1.题目描述:

1175. Strange Sequence

Time limit: 1.0 second
Memory limit: 2 MB
You have been asked to discover some important properties of one strange sequences set. Each sequence of the parameterized set is given by a recurrent formula:
Xn+1 = F(Xn-1, Xn),
where n > 1, and the value of F(X,Y) is evaluated by the following algorithm:
  1. find H = (A1*X*Y + A2*X + A3*Y + A4);
  2. if H > B1 then H is decreased by C until H ≤ B2;
  3. the resulting value of H is the value of function F.
The sequence is completely defined by nonnegative constants A1, A2, A3, A4, B1, B2 and C.
One may easily verify that such sequence possess a property that Xp+n = Xp+q+n for appropriate large enough positive integers p and q and for all n ≥ 0. You task is to find the minimal p and q for the property above to hold. Pay attention that numbers p and q are well defined and do not depend on way minimization is done.

Input

The first line contains seven integers: A1, A2, A3, A4, B1, B2 and C. The first two members of sequence (X1 and X2) are placed at the second line. You may assume that all intermediate values of H and all values of F fit in range [0..100000].

Output

An output should consist of two integers (p and q) separated by a space.

Sample

input output
0 0 2 3 20 5 7
0 1
2 3

2.解题思路

题目要求确定p和q,我们先确定q再确定p,有了q只需要从x1,x2开始逐个尝试就能得出p。如何确定q呢?既然题目说了这个序列以q为循环节,我们可以假定序列在迭代足够多次之后就进入了循环,设这个次数为max。这样迭代max后再从1开始寻找q就可以啦~

3.代码:

#include <iostream>
#define max 300000
using namespace std; int a1,a2,a3,a4,b1,b2,c,x1,x2;
int p,q; int cal(int x, int y) {
int h = a1*x*y+a2*x+a3*y+a4;
if (h > b1) {
while (h > b2) {
h -= c;
}
}
return h;
} void iter(int& x, int& y) {
int tmp = cal(x, y);
x = y;
y = tmp;
} //假定迭代max次后处于稳定状态
//思路:迭代max次;继续迭代找到q;x1,x2迭代q次得到x3,x4;同时迭代直到x1==x3 && x2 == x4,找到p
int main() {
cin >> a1 >> a2 >> a3 >> a4 >> b1 >> b2 >> c >> x1 >> x2;
int i, tmp1, tmp2, tmp3, tmp4;
tmp3 = x1;
tmp4 = x2;
for (i = ; i < max; i++) {
iter(x1, x2);
}
tmp1 = x1;
tmp2 = x2;
q = ;
iter(x1, x2);
while (x1 != tmp1 || x2 != tmp2) {
q++;
iter(x1, x2);
}
tmp1 = tmp3;
tmp2 = tmp4;
iter(tmp1,tmp2);
i = ;
while (i != q) {
i++;
iter(tmp1, tmp2);
}
p = ;
while (tmp1 != tmp3 || tmp2 != tmp4) {
p++;
iter(tmp1, tmp2);
iter(tmp3, tmp4);
}
cout << p << " " << q << endl;
}

4.复杂度:

空间:O(1)

时间:O(p+q+max)

5.心得:

把这道题写出来,是因为我第一次尝试时思路是错的:把<x0,x1>作为一个pair,pos是迭代次数,用一个map<pair,pos> result存储,一直迭代<x0,x1>,递增pos,并把结果存入result,当重复插入时(x.pair == y.pair,x.pos < y.pos),就求出了p,q: p = x.pos, q = y.pos - x.pos。

。。。

没错我当时就是这么写的,然后华丽丽的爆了内存。

这种做法的复杂度:

时间:O(p+q)

空间:O(p+q)

然而根据题目要求,空间限制2M,是很严格的。。以后一定要好好看题目。。

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