算法导论 第十四章：区间树

区间树是一种对动态集合进行维护的红黑树，具体设计如下：

step1:基础数据结构

   我们选择的基础数据结构式红黑树，其中每个节点x包含一个区间域x.int,x的关键字为区间的低端点 x.int.low.

step2:附加信息

    每个节点x除了区间信息外，还包含一个值x.max，即以x为根的的子树中所有区间的断点的最大值

step3:对信息维护

   必须验证对含有n个节点的区间树的插入和删除都能在O(lgn)时间内完成，给定区间x.int 和 x.max 则有：x.max = max(x.int.high,x.left.max,x.right.max)

step4:设计新的操作

   我们唯一需要的新操作是INTERVAL-SEARCH(T，i),用以找出树T中覆盖区间i的那个节点。伪代码如下：

运行时间：O(lgn).

构建的区间树可以表示成如下：

完整代码如下：

#include<iostream>#include<iomanip>using namespace std;#define BLACK 0#define RED 1typedef struct interval{int low;int high;}interval;typedef struct IntervalTNode{int key;bool color;IntervalTNode *parent;IntervalTNode *left;IntervalTNode *right;interval inte;  //additional informationint max;        //additional information}IntervalTNode;typedef struct IntervalTree{IntervalTNode *root;}IntervalTree;//init sentine NILinterval interval0={-1,-1};IntervalTNode NILL={-1,BLACK,NULL,NULL,NULL,interval0,-1};IntervalTNode *NIL=&NILL;          /*-----------------------------------------------------------------------*/int Max(int a,int b,int c){if(a>b)return a>c ? a:c;elsereturn b>c ? b:c;}bool Overlap(interval a,interval b){if(a.high < b.low || a.low > b.high)     // a & b do not overlapreturn 0;return 1;}IntervalTNode *IntervalT_Search(IntervalTree *T,interval i){IntervalTNode *x=T->root;while(x!=NIL && !Overlap(i,x->inte)){ if(x->left !=NIL && x->left->max>= i.low)x=x->left;elsex=x->right;}return x;}/*-----------------------------------------------------------------------*/void IntervalT_InorderWalk(IntervalTNode *x){if(x!=NIL)  {  IntervalT_InorderWalk(x->left);cout<<"["<<setw(3)<<x->inte.low<<setw(3)<<x->inte.high<<"  ]";if(x->color==1)cout<<"     Red       "<<x->max<<endl;elsecout<<"     Black     "<<x->max<<endl;IntervalT_InorderWalk(x->right); } }IntervalTNode *IntervalT_Minimum(IntervalTNode *x){while(x->left != NIL)x=x->left;return x;}IntervalTNode *IntervalT_Successor(IntervalTNode *x){     if(x->right != NIL)         return IntervalT_Minimum(x->right);     IntervalTNode *y = x->parent;      while(y != NIL && x  == y->right){         x = y;         y = y->parent;    }     return y;}void Left_Rotate(IntervalTree *T,IntervalTNode *x){IntervalTNode *y=x->right;    //set yx->right=y->left;       //turn y's left subtree into x's right subtreeif(y->left!=NIL)y->left->parent=x;y->parent=x->parent;     //link x's parent to y;if(x->parent == NIL)T->root=y;else if(x==x->parent->left)x->parent->left=y;elsex->parent->right=y;y->left=x;               //put x on y's leftx->parent=y;//maitaining additional informationy->max=x->max;x->max=Max(x->inte.high,x->left->max,x->right->max);}void Right_Rotate(IntervalTree *T,IntervalTNode *x){IntervalTNode *y=x->left;      //set yx->left=y->right;   //link x's left tree into y's right subtree;if(y->right !=NIL)y->right->parent=x;y->parent=x->parent;    //link x's parent to yif(x->parent == NIL)T->root=y;else if(x == x->parent->left)x->parent->left=y;elsex->parent->right=y;y->right=x;         //put x on y's rightx->parent=y;//Maintaining additional informationy->max=x->max;x->max=Max(x->inte.high,x->left->max,x->right->max);}void IntervalT_InsertFixup(IntervalTree *T,IntervalTNode *z){while(z->parent->color==RED){if(z->parent == z->parent->parent->left)    {IntervalTNode *y=z->parent->parent->right;   if(y->color==RED){  z->parent->color=BLACK;            //case 1y->color=BLACK;                    //case 1z->parent->parent->color=RED;      //case 1z=z->parent->parent;               //case 1}else{   if(z==z->parent->right){ z=z->parent;                    //case 2Left_Rotate(T,z);               //case 2 }z->parent->color=BLACK;             //case 3z->parent->parent->color=RED;       //case 3Right_Rotate(T,z->parent->parent);  //case 3 }  }else{//a me as then clause with "right" and "left" exchanged  IntervalTNode *y=z->parent->parent->left;if(y->color==RED){z->parent->color==BLACK;y->color=BLACK;z->parent->parent->color=RED;z=z->parent->parent;  }else{if(z==z->parent->left){z=z->parent;Right_Rotate(T,z);  }z->parent->color=BLACK;z->parent->parent->color=RED;Left_Rotate(T,z->parent->parent);  } } }   T->root->color=BLACK;      //turn the root to BLACK}void IntervalT_Insert(IntervalTree *T,interval inte){IntervalTNode *z=new IntervalTNode();z->key=inte.low;z->max=inte.high;z->inte=inte;z->color =RED;   z->parent=NIL;z->left=NIL;z->right=NIL;IntervalTNode *y=NIL;        //y is the parent of xIntervalTNode *x=T->root;while(x != NIL){ x->max=max(x->max,z->max);     //Maintaining the max value of each node from z up to rooty=x;if(z->key < x->key)x=x->left;elsex=x->right;}   z->parent=y;   //link new node's parent node to y(y's child is NIL)if(y==NIL)T->root=z;else if(z->key < y->key)y->left=z;elsey->right =z;IntervalT_InsertFixup(T,z);}void IntervalT_DeleteFixup(IntervalTree *T,IntervalTNode *x){IntervalTNode *w;while(x!=T->root && x->color==BLACK){if(x==x->parent->left) {w=x->parent->right;  //set w to x's siblingif(w->color==RED)      //case 1:x's sibling w is red{w->color=BLACK;x->parent->color=RED;Left_Rotate(T,x->parent);w=x->parent->right; }if(w->left->color==BLACK && w->right->color==BLACK)  { //case 2:x's sibling w is black and both of w's children are blackw->color=RED;x=x->parent;}else{ if(w->right->color==BLACK)     {//case 3:x's sibling w is black,w's left child is red, and w's right child is blackw->left->color=BLACK;w->color=RED;Right_Rotate(T,w);w=x->parent->right; }w->color=x->parent->color;      //case 4: x's sibling w is black,and w's right child is redx->parent->color=BLACK;         //.w->right->color=BLACK;         // .Left_Rotate(T,x->parent);      // .x=T->root;                     //case 4}}else{//Same as then clause with "right" and "left" exchangedw=x->parent->left;if(w->color==RED){w->color=BLACK;x->parent->color=RED;Right_Rotate(T,x->parent);w=x->parent->left;} if(w->left->color==BLACK && w->right->color==BLACK){ w->color=RED;x=x->parent;}else{if(w->left->color==BLACK){w->right->color=BLACK;w->color=RED;Left_Rotate(T,w);w=x->parent->left;  }w->color=x->parent->color;x->parent->color=BLACK;w->left->color=BLACK;Right_Rotate(T,x->parent);x=T->root; }  }  } x->color=BLACK;}void IntervalT_Delete(IntervalTree *T,IntervalTNode *z){IntervalTNode *x=NULL,*y=NULL,*g=NULL;if(z->left == NIL || z->right==NIL)y=z;elsey=IntervalT_Successor(z);//maintaining additional informationg=y->parent;g->max=g->inte.high;g=g->parent;while(g->max==y->max){            g->max=Max(g->max,g->left->max,g->right->max);g=g->parent;}//delete y nodeif(y->left !=NIL)x=y->left;elsex=y->right;x->parent=y->parent;if(y->parent==NIL)T->root=x;else if(y==y->parent->left)y->parent->left=x;elsey->parent->right=x;if(y != z)z->key=y->key;if(y->color==BLACK)IntervalT_DeleteFixup(T,x);}int main(){interval A[]={{16,21},{8,9},{25,30},{5,8},{15,23},{17,19},{26,26},{0,3},{6,10},{19,20}};int n=sizeof(A)/sizeof(interval);cout<<"/*---------------------Create Interval Tree-------------------*/"<<endl;IntervalTree *T=new IntervalTree();T->root=NIL;for(int i=0;i<n;i++)IntervalT_Insert(T,A[i]);cout<<"The interval tree is:"<<endl;IntervalT_InorderWalk(T->root);cout<<"The root of the tree is:"<<T->root->inte.low<<"   "<<T->root->inte.high<<endl;cout<<"/*-------------------------------------------------------------*/"<<endl;cout<<"/*--------------------Searching Interval Tree------------------*/"<<endl;interval sInt;cout<<"Please input the searching interval:";cin>>sInt.low>>sInt.high;IntervalTNode *sITNode=NIL;sITNode=IntervalT_Search(T,sInt);if(sITNode==NIL)cout<<"The searching interval doesn't exist in the tree."<<endl;else{cout<<"The overlap interval is:"<<endl;cout<<"["<<sITNode->inte.low<<"  "<<sITNode->inte.high<<"]";if(sITNode->color==0)cout<<"   color:RED     ";elsecout<<"   color:BLACK   ";cout<<"Max:"<<sITNode->max<<endl;}cout<<"/*------------------Deleting INterval Tree--------------------*/"<<endl;interval dInt;cout<<"Please input the deleting interval:";cin>>dInt.low>>dInt.high;IntervalTNode  *dITNode=NIL;dITNode=IntervalT_Search(T,dInt);if(dITNode==NIL)cout<<"The deleting interval doesn't exist in the tree."<<endl;else{ IntervalT_Delete(T,dITNode);cout<<"After deleting ,the interval tree is:"<<endl;IntervalT_InorderWalk(T->root);cout<<"The root of the tree is:"<<T->root->inte.low<<"   "<<T->root->inte.high<<endl;}cout<<"/*------------------------------------------------------------*/"<<endl;return 0;}

运行结果:

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