非旋treap

Posted 2021-03-25 chy-2003

目录

• 核心思想
• 核心操作
• 其他操作
• 参考程序

核心思想

主要的思想与treap是一样的。通过让二叉查找树同时满足堆（随机参数）的性质来防止深度过大。与普通treap不同的是非旋treap通过树的分裂与合并来实现这一点，而非旋转。

核心操作

Update
如果是要实现类似于 set<int> 的功能，可以不用这一部分。本文以 loj104 为例，我们需要在这里更新节点的 (Size) 信息。

void node::Update()
{
    Size = Count;
    if (LeftChild != NULL)
        Size += LeftChild->Size;
    if (RightChild != NULL)
        Size += RightChild->Size;
    return;
}

Split
将一个非旋treap按关键字 (x) 分成两个，其中一个树中关键字大小不超过 (b) ，另一个树中关键字严格大于 (x)。

std::pair<node *, node *> Split(node *Rt, int x)
{
    if (Rt == NULL)
        return std::pair<node *, node *>(NULL, NULL);
    if (x < Rt->Value)
    {
        std::pair<node *, node *> Temp = Split(Rt->LeftChild, x);
        Rt->LeftChild = Temp.second;
        Rt->Update();
        return std::pair<node *, node *>(Temp.first, Rt);
    }
    else
    {
        std::pair<node *, node *> Temp = Split(Rt->RightChild, x);
        Rt->RightChild = Temp.first;
        Rt->Update();
        return std::pair<node *, node *>(Rt, Temp.second);
    }
}

Merge
合并两棵非旋treap，其中一棵中关键字严格小于另外一棵，使得新的非旋treap同时满足二叉查找树和堆得性质。
可以递归实现，每次合并使随机的 priority 小的（或大的）在上即可。

node *Merge(node *l, node *r)
{
    if (l == NULL)
        return r;
    if (r == NULL)
        return l;
    if (l->Priority <= r->Priority)
    {
        l->RightChild = Merge(l->RightChild, r);
        l->Update();
        return l;
    }
    else
    {
        r->LeftChild = Merge(l, r->LeftChild);
        r->Update();
        return r;
    }
}

其他操作

Insert & Delete
首先查询是否需要改变节点的数量。如果不需要，直接修改 Size 即可。否则：

Insert: 将树分成 小于 (x) 和 大于 (x) 两部分，然后合并这两棵树和新节点 (x) 。
Delete：将树分成 小于 (x) 、等于 (x) 和大于 (x) 三个部分，然后删除等于 (x) 的部分并且合并 小于 (x) 的部分和 大于 (s) 的部分。

void Insert(int x)
{
    node *T = Find(x);
    if (T != NULL)
    {
        Update(x, 1);
        return;
    }
    std::pair<node *, node *> Temp = Split(Root, x);
    Temp.first = Merge(Temp.first, new node(x));
    Root = Merge(Temp.first, Temp.second);
    return;
}

int Delete(int x)
{
    node *T = Find(x);
    if (T == NULL)
        return 1;
    if (T->Count > 1)
    {
        Update(x, -1);
        return 0;
    }
    std::pair<node *, node *> Temp1 = Split(Root, x - 1);
    std::pair<node *, node *> Temp2 = Split(Temp1.second, x);
    delete Temp2.first;
    Root = Merge(Temp1.first, Temp2.second);
    return 0;
}

Rank & Query & Precursor & Successor
这些就和一般的二叉查找树差不多，就不赘述了。

参考程序

loj 104

#include <cstdio>
#include <algorithm>

const int INF = 1e7 + 10;

struct node
{
    int Value, Priority, Size, Count;
    node *LeftChild, *RightChild;
    node()
    {
        Value = Priority = Size = Count = 0;
        LeftChild = RightChild = NULL;
        return;
    }
    node(int _Value)
    {
        Value = _Value;
        Priority = rand();
        Size = Count = 1;
        LeftChild = RightChild = NULL;
        return;
    }
    inline void Update();
};
struct noneRotateTreap
{
    node *Root;
    noneRotateTreap()
    {
        Root = NULL;
        return;
    }
    inline std::pair<node *, node *> Split(node *Rt, int x);
    inline node *Merge(node *x, node *y);
    inline node *Find(int x);
    inline void Update(int x, int State);
    inline void Insert(int x);
    inline int Delete(int x);
    inline int Rank(int x);
    inline int Query(int x);
    inline int Precursor(int x);
    inline int Successor(int x);
};
noneRotateTreap NoneRotateTreap;

int main()
{
    srand((unsigned long long)"非旋treap呀");
    int n, Opt, x;
    scanf("%d", &n);
    for (; n; --n)
    {
        scanf("%d%d", &Opt, &x);
        switch (Opt)
        {
        case 1:
            NoneRotateTreap.Insert(x);
            break;
        case 2:
            NoneRotateTreap.Delete(x);
            break;
        case 3:
            printf("%d
", NoneRotateTreap.Rank(x));
            break;
        case 4:
            printf("%d
", NoneRotateTreap.Query(x));
            break;
        case 5:
            printf("%d
", NoneRotateTreap.Precursor(x));
            break;
        case 6:
            printf("%d
", NoneRotateTreap.Successor(x));
        default:
            break;
        }
    }
    return 0;
}

inline void node::Update()
{
    Size = Count;
    if (LeftChild != NULL)
        Size += LeftChild->Size;
    if (RightChild != NULL)
        Size += RightChild->Size;
    return;
}

inline std::pair<node *, node *> noneRotateTreap::Split(node *Rt, int x)
{
    if (Rt == NULL)
        return std::pair<node *, node *>(NULL, NULL);
    if (x < Rt->Value)
    {
        std::pair<node *, node *> Temp = Split(Rt->LeftChild, x);
        Rt->LeftChild = Temp.second;
        Rt->Update();
        return std::pair<node *, node *>(Temp.first, Rt);
    }
    else
    {
        std::pair<node *, node *> Temp = Split(Rt->RightChild, x);
        Rt->RightChild = Temp.first;
        Rt->Update();
        return std::pair<node *, node *>(Rt, Temp.second);
    }
}

inline node *noneRotateTreap::Merge(node *l, node *r)
{
    if (l == NULL)
        return r;
    if (r == NULL)
        return l;
    if (l->Priority <= r->Priority)
    {
        l->RightChild = Merge(l->RightChild, r);
        l->Update();
        return l;
    }
    else
    {
        r->LeftChild = Merge(l, r->LeftChild);
        r->Update();
        return r;
    }
}

inline node *noneRotateTreap::Find(int x)
{
    node *Rt = Root;
    while (Rt)
    {
        if (Rt->Value == x)
            return Rt;
        if (x < Rt->Value)
            Rt = Rt->LeftChild;
        else
            Rt = Rt->RightChild;
    }
    return NULL;
}

inline void noneRotateTreap::Update(int x, int State)
{
    node *Rt = Root;
    while (Rt)
    {
        Rt->Size += State;
        if (Rt->Value == x)
        {
            Rt->Count += State;
            return;
        }
        if (x < Rt->Value)
            Rt = Rt->LeftChild;
        else
            Rt = Rt->RightChild;
    }
    return;
}

inline void noneRotateTreap::Insert(int x)
{
    node *T = Find(x);
    if (T != NULL)
    {
        Update(x, 1);
        return;
    }
    std::pair<node *, node *> Temp = Split(Root, x);
    Temp.first = Merge(Temp.first, new node(x));
    Root = Merge(Temp.first, Temp.second);
    return;
}

inline int noneRotateTreap::Delete(int x)
{
    node *T = Find(x);
    if (T == NULL)
        return 1;
    if (T->Count > 1)
    {
        Update(x, -1);
        return 0;
    }
    std::pair<node *, node *> Temp1 = Split(Root, x - 1);
    std::pair<node *, node *> Temp2 = Split(Temp1.second, x);
    delete Temp2.first;
    Root = Merge(Temp1.first, Temp2.second);
    return 0;
}

#define LCS (Rt->LeftChild ? Rt->LeftChild->Size : 0)

inline int noneRotateTreap::Rank(int x)
{
    node *Rt = Root;
    int Ans = 0;
    while (Rt)
    {
        if (Rt->Value == x)
            return Ans + LCS + 1;
        if (x < Rt->Value)
            Rt = Rt->LeftChild;
        else
            Ans += LCS + Rt->Count, Rt = Rt->RightChild;
    }
    return Ans;
}

inline int noneRotateTreap::Query(int x)
{
    node *Rt = Root;
    while (Rt)
    {
        if (LCS < x && x <= LCS + Rt->Count)
            return Rt->Value;
        if (x <= LCS)
            Rt = Rt->LeftChild;
        else
            x -= LCS + Rt->Count, Rt = Rt->RightChild;
    }
    return 0;
}

#undef LCS

inline int noneRotateTreap::Precursor(int x)
{
    int Ans = INF;
    node *Rt = Root;
    while (Rt)
    {
        if (Rt->Value < x)
            Ans = Rt->Value, Rt = Rt->RightChild;
        else
            Rt = Rt->LeftChild;
    }
    return Ans;
}

inline int noneRotateTreap::Successor(int x)
{
    int Ans = -INF;
    node *Rt = Root;
    while (Rt)
    {
        if (Rt->Value > x)
            Ans = Rt->Value, Rt = Rt->LeftChild;
        else
            Rt = Rt->RightChild;
    }
    return Ans;
}