区间树

《算法导论》描述了一个关于区间树的重叠搜索，这里简单描述下原理，然后给出代码。

区间树是建立在红黑树的基础上，额外维护了一个信息域。在《算法导论》中，已经给出了任何额外信息域的维护，是相似的证明。所以，建议不懂得，先试着实现一个基本的，带size域的红黑树（书上已经给出原理），然后再扩展到区间树。下面是代码。

定义区间树

class rb_tree {//区间树
public:
    typedef struct _rb_interval {
        _rb_interval(int _low, int _high):low(_low), high(_high){}
        int low;
        int high;
    }rb_interval, *prb_interval;
    typedef struct _rb_type {
        _rb_type(_rb_type *_left, _rb_type *_right, _rb_type *_p, bool cl, _rb_interval _inte) :
            left(_left), right(_right), p(_p), color(cl), inte(_inte), max(_inte.high) {}
        bool color;//true for red, false for black
        int max;//区间上限
        rb_interval inte;//区间范围
        _rb_type *left, *right, *p;
    }rb_type, *prb_type;
    rb_tree(_rb_interval *A, int n) :root(NULL) {
        for (int i = 0; i < n; i++)
            this->rb_insert(A[i]);
    }
    ~rb_tree() {
        rb_empty(root);
    }
    void left_rotate(prb_type x);
    void right_rotate(prb_type x);
    void rb_insert(rb_interval _inte);
    prb_type rb_max(prb_type x);
    prb_type rb_min(prb_type x);
    prb_type rb_search(rb_interval _inte);//《算法导论》给出的重叠查找
    prb_type rb_search_exact(rb_interval _inte);//精确查找，删除节点需要
    prb_type rb_next(rb_interval _inte);
    prb_type rb_prev(rb_interval _inte);
    void rb_delete(rb_interval _inte);
    void rb_empty(prb_type x);//后续全部删除
    prb_type rb_root();
    void rb_show(prb_type x);
private:
    bool overlap(rb_interval _x, rb_interval _y);
    int max(int a, int b, int c);
    int max(int a, int b);
    void rb_insert_fixup(prb_type x);
    void rb_delete_fixup(prb_type x);
    //测试使用
    int rb_max_depth(prb_type x);
    int rb_min_depth(prb_type x);
    prb_type root;
};

各成员函数实现

left_rotate、right_rotate成员函数，在本身的红黑树基础上，多了一个max域的维护

void rb_tree::left_rotate(typename rb_tree::prb_type x) {
    prb_type y = x->right;//y非空
    x->right = y->left;
    if (y->left) y->left->p = x;//交换子节点
    y->p = x->p;//更新父节点
    if (x->p == NULL)//将y连接到x的父节点
        root = y;
    else {
        if (x == x->p->left)
            x->p->left = y;
        else
            x->p->right = y;
    }
    y->left = x;
    x->p = y;
    //阶段二更新max
    y->max = x->max;
    x->max = this->max(x->inte.high, x->left ? x->left->max : 0, x->right ? x->right->max : 0);
}

void rb_tree::right_rotate(typename rb_tree::prb_type x) {
    prb_type y = x->left;
    x->left = y->right;
    if (y->right) y->right->p = x;
    y->p = x->p;
    if (x->p == NULL)
        root = y;
    else {
        if (x == x->p->left)
            x->p->left = y;
        else
            x->p->right = y;
    }
    y->right = x;
    x->p = y;
    //阶段二更新max
    y->max = x->max;
    x->max = this->max(x->inte.high, x->left ? x->left->max : 0, x->right ? x->right->max : 0);
}

rb_min、rb_max成员函数，相比红黑树，没什么变化

typename rb_tree::prb_type rb_tree::rb_max(typename rb_tree::prb_type x) {
    if (x == NULL) return NULL;
    while (x->right) x = x->right;
    return x;
}

typename rb_tree::prb_type rb_tree::rb_min(typename rb_tree::prb_type x) {
    if (x == NULL) return NULL;
    while (x->left) x = x->left;
    return x;
}

rb_search成员函数，此函数原理，由《算法导论》给出描述。这个可以用于实际应用，但是不能用于删除，因为这个函数只检测重叠的区间。

typename rb_tree::prb_type rb_tree::rb_search(typename rb_tree::rb_interval _inte) {
    prb_type x = root;
    while (x && !overlap(_inte, x->inte)) {
        if (x->left && x->left->max >= _inte.low)
            x = x->left;
        else
            x = x->right;
    }
    return x;
}

rb_search_exact成员函数，基于上面rb_search函数的描述，为了之后能精准删除所有节点，再实现一个精准查找。

typename rb_tree::prb_type rb_tree::rb_search_exact(typename rb_tree::rb_interval _inte) {
    prb_type x = root;
    while (x && !(x->inte.low == _inte.low && x->inte.high == _inte.high)) {
        if (_inte.low < x->inte.low)
            x = x->left;
        else
            x = x->right;
    }
    return x;
}

rb_next、rb_prev成员函数

typename rb_tree::prb_type rb_tree::rb_next(typename rb_tree::rb_interval _inte) {
    prb_type x = rb_search_exact(_inte), y;
    if (x == NULL) return NULL;
    if (x->right)
        return rb_min(x->right);
    y = x->p;
    while (y != NULL && y->right == x) {//没有则返回NULL
        x = y;
        y = y->p;
    }
    return y;
}

typename rb_tree::prb_type rb_tree::rb_prev(typename rb_tree::rb_interval _inte) {
    prb_type x = rb_search_exact(_inte), y;
    if (x == NULL) return NULL;
    if (x->left)
        return rb_max(x->left);
    y = x->p;
    while (y != NULL && y->left == x) {
        x = y;
        y = y->p;
    }
    return y;
}

rb_insert函数，有第一阶段额外信息域的维护

void rb_tree::rb_insert(typename rb_tree::rb_interval _inte) {
    prb_type y = NULL, x = root, z = new rb_type(NULL, NULL, NULL, true,_inte);
    while (x != NULL) {
        y = x;
        x->max = this->max(x->max, z->max);//阶段一更新max
        if (_inte.low < x->inte.low)
            x = x->left;
        else
            x = x->right;
    }
    z->p = y;
    if (y == NULL)
        root = z;
    else {
        if (_inte.low < y->inte.low)
            y->left = z;
        else
            y->right = z;
    }
    rb_insert_fixup(z);
}

rb_insert_fixup成员函数，插入后修复，和红黑树相比，没有变化，原因参考《算法导论》

void rb_tree::rb_insert_fixup(typename rb_tree::prb_type x) {
    prb_type y;
    while (x->p && x->p->color) {//红色
        if (x->p == x->p->p->left) {//父节点存在，一定存在祖父节点
            y = x->p->p->right;
            //无法满足性质4
            if (!y || y->color) {//若y为NULL，默认不存在的节点是黑色
                x->p->color = false;
                if (y) y->color = false;
                x->p->p->color = true;
                x = x->p->p;//重新设置z节点
            }
            else if (x == x->p->right) { //无法满足性质5
                x = x->p;
                left_rotate(x);
            }
            if (x->p && x->p->p) {//保证存在
                x->p->color = false;
                x->p->p->color = true;
                right_rotate(x->p->p);
            }
        }
        else {//和上面左节点相反
            y = x->p->p->left;
            if (!y || y->color) {
                x->p->color = false;
                if (y) y->color = false;
                x->p->p->color = true;
                x = x->p->p;//重新设置z节点
            }
            else if (x == x->p->left) {
                x = x->p;
                right_rotate(x);
            }
            if (x->p && x->p->p) {
                x->p->color = false;
                x->p->p->color = true;
                left_rotate(x->p->p);
            }
        }
    }
    root->color = false;
}

rb_delete函数，有第一阶段，额外信息域的维护

void rb_tree::rb_delete(typename rb_tree::rb_interval _inte) {
    prb_type z = rb_search_exact(_inte), y, x;
    if (z == NULL) return;
    if (z->left == NULL || z->right == NULL)//y是待删除的节点
        y = z;//z有一个子节点
    else
        y = rb_next(_inte);//z有两个子节点，后继和前趋保证了y有一个或没有子节点
    if (y->left != NULL)
        x = y->left;
    else
        x = y->right;
    if (x != NULL) //存在一个子节点，先更正父子关系
        x->p = y->p;
    if (y->p == NULL)//再决定是在左或者右节点
        root = x;
    else {
        if (y->p->left == y)
            y->p->left = x;
        else
            y->p->right = x;
    }
    if (y != z)//处理两个子节点的交换
        z->inte = y->inte;
    //更新max
    z = y->p;
    while (z) {
        z->max = this->max(z->max, z->left ? z->left->max : 0, z->right ? z->right->max : 0);
        z = z->p;
    }
    if (!y->color)//黑色
        rb_delete_fixup(x);
    delete y;
}

rb_delete_fixup成员函数，没有任何变化

void rb_tree::rb_delete_fixup(typename rb_tree::prb_type x) {
    prb_type w;
    while (x && x != root && !x->color) {//黑色
        if (x == x->p->left) {
            w = x->p->right;
            if (w->color) {//红色
                w->color = false;
                x->p->color = true;
                left_rotate(x->p);
                w = x->p->right;
            }
            if ((!w->left && !w->right) || (!w->left->color && !w->right->color)) {//双黑
                w->color = true;
                x = x->p;
            }
            else {
                if (!w->right->color) {//单黑
                    w->left->color = false;
                    w->color = true;
                    right_rotate(w);
                    w = x->p->right;
                }
                w->color = x->p->color;
                x->p->color = false;
                w->right->color = false;
                left_rotate(x->p);
                x = root;
            }
        }
        else {//相反的情况
            w = x->p->left;
            if (w->color) {//红色
                w->color = false;
                x->p->color = true;
                right_rotate(x->p);
                w = x->p->left;
            }
            if ((!w->left && !w->right) || (!w->left->color && !w->right->color)) {//双黑
                w->color = true;
                x = x->p;
            }
            else {
                if (!w->left->color) {//单黑
                    w->right->color = false;
                    w->color = true;
                    left_rotate(w);
                    w = x->p->left;
                }
                w->color = x->p->color;
                x->p->color = false;
                w->left->color = false;
                right_rotate(x->p);
                x = root;
            }
        }
    }
    if (x) x->color = false;//巧妙处理，默认黑
}

rb_empty成员函数，清空所有节点

void rb_tree::rb_empty(typename rb_tree::prb_type x) {
    if (x != NULL) {
        rb_empty(x->left);
        rb_empty(x->right);
        printf("
--------------[%d,%d]---------
",x->inte.low,x->inte.high);
        rb_delete(x->inte);//后续保证子叶为空
        rb_show(root);
    }
}

typename rb_tree::prb_type rb_tree::rb_root() {
    return root;
}

三个辅助函数，overlap,max（有重载）

bool rb_tree::overlap(typename rb_tree::rb_interval _x, typename rb_tree::rb_interval _y) {//闭区间
    if (_x.high < _y.low || _x.low > _y.high)     // _x 和 _y 没有重叠
        return false;
    return true;
}

int rb_tree::max(int a, int b, int c) {
    if (a>b)
        return a>c ? a : c;
    else
        return b>c ? b : c;
}

int rb_tree::max(int a, int b) {
    return a > b ? a : b;
}

用于测试各成员函数是否正确的相关成员函数

void rb_tree::rb_show(typename rb_tree::prb_type x) {
    if (x != NULL) {
        rb_show(x->left);
        if (x == root)
            printf("root: (%s)[%d,%d], max=%d, (%d,%d)
", root->color ? "red" : "black", x->inte.low, x->inte.high, x->max,
                rb_max_depth(x), rb_min_depth(x));
        else
            printf("(%s)[%d,%d], max=%d, (%d,%d)
", x->color ? "red" : "black", x->inte.low, x->inte.high, x->max,
                rb_max_depth(x), rb_min_depth(x));
        rb_show(x->right);
    }
}

int rb_tree::rb_max_depth(typename rb_tree::prb_type x) {
    if (x == NULL)
        return 0;
    int l = rb_max_depth(x->left);
    int r = rb_max_depth(x->right);
    return (l > r ? l : r) + 1;
}

int rb_tree::rb_min_depth(typename rb_tree::prb_type x) {
    if (x == NULL)
        return 0;
    int l = rb_min_depth(x->left);
    int r = rb_min_depth(x->right);
    return (l < r ? l : r) + 1;
}

所有代码均经过测试，结果正确！！！