二叉堆(以最小堆为例),其具有结构性质和堆序性质
结构性质: 堆是一棵完全的二叉树,一颗高为h的完全二叉树有2^h到2^h-1个节点,高度为log N
而且该结构可以很容易的使用数组来表示:对于数组中任一位置i上的元素,其左儿子在位置2i上,右儿子在2i+1,其父节点在[x/2]处
堆序性质:在一个堆中,对于每一个节点X,X的父亲中的关键字小于或等于X中的关键字
也就是说:最小元总可以在根处找到
主要的操作为插入和删除:
以数组存储为例,算法在代码中体现:
/**
* 向堆中插入元素x,
* 利用堆的性质,在一个堆中,对于每一个节点X,X的父亲中的关键字都小于或者等于X中的关键字,
* ps根节点除外(根节点没有父节点),时间复杂度为logN
* step1:如果堆没有满,在完全二叉树的下一个位置插入一个空穴
* step2:判断空穴是否存在父节点,如果不存在,直接插入;否则,step3;
* step3:(x.value>=[x/2].value)?step4:step5;
* step4:将X直接放在该空穴,return
* step5:将父节点的值移入空穴中,空穴就朝着根的方向上前进,回到step2;
* @param x
*/
public void insert(Comparable x){
if(cursize == array.length-1){
//堆已经满了,需要重新调整
rebuild();
}
if(cursize==0){
//没有父节点
array[1] = x;
cursize++;
}else{
int temp = ++cursize;
while(temp>1 && x.compareTo(array[temp/2])<0){
//父节点下移
array[temp] =array[temp/2];
temp/=2;
}
//空穴插入
array[temp] = x;
}
}
删除操作:
/**
* 删除堆中的最小元素并返回,方式与插入向反,时间复杂度为logN
* step1:将根节点出视为空穴X
* step2:if(空穴X的左右子树都存在) step3;else if(空穴只存在左子树) step4 ;else step5
* step3:if(空穴的X的左子树2X>X的右子树2X+1) 空穴<-->右子树;else 空穴<-->左子树;finally 继续step2
* step4:空穴和左子树交换,空穴已就位,且满足完全二叉树的要求
* step5:空穴和最后一个元素交换位置
* @return
*/
public Comparable deleteMin(){
if(isEmpty())
return null;
Comparable min = array[1];
int temp = 1;
while(2*temp<=cursize){
if(2*temp+1<=cursize){
//左右子树都存在
if(array[2*temp].compareTo(array[2*temp+1])>0){
array[temp] = array[2*temp+1];
temp = 2*temp+1;
}else{
array[temp] = array[2*temp];
temp = 2*temp;
}
}else{
//只存在左子树
array[temp] = array[2*temp];
temp = 2*temp;
}
}
array[temp] = array[cursize--];
return min;
}
测试代码:
@Test
public void test() {
BinaryHeap heap = new BinaryHeap();
for(int i = 19;i>1;i--){
heap.insert(i);
// heap.printHeap();
}
while(!heap.isEmpty()){
System.out.print(heap.deleteMin()+"\t");
}
}
结果: 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" alt="" />
好饿啊。。。。。。。。。。。。。。