BZOJ 4827 循环卷积
题意：求两个手环任意旋转对应位置的差值+c的平方最小
设b旋转到k最小，那么先将b扩张一倍构成一圈，那么答案式子就是
将这个式子展开一下，事情就变得有趣了起来
这个式子将a[ ]翻转可以化成卷积形式
直接套上一个FFT就可以了
然后枚举C就行了（C的范围比较小，主要是写起来容易，追求效率可以用求根公式算出这个二次函数取得最小值的时候，C的大小）
1 #include <cstdio> 2 #include <cstring> 3 #include <queue> 4 #include <cmath> 5 #include <algorithm> 6 #include <set> 7 #include <iostream> 8 #include <map> 9 #include <stack> 10 #include <string> 11 #include <vector> 12 #define pi acos(-1.0) 13 #define eps 1e-9 14 #define fi first 15 #define se second 16 #define rtl rt<<1 17 #define rtr rt<<1|1 18 #define bug printf("******\n") 19 #define mem(a,b) memset(a,b,sizeof(a)) 20 #define name2str(x) #x 21 #define fuck(x) cout<<#x" = "<<x<<endl 22 #define f(a) a*a 23 #define sf(n) scanf("%d", &n) 24 #define sff(a,b) scanf("%d %d", &a, &b) 25 #define sfff(a,b,c) scanf("%d %d %d", &a, &b, &c) 26 #define sffff(a,b,c,d) scanf("%d %d %d %d", &a, &b, &c, &d) 27 #define pf printf 28 #define FRE(i,a,b) for(i = a; i <= b; i++) 29 #define FREE(i,a,b) for(i = a; i >= b; i--) 30 #define FRL(i,a,b) for(i = a; i < b; i++)+ 31 #define FRLL(i,a,b) for(i = a; i > b; i--) 32 #define FIN freopen("data.txt","r",stdin) 33 #define gcd(a,b) __gcd(a,b) 34 #define lowbit(x) x&-x 35 #define rep(i,a,b) for(int i=a;i<b;++i) 36 #define per(i,a,b) for(int i=a-1;i>=b;--i) 37 using namespace std; 38 typedef long long LL; 39 typedef unsigned long long ULL; 40 const int maxn = 3e5 + 7; 41 const int maxm = maxn * 4; 42 const int mod = 1e9 + 7; 43 int n, m, a[maxn], b[maxn]; 44 int len, res[maxm], mx; //开大4倍 45 struct cpx { 46 long double r, i; 47 cpx ( long double r = 0, long double i = 0 ) : r ( r ), i ( i ) {}; 48 cpx operator+ ( const cpx &b ) { 49 return cpx ( r + b.r, i + b.i ); 50 } 51 cpx operator- ( const cpx &b ) { 52 return cpx ( r - b.r, i - b.i ); 53 } 54 cpx operator* ( const cpx &b ) { 55 return cpx ( r * b.r - i * b.i, i * b.r + r * b.i ); 56 } 57 } va[maxm], vb[maxm]; 58 void rader ( cpx F[], int len ) { //len = 2^M,reverse F[i] with F[j] j为i二进制反转 59 int j = len >> 1; 60 for ( int i = 1; i < len - 1; ++i ) { 61 if ( i < j ) swap ( F[i], F[j] ); // reverse 62 int k = len >> 1; 63 while ( j >= k ) j -= k, k >>= 1; 64 if ( j < k ) j += k; 65 } 66 } 67 void FFT ( cpx F[], int len, int t ) { 68 rader ( F, len ); 69 for ( int h = 2; h <= len; h <<= 1 ) { 70 cpx wn ( cos ( -t * 2 * pi / h ), sin ( -t * 2 * pi / h ) ); 71 for ( int j = 0; j < len; j += h ) { 72 cpx E ( 1, 0 ); //旋转因子 73 for ( int k = j; k < j + h / 2; ++k ) { 74 cpx u = F[k]; 75 cpx v = E * F[k + h / 2]; 76 F[k] = u + v; 77 F[k + h / 2] = u - v; 78 E = E * wn; 79 } 80 } 81 } 82 if ( t == -1 ) //IDFT 83 for ( int i = 0; i < len; ++i ) F[i].r /= len; 84 } 85 void Conv ( cpx a[], cpx b[], int len ) { //求卷积 86 FFT ( a, len, 1 ); 87 FFT ( b, len, 1 ); 88 for ( int i = 0; i < len; ++i ) a[i] = a[i] * b[i]; 89 FFT ( a, len, -1 ); 90 } 91 void gao () { 92 len = 1; 93 mx = n + m; 94 while ( len <= mx ) len <<= 1; //mx为卷积后最大下标 95 for ( int i = 0; i < len; i++ ) va[i].r = va[i].i = vb[i].r = vb[i].i = 0; 96 for ( int i = 0; i < n; i++ ) va[i].r = a[i]; //根据题目要求改写 97 for ( int i = 0; i < m; i++ ) vb[i].r = b[i]; //根据题目要求改写 98 Conv ( va, vb, len ); 99 for ( int i = 0; i < len; ++i ) res[i] += ( LL ) floor ( va[i].r + 0.5 ); 100 } 101 102 int main() { 103 // FIN; 104 sff ( n, m ); 105 int suma = 0, sumb = 0, sum1 = 0, sum2 = 0; 106 for ( int i = 0 ; i < n ; i++ ) sf ( a[n - 1 - i] ), suma += a[n - 1 - i], sum1 += a[n - 1 - i] * a[n - 1 - i]; 107 for ( int i = 0 ; i < n ; i++ ) sf ( b[i] ), b[n + i] = b[i], sumb += b[i], sum2 += b[i] * b[i]; 108 m = 2 * n; 109 gao(); 110 LL cnt = 0, ans = 0x3f3f3f3f3f3fLL; 111 for ( int i = n ; i < 2 * n ; i++ ) cnt = max ( cnt, 1LL * res[i] ); 112 //fuck ( cnt ); 113 for ( int i = -m ; i <= m ; i++ ) 114 ans = min ( ans, 1LL * i * i * n + 1LL * 2 * i * ( suma - sumb ) + sum1 + sum2 - 1LL * 2 * cnt ); 115 printf ( "%lld\n", ans ); 116 return 0; 117 }
