http://poj.org/problem?id=1995
Raising Modulo Numbers
Time Limit:1000MS |
|
Memory Limit:30000K |
Description
People are different. Some secretly read magazines full of interesting girls' pictures, others create an A-bomb in their cellar, others like using Windows, and some like difficult mathematical games. Latest marketing research shows, that this market segment was so far underestimated and that there is lack of such games. This kind of game was thus included into the KOKODáKH. The rules follow:
Each player chooses two numbers Ai and Bi and writes them on a slip of paper. Others cannot see the numbers. In a given moment all players show their numbers to the others. The goal is to determine the sum of all expressions AiBifrom all players including oneself and determine the remainder after division by a given number M. The winner is the one who first determines the correct result. According to the players' experience it is possible to increase the difficulty by choosing higher numbers.
You should write a program that calculates the result and is able to find out who won the game.
Input
The input consists of Z assignments. The number of them is given by the single positive integer Z appearing on the first line of input. Then the assignements follow. Each assignement begins with line containing an integer M (1 <= M <= 45000). The sum will be divided by this number. Next line contains number of players H (1 <= H <= 45000). Next exactly H lines follow. On each line, there are exactly two numbers Ai and Bi separated by space. Both numbers cannot be equal zero at the same time.
Output
For each assingnement there is the only one line of output. On this line, there is a number, the result of expression
(A1B1+A2B2+ ... +AHBH)mod M.
Sample Input
3
16
4
2 3
3 4
4 5
5 6
36123
1
2374859 3029382
17
1
3 18132
Sample Output
2
13195
13
求a^b%c(这就是著名的RSA公钥的加密方法)
当a,b很大时,直接求解这个问题不太可能
你能想到哪些优化呢?
算法1:直观上,也许最容易想到的是利用a*b%c=((a%c)*b)%c,这样每一步都进行这种处理,这就解决了a^b可能太大存不下的问题,但这个算法的时间复杂度依然是O(n),根本没有得到优化。当b很大时运行时间会很长
算法2:另一种算法利用了二分的思想,可以达到O(logn)。
可以把b按二进制展开为b=p(n)*2^n+p(n-1)*2^(n-1)+…+p(1)*2+p(0)
其中p(i) (0<=i<=n)为0或1
这样a^b=a^(p(n)*2^n+p(n-1)*2^(n-1)+...+p(1)*2+p(0))
=a^(p(n)*2^n)*a^(p(n-1)*2^(n-1))*...*a^(p(1)*2)*a^p(0)
对于p(i)=0的情况,a^p(i)*2^(i-1)=a^0=1,不用处理
我们要考虑的仅仅是p(i)=1的情况
a^(2^i)=(a^(p(i)*2(i-1)))^2
利用这一点,我们可以递推地算出所有的a^(2^i)
当然由算法1的结论,我们加上取模运算a^(2^i)%c=((a^(2(i-1))%c)*a^(2(i-1)))%c
于是再把所有满足p(i)=1的a^(2^i)%c按照算法1乘起来再%c就是结果
即二进制扫描从最高位一直扫描到最低位。
模板:
int modexp(long long a,long long b,long long n)
{
int ret=1;
long long tmp=a;
while(b)
{
//基数存在
if(b&0x1) ret=ret*tmp%n;
tmp=tmp*tmp%n;
b>>=1;
}
return ret;
}
分享到:
相关推荐
北大POJ3252-Round Numbers 解题报告+AC代码
C语言 poj npu 西工大 C语言Poj答案全完整打包,给有需要的朋友
如题所示,亲测可用。求高精度幂,不会的同学可以参考下,会做的同学可以给挑挑毛病!大家以代码会友!
北大POJ1016-Numbers That Count【字符串处理】 解题报告+AC代码
POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类POJ分类
Businesses like to have memorable telephone numbers. One way to make a telephone number memorable is to have it spell a memorable word or phrase. For example, you can call the University of Waterloo ...
北大POJ2739-Sum of Consecutive Prime Numbers 解题报告+AC代码
2遍dp poj_3613解题报告 poj_3613解题报告
poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题报告poj 解题...
北大POJ3292-Semi-prime H-numbers 解题报告+AC代码
poj 2196 Specialized Four-Digit Numbers.md
POJ第1861题源码 POJ第1861题源码 POJ第1861题源码
北大POJ1159-Palindrome 解题报告+AC代码
poj分类poj分类poj分类poj分类
poj 3414解题报告poj 3414解题报告poj 3414解题报告poj 3414解题报告
poj 1012解题报告poj 1012解题报告poj 1012解题报告poj 1012解题报告
poj 2329解题报告poj 2329解题报告poj 2329解题报告poj 2329解题报告
poj 1659解题报告poj 1659解题报告poj 1659解题报告poj 1659解题报告
POJ1503解答 POJ1503解答,正确答案(已通过POJ)