【学习笔记】基础数据结构：猫树

Vsinger_洛天依2024-04-08 10:33

猫树是线段树的一个特殊版本，猫树不再支持修改操作，类似 \(\text{ST}\) 表

猫树支持高速区间查询，每次查询都只需要进行 \(1\) 次合并操作，设单次合并操作的复杂度为 \(O(k)\)，建立猫树的复杂度是 \(O(kn \log n)\) 的，而查询的复杂度是 \(O(k)\) 的

一般单次查询的复杂度是 \(O(1)\) ，所以建立猫树的复杂度一般是 \(O(n \log n)\)，查询是 \(O(1)\) 的

原理

考虑一个区间 \([l\sim r]\)，我们找到代表 \([l\sim l]\) 和代表 \([r\sim r]\)，然后对于这两个节点求 \(\text{LCA}\)

设 \(p=\text{LCA}(l,r)\) ，对于其维护的区间 \([ll \sim rr]\)，我们发现了一些性质

  • \([ll\sim rr]\) 同时包含 \([l\sim l]\) 和 \([r\sim r]\)

  • \([l\sim r]\) 区间一定跨越 \([ll\sim rr]\) 的中点，也就是 \([l \sim r]\) 一定包含 \(\dfrac{ll+rr}{2}\) 这个节点

  • 节点 \(l\) 一定在 \([ll\sim mid]\) 内，而 \(r\) 一定在 \([mid+1,rr]\) 内

建树时对于树上的一个节点，设它代表的区间为 \((l,r]\)，我们在这个节点里面额外保存 \((l,mid]\) 的后缀和数组和 \(({mid},r]\) 的前缀和数组。

这样建树的复杂度就从 \(O(n)\) 变成了 \(O(n \log n)\)，如果我们询问的区间是 \([l,r]\) 那么我们求出 \([l,l]\) 的节点和 \([r,r]\) 的节点的 LCA ，记为 \(p\)

根据刚才的性质， \([l,r]\) 在 \(p\) 所包含的区间之内并且一定跨越了 \(p\) 的中点。

因此可以用 \(p\) 的前缀和和后缀和数组，将 \([l,r]\) 拆成 \([l,\mathit{mid}]+(\mathit{mid},r]\) ，拼出 \([l,r]\) 这个区间，只需要一次合并操作

但是求 LCA 不是 \(O(1)\) 的啊，那怎么办呢？

我们可以 堆式建树 ，把这个序列补成 \(2\) 的整数次幂后建猫树

此时猫树上两个节点的 LCA 编号，就是两个节点二进制编号的 LCP，而且可以发现 \(\text {LCP}(x,y)=x>>\log x^y\)

所以只要预处理一个 \(\log\) 数组即可实现高速求 LCA

这里贴一个Can you answer these queries I的代码当做板子题
点击查看代码

cpp 复制代码 
#include<bits/stdc++.h>
#define fire(a) input.reset(new InputFile(a".in")),output.reset(new OutputFile(a".out"))
#define text()  input.reset(new InputFile()),output.reset(new OutputFile())
#define online() input.reset(new InputFile(stdin,false)),output.reset(new OutputFile())
using namespace std;
template<class T>struct is_iterator{template<class U,typename enable_if<!is_convertible<U,const char*>::value,int>::type=0>constexpr static auto has_indirection(int)->decltype(*declval<U>(),bool()){return true;}template<class>constexpr static bool has_indirection(long){return false;}constexpr static bool value=has_indirection<T>(0);};using uint=unsigned int;const uint BUFFER_SIZE=1<<12;const uint MAX_LENGTH=1<<7;namespace Detail{struct Width{uint value;};struct Fill{char value;};struct Base{uint value;};struct Precision{uint value;};struct Delimiter{const char*value;};}Detail::Width setWidth(uint value=0){return{value};}Detail::Fill setFill(char value=' '){return{value};}Detail::Base setBase(uint value=10){assert(2<=value&&value<=36);return{value};}Detail::Precision setPrecision(uint value=9){assert(value<MAX_LENGTH);return{value};}Detail::Delimiter setDelimiter(const char*value=" "){return{value};}class InputDevice{protected:const char*head;const char*tail;InputDevice(const char*head,const char*tail):head(head),tail(tail),base(setBase().value){}virtual void fillInput()=0;inline char nextChar(){if(__builtin_expect(head>=tail,false))fillInput();return*head++;}template<class I>int readUnsignedIntGeneral(I&arg,char c){I value=0;int length=0;for(;;++length,c=nextChar()){if(isDigit(c))c-='0';else if(isUpper(c))c-='A'-10;else if(isLower(c))c-='a'-10;else c=base;if(c>=base)break;value=base*value+c;}arg=value;return--head,length;}template<class I>inline int readUnsignedInt(I&arg,char c){if(__builtin_expect(base>10,false))return readUnsignedIntGeneral(arg,c);I value=0;int length=0;for(;static_cast<unsigned char>(c-'0')<base;++length,c=nextChar())value=base*value+c-'0';arg=value;return--head,length;}template<class I>inline bool readSignedInt(I&arg,char c){bool negative=c=='-';if(negative)c=nextChar();typename make_unsigned<I>::type unsignedArg;if(readUnsignedInt(unsignedArg,c)==0)return false;arg=negative?~static_cast<I>(unsignedArg-1):static_cast<I>(unsignedArg);return true;}template<class F>bool readFloatingPoint(F&arg,char c){bool negative=c=='-';if(negative)c=nextChar();unsigned long long integerPart;if(readUnsignedInt(integerPart,c)==0)return false;arg=static_cast<F>(integerPart);if(nextChar()=='.'){unsigned long long fractionalPart=0;int fractionalLength=readUnsignedInt(fractionalPart,nextChar());if(fractionalLength>0){unsigned long long basePower=1;for(;fractionalLength;--fractionalLength)basePower*=base;arg+=static_cast<F>(fractionalPart)/basePower;}}else--head;if(negative)arg=-arg;return true;}public:uint base;InputDevice(InputDevice const&)=delete;InputDevice&operator=(InputDevice const&)=delete;static inline bool isSpace(char c){return static_cast<unsigned char>(c-'\t')<5||c==' ';}static inline bool isDigit(char c){return static_cast<unsigned char>(c-'0')<10;}static inline bool isUpper(char c){return static_cast<unsigned char>(c-'A')<26;}static inline bool isLower(char c){return static_cast<unsigned char>(c-'a')<26;}static inline bool isOneOf(char c,const char*str){return strchr(str,c)!=nullptr;}void putBack(){--head;}inline bool readChar(char&arg){if(__builtin_expect(head>=tail,false)){fillInput();if(__builtin_expect(head>=tail,false))return arg='\0',false;}return arg=*head++,true;}template<class UnaryPredicate>inline char skipCharacters(UnaryPredicate isSkipped){char c;do{c=nextChar();}while(isSkipped(c));return c;}inline char skipCharacters(){return skipCharacters(isSpace);}template<class UnaryPredicate>inline int readString(char*arg,int limit,UnaryPredicate isTerminator){skipCharacters(isTerminator);int charsRead=0;for(--head,--limit;head<tail;fillInput()){ptrdiff_t chunkSize=find_if(head,min(tail,head+limit-charsRead),isTerminator)-head;arg=copy_n(head,chunkSize,arg);head+=chunkSize;charsRead+=chunkSize;if(chunkSize==0||head<tail)break;}return*arg='\0',charsRead;}inline int readString(char*arg,int limit,const char*terminators){if(!*terminators)return readString(arg,limit,InputDevice::isSpace);return readString(arg,limit,[terminators](char c){return InputDevice::isOneOf(c,terminators);});}inline bool read(Detail::Base newBase){base=newBase.value;return true;}inline bool read(){return true;}inline bool read(char&arg){return readChar(arg);}template<class I>inline typename enable_if<is_integral<I>::value&&is_unsigned<I>::value,bool>::type read(I&arg){return readUnsignedInt(arg,skipCharacters())>0;}template<class I>inline typename enable_if<is_integral<I>::value&&is_signed<I>::value,bool>::type read(I&arg){return readSignedInt(arg,skipCharacters());}template<class F>inline typename enable_if<is_floating_point<F>::value,bool>::type read(F&arg){return readFloatingPoint(arg,skipCharacters());}inline bool read(const char&arg){skipCharacters([arg](char c){return arg!=c;});return true;}inline bool read(const char*arg){if(*arg)skipCharacters([arg](char c){return InputDevice::isOneOf(c,arg);});else skipCharacters();return putBack(),true;}inline bool read(bool(*isSkipped)(char)){skipCharacters(isSkipped);putBack();return true;}template<class I,class Terminator,class...Ts>inline typename enable_if<is_integral<I>::value,bool>::type read(char*arg,I limit,Terminator terminator,Ts&&...args){readString(arg,static_cast<int>(limit),terminator);return read(forward<Ts>(args)...);}template<class I>inline typename enable_if<is_integral<I>::value,bool>::type read(char*arg,I limit){return read(arg,limit,"");}template<class...Ts>inline bool read(char*first,char*last,Ts&&...args){return read(first,static_cast<int>(last-first),forward<Ts>(args)...);}template<int N,class...Ts>inline bool read(char(&arg)[N],Ts&&...args){return read(static_cast<char*>(arg),N,forward<Ts>(args)...);}template<class Terminator,class...Ts>inline bool read(string&arg,Terminator terminator,Ts&&...args){for(int length=16,last=0;;last+=length,length<<=1){arg.resize(last+length);int charsRead=readString(&arg[last],length+1,terminator);if(charsRead<length){arg.resize(last+charsRead);return read(forward<Ts>(args)...);}}}inline bool read(string&arg){return read(arg,"");}template<class T1,class T2>inline bool read(pair<T1,T2>&arg){return read(arg.first,arg.second);}template<class T>inline bool read(complex<T>&arg){T real,imag;if(!read(real,imag))return false;arg.real(real),arg.imag(imag);return true;}template<class T>inline bool read(vector<T>&arg){uint n;if(!read(n))return false;arg.resize(n);return read(arg.begin(),arg.end());}template<class Iterator,class...Ts>inline typename enable_if<is_iterator<Iterator>::value,bool>::type read(Iterator first,Iterator last,Ts&&...args){for(;first!=last;++first)if(!read(*first))return false;return read(forward<Ts>(args)...);}template<class Iterator,class I,class...Ts>inline typename enable_if<is_iterator<Iterator>::value&&is_integral<I>::value,bool>::type read(Iterator first,I count,Ts&&...args){return read(first,first+count,forward<Ts>(args)...);}template<class T>inline auto read(T&arg)->decltype(arg.read(*this)){return arg.read(*this);}template<class T0,class T1,class...Ts>inline typename enable_if<!is_iterator<T0>::value&&!is_convertible<T0,char*>::value,bool>::type read(T0&&arg0,T1&&arg1,Ts&&...args){return read(forward<T0>(arg0))&&read(forward<T1>(arg1),forward<Ts>(args)...);}};class InputFile:public InputDevice{FILE*file;bool lineBuffered;bool owner;char buffer[BUFFER_SIZE];void fillInput()override{head=buffer;*buffer='\0';if(__builtin_expect(!lineBuffered,true)){tail=head+fread(buffer,1,BUFFER_SIZE,file);}else{tail=head;if(fgets(buffer,BUFFER_SIZE,file))while(*tail)++tail;}}public:InputFile(FILE*file=stdin,bool lineBuffered=true,bool takeOwnership=false):InputDevice(buffer,buffer),file(file),lineBuffered(lineBuffered),owner(takeOwnership){}InputFile(const char*fileName):InputFile(fopen(fileName,"r"),false,true){}~InputFile(){if(owner)fclose(file);}};class InputString:public InputDevice{void fillInput()override{while(*tail)++tail;}public:InputString(const string&s):InputDevice(s.data(),s.data()+s.size()){}InputString(const char*s):InputDevice(s,s+strlen(s)){}};class OutputDevice{protected:char buffer[BUFFER_SIZE+MAX_LENGTH];char*output;char*end;bool separate;OutputDevice():output(buffer),end(buffer+BUFFER_SIZE+MAX_LENGTH),separate(false),width(setWidth().value),fill(setFill().value),base(setBase().value),precision(setPrecision().value),delimiter(setDelimiter().value){computeBasePower();}virtual void writeToDevice(uint count)=0;inline void flushMaybe(){if(__builtin_expect(output>=buffer+BUFFER_SIZE,false)){writeToDevice(BUFFER_SIZE);output=copy(buffer+BUFFER_SIZE,output,buffer);}}void computeBasePower(){basePower=1;for(uint i=0;i<precision;++i)basePower*=base;}template<class I>inline char*writeUnsignedInt(I arg,char*last){if(__builtin_expect(arg==0,false))*--last='0';if(__builtin_expect(base==10,true)){for(;arg;arg/=10)*--last='0'+arg%10;}else for(;arg;arg/=base){I digit=arg%base;*--last=digit<10?'0'+digit:'A'-10+digit;}return last;}template<class I>inline char*writeSignedInt(I arg,char*last){auto unsignedArg=static_cast<typename make_unsigned<I>::type>(arg);if(arg<0){last=writeUnsignedInt(~unsignedArg+1,last);*--last='-';return last;}return writeUnsignedInt(unsignedArg,last);}template<class F>char*writeFloatingPoint(F arg,char*last){bool negative=signbit(arg);if(negative)arg=-arg;if(isnan(arg))for(int i=0;i<3;++i)*--last=i["NaN"];else if(isinf(arg))for(int i=0;i<3;++i)*--last=i["fnI"];else{auto integerPart=static_cast<unsigned long long>(arg);auto fractionalPart=static_cast<unsigned long long>((arg-integerPart)*basePower+F(0.5));if(fractionalPart>=basePower)++integerPart,fractionalPart=0;char*point=last-precision;if(precision>0){::fill(point,writeUnsignedInt(fractionalPart,last),'0');*--point='.';}last=writeUnsignedInt(integerPart,point);}if(negative)*--last='-';return last;}inline int writeT(char*first){int delimiterLenght=separate?writeDelimiter():0;separate=true;uint charsWritten=static_cast<uint>(end-first);if(__builtin_expect(charsWritten<width,false))charsWritten+=writeFill(width-charsWritten);output=copy(first,end,output);flushMaybe();return delimiterLenght+static_cast<int>(charsWritten);}inline int writeFill(uint count){int charsWritten=static_cast<int>(count);if(__builtin_expect(output+count+MAX_LENGTH<end,true)){if(count==1)*output++=fill;else output=fill_n(output,count,fill);}else for(uint chunkSize=static_cast<uint>(buffer+BUFFER_SIZE-output);;chunkSize=BUFFER_SIZE){if(chunkSize>count)chunkSize=count;output=fill_n(output,chunkSize,fill);flushMaybe();if((count-=chunkSize)==0)break;}return charsWritten;}public:uint width;char fill;uint base;uint precision;unsigned long long basePower;string delimiter;OutputDevice(OutputDevice const&)=delete;OutputDevice&operator=(OutputDevice const&)=delete;virtual~OutputDevice(){};inline int writeChar(char arg){separate=false;*output++=arg;flushMaybe();return 1;}inline int writeString(const char*arg,size_t length,bool checkWidth=true){separate=false;uint count=static_cast<uint>(length);int charsWritten=static_cast<int>(count)+(checkWidth&&count<width?writeFill(width-count):0);if(__builtin_expect(output+count+MAX_LENGTH<end,true)){if(count==1)*output++=*arg;else output=copy_n(arg,count,output);}else for(uint chunkSize=static_cast<uint>(buffer+BUFFER_SIZE-output);;chunkSize=BUFFER_SIZE){if(chunkSize>count)chunkSize=count;output=copy_n(arg,chunkSize,output);flushMaybe();if((count-=chunkSize)==0)break;arg+=chunkSize;}return charsWritten;}inline int writeDelimiter(){return writeString(delimiter.c_str(),delimiter.size(),false);}inline void flush(){writeToDevice(static_cast<uint>(output-buffer));output=buffer;}inline int write(Detail::Width newWidth){width=newWidth.value;return 0;}inline int write(Detail::Fill newFill){fill=newFill.value;return 0;}inline int write(Detail::Base newBase){base=newBase.value;computeBasePower();return 0;}inline int write(Detail::Precision newPrecision){precision=newPrecision.value;computeBasePower();return 0;}inline int write(Detail::Delimiter newDelimiter){delimiter=newDelimiter.value;return 0;}inline int write(){return 0;}inline int write(char arg){return writeChar(arg);}template<class I>inline typename enable_if<is_integral<I>::value&&is_unsigned<I>::value,int>::type write(I arg){return writeT(writeUnsignedInt(arg,end));}template<class I>inline typename enable_if<is_integral<I>::value&&is_signed<I>::value,int>::type write(I arg){return writeT(writeSignedInt(arg,end));}template<class F>inline typename enable_if<is_floating_point<F>::value,int>::type write(F arg){return writeT(writeFloatingPoint(arg,end));}inline int write(const char*arg){return writeString(arg,strlen(arg));}template<int N>inline int write(char(&arg)[N]){return writeString(arg,strlen(arg));}inline int write(const string&arg){return writeString(arg.c_str(),arg.size());}template<class T1,class T2>inline int write(const pair<T1,T2>&arg){int charsWritten=write(arg.first);charsWritten+=writeDelimiter();return charsWritten+write(arg.second);}template<class T>inline int write(const complex<T>&arg){return write(real(arg),imag(arg));}template<class Iterator,class...Ts>inline typename enable_if<is_iterator<Iterator>::value,int>::type write(Iterator first,Iterator last,Ts&&...args){int charsWritten=0;for(;first!=last;charsWritten+=++first==last?0:writeDelimiter())charsWritten+=write(*first);return charsWritten+write(forward<Ts>(args)...);}template<class Iterator,class I,class...Ts>inline typename enable_if<is_iterator<Iterator>::value&&is_integral<I>::value,int>::type write(Iterator first,I count,Ts&&...args){return write(first,first+count,forward<Ts>(args)...);}template<class T>inline auto write(const T&arg)->decltype(arg.write(*this)){return arg.write(*this);}template<class T0,class T1,class...Ts>inline typename enable_if<!is_iterator<T0>::value,int>::type write(T0&&arg0,T1&&arg1,Ts&&...args){int charsWritten=write(forward<T0>(arg0));return charsWritten+write(forward<T1>(arg1),forward<Ts>(args)...);}};class OutputFile:public OutputDevice{FILE*file;bool owner;void writeToDevice(uint count)override{fwrite(buffer,1,count,file);fflush(file);}public:OutputFile(FILE*file=stdout,bool takeOwnership=false):file(file),owner(takeOwnership){}OutputFile(const char*fileName):OutputFile(fopen(fileName,"w"),true){}~OutputFile()override{flush();if(owner)fclose(file);}};class OutputString:public OutputDevice{string&str;void writeToDevice(uint count)override{str.append(buffer,count);}public:OutputString(string&str):OutputDevice(),str(str){}~OutputString()override{flush();}};unique_ptr<InputDevice>input;unique_ptr<OutputDevice>output;template<class...Ts>inline bool read(Ts&&...args){return input->read(forward<Ts>(args)...);}template<class...Ts>inline int write(Ts&&...args){return output->write(forward<Ts>(args)...);}template<class...Ts>inline int writeln(Ts&&...args){return write(forward<Ts>(args)...,'\n');}void flush(){output->flush();}
#define ull unsigned long long
#define int long long
#define INF (0x66ccff0712ll)
#define inf LLONG_MAX
#define M (0x6cf)
#define N (0x66ccff)
#define MAXN (int)(1e6+5)
#define MAXM (int)(1e5+5)
#define re register
#define it map<int,int>::iterator
#define endl "\n"
#define PII pair<int,int>
#define for_(a,b,c) for(int a=b;a<=c;a++)
#define _for(a,b,c) for(int a=b;a>=c;a--)
#define lc(q) (son[q][0])
#define rc(q) (son[q][1])
#define mid(l,r) ((l+r)>>1)
namespace solve{
    int a[70005],Log[140005],n,len=1,dep=0,q;
    int loc[70005],sum[18][70005],ans[18][70005];
    namespace CatTree{
        inline void build(int n,int d){
            for_(i,1,n)
                loc[i]=i+n-1;
            for(int len=2,q=d-1;len<=n;len<<=1,q--){
                for(int l=1;l<=n;l+=len){
                    int r=l+len-1;
                    sum[q][mid(l,r)]=a[mid(l,r)];
                    sum[q][mid(l,r)+1]=a[mid(l,r)+1];
                    _for(i,mid(l,r)-1,l)
                        sum[q][i]=sum[q][i+1]+a[i];
                    for_(i,mid(l,r)+2,r)
                        sum[q][i]=sum[q][i-1]+a[i];
                    _for(i,mid(l,r)-1,l)
                        sum[q][i]=max(sum[q][i],sum[q][i+1]);
                    for_(i,mid(l,r)+2,r)
                        sum[q][i]=max(sum[q][i],sum[q][i-1]);
                    ans[q][mid(l,r)]=a[mid(l,r)],ans[q][mid(l,r)+1]=a[mid(l,r)+1];
                    _for(i,mid(l,r)-1,l)
                        ans[q][i]=max(ans[q][i+1],0ll)+a[i];
                    for_(i,mid(l,r)+2,r)
                        ans[q][i]=max(ans[q][i-1],0ll)+a[i];
                    _for(i,mid(l,r)-1,l)
                        ans[q][i]=max(ans[q][i+1],ans[q][i]);
                    for_(i,mid(l,r)+2,r)
                        ans[q][i]=max(ans[q][i-1],ans[q][i]);
                }
            }
        }
        int ask(int l,int r){
            if(l==r)
                return a[l];
            int q=Log[loc[l]]-Log[loc[l]^loc[r]]-1;
            return max({ans[q][l],ans[q][r],sum[q][l]+sum[q][r]});
        }
    };
    using namespace CatTree;
    inline void In(){
        read(n);
        while(len<n){
            len<<=1;
            dep++;
        }
        for_(i,1,n)
            read(a[i]);
        Log[1]=0;
        for_(i,2,len<<1) 
            Log[i]=Log[i>>1]+1;
        build(len,dep);
        read(q);
        for_(i,1,q){
            int l,r;
            read(l,r);
            writeln(ask(l,r));
        }
    }
}
using namespace solve;
signed main(){
    fire("data");
    In();
}
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