timus 1175. Strange Sequence 解题报告
1.题目描述:
1175. Strange Sequence
Time limit: 1.0 second
Memory limit: 2 MB
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:
- find H = (A1*X*Y + A2*X + A3*Y + A4);
- if H > B1 then H is decreased by C until H ≤ B2;
- 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 = 0; i < max; i++) { iter(x1, x2); } tmp1 = x1; tmp2 = x2; q = 1; iter(x1, x2); while (x1 != tmp1 || x2 != tmp2) { q++; iter(x1, x2); } tmp1 = tmp3; tmp2 = tmp4; iter(tmp1,tmp2); i = 1; while (i != q) { i++; iter(tmp1, tmp2); } p = 1; 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。。