【学习笔记】 - 可持久化数据结构初步：可持久化线段树

前置知识：

• 权值线段树

  权值线段树每个节点不再是区间，而是值在某个范围内的个数

  可以用于求区间第 \(k\) 大值

• 动态开点

  一个点只有在需要时才被创建

正文

什么是可持久化数据结构

就是说这个数据结构能保留每一个历史版本且支持操作

可持久化线段树

又称函数式线段树/主席树

朴素的想法是对于每次修改，都建立一颗新的线段树，复杂度较高

如图所示，可持久化线段树通过创建新的节点来记录历史版本，每次都基于原本的线段树创建一条新的链，时间复杂度较优\(\text O((n+m)\log n)\)，空间 \(\text O((2n-1)+n(\lceil\log_2{n}\rceil+1))\)

特殊的，主席树可以维护查询静态区间 \(k\) 小值操作

大概是这样的一张图？

最开始只有根节点(\(1\))

插入\(2\)后建立包括 \(\{1',2\}\) 的历史版本 \(1'\)

插入\(4\)后建立包括 \(\{1'',2',4\}\) 的历史版本 \(1''\)

插入\(5\)后建立包括 \(\{1''',2'',4,5\}\) 的历史版本 \(1'''\)

插入\(3\)后建立包括 \(\{1'''',2'',4,5,3\}\) 的历史版本 \(1''''\)

插入\(6\)后建立包括 \(\{1''''',2'',4,5,3',6\}\) 的历史版本 \(1'''''\)

插入\(7\)后建立包括 \(\{1'''''',2'',4,5,3'',6，7\}\) 的历史版本 \(1''''''\)

这么多的权值线段树就可以大力维护区间第 \(k\) 小值了

对于查询\(l \sim r\)区间，我们可以用\(1\sim r\)减去\(1\sim l-1\)

天依的代码

#include<bits/stdc++.h>
#define N 0x66ccff
#define sort std::stable_sort
namespace Fast_I {char *_Buf, *_Start_ptr, *_End_ptr;std::streambuf* inbuf;unsigned int Size;bool _Ok;struct Fast_Istream {operator bool(){return _Ok; }Fast_Istream(std::streambuf*, unsigned int);Fast_Istream(unsigned int);Fast_Istream(const char*, unsigned int);Fast_Istream& operator>>(char&);Fast_Istream& operator>>(char*);Fast_Istream& operator>>(bool&);Fast_Istream& operator>>(short&);Fast_Istream& operator>>(int&);Fast_Istream& operator>>(long&);Fast_Istream& operator>>(long long&);Fast_Istream& operator>>(unsigned short&);Fast_Istream& operator>>(unsigned int&);Fast_Istream& operator>>(unsigned long&);Fast_Istream& operator>>(unsigned long long&);Fast_Istream& operator>>(float&);Fast_Istream& operator>>(double&);Fast_Istream& operator>>(long double&);Fast_Istream& operator>>(std::string&);template <typename Typex>void operator()(Typex& _Val) { *this >> _Val; }template <typename Typex, typename... More>void operator()(Typex&, More&...);std::streambuf* rdbuf() { return inbuf; }void rdbuf(std::streambuf* _inbuf) { inbuf = _inbuf; }void rdbuf(const char*);void pop();char peek();};}
namespace Fast_O {std::string buf;std::streambuf* outbuf;struct Fast_Ostream {Fast_Ostream(std::streambuf*, unsigned int);Fast_Ostream(std::streambuf* out) { outbuf = out; }Fast_Ostream(const char*, unsigned int);Fast_Ostream(unsigned int);void flush();~Fast_Ostream();void endl() { buf.push_back('\n'); }template <typename Typex>void endl(const Typex& _Val);template <typename Typex, typename... More>void endl(const Typex&, const More&...);template <typename Typex>void operator()(const Typex& _Val);template <typename Typex, typename... More>void operator()(const Typex&, const More&...);Fast_Ostream& operator<<(char);Fast_Ostream& operator<<(const char*);Fast_Ostream& operator<<(const std::string&);Fast_Ostream& operator<<(bool);Fast_Ostream& operator<<(short);Fast_Ostream& operator<<(int);Fast_Ostream& operator<<(long);Fast_Ostream& operator<<(long long);Fast_Ostream& operator<<(unsigned short);Fast_Ostream& operator<<(unsigned int);Fast_Ostream& operator<<(unsigned long);Fast_Ostream& operator<<(unsigned long long);std::streambuf* rdbuf() { return outbuf; }void rdbuf(std::streambuf* _outbuf) { outbuf = _outbuf; }void rdbuf(const char*);};}
namespace Fast_IO {Fast_I::Fast_Istream fin(std::cin.rdbuf(), 1048576);Fast_O::Fast_Ostream fout(std::cout.rdbuf()); }
#define cin Fast_IO::fin
#define cout Fast_IO::fout
namespace Fast_I {Fast_Istream::Fast_Istream(std::streambuf* in, unsigned int Sz) {_Ok = 1;Fast_I::Size = Sz;inbuf = in;_Start_ptr = _End_ptr = _Buf = new char[Sz];}Fast_Istream::Fast_Istream(const char* in, unsigned int Sz) {_Ok = 1;Fast_I::Size = Sz;rdbuf(in);_Start_ptr = _End_ptr = _Buf = new char[Sz];}Fast_Istream::Fast_Istream(unsigned int Sz) {_Ok = 1;Fast_I::Size = Sz;_Start_ptr = _End_ptr = _Buf = new char[Sz];}void Fast_Istream::rdbuf(const char* File) {static std::ifstream __In__(File);rdbuf(__In__.rdbuf());}void Get_Char(char& _Val) {if (_Start_ptr == _End_ptr) {_Start_ptr = _Buf;_End_ptr = _Buf + inbuf->sgetn(_Buf, Size);}if (_Start_ptr == _End_ptr) {_Val = -1;_Ok = 0;} else {_Val = *_Start_ptr++;}}Fast_Istream& Fast_Istream::operator>>(char& _Val) {if(_Ok){Get_Char(_Val);while (_Val == 32 || _Val == 10 || _Val == 13 || _Val == 8 || _Val == 9 || _Val == 7 || _Val == 12 || _Val == 11) {Get_Char(_Val);}}return *this;}Fast_Istream& Fast_Istream::operator>>(char* _Val) {if (_Ok) {Get_Char(*_Val);while (*_Val == 32 || *_Val == 10 || *_Val == 13 || *_Val == 8 ||*_Val == 9 || *_Val == 7 || *_Val == 12 || *_Val == 11) {Get_Char(*_Val);}while (*_Val != 32 && *_Val != 10 && *_Val && *_Val != -1 && *_Val != 9 &&*_Val != 11 && *_Val != 12) {Get_Char(*++_Val);}*_Val = 0;--_Start_ptr;}return *this;}Fast_Istream& Fast_Istream::operator>>(std::string& _Val) {if (_Ok) {char c;Get_Char(c);while (c == 32 || c == 10 || c == 13 || c == 8 || c == 9 || c == 7 ||c == 12 || c == 11) {Get_Char(c);}for (_Val.clear();c != 32 && c != 10 && c && c != -1 && c != 9 && c != 11 && c != 12;Get_Char(c)) {_Val.push_back(c);}--_Start_ptr;}return *this;}template <typename Typex>void Get_Int(Typex& _Val) {if (_Ok) {char ch;bool _F = 0;for (Get_Char(ch); (ch < 48 || ch > 57) && ch != -1; Get_Char(ch)) {_F = ch == 45;}for (_Val = 0; ch > 47 && ch < 58 && ch != -1; Get_Char(ch)) {_Val = _Val * 10 + (ch ^ 48);}if (_F) {_Val = ~_Val + 1;}--_Start_ptr;}}template <typename Typex>void Get_Unsigned(Typex& _Val) {if (_Ok) {char ch;Get_Char(ch);while ((ch < 48 || ch > 57) && ch != -1) {Get_Char(ch);}for (_Val = 0; ch > 47 && ch < 58 && ch != -1; Get_Char(ch)) {_Val = _Val * 10 + (ch ^ 48);}--_Start_ptr;}}template <typename Typex>void Get_Double(Typex& _Val) {if(_Ok){char ch;bool _F = 0;for (Get_Char(ch); (ch < 48 || ch > 57) && ch != -1; Get_Char(ch)) {_F = ch == 45;}for (_Val = 0; ch > 47 && ch < 58 && ch != -1; Get_Char(ch)) {_Val = _Val * 10 + (ch ^ 48);}if (ch == 46) {unsigned long long _Pow = 1;for (Get_Char(ch); ch > 47 && ch < 58 && ch != -1; Get_Char(ch)) {_Val += Typex((ch ^ 48) * 1.0 / (_Pow *= 10));}}if (_F) {_Val = -_Val;}--_Start_ptr;}}Fast_Istream& Fast_Istream::operator>>(bool& _Val) {if(_Ok){char ch;Get_Char(ch);while (ch == 32 || ch == 10 || ch == 13 || ch == 8 || ch == 9 || ch == 7 ||ch == 12 || ch == 11) {Get_Char(ch);}while (ch != 32 && ch != 10 && ch && ch != -1 && ch != 9 && ch != 11 &&ch != 12) {_Val |= ch != 48;Get_Char(ch);}--_Start_ptr;}return *this;}Fast_Istream& Fast_Istream::operator>>(short& _Val) {Get_Int(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(int& _Val) {Get_Int(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(long& _Val) {Get_Int(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(long long& _Val) {Get_Int(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(unsigned short& _Val) {Get_Unsigned(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(unsigned int& _Val) {Get_Unsigned(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(unsigned long& _Val) {Get_Unsigned(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(unsigned long long& _Val) {Get_Unsigned(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(float& _Val) {Get_Double(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(double& _Val) {Get_Double(_Val);return *this;}Fast_Istream& Fast_Istream::operator>>(long double& _Val) {Get_Double(_Val);return *this;}template <typename Typex, typename... More>void Fast_Istream::operator()(Typex& _Val, More&... _More) {*this >> _Val;operator()(_More...);}void Fast_Istream::pop() {char ch;Get_Char(ch);}char Fast_Istream::peek() {if (_Start_ptr == _End_ptr) {_Start_ptr = _Buf;_End_ptr = _Buf + inbuf->sgetn(_Buf, Size);}if (_Start_ptr == _End_ptr) {_Ok = 0;return -1;} else {return *_Start_ptr;}}}
namespace Fast_O {Fast_Ostream::Fast_Ostream(std::streambuf* out, unsigned int Size) {buf.reserve(Size);outbuf = out;}Fast_Ostream::Fast_Ostream(const char* File, unsigned int Size) {buf.reserve(Size);rdbuf(File);}void Fast_Ostream::rdbuf(const char* File) {static std::ofstream __Out__(File);rdbuf(__Out__.rdbuf());}Fast_Ostream::Fast_Ostream(unsigned int Size) {buf.reserve(Size);}void Fast_Ostream::flush() {outbuf->sputn(buf.data(), buf.size());buf.clear();}Fast_Ostream::~Fast_Ostream() {flush();}Fast_Ostream& Fast_Ostream::operator<<(char _Val) {buf.push_back(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(const char* _Val) {while (*_Val) {buf.push_back(*_Val++);}return *this;}Fast_Ostream& Fast_Ostream::operator<<(const std::string& _Val) {for (auto&& i : _Val) {buf.push_back(i);}return *this;}template <typename Typex>void Put_Unsigned(Typex _Val) {char* _Stack = (char*)malloc(sizeof(Typex) * 3);unsigned S_top = 0;while (_Val) {_Stack[++S_top] = (_Val % 10) ^ 48;_Val /= 10;}if (!S_top) {buf.push_back('0');}while (S_top) {buf.push_back(_Stack[S_top--]);}free(_Stack);}void Put_Int(long long _Val) {if (_Val < 0) {buf.push_back('-');Put_Unsigned(~_Val + 1);} else {Put_Unsigned(_Val);}}Fast_Ostream& Fast_Ostream::operator<<(bool _Val) {buf.push_back(_Val ? '1' : '0');return *this;}Fast_Ostream& Fast_Ostream::operator<<(short _Val) {Put_Int(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(int _Val) {Put_Int(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(long _Val) {Put_Int(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(long long _Val) {Put_Int(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(unsigned short _Val) {Put_Unsigned(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(unsigned int _Val) {Put_Unsigned(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(unsigned long _Val) {Put_Unsigned(_Val);return *this;}Fast_Ostream& Fast_Ostream::operator<<(unsigned long long _Val) {Put_Unsigned(_Val);return *this;}template <typename Typex>void Fast_Ostream::endl(const Typex& _Val) {*this << _Val << '\n';}template <typename Typex, typename... More>void Fast_Ostream::endl(const Typex& _Val, const More&... _More) {*this << _Val;endl(_More...);}template <typename Typex>void Fast_Ostream::operator()(const Typex& _Val) {*this << _Val;}template <typename Typex, typename... More>void Fast_Ostream::operator()(const Typex& _Val, const More&... _More) {*this << _Val;operator()(_More...);}}
#define lower_bound std::lower_bound
#define int long long
#define lc ls[q]
#define rc rs[q]
#define mid ((l+r)>>1)
int n,m,q[N<<1],len,a[N],tot;
int root[N],ls[N],rs[N<<1],sum[N<<1];
inline void build(int &q,int l,int r){
    q=++tot;
    if(l==r) return;
    build(lc,l,mid);
    build(rc,mid+1,r);
}
inline void newnode(int x,int y){
    ls[x]=ls[y];
    rs[x]=rs[y];
    sum[x]=sum[y]+1;
}
inline int change(int q,int l,int r,int x){
    int t=++tot;
    newnode(t,q);
    if(l==r) return t;
    if(mid>=x) ls[t]=change(ls[q],l,mid,x);
    else rs[t]=change(rs[q],mid+1,r,x);
    return t;
}
inline int ask(int rt,int q,int l,int r,int k){
    if(l==r) return l;
    int x=sum[lc]-sum[ls[rt]];
    if(x>=k) return ask(ls[rt],lc,l,mid,k);
    else return ask(rs[rt],rc,mid+1,r,k-x);
}
signed main(){
#ifndef ONLINE_JUDGE
    freopen("1.in","r",stdin);
    freopen("1.out","w",stdout);
#endif
    cin(n,m);
    for(int i=1;i<=n;++i)
        cin(a[i]),q[++len]=a[i];
    sort(q+1,q+len+1);
    len=std::unique(q+1,q+len+1)-q-1;
    build(root[0],1,len);
    for(int i=1;i<=n;++i){
        int t=lower_bound(q+1,q+len+1,a[i])-q;
        root[i]=change(root[i-1],1,len,t);
    }
    for(int i=1;i<=m;++i){
        int x,y,k;cin(x,y,k);
        cout(q[ask(root[x-1],root[y],1,len,k)],"\n");
    }
}

我们可以借可持久化线段树实现一些较简单的可持久化数据结构

• 可持久化数组

  需要支持的操作：修改某一历史版本的某个值，查询某一历史版本的某个值

  • 代码

    天依的代码

    #include<bits/stdc++.h>
    #define int long long
    #define lc T[q].ls
    #define rc T[q].rs
    #define mid ((l+r)>>1)
    using namespace std;
    const int N=0x66CCFF*10;
    int a[N],n,q,rt[N],ans,m,v;
    class{
    public:
        int ls,rs,val;
    }T[N];
    inline int read(){
        int f=1,x=0;char ch;
        while(ch<'0'||ch>'9'){ch=getchar();if(ch=='-')f=-1;};
        while(ch>='0'&&ch<='9'){x=x*10+ch-'0';ch=getchar();};
        return f*x;
    }
    inline void build(int &q,int l,int r){
        q=++ans;
        if(l==r){T[q].val=a[l];return;}
        build(lc,l,mid);
        build(rc,mid+1,r);
    }
    inline void change(int &q,int tot,int l,int r,int m,int v){
        q=++ans;
        lc=T[tot].ls;rc=T[tot].rs;
        T[q].val=T[tot].val;
        if(l==r){T[q].val=v;return;}
        if(m<=mid) change(lc,T[tot].ls,l,mid,m,v);
        else change(rc,T[tot].rs,mid+1,r,m,v);
    }
    inline int ask(int q,int l,int r,int ll,int rr){
        if(ll<=l&&rr>=r)return T[q].val;
        int val=0;
        if(ll<=mid)val+=ask(lc,l,mid,ll,rr);
        if(rr>mid)val+=ask(rc,mid+1,r,ll,rr);
        return val;
    }
    signed main(){
        // freopen("1.in","r",stdin);
        n=read(),m=read();
        for(int i=1;i<=n;i++)
            a[i]=read();
        build(rt[0],1,n);
        for(int i=1;i<=m;i++){
            int tot=read(),opt=read(),x=read();
            if(opt==1)
                v=read();
                change(rt[i],rt[tot],1,n,x,v);
            if(opt==2){
                cout<<ask(rt[tot],1,n,x,x)<<"\n";
                rt[i]=rt[tot];
            }
        }
    }
    

• 可持久化并查集

  注意不能进行路径压缩，因为路径压缩一次会修改大量数组

  我们可以选择启发式合并