Brief descrption :
给定一组数列,你要求回答询问 [l, r] 区间内的最大子学列和是多少。
这里的最大子序列和被重定义为,对于区间内重复出现的数字,只会统计一次,另外,允许不选。
Analysis :
Tags: GSS、线段树、离线、Hash、双标记。
fgd 的数据结构 S 级神题。。。无论从建模还是从实现上都有一定难度.. .
首先,你要对 GSS1 的离线算法有了解,方法是对询问的右端点进行排序,然后向右扫描,
同时处理一个单点修改,区间成段询问的线段树问题,对于区间成段询问(就是求最大子序列。。),线段树是没有办法做的,
但是可以类比树状数组中处理区间修改,单点询问时的策略,将之转换成区间修改,区间单点最值。
那么在 GSS2 中,我们就只能使用这种方法了,因为可以通过开一个 Hash 记录这个数字上次出现的的位置,来处理掉重复数字只记录一次的问题。
此外,需要查询区间中历史存在的最值,同时需要再引入一个标记来维护它。
都清楚之后级可以动笔了。。。。数据结构都是水题。。。
(对于至少选一个数字的问题(例如 GSS1,但是我用离线超时了撒。。)。。
。。将下文中的的 SS、DD 初始化成 -INF 和 DD 每次 Release 掉之后也赋值成 -INF 即可了。)
/** ` Micro Mezzo Macro Flation -- Overheated Economy ., **/ #include <algorithm> #include <iostream> #include <iomanip> #include <sstream> #include <cstring> #include <cstdio> #include <string> #include <vector> #include <bitset> #include <queue> #include <cmath> #include <ctime> #include <list> #include <set> #include <map> using namespace std; #define REP(i, n) for (int i=0;i<int(n);++i) #define FOR(i, a, b) for (int i=int(a);i<int(b);++i) #define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i) #define REP_1(i, n) for (int i=1;i<=int(n);++i) #define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i) #define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i) #define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i) #define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i) #define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i) #define REP_N(i, n) for (i=0;i<int(n);++i) #define FOR_N(i, a, b) for (i=int(a);i<int(b);++i) #define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i) #define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i) #define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i) #define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i) #define REP_1_N(i, n) for (i=1;i<=int(n);++i) #define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i) #define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i) #define REP_C_N(i, n) for (n____=int(n),i=0;i<n____;++i) #define FOR_C_N(i, a, b) for (b____=int(b),i=a;i<b____;++i) #define DWN_C_N(i, b, a) for (a____=int(a),i=b-1;i>=a____;--i) #define REP_1_C_N(i, n) for (n____=int(n),i=1;i<=n____;++i) #define FOR_1_C_N(i, a, b) for (b____=int(b),i=a;i<=b____;++i) #define DWN_1_C_N(i, b, a) for (a____=int(a),i=b;i>=a____;--i) #define DO(n) while(n--) #define DO_C(n) int n____ = n; while(n____--) #define TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a;i!=b_;i+=s_) #define TO_1(i, a, b) int s_=a<b?1:-1,b_=b;for(int i=a;i!=b_;i+=s_) #define SQZ(i, j, a, b) for (int i=int(a),j=int(b)-1;i<j;++i,--j) #define SQZ_1(i, j, a, b) for (int i=int(a),j=int(b);i<=j;++i,--j) #define REP_2(i, j, n, m) REP(i, n) REP(j, m) #define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m) #define ALL(A) A.begin(), A.end() #define LLA(A) A.rbegin(), A.rend() #define CPY(A, B) memcpy(A, B, sizeof(A)) #define INS(A, P, B) A.insert(A.begin() + P, B) #define ERS(A, P) A.erase(A.begin() + P) #define BSC(A, X) find(ALL(A), X) // != A.end() #define CTN(T, x) (T.find(x) != T.end()) #define SZ(A) int(A.size()) #define PB push_back #define MP(A, B) make_pair(A, B) #define Rush int T____; RD(T____); DO(T____) #pragma comment(linker, "/STACK:36777216") #pragma GCC optimize ("O2") #define Ruby system("ruby main.rb") #define Haskell system("runghc main.hs") #define Pascal system("fpc main.pas") typedef long long LL; typedef double DB; typedef unsigned UINT; typedef unsigned long long ULL; typedef vector<int> VI; typedef vector<char> VC; typedef vector<string> VS; typedef vector<LL> VL; typedef vector<DB> VD; typedef set<int> SI; typedef set<string> SS; typedef set<LL> SL; typedef set<DB> SD; typedef map<int, int> MII; typedef map<string, int> MSI; typedef map<LL, int> MLI; typedef map<DB, int> MDI; typedef map<int, bool> MIB; typedef map<string, bool> MSB; typedef map<LL, bool> MLB; typedef map<DB, bool> MDB; typedef pair<int, int> PII; typedef pair<int, bool> PIB; typedef vector<PII> VII; typedef vector<VI> VVI; typedef vector<VII> VVII; typedef set<PII> SII; typedef map<PII, int> MPIII; typedef map<PII, bool> MPIIB; /** I/O Accelerator **/ /* ... :" We are I/O Accelerator ... Use us at your own risk ;) ... " .. */ template<class T> inline void RD(T &); template<class T> inline void OT(const T &); template<class T> inline T& _RD(T &x){ RD(x); return x;} inline int RD(){ int x; RD(x); return x;} inline LL RD_LL(){ LL x; RD(x); return x;} inline DB RD_DB(){ DB x; RD(x); return x;} template<class T0, class T1> inline void RD(T0 &x0, T1 &x1){RD(x0), RD(x1);} template<class T0, class T1, class T2> inline void RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2);} template<class T0, class T1, class T2, class T3> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3);} template<class T0, class T1, class T2, class T3, class T4> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6);} template<class T0, class T1> inline void OT(T0 &x0, T1 &x1){OT(x0), OT(x1);} template<class T0, class T1, class T2> inline void OT(T0 &x0, T1 &x1, T2 &x2){OT(x0), OT(x1), OT(x2);} template<class T0, class T1, class T2, class T3> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);} template<class T0, class T1, class T2, class T3, class T4> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);} template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));} template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);} template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);} template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);} template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);} template<class T> inline void CLR(T &A){A.clear();} template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);} template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);} template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);} template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);} template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);} template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));} template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);} template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2){FLC(A0), FLC(A1), FLC(A2);} template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3){FLC(A0), FLC(A1), FLC(A2), FLC(A3);} template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4);} template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5);} template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5), FLC(A6);} template<class T> inline void SRT(T &A){sort(ALL(A));} template<class T, class C> inline void SRT(T &A, C B){sort(ALL(A), B);} /** Add - On **/ const int MOD = 1000000007; const LL INF = 0x7fffffff; const DB PI = acos(-1.0); const DB EPS = 1e-6; const DB OO = 1e15; // <<= ` 0. Daily Use ., template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;} template<class T> inline void checkMax(T &a,const T b){if (b>a) a=b;} template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;} template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;} template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);} template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);} template<class T> inline T sqr(T a){return a*a;} template<class T> inline T cub(T a){return a*a*a;} int Ceil(int x, int y){return (x - 1) / y + 1;} // <<= ` 1. Bitwise Operation ., inline bool _1(int x, int i){return x & 1<<i;} inline int _1(int i){return 1<<i;} inline int _U(int i){return _1(i) - 1;}; inline int count_bits(int x){ x = (x & 0x55555555) + ((x & 0xaaaaaaaa) >> 1); x = (x & 0x33333333) + ((x & 0xcccccccc) >> 2); x = (x & 0x0f0f0f0f) + ((x & 0xf0f0f0f0) >> 4); x = (x & 0x00ff00ff) + ((x & 0xff00ff00) >> 8); x = (x & 0x0000ffff) + ((x & 0xffff0000) >> 16); return x; } template<class T> inline T low_bit(T x) { return x & -x; } template<class T> inline T high_bit(T x) { T p = low_bit(x); while (p != x) x -= p, p = low_bit(x); return p; } // <<= ` 2. Modular Arithmetic Basic ., inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;} inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;} inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;} inline int dff(int a, int b){a -= b; if (a < 0) a += MOD; return a;} inline void MUL(int &a, int b){a = (LL)a * b % MOD;} inline int pdt(int a, int b){return (LL)a * b % MOD;} // <<= ' 0. I/O Accelerator interface ., template<class T> inline void RD(T &x){ //cin >> x; scanf("%d", &x); //char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0'; //char c; c = getchar(); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0'; } //int ____Case; template<class T> inline void OT(const T &x){ //cout << x << endl; printf("%lld\n", x); //printf("%.2lf\n", x); //printf("Case %d: %d\n", ++____Case, x); } #define For_each(it, A) for (SII::iterator it = A.begin(); it != A.end(); ++it) /* .................................................................................................................................. */ const int N = 100001; VI Q[N], ID[N]; // 询问.. int A[N]; LL B[N]; int n, m, a, b; LL d; // A[] 存初始数组,B[] 存答案, a, b 表示询问区间。 LL K[4*N], D[4*N], KK[4*N], DD[4*N]; // 线段树,单点加减,区间成段最值 -> 成段加减,区间最值 // 当前的最大值,延迟标记,历史最大值,历史延迟标记。 int ____[N], Hash[N]; #define root 1, 1, n #define lx (x << 1) #define rx (lx | 1) #define lc lx, l, m #define rc rx, m+1, r #define Update KK[x] = max(max(KK[lx], K[lx] + DD[lx]), max(KK[rx], K[rx] + DD[rx])), K[x] = max(K[lx] + D[lx], K[rx] + D[rx]) #define Release checkMax(DD[lx], D[lx] + DD[x]), checkMax(DD[rx], D[rx] + DD[x]), D[lx] += D[x], D[rx] += D[x], DD[x] = -INF, D[x] = 0 void Insert(int x, int l, int r){ if (a <= l && r <= b){ D[x] += d, checkMax(DD[x], D[x]); } else { int m = (l + r) >> 1; Release; if (a <= m) Insert(lc); if (m < b) Insert(rc); Update; } } LL Query(int x, int l, int r){ if (a <= l && r <= b){ return max(KK[x], K[x] + DD[x]); } else { int m = (l + r) >> 1; Release, Update; return max((a <= m ? Query(lc) : -INF), (m < b ? Query(rc) : -INF)); } } int main(){ //freopen("in.txt", "r", stdin); REP_1_C(i, _RD(n)) RD(A[i]); REP_C(i, _RD(m)) RD(a, b), Q[b].PB(a), ID[b].PB(i); REP_1_N(b, n){ a = Hash[A[b]] + 1, d = A[b], Insert(root), Hash[d] = b; REP(i, SZ(Q[b])){ a = Q[b][i], B[ID[b][i]] = Query(root); } } REP(i, m) OT(B[i]); }