2019南昌邀请赛 C. Angry FFF Party 大数矩阵快速幂+分类讨论

题目链接 https://nanti.jisuanke.com/t/38222

题意：

定义函数：

$$F(n)=\left\{
\begin{aligned}
1, \quad n=1,2 \\
F(n-1)+F(n-2),\quad n\geq3 \quad
\end{aligned}
\right.
$$

给定一个W 找到一个字典序最小的集合S使得

$$W=\sum_{f\in S}F(F(f))$$

$1\leq  T\leq 10$

$1\leq  W\leq 10^{100,000}$

解析：java大数打表可以发现 当f >28时已经超过W上界了，所以快速幂求出来前28项就好了，数增加的非常快，只有当W<=10时才有 多个解的情况

所以从大的开始减，当减到第五项的时候 再分类讨论一下就好了。

AC代码

import java.util.*;
import java.math.*;

public class Main {

    static class Matrix {
        public static  int maxn = ;
        BigInteger a[][] = new BigInteger [maxn][maxn];
        public void init() {
            for (int i = ; i < maxn; ++i) for (int j = ; j < maxn; ++j) a[i][j] = BigInteger.ZERO;
        }
        public void _init() {
            init();
            for (int i = ; i < maxn; ++i) a[i][i] = BigInteger.ONE;
        }
        public static Matrix mul(Matrix A, Matrix B) {
            Matrix res = new Matrix();
            res.init();
            for (int i = ; i < maxn; ++i) {
                for (int j = ; j < maxn; ++j) {
                    for (int k = ; k < maxn; ++k) {
                        res.a[i][k] = res.a[i][k].add(A.a[i][j].multiply(B.a[j][k]));
                    }
                }
            }
            return res;
        }
        public static Matrix q_pow(Matrix A, BigInteger k) {
            Matrix res = new Matrix();
            res._init();
            while(k.compareTo(BigInteger.ZERO) > ) {
                if(k.mod(BigInteger.valueOf()).compareTo(BigInteger.ZERO) > ) res = mul(res, A);
                A = mul(A, A);
                k = k.shiftRight();
            }
            return res;
        }
    }
    public static BigInteger get_fib(BigInteger n) {
        if(n.compareTo(BigInteger.ONE) == ) return BigInteger.ONE;
        if(n.compareTo(BigInteger.valueOf()) == ) return BigInteger.ONE;
        Matrix A = new Matrix();
        A.a[][] = BigInteger.ZERO;
        A.a[][] = A.a[][] = A.a[][] = BigInteger.ONE;
        A = Matrix.q_pow(A, n.subtract(BigInteger.valueOf()));
        return A.a[][].add(A.a[][]);
    }

    public static void main(String[] args) {

        BigInteger f[] = new BigInteger[];
        int ans[] = new int[];
        Scanner cin = new Scanner(System.in);
        int T = cin.nextInt();
        for (int i = ; i <= ; ++i) {
            f[i] = Main.get_fib(Main.get_fib(BigInteger.valueOf(i)));
        }
        while(T--> ) {
            BigInteger W = cin.nextBigInteger();
            int cnt = ;
            for (int i = ; i >= ; --i) {
                if(f[i].compareTo(W) <= ) {
                    ans[++cnt] = i;
                    W = W.subtract(f[i]);
                }
            }
            if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ){
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.valueOf()) == ) {
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
                ans[++cnt] = ;
            }
            else if(W.compareTo(BigInteger.ZERO)!=){
                System.out.println(-);
                continue;
            }
            for (int i = cnt; i >= ; --i) {
                if(i == )
                    System.out.println(ans[i]);
                else
                    System.out.print(ans[i]+" ");
            }
        }
    }

}

代码参考 https://www.cnblogs.com/widsom/p/10742707.html