HDU 4549 m Fibonacci sequence (Fast power matrix fast Power Fermat theorem)

Topic link http://acm.hdu.edu.cn/showproblem.php?pid=4549;

It is easy to understand the topic of Chinese. But at first I read the wrong question, this problem haha. At first I read the wrong question, used a quick power to solve, needless to say, WA, see the passage rate of the topic is not high, I think there will be what pit ah. But I'm the big pit, haha.

Do not say, directly to say the question, first discuss k=1,2,3; This should be solved, not much to say;

Starting from item fourth F (4) =a^1+b^2;f (5) =a^2+b^3;f (6) =a^3+b^5 ...; See, the index on a is Cheng Feibo, and B is the same, and the index of a is the previous item of B;

Then the table to find the Fibonacci sequence, dare not, the scope is too big, and burst memory and overtime.

F (n) =a^k (n-4) +b^k (n-3), wherein n>=4;k (0) =1;k (1) = 2;

A matrix is used to find the adjacent two items of the Fibonacci sequence [K[n-2],k[n-1]]*[0 1]

[1 1]

The above formula equals [K[n-1],k[n]];

Then request [K[n-4],k[n-3]]=[1,2]*[0 1]^ (n-3)

[1 1]

In this way, the matrix can be used to quickly power the k[n-4],k[n-3];

But the light is not enough, in the matrix fast power process also has to take the mold, otherwise it will overflow.

The Fermat theorem has x^p==x for any integer x in the case of P as a prime number (mod p)

If x cannot be divisible by P with x^ (p-1) =1 (mod p), because a,b<1e9; can not be divisible by 1e9+7, K[n], then a^k[n]%p=a^ (k[n]% (p-1))%p;

The proof is as follows: k[n]=m* (p-1) +d; then a^k[n]%p=a^[(m* (p-1)) +d]%p= (a^[m* (p-1)]%p*a^ (d)%p)%p;

The a^ (m* (p-1))%p=1 is known by the Fermat theorem; and d=k[n]% (p-1);

So the matrix of the fast power to 1e+6 to take the mold on the line; when K (n) is obtained is then a fast power operation can be, the complexity of 2*LOG2 (n); Be careful to open longlong or it will overflow.

Here's a look at the code:

1#include <stdio.h>2#include <string.h>3#include <iostream>4#include <algorithm>5#include <stdlib.h>6#include <math.h>7 Const Long Longn=1e9+7;8 Const Long Longm=1e9+6;9typedefLong Longll;Ten voidKK (ll y); One ll pp (ll x,ll y); All b[2][2]; -ll a[2][2]; - using namespacestd; the intMainvoid) - { - ll I,j,k,p,q,n,m; -  +      while(SCANF ("%lld%lld%lld", &p,&q,&k)! =EOF) -     { +         if(k==0) A         { atprintf"%lld\n", p%N); -  -         } -         Else if(k==1) -         { -printf"%lld\n", q%N); in  -         } to         Else if(k==2) +         { -printf"%lld\n", (p%n*q%n)%N); the         } *         Else $         {Panax Notoginsengk=k-3; - KK (k); thell p1= (b[0][0]%m+2*b[1][0]%M)%M;//Request K (n-4) +ll P2= (b[0][1]%m+2*b[1][1]%M)%M;//K[n-3] All x1=pp (P,P1);//a^k (n-4); thell x2=pp (Q,P2);//b^k (n-3); +ll dd= (x1*x2)%N;//d= (A^k (n-4) *b^k (n-3))%N; -printf"%lld\n", DD); $  $         } -  -     } the  -     return 0;Wuyi  the } -  WuLL pp (ll X,ll y)//Fast Power - { Aboutll p=x; $ll q=1; -  -      while(y) -     { A         if(y&1) +         { theq= (q%n*p%n)%N; -         } $p= (p%n*p%n)%N; the  they=y/2; the     } the     returnQ; -  in } the  the  About voidKK (ll y)//The fast power form of matrix is the same as that of fast power.  the { the ll I,j,k; the ll x1,x2,x3,x4; +a[0][0]=0;//A is a transformation matrix; -a[0][1]=1; thea[1][0]=1;Bayia[1][1]=1; theb[0][0]=1;//b is the unit array; theb[0][1]=0; -b[1][0]=0; -b[1][1]=1; the  the      while(y) the     { the         if(y&1) -         { theX1= ((b[0][0]*a[0][0])%m+ (b[0][1]*a[1][0])%M)%M; theX2= ((b[0][0]*a[0][1])%m+ (b[0][1]*a[1][1])%M)%M; theX3= ((b[1][0]*a[0][0])%m+ (b[1][1]*a[1][0])%M)%M;94X4= ((b[1][0]*a[0][1])%m+ (b[1][1]*a[1][1])%M)%M; theb[0][0]=X1; theb[0][1]=x2; theb[1][0]=X3;98b[1][1]=x4; About         } -X1= ((a[0][0]*a[0][0])%m+ (a[0][1]*a[1][0])%M)%M;101X2= ((a[0][0]*a[0][1])%m+ (a[0][1]*a[1][1])%M)%M;102X3= ((a[1][0]*a[0][0])%m+ (a[1][1]*a[1][0])%M)%M;103X4= ((a[1][0]*a[0][1])%m+ (a[1][1]*a[1][1])%M)%M;104a[0][0]=X1; thea[0][1]=x2;106a[1][0]=X3;107a[1][1]=x4;108Y/=2;109  the     }111  the 113  the  the}

HDU 4549 m Fibonacci sequence (Fast power matrix fast Power Fermat theorem)