hdu 5751 Eades

题意：对于整数序列$A[1...n]$定义$f(l, r)$为区间$[l, r]$内等于区间最大值元素的个数，定义$z[i]$为所有满足$f(l, r)=i$的区间总数。对于所有的$1 \leq i \leq n$，计算$z[i]$。

分析：考虑由大往小枚举最大值，对于某一最大值为$M$的区间$[l, r)$，满足$a[p_i]=M$的元素将区间切割为若干子区间，那么这些子区间对长度的积对答案的某一项有等值的贡献，暴力枚举需要$O(n^2)$的时间，整体考虑那些对答案某一项有贡献的子区间对的积，它们满足卷积的性质。因此只需将长度序列与其反序列作类似多项式乘法的fft，即可将时间优化为$O(nlog(n))$。子区间递归处理即可。代码如下：

 #include <algorithm>
 #include <cstdio>
 #include <cstring>
 #include <string>
 #include <queue>
 #include <map>
 #include <set>
 #include <stack>
 #include <ctime>
 #include <cmath>
 #include <iostream>
 #include <assert.h>
 #pragma comment(linker, "/STACK:102400000,102400000")
 #define max(a, b) ((a) > (b) ? (a) : (b))
 #define min(a, b) ((a) < (b) ? (a) : (b))
 #define mp std :: make_pair
 #define st first
 #define nd second
 #define keyn (root->ch[1]->ch[0])
 #define lson (u << 1)
 #define rson (u << 1 | 1)
 #define pii std :: pair<int, int>
 #define pll pair<ll, ll>
 #define pb push_back
 #define type(x) __typeof(x.begin())
 #define foreach(i, j) for(type(j)i = j.begin(); i != j.end(); i++)
 #define FOR(i, s, t) for(int i = (s); i <= (t); i++)
 #define ROF(i, t, s) for(int i = (t); i >= (s); i--)
 #define dbg(x) std::cout << x << std::endl
 #define dbg2(x, y) std::cout << x << " " << y << std::endl
 #define clr(x, i) memset(x, (i), sizeof(x))
 #define maximize(x, y) x = max((x), (y))
 #define minimize(x, y) x = min((x), (y))
 using namespace std;
 typedef long long ll;
 const int int_inf = 0x3f3f3f3f;
 const ll ll_inf = 0x3f3f3f3f3f3f3f3f;
 const int INT_INF = (int)((1ll << ) - );
 const double double_inf = 1e30;
 const double eps = 1e-;
 typedef unsigned long long ul;
 typedef unsigned int ui;
 inline int readint(){
     int x;
     scanf("%d", &x);
     return x;
 }
 inline int readstr(char *s){
     scanf("%s", s);
     return strlen(s);
 }
 
 class cmpt{
 public:
     bool operator () (const int &x, const int &y) const{
         return x > y;
     }
 };
 
 int Rand(int x, int o){
     //if o set, return [1, x], else return [0, x - 1]
     if(!x) return ;
     int tem = (int)((double)rand() / RAND_MAX * x) % x;
     return o ? tem +  : tem;
 }
 void data_gen(){
     srand(time());
     freopen("in.txt", "w", stdout);
     int kases = ;
     printf("%d\n", kases);
     while(kases--){
         int sz = 6e4;
         printf("%d\n", sz);
         FOR(i, , sz) printf("%d ", Rand(sz, ));
         printf("\n");
     }
 }
 
 struct cmpx{
     bool operator () (int x, int y) { return x > y; }
 };
 const int maxn = 6e4 + ;
 int a[maxn];
 int n;
 struct Seg{
     int l, r, v;
 }seg[maxn << ];
 void build(int u, int l, int r){
     seg[u].l = l, seg[u].r = r;
     if(seg[u].r - seg[u].l < ){
         seg[u].v = l;
         return;
     }
     int mid = (l + r) >> ;
     build(lson, l, mid), build(rson, mid, r);
     if(a[seg[rson].v] > a[seg[lson].v]) seg[u].v = seg[rson].v;
     else seg[u].v = seg[lson].v;
 }
 int query(int u, int l, int r){
     if(seg[u].l == l && seg[u].r == r) return seg[u].v;
     int mid = (seg[u].l + seg[u].r) >> ;
     if(r <= mid) return query(lson, l, r);
     else if(l >= mid) return query(rson, l, r);
     int lhs = query(lson, l, mid), rhs = query(rson, mid, r);
     if(a[rhs] > a[lhs]) return rhs;
     return lhs;
 }
 int maxi;
 vector<int> pos[maxn];
 int idx[maxn];
 void init(){
     maxi = -;
     FOR(i, , n) maximize(maxi, a[i]);
     FOR(i, , maxi) pos[i].clear();
     FOR(i, , n){
         int sz = pos[a[i]].size();
         idx[i] = sz;
         pos[a[i]].pb(i);
     }
 }
 ll z[maxn];
 ll c[maxn], d[maxn], e[maxn << ];
 int k;
 const double PI =  * asin(.);
 struct Complex{
     double x, y;
     Complex(double x = , double y = ) : x(x), y(y) {}
 };
 Complex operator + (const Complex &lhs, const Complex &rhs){
     return Complex(lhs.x + rhs.x, lhs.y + rhs.y);
 }
 Complex operator - (const Complex &lhs, const Complex &rhs){
     return Complex(lhs.x - rhs.x, lhs.y - rhs.y);
 }
 Complex operator * (const Complex &lhs, const Complex &rhs){
     double tl = lhs.x * rhs.x, tr = lhs.y * rhs.y, tt = (lhs.x + lhs.y) * (rhs.x + rhs.y);
     return Complex(lhs.x * rhs.x - lhs.y * rhs.y, lhs.x * rhs.y + lhs.y * rhs.x);
 }
 Complex w[][maxn << ], x[maxn << ], y[maxn << ];
 
 void fft(Complex x[], int k, int v){
     int i, j, l;
     Complex tem;
     for(i = j = ; i < k; i++){
         if(i > j) tem = x[i], x[i] = x[j], x[j] = tem;
         for(l = k >> ; (j ^= l) < l; l >>= ) ;
     }
     for(i = ; i <= k; i <<= ) for(j = ; j < k; j += i) for(l = ; l < i >> ; l++){
         tem = x[j + l + (i >> )] * w[v][k / i * l];
         x[j + l + (i >> )] = x[j + l] - tem;
         x[j + l] = x[j + l] + tem;
     }
 }
 int tot;
 void solve(int l, int r){
     if(l >= r) return;
     if(r - l == ){
         ++z[];
         return;
     }
     int p = query(, l, r);
     int tem = idx[p];
     int sz = pos[a[p]].size();
     k = ;
     c[k++] = p - l + ;
     while(tem  +  < sz && pos[a[p]][tem + ] < r) c[k++] = pos[a[p]][tem + ] - pos[a[p]][tem], ++tem;
     c[k++] = r - pos[a[p]][tem];
     ROF(i, k - , ) d[i] = c[k -  - i];
     int len;
     for(len = ; len < (k << ); len <<= ) ;
     FOR(i, , len) w[][len - i] = w[][i] = Complex(cos(PI *  * i / len), sin(PI *  * i / len));
     FOR(i, , k - ) x[i] = Complex(c[i], );
     FOR(i, k, len - ) x[i] = Complex(, );
     fft(x, len, );
     FOR(i, , k - ) y[i] = Complex(d[i], );
     FOR(i, k, len - ) y[i] = Complex(, );
     fft(y, len, );
     FOR(i, , len - ) x[i] = x[i] * y[i];
     fft(x, len, );
     FOR(i, ,  * k - ) e[i] = (ll)(x[i].x / len + .);
     FOR(i, , k - ) z[k -  - i] += e[i];
     tem = idx[p];
     solve(l, p);
     while(tem +  < sz && pos[a[p]][tem + ] < r) solve(pos[a[p]][tem] + , pos[a[p]][tem + ]), ++tem;
     solve(pos[a[p]][tem] + , r);
 }
 int main(){
     //data_gen(); return 0;
     //C(); return 0;
     int debug = ;
     if(debug) freopen("in.txt", "r", stdin);
     //freopen("out.txt", "w", stdout);
     int T = readint();
     while(T--){
         n = readint();
         FOR(i, , n) a[i] = readint();
         build(, , n + );
         init();
         clr(z, );
         tot = ;
         solve(, n + );
         ll ans = ;
         FOR(i, , n) ans += z[i] ^ i;
         printf("%lld\n", ans);
     }
     return ;
 }

code：