演算法導論 第十四章：區間樹
阿新 • • 發佈：2019-01-01
區間樹是一種對動態集合進行維護的紅黑樹，具體設計如下：
step1:基礎資料結構
我們選擇的基礎資料結構式紅黑樹，其中每個節點x包含一個區間域x.int,x的關鍵字為區間的低端點 x.int.low.
step2:附加資訊
每個節點x除了區間資訊外，還包含一個值x.max，即以x為根的的子樹中所有區間的斷點的最大值
step3:對資訊維護
必須驗證對含有n個節點的區間樹的插入和刪除都能在O(lgn)時間內完成，給定區間x.int 和 x.max 則有：x.max = max(x.int.high,x.left.max,x.right.max)
step4:設計新的操作
我們唯一需要的新操作是INTERVAL-SEARCH(T，i),用以找出樹T中覆蓋區間i的那個節點。虛擬碼如下：
執行時間：O(lgn).
構建的區間樹可以表示成如下：
完整程式碼如下：
#include<iostream> #include<iomanip> using namespace std; #define BLACK 0 #define RED 1 typedef struct interval{ int low; int high; }interval; typedef struct IntervalTNode{ int key; bool color; IntervalTNode *parent; IntervalTNode *left; IntervalTNode *right; interval inte; //additional information int max; //additional information }IntervalTNode; typedef struct IntervalTree{ IntervalTNode *root; }IntervalTree; //init sentine NIL interval 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; else return b>c ? b:c; } bool Overlap(interval a,interval b) { if(a.high < b.low || a.low > b.high) // a & b do not overlap return 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; else x=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; else cout<<" 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 y x->right=y->left; //turn y's left subtree into x's right subtree if(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; else x->parent->right=y; y->left=x; //put x on y's left x->parent=y; //maitaining additional information y->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 y x->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 y if(x->parent == NIL) T->root=y; else if(x == x->parent->left) x->parent->left=y; else x->parent->right=y; y->right=x; //put x on y's right x->parent=y; //Maintaining additional information y->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 1 y->color=BLACK; //case 1 z->parent->parent->color=RED; //case 1 z=z->parent->parent; //case 1 } else { if(z==z->parent->right) { z=z->parent; //case 2 Left_Rotate(T,z); //case 2 } z->parent->color=BLACK; //case 3 z->parent->parent->color=RED; //case 3 Right_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 x IntervalTNode *x=T->root; while(x != NIL) { x->max=max(x->max,z->max); //Maintaining the max value of each node from z up to root y=x; if(z->key < x->key) x=x->left; else x=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; else y->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 sibling if(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 black w->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 black w->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 red x->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" exchanged w=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; else y=IntervalT_Successor(z); //maintaining additional information g=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 node if(y->left !=NIL) x=y->left; else x=y->right; x->parent=y->parent; if(y->parent==NIL) T->root=x; else if(y==y->parent->left) y->parent->left=x; else y->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 "; else cout<<" 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; }
執行結果:
【注：若有錯誤，請指正~~~】