Splay树小结

http://blog.csdn.net/acm_cxlove/article/details/7815019 附上cxlove大神的博客链接，最近是跟着上面的题目做的。

刚学的时候感觉好难，显示照着大白皮上面敲的模板，然后网上各种看论文，感觉指针写还是太繁琐，还是直接数组写的精炼。感觉Splay树确实强大，可以解决好多序列问题。感觉Splay树的核心是Key_Tree = ch[ch[root][1]][0]这个东西的使用，因为这棵子树就代表了你当前要修改，或者查询的区间，当然单点修改也可以这样做。

Splay树的核心操作有两个：一个是Splay操作（把当前点转到根节点的操作），一个是把第k个点转到根（其实不一定是根）。

下面附上这两种操作的模板：

kind == 0 表示左旋，kind == 1表示右旋

    void Rotate(int x,int kind){
        int y = pre[x];
        Push_Down(y);
        Push_Down(x);
        ch[y][!kind] = ch[x][kind];
        pre[ch[x][kind]] = y;
        if(pre[y]){
            ch[pre[y]][ch[pre[y]][1] == y] = x;
        }
        pre[x] = pre[y];
        ch[x][kind] = y;
        pre[y] = x;
        Push_Up(y);
    }

    //把x节点旋转到goal下面
    void Splay(int x,int goal){
        Push_Down(x);
        while(pre[x] != goal){
            Push_Down(pre[pre[x]]); Push_Down(pre[x]);  Push_Down(x);
            if(pre[pre[x]] == goal){
                Rotate(x,ch[pre[x]][0] == x);
            }
            else{
                int y = pre[x];
                int kind = (ch[pre[y]][0] == y);
                if(ch[y][kind] == x){
                    Rotate(x,!kind);
                    Rotate(x,kind);
                }
                else{
                    Rotate(y,kind);
                    Rotate(x,kind);
                }
            }
        }
        Push_Up(x);
        if(!goal)   root = x;
    }

    ///把第k个节点转到goal下面
    void RotateTo(int k,int goal){
        int x = root;
        Push_Down(x);
        while(sz[ch[x][0]] != k){
            if(k < sz[ch[x][0]]){
                x = ch[x][0];
            }
            else{
                k -= (sz[ch[x][0]]+1);
                x = ch[x][1];
            }
            Push_Down(x);
        }
        Splay(x,goal);
    }

上面的模板入门不能好好理解得话，画几张图就会了。

区间操作的时候，查询或者修改[l,r]这段区间，那么我们把RotateTo(l-1,0),RotateTo(r+1,root)那么，此时Key_Tree的子树就代表这段要查询或者修改的区间。这个是伸展树里面区间操作的核心的核心！！！

然后区间操作还有一个关键的地方就是：Push_Up,Push_Down操作，和线段树很像。还有就是初始化了，一般我都是在序列头，序列尾多加一个节点，这样就能保证不会越界了，还有就是初始化的时候最好把0这种节点的值附为一个不会影响答案的值。

下面附上几道题目（可以跟着练手）：

HYSBZ 1588 营业额统计：

营业额统计

HDU 3436 Queue_jumpers

这题主要要把一个区间缩成一个点。写起来可能稍微麻烦一点

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
#include <map>
#define LL long long
#define FOR(i,x,y)  for(int i = x;i < y;i ++)
#define IFOR(i,x,y) for(int i = x;i > y;i --)
#define Key_Tree ch[ch[root][1]][0]
#define MAXN 110000

using namespace std;

int n,m;
int TQ[MAXN],cnt_TQ;
int N,ssz[MAXN],l[MAXN];
map <int,int> mat;

struct Commends{
    char str[10];
    int num;
}cmd[MAXN];

struct SplayTree{
    int ch[MAXN][2],sz[MAXN],pre[MAXN],key[MAXN],root,tot,sum[MAXN],val[MAXN],pos[MAXN];
    int flip[MAXN];

    void Rotate(int x,int kind){
        int y = pre[x];
        Push_Down(y);
        Push_Down(x);
        ch[y][!kind] = ch[x][kind];
        pre[ch[x][kind]] = y;
        if(pre[y]){
            ch[pre[y]][ch[pre[y]][1] == y] = x;
        }
        pre[x] = pre[y];
        ch[x][kind] = y;
        pre[y] = x;
        Push_Up(y);
    }

    void Splay(int x,int goal){
        Push_Down(x);
        while(pre[x] != goal){
            Push_Down(pre[pre[x]]); Push_Down(pre[x]);  Push_Down(x);
            if(pre[pre[x]] == goal){
                Rotate(x,ch[pre[x]][0] == x);
            }
            else{
                int y = pre[x];
                int kind = (ch[pre[y]][0] == y);
                if(ch[y][kind] == x){
                    Rotate(x,!kind);
                    Rotate(x,kind);
                }
                else{
                    Rotate(y,kind);
                    Rotate(x,kind);
                }
            }
        }
        Push_Up(x);
        if(!goal)   root = x;
    }

    void RotateTo(int k,int goal){
        int x = root;
        Push_Down(x);
        while(sz[ch[x][0]] != k){
            if(k < sz[ch[x][0]]){
                x = ch[x][0];
            }
            else{
                k -= (sz[ch[x][0]]+1);
                x = ch[x][1];
            }
            Push_Down(x);
        }
        Splay(x,goal);
    }


    //debug部分copy from hh
    void Treaval(int x) {
        if(x) {
            Treaval(ch[x][0]);
            printf("结点%2d:左儿子 %2d 右儿子 %2d 父结点 %2d size = %2d ,key = %2d,sum = %2d,val = %2d 
",x,ch[x][0],ch[x][1],pre[x],sz[x],key[x],sum[x],val[x]);
            Treaval(ch[x][1]);
        }
    }
    void debug() {printf("%d
",root);Treaval(root);}
    //以上debug

    void NewNode(int& x,int father,int k,int v){
        x = ++tot;
        pre[x] = father;
        key[x] = k;
        val[x] = v;
        ch[x][0] = ch[x][1] = 0;
        sz[x] = 1;
        flip[x] = 0;
    }

    void Push_Up(int x){
        sz[x] = sz[ch[x][0]] + sz[ch[x][1]] + 1;
        sum[x] = sum[ch[x][0]]+ sum[ch[x][1]] + val[x];
    }

    void Push_Down(int x){
        if(flip[x]){
            flip[ch[x][0]] ^= 1;
            flip[ch[x][1]] ^= 1;
            flip[x] = 0;
            swap(ch[x][1],ch[x][0]);
        }
    }

    void Build(int& x,int l,int r,int father){
        if(l > r)   return;
        int mid = (l+r) >> 1;
        NewNode(x,father,mid,ssz[mid]);
        pos[mid] = x;
        Build(ch[x][0],l,mid-1,x);
        Build(ch[x][1],mid+1,r,x);
        Push_Up(x);
    }

    void Init(){
        root = tot = 0;
        ch[0][1] = ch[0][0] = pre[0] = sz[0] = 0;
        NewNode(root,0,0,1);
        NewNode(ch[root][1],root,0,1);
        sz[root] = 2;
        Build(Key_Tree,1,N,ch[root][1]);
        Push_Up(ch[root][1]);
        Push_Up(root);
    }

    void UpDate(int x){
        ch[x][1] = ch[x][0] = 0;
        sz[x] = 1;
        sum[x] = val[x];
    }

    void Top(int x){
        RotateTo(0,0);
        Splay(x,root);
        int a = Get_Pre(x);
        if(a == -1) return;
        Splay(a,x);
        pre[Key_Tree] = root;
        pre[ch[ch[root][1]][1]] = Key_Tree;
        ch[Key_Tree][1] = ch[ch[root][1]][1];
        ch[root][1] = Key_Tree;
        Push_Up(ch[root][1]);
        Push_Up(root);
        UpDate(x);
        RotateTo(1,root);
        pre[x] = ch[root][1];
        Key_Tree = x;
        Push_Up(ch[root][1]);
        Push_Up(root);
    }

    int Query(int x){
        //debug();
        Splay(x,root);
        return sum[Key_Tree]+1;
    }

    int Rank(int x,int k){
        if(k >= sum[ch[x][0]] && k < sum[ch[x][0]] + val[x])
            return (l[key[x]] + k - (sum[ch[x][0]]));
        if(k < sum[ch[x][0]])
            return Rank(ch[x][0],k);
        return Rank(ch[x][1],k - (sum[ch[x][0]] + val[x]));
    }

    void solve(){
        //debug();
        FOR(i,0,m){
            if(cmd[i].str[0] == 'T')    {Top(pos[mat[cmd[i].num]]);}
            else if(cmd[i].str[0] == 'Q')   {printf("%d
",Query(pos[mat[cmd[i].num]]));}
            else {printf("%d
",Rank(root,cmd[i].num));}
        }
    }

    int Get_Pre(int x){
        Push_Down(x);
        x = ch[x][0];
        while(x){
            Push_Down(x);
            while(ch[x][1]){
                x = ch[x][1];
                Push_Down(x);
            }
            return x;
        }
        return -1;
    }

    int Get_Next(int x){
        Push_Down(x);
        x = ch[x][1];
        while(x){
            Push_Down(x);
            while(ch[x][0]){
                x = ch[x][0];
                Push_Down(x);
            }
            return x;
        }
        return -1;
    }
}spt;

void Read(){
    cnt_TQ = 0;
    FOR(i,0,m){
        scanf("%s%d",cmd[i].str,&cmd[i].num);
        if(cmd[i].str[0] != 'R'){
            TQ[cnt_TQ ++] = cmd[i].num;
        }
    }
    sort(TQ,TQ+cnt_TQ);
    N = 0;
    if(TQ[0] == 1){
        mat[1] = 1;
        ssz[++N] = 1;
        l[N] = 1;
    }
    else{
        ssz[++N] = TQ[0]-1;
        l[N] = 1;
        mat[TQ[0]] = ++N;
        ssz[N] = 1;
        l[N] = TQ[0];
    }
    FOR(i,1,cnt_TQ){
        if(TQ[i] == TQ[i-1])    continue;
        if(TQ[i] == TQ[i-1] + 1){
            mat[TQ[i]] = ++N;
            ssz[N] = 1;
            l[N] = TQ[i];
        }
        else{
            ssz[++N] = TQ[i] - TQ[i-1]-1;
            l[N] = TQ[i-1] + 1;
            mat[TQ[i]] = ++N;
            ssz[N] = 1;
            l[N] = TQ[i];
        }
    }
    if(TQ[cnt_TQ-1] != n){
        ssz[++N] = n - TQ[cnt_TQ-1];
        l[N] = TQ[cnt_TQ-1] + 1;
    }
}

int main()
{
    //freopen("test.in","r",stdin);
    int T,tCase = 0;
    scanf("%d",&T);
    while(T--){
        printf("Case %d:
",++tCase);
        scanf("%d%d",&n,&m);
        Read();
        spt.Init();
        spt.solve();
    }
    return 0;
}

HDU 1890 Robotic Sort

Robotic Sort

HDU 3487 Play With Chain

没什么难的地方：

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <cmath>
#include <cstring>
#define LL long long
#define FOR(i,x,y)  for(int i = x;i < y;i ++)
#define IFOR(i,x,y) for(int i = x;i > y;i --)
#define Key_Tree ch[ch[root][1]][0]
#define MAXN 330000

using namespace std;

int n,m,num[MAXN],cnt;
char cmd[10];

struct SplayTree{
    int ans[MAXN];
    int ch[MAXN][2],sz[MAXN],pre[MAXN],key[MAXN],root,tot;
    int flip[MAXN];

    void Rotate(int x,int kind){
        int y = pre[x];
        Push_Down(y);
        Push_Down(x);
        ch[y][!kind] = ch[x][kind];
        pre[ch[x][kind]] = y;
        if(pre[y]){
            ch[pre[y]][ch[pre[y]][1] == y] = x;
        }
        pre[x] = pre[y];
        ch[x][kind] = y;
        pre[y] = x;
        Push_Up(y);
    }

    void Splay(int x,int goal){
        Push_Down(x);
        while(pre[x] != goal){
            Push_Down(pre[pre[x]]); Push_Down(pre[x]);  Push_Down(x);
            if(pre[pre[x]] == goal){
                Rotate(x,ch[pre[x]][0] == x);
            }
            else{
                int y = pre[x];
                int kind = (ch[pre[y]][0] == y);
                if(ch[y][kind] == x){
                    Rotate(x,!kind);
                    Rotate(x,kind);
                }
                else{
                    Rotate(y,kind);
                    Rotate(x,kind);
                }
            }
        }
        Push_Up(x);
        if(!goal)   root = x;
    }

    void RotateTo(int k,int goal){
        int x = root;
        Push_Down(x);
        while(sz[ch[x][0]] != k){
            if(k < sz[ch[x][0]]){
                x = ch[x][0];
            }
            else{
                k -= (sz[ch[x][0]]+1);
                x = ch[x][1];
            }
            Push_Down(x);
        }
        Splay(x,goal);
    }


    //debug部分copy from hh
    void Treaval(int x) {
        if(x) {
            Treaval(ch[x][0]);
            printf("结点%2d:左儿子 %2d 右儿子 %2d 父结点 %2d size = %2d ,val = %2d 
",x,ch[x][0],ch[x][1],pre[x],sz[x],key[x]);
            Treaval(ch[x][1]);
        }
    }
    void Debug() {printf("%d
",root);Treaval(root);}
    //以上debug


    void Build(int& x,int l,int r,int father){
        if(l > r)   return;
        int mid = (l+r) >> 1;
        NewNode(x,father,num[mid]);
        Build(ch[x][0],l,mid-1,x);
        Build(ch[x][1],mid+1,r,x);
        Push_Up(x);
    }

    void Init(){
        root = tot = 0;
        ch[0][1] = ch[0][0] = pre[0] = sz[0] = 0;
        NewNode(root,0,0);
        NewNode(ch[root][1],root,0);
        sz[root] = 2;
        Build(Key_Tree,0,n-1,ch[root][1]);
        Push_Up(ch[root][1]);
        Push_Up(root);
    }


    int Get_Pre(int x){
        Push_Down(x);
        x = ch[x][0];
        while(x){
            Push_Down(x);
            while(ch[x][1]){
                x = ch[x][1];
                Push_Down(x);
            }
            return x;
        }
        return -1;
    }

    int Get_Next(int x){
        Push_Down(x);
        x = ch[x][1];
        while(x){
            Push_Down(x);
            while(ch[x][0]){
                x = ch[x][0];
                Push_Down(x);
            }
            return x;
        }
        return -1;
    }

    void NewNode(int& x,int father,int k){
        x = ++tot;
        pre[x] = father;
        key[x] = k;
        ch[x][0] = ch[x][1] = 0;
        sz[x] = 1;
        flip[x] = 0;
    }

    void Push_Up(int x){
        sz[x] = sz[ch[x][0]] + sz[ch[x][1]] + 1;
    }

    void Push_Down(int x){
        if(flip[x]){
            flip[ch[x][0]] ^= 1;
            flip[ch[x][1]] ^= 1;
            flip[x] = 0;
            swap(ch[x][1],ch[x][0]);
        }
    }

    void Flip(int l,int r){
        RotateTo(l-1,0);
        RotateTo(r+1,root);
        flip[Key_Tree] ^= 1;
    }

    void Cut(int l,int r,int k){
        RotateTo(l-1,0);
        RotateTo(r+1,root);
        int p = Key_Tree;
        ch[ch[root][1]][0] = 0;
        sz[ch[root][1]] = sz[ch[ch[root][1]][1]] + 1;
        Push_Up(root);
        //Debug();
        RotateTo(k,0);
        RotateTo(k+1,root);
        pre[p] = ch[root][1];
        ch[ch[root][1]][0] = p;
        Push_Up(ch[root][1]);
        Push_Up(root);
    }

    void Print(int x){
        Push_Down(x);
        if(ch[x][0])    {Print(ch[x][0]);}
        if(cnt == 0 && key[x] != 0)  {printf("%d",key[x]);cnt++;}
        else if(key[x] != 0)
            {printf(" %d",key[x]);cnt++;}
        if(ch[x][1])    {Print(ch[x][1]);}
    }

}spt;

int main()
{
    //freopen("test.in","r",stdin);
    while(~scanf("%d%d",&n,&m) && (n >= 0 || m >= 0)){
        FOR(i,0,n)  num[i] = i+1;
        spt.Init();
        //spt.Debug();
        FOR(i,0,m){
            scanf("%s",cmd);
            int l,r,k;
            if(cmd[0] == 'C'){
                scanf("%d%d%d",&l,&r,&k);
                spt.Cut(l,r,k);
                //spt.Debug();
            }
            else{
                scanf("%d%d",&l,&r);
                spt.Flip(l,r);
            }
        }
        cnt = 0;
        spt.Print(spt.root);
        printf("
");
    }
    return 0;
}

POJ 3580 SuperMemo

SuperMemo

NOI 2005 维修数列

非常BT的一题

维修数列

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