次优查找树

在有序序列的查找中，如果各个元素的查找概率都是一样的，那么二分查找是最快的查找算法，但是如果查找元素的查找概率是不一样的，那么用二分查找就不一定是最快的查找方法了，可以通过计算ASL来得知。

所以基于这种查找元素概率不想等的有序序列，可以通过构造最优二叉树的方法，使得该二叉树的带权路径长度最小，这样的二叉树的构造代价是非常大的，所以用一种近似的算法，构造次优查找树，该树的带权路径长度近似达到最小。

次优查找数的算法描述如下

下面是次优二叉树构造的一个算法

#include <iostream>#include <cmath>using namespace std;typedef struct treenode{char data;    int weight;    treenode *left;    treenode *right;}Treenode,*Treep;//初始化二叉树void init_tree(Treep &root){    root=NULL;    cout<<"初始化成功!"<<endl;}//创建二叉树void SecondOptimal(Treep &rt, char R[],int sw[], int low, int high){    //由有序表R[low....high]及其累积权值表sw(其中sw[0]==0)递归构造次优查找树T    int i=low;    int min = fabs(sw[high] - sw[low]);    int dw = sw[high] + sw[low-1];    for(int j=low+1; j<=high; ++j)        //选择最小的ΔPi值    {        if(fabs(dw-sw[j]-sw[j-1]) < min)        {            i=j;            min=fabs(dw-sw[j]-sw[j-1]);        }    }    rt=new Treenode;    rt->data=R[i];        //生成节点    if(i==low)            //左子树为空        rt->left = NULL;    else                //构造左子树        SecondOptimal(rt->left, R, sw, low, i-1);    if(i==high)            //右子树为空        rt->right = NULL;    else                //构造右子树        SecondOptimal(rt->right, R, sw, i+1, high);}//SecondOptimal//前序遍历二叉树void pre_order(Treep rt){    if(rt)    {        cout<<rt->data<<"  ";        pre_order(rt->left);        pre_order(rt->right);    }}//中序遍历二叉树void in_order(Treep rt){if(rt){in_order(rt->left);cout<<rt->data<<"  ";in_order(rt->right);}}//后序遍历二叉树void post_order(Treep rt){if(rt){post_order(rt->left);post_order(rt->right);cout<<rt->data<<"  ";}}//查找二叉树中是否存在某元素int seach_tree(Treep &rt,char key){    if(rt==NULL)         return 0;     else     {         if(rt->data==key)         {            return 1;        }        else        {            if(seach_tree(rt->left,key) || seach_tree(rt->right,key))                return 1;    //如果左右子树有一个搜索到，就返回1            else                return 0;    //如果左右子树都没有搜索到，返回0        }    }}

#include "tree.h"#include <iomanip>#include <iostream>using namespace std;int main(){    Treep root;    init_tree(root);        //初始化树int low=1,high=10;int *weight,*sw;char *R;    R=new char[high];    for(int i=low; i<high; i++)        R[i]='A'+i-1;    cout<<"构造次优查找树的点R[]:"<<endl;    for(i=low; i<high; i++)        cout<<setw(3)<<R[i]<<"  ";    cout<<endl;        weight=new int[high];    weight[0]=0;    weight[1]=1;    weight[2]=1;    weight[3]=2;    weight[4]=5;    weight[5]=3;    weight[6]=4;    weight[7]=4;    weight[8]=3;    weight[9]=5;    cout<<"构造次优查找树的点的权值weight[]:"<<endl;    for(i=low; i<high; i++)        cout<<setw(3)<<weight[i]<<"  ";    cout<<endl;        sw=new int[high];    sw[0]=0;    for(i=low; i<high; i++)    {        sw[i]=sw[i-1]+weight[i];    }    cout<<"构造次优查找树的点累积权值sw[]:"<<endl;    for(i=low; i<high; i++)        cout<<setw(3)<<sw[i]<<"  ";    cout<<endl;    //创建二叉树    SecondOptimal(root, R, sw, low, high-1);    cout<<"前序遍历序列是:"<<endl;    pre_order(root);    cout<<endl;    cout<<"中序遍历序列是:"<<endl;    in_order(root);    cout<<endl;cout<<"后序遍历序列是:"<<endl;    post_order(root);    cout<<endl;    //查找二叉树中是否存在某元素    cout<<"输入要查找的元素!"<<endl;    char ch;    cin>>ch;    if(seach_tree(root,ch)==1)        cout<<"yes!"<<endl;    else        cout<<"no!"<<endl;    return 0;}

运行结果

初始化成功!构造次优查找树的点R[]:  A    B    C    D    E    F    G    H    I构造次优查找树的点的权值weight[]:  1    1    2    5    3    4    4    3    5构造次优查找树的点累积权值sw[]:  1    2    4    9   12   16   20   23   28前序遍历序列是:F  D  B  A  C  E  H  G  I中序遍历序列是:A  B  C  D  E  F  G  H  I后序遍历序列是:A  C  B  E  D  G  I  H  F输入要查找的元素!Fyes!Press any key to continue