szoj657 【AHSDFZNOI 7.2 WuHongxun】Odd

【题目大意】

给出$n$个数$a_1, a_2, ..., a_n$，求有多少个区间$[l, r]$，满足每个数都出现了奇数次。

$1 \leq n \leq 2 * 10^5, 0 \leq a_i \leq 10^6$

【题解】

稳爷爷出的noi模拟题（原题来自某地区ioi选拔赛）

给每个数赋一个随机权值$H_x$，那么问题转化为：

有多少个区间，使得区间内的数的异或和=区间内出现过的数的异或和。

这是可以分块的，记$lst_i$表示数$i$上一次出现的位置，令$x = H_{a_i}$那么每次就是给$[1, lst_x]$异或一个数，并且询问$[1, i]$中为0的数的个数。

分块+map可以做到$O(n\sqrt{nlogn})$。但是会被卡。

分块+hash就能做到$O(n\sqrt{n})$了，把hash表的取模开到1000~3000较优。每次暴力就重构一次。重构的时空复杂度是正确的。

# include <map>
# include <vector>
# include <stdio.h>
# include <string.h>
# include <iostream>
# include <algorithm>
// # include <bits/stdc++.h>

using namespace std;

typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
const int N = 2e5 + 10, M = 1e6 + 10, NB = 500 + 5;
const int mod = 1e9+7, BLOCK = 400;
const int MOD = 2339;

int n, m, B, a[N], bl[N], L[N], R[N];
vector<int> ps;
ull H[N];
int lst[N];

struct hasher {
    int head[MOD + 5], w[MOD + 5], nxt[MOD + 5], tot, id, hid[MOD + 5]; ull to[MOD + 5];
    inline void set() {
        tot = 0; id = 0;
        memset(head, 0, sizeof head);
        memset(hid, 0, sizeof hid); 
    }
    inline void reset() {
        tot = 0; ++id;
    }
    inline void add(int u, ull v, int _w) {
        ++tot; nxt[tot] = head[u]; 
        head[u] = tot; to[tot] = v; w[tot] = _w;
    }
    inline void init(int L, int R) {
        add(0, 0, R-L+1);
    }
    
    inline int bg(int x) {
        if(hid[x] == id) return head[x];
        else {
            hid[x] = id;
            return head[x] = 0;
        }
    }
    
    inline void ins(ull x) {
        int hx = x % MOD; 
        for (int i=bg(hx); i; i=nxt[i]) 
            if(to[i] == x) {
                ++w[i];
                return ;
            }
        add(hx, x, 1);
    }
    
    inline int query(ull x) {
        int hx = x % MOD;
        for (int i=bg(hx); i; i=nxt[i]) 
            if(to[i] == x) return w[i];
        return 0;
    }
}solver[NB];

inline ull irand() {
    ull ret = 0;
    for (int i=1; i<=8; ++i) ret = (ret << 15) + rand();
    return ret;
}

ull tag[NB], c[N];
inline void dealxor(int x, ull v) {
    if(!x) return ;
    int p = bl[x];
    for (int i=1; i<p; ++i) tag[i] ^= v;
    solver[p].reset(); 
    for (int i=L[p]; i<=x; ++i) {
        c[i] ^= v;
        solver[p].ins(c[i]);
    }
    for (int i=x+1; i<=R[p]; ++i) solver[p].ins(c[i]); 
}

inline int query(int x) {
    int p = bl[x], ret = 0;
    for (int i=1; i<p; ++i) ret += solver[i].query(tag[i]);
    for (int i=L[p]; i<=x; ++i) if(c[i] == tag[p]) ++ret;
    return ret;
}

int main() {
//    freopen("odd2.in", "r", stdin);
    cin >> n;
    for (int i=1; i<=n; ++i) {
        scanf("%d", a+i);
        ps.push_back(a[i]);
    }
    sort(ps.begin(), ps.end());
    ps.erase(unique(ps.begin(), ps.end()), ps.end());
    m = ps.size();
    for (int i=1; i<=n; ++i) a[i] = lower_bound(ps.begin(), ps.end(), a[i]) - ps.begin() + 1;
    for (int i=1; i<=m; ++i) H[i] = irand();
    for (int i=1; i<=n; ++i) bl[i] = (i-1)/BLOCK + 1;
    B = bl[n];
    for (int i=1; i<=B; ++i) L[i] = (i-1) * BLOCK + 1, R[i] = i * BLOCK;
    R[B] = min(R[B], n);
    for (int i=1; i<=B; ++i) solver[i].set(), solver[i].init(L[i], R[i]);
    ll ans = 0;
    for (int i=1; i<=n; ++i) {
        int pre = lst[a[i]]; lst[a[i]] = i;
//        cerr << i << ' ' << pre << endl;
        dealxor(pre, H[a[i]]);
        ans += query(i);
    }
    cout << ans;
    return 0;
}
/*
4
2 2 2 3
Ans = 7
*/