertex source;
private final Vertex destination;
private final int weight;
public Edge(String id, Vertex source, Vertex destination, int weight) {
this.id = id;
this.source = source;
this.destination = destination;
this.weight = weight;
}
public String getId() {
return id;
}
public Vertex getDestination() {
return destination;
}
public Vertex getSource() {
return source;
}
public int getWeight() {
return weight;
}
@Override
public String toString() {
return source + " " + destination;
}
}
public class Graph {
private final List<Vertex> vertexes;
private final List<Edge> edges;
public Graph(List<Vertex> vertexes, List<Edge> edges) {
this.vertexes = vertexes;
this.edges = edges;
}
public List<Vertex> getVertexes() {
return vertexes;
}
public List<Edge> getEdges() {
return edges;
}
}
公共类图{
私有最终列表顶点；
私人最终名单边缘；
公共图（列出顶点、列出边）{
this.vertexs=顶点；
这个。边=边；
}
公共列表getVertex（）{
返回顶点；
}
公共列表getEdge（）{
返回边；
}
}
public class DijkstraAlgorithm {
private final List<Vertex> nodes;
private final List<Edge> edges;
private Set<Vertex> settledNodes;
private Set<Vertex> unSettledNodes;
private Map<Vertex, Vertex> predecessors;
private Map<Vertex, Integer> distance;
public DijkstraAlgorithm(Graph graph) {
// create a copy of the array so that we can operate on this array
this.nodes = new ArrayList<Vertex>(graph.getVertexes());
this.edges = new ArrayList<Edge>(graph.getEdges());
}
public void execute(Vertex source) {
settledNodes = new HashSet<Vertex>();
unSettledNodes = new HashSet<Vertex>();
distance = new HashMap<Vertex, Integer>();
predecessors = new HashMap<Vertex, Vertex>();
distance.put(source, 0);
unSettledNodes.add(source);
while (unSettledNodes.size() > 0) {
Vertex node = getMinimum(unSettledNodes);
settledNodes.add(node);
unSettledNodes.remove(node);
findMinimalDistances(node);
}
}
private void findMinimalDistances(Vertex node) {
List<Vertex> adjacentNodes = getNeighbors(node);
for (Vertex target : adjacentNodes) {
if (getShortestDistance(target) > getShortestDistance(node)
+ getDistance(node, target)) {
distance.put(target, getShortestDistance(node)
+ getDistance(node, target));
predecessors.put(target, node);
unSettledNodes.add(target);
}
}
}
private int getDistance(Vertex node, Vertex target) {
for (Edge edge : edges) {
if (edge.getSource().equals(node)
&& edge.getDestination().equals(target)) {
return edge.getWeight();
}
}
throw new RuntimeException("Should not happen");
}
private List<Vertex> getNeighbors(Vertex node) {
List<Vertex> neighbors = new ArrayList<Vertex>();
for (Edge edge : edges) {
if (edge.getSource().equals(node)
&& !isSettled(edge.getDestination())) {
neighbors.add(edge.getDestination());
}
}
return neighbors;
}
private Vertex getMinimum(Set<Vertex> vertexes) {
Vertex minimum = null;
for (Vertex vertex : vertexes) {
if (minimum == null) {
minimum = vertex;
} else {
if (getShortestDistance(vertex) < getShortestDistance(minimum)) {
minimum = vertex;
}
}
}
return minimum;
}
private boolean isSettled(Vertex vertex) {
return settledNodes.contains(vertex);
}
private int getShortestDistance(Vertex destination) {
Integer d = distance.get(destination);
if (d == null) {
return Integer.MAX_VALUE;
} else {
return d;
}
}
/*
* This method returns the path from the source to the selected target and
* NULL if no path exists
*/
public LinkedList<Vertex> getPath(Vertex target) {
LinkedList<Vertex> path = new LinkedList<Vertex>();
Vertex step = target;
// check if a path exists
if (predecessors.get(step) == null) {
return null;
}
path.add(step);
while (predecessors.get(step) != null) {
step = predecessors.get(step);
path.add(step);
}
// Put it into the correct order
Collections.reverse(path);
return path;
}
}
公共类dijkstra算法{
私有最终列表节点；
私人最终名单边缘；
私有集合结算节点；
私有集未结算节点；
私有地图；
私有地图距离；
公共Dijkstra算法（图）{
//创建阵列的副本，以便我们可以在此阵列上操作
this.nodes=newArrayList（graph.getVertexs（））；
this.edges=newarraylist（graph.getEdges（））；
}
公共void执行（顶点源）{
settledNodes=新HashSet（）；
unsetledNodes=新HashSet（）；
距离=新HashMap（）；
前辈=新的HashMap（）；
距离。放置（源，0）；
未设置的数据节点。添加（源）；
while（unsetledNodes.size（）>0）{
顶点节点=getMinimum（未设置的节点）；
结算节点。添加（节点）；
未设置的节点。删除（节点）；
findminimaldistance（节点）；
}
}
私有无效findMinimalDi