BZOJ3509: [CodeChef] COUNTARI(生成函数 分块)
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2022-04-28 14:01:51
题意 "链接" Sol 这都能分块。。。。 首先移一下项,变为统计多少$i define Pair pair define MP(x, y) make_pair(x, y) define fi first define se second define LL long long define ull ......
题意
sol
这都能分块。。。。
首先移一下项,变为统计多少\(i < j < k\),满足\(2a[j] = a[i] + a[k]\)
发现\(a[i] \leqslant 30000\),那么有一种暴力思路是枚举\(j\),对于之前出现过的数构造一个生成函数,对于之后出现过的数构造一个生成函数,求一下第\(2a[j]\)项的值。复杂度\(o(nvlogv)\)
每次枚举\(j\)暴力卷积显然太zz了,我们可以分一下块,对于每一块之前之后的数分别构造生成函数暴力卷积算,对于块内的直接暴力(这里的暴力不只是统计块内的\((i, j, k)\),还要考虑\(j, k\)在块内\(i\)在块外,以及\(i, j\)在块内,\(k\)在块外的情况,但都是可以暴力的)
如果分成\(b\)块的话,复杂度是\(\frac{n}{b} vlogv + b^2\),假设\(n\)与\(v\)同阶的话,\(b\)大概取\(nlogn\)是最优的。此时复杂度为\(o(n \sqrt{nlogn})\)
下面的代码在原bzoj上可能会t
#include<bits/stdc++.h> #define pair pair<int, int> #define mp(x, y) make_pair(x, y) #define fi first #define se second #define ll long long #define ull unsigned long long #define fin(x) {freopen(#x".in","r",stdin);} #define fout(x) {freopen(#x".out","w",stdout);} using namespace std; const int maxn = 1e6 + 10, inf = 1e9 + 1; const double eps = 1e-9, pi = acos(-1); inline int read() { char c = getchar(); int x = 0, f = 1; while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();} while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar(); return x * f; } namespace poly { int rev[maxn], gpow[maxn], a[maxn], b[maxn], c[maxn], lim, inv2; const int g = 3, mod = 1004535809, mod2 = 1004535808; template <typename a, typename b> inline ll add(a x, b y) {if(x + y < 0) return x + y + mod; return x + y >= mod ? x + y - mod : x + y;} template <typename a, typename b> inline void add2(a &x, b y) {if(x + y < 0) x = x + y + mod; else x = (x + y >= mod ? x + y - mod : x + y);} template <typename a, typename b> inline ll mul(a x, b y) {return 1ll * x * y % mod;} template <typename a, typename b> inline void mul2(a &x, b y) {x = (1ll * x * y % mod + mod) % mod;} template <typename a, typename b> inline bool chmax(a &x, b y) {return x < y ? x = y, 1 : 0;} template <typename a, typename b> inline bool chmin(a &x, b y) {return x > y ? x = y, 1 : 0;} int fp(int a, int p, int p = mod) { int base = 1; for(; p > 0; p >>= 1, a = 1ll * a * a % p) if(p & 1) base = 1ll * base * a % p; return base; } int inv(int x) { return fp(x, mod - 2); } int getlen(int x) { int lim = 1; while(lim < x) lim <<= 1; return lim; } void init(/*int p,*/ int lim) { inv2 = fp(2, mod - 2); for(int i = 1; i <= lim; i++) gpow[i] = fp(g, (mod - 1) / i); } void ntt(int *a, int lim, int opt) { int len = 0; for(int n = 1; n < lim; n <<= 1) ++len; for(int i = 1; i <= lim; i++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (len - 1)); for(int i = 0; i <= lim; i++) if(i < rev[i]) swap(a[i], a[rev[i]]); for(int mid = 1; mid < lim; mid <<= 1) { int wn = gpow[mid << 1]; for(int i = 0; i < lim; i += (mid << 1)) { for(int j = 0, w = 1; j < mid; j++, w = mul(w, wn)) { int x = a[i + j], y = mul(w, a[i + j + mid]); a[i + j] = add(x, y), a[i + j + mid] = add(x, -y); } } } if(opt == -1) { reverse(a + 1, a + lim); int inv = fp(lim, mod - 2); for(int i = 0; i <= lim; i++) mul2(a[i], inv); } } void mul(int *a, int *b, int n, int m) { memset(a, 0, sizeof(a)); memset(b, 0, sizeof(b)); int lim = 1, len = 0; while(lim <= n + m) len++, lim <<= 1; for(int i = 0; i <= n; i++) a[i] = a[i]; for(int i = 0; i <= m; i++) b[i] = b[i]; ntt(a, lim, 1); ntt(b, lim, 1); for(int i = 0; i <= lim; i++) b[i] = mul(b[i], a[i]); ntt(b, lim, -1); for(int i = 0; i <= n + m; i++) b[i] = b[i]; memset(a, 0, sizeof(a)); memset(b, 0, sizeof(b)); } }; using namespace poly; int n, a[maxn], mx, block, ll[maxn], rr[maxn], belong[maxn], mxblock, num[maxn]; ll solve1(int l, int r) { for(int i = 1; i < l; i++) num[a[i]]++; ll ret = 0; for(int j = l; j <= r; j++) for(int k = j + 1; k <= r; k++) if(2 * a[j] - a[k] >= 0) ret += num[2 * a[j] - a[k]]; for(int i = 1; i < l; i++) num[a[i]]--; for(int i = n; i > r; i--) num[a[i]]++; for(int j = r; j >= l; j--) for(int k = j - 1; k >= l; k--) if(2 * a[j] - a[k] >= 0) ret += num[2 * a[j] - a[k]]; for(int i = n; i > r; i--) num[a[i]]--; for(int j = l; j <= r; j++) { for(int i = j - 1; i >= l; i--) num[a[i]]++; for(int k = j + 1; k <= r; k++) if(2 * a[j] - a[k] >= 0) ret += num[2 * a[j] - a[k]]; for(int i = j - 1; i >= l; i--) num[a[i]]--; } return ret; } int ta[maxn], tb[maxn], lim; ll solve2(int l, int r) { memset(ta, 0, sizeof(ta)); memset(tb, 0, sizeof(tb)); for(int i = l - 1; i >= 1; i--) ta[a[i]]++; for(int i = r + 1; i <= n; i++) tb[a[i]]++; mul(ta, tb, mx, mx); ll ret = 0; for(int i = l; i <= r; i++) ret += tb[2 * a[i]]; return ret; } signed main() { //freopen("a.in", "r", stdin); n = read(); block = sqrt(8 * n * log2(n)); memset(ll, 0x3f, sizeof(ll)); for(int i = 1; i <= n; i++) { belong[i] = (i - 1) / block + 1; chmax(mxblock, belong[i]); chmin(ll[belong[i]], i); chmax(rr[belong[i]], i); a[i] = read(), chmax(mx, a[i]); } lim = getlen(mx); init(4 * lim); ll ans = 0; for(int i = 1; i <= mxblock; i++) { ans += solve1(ll[i], rr[i]); ans += solve2(ll[i], rr[i]); } cout << ans; return 0; } /* 7 7 0 4 7 0 8 8 */
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