我正在尝试通过递归调用实现Karatsuba multiplication。下面的代码应该可以工作,但是我总是得到错误的答案。有什么想法吗?

    public static long karatsuba(long x, long y){
        //base case:
        if (x < 10 || y < 10) return x * y;

        //length of digits:
        int xSize = String.valueOf(x).length();
        int ySize = String.valueOf(y).length();
        int N     = Math.max(xSize, ySize);

        //split each number in half (by length of digits):
        long numX_hi = Long.valueOf((String.valueOf(x).substring(0, N/2)));
        long numX_lo = Long.valueOf((String.valueOf(x).substring(N/2, xSize)));
        long numY_hi = Long.valueOf((String.valueOf(y).substring(0, N/2)));
        long numY_lo = Long.valueOf((String.valueOf(y).substring(N/2, ySize)));

        //solve multiplications recursively:
        long z0 = karatsuba(numX_lo,numY_lo);
        long z1 = karatsuba((numX_hi+numX_lo),(numY_hi+numY_lo));
        long z2 = karatsuba(numX_hi,numY_hi);

        //answer:
        return  (long)(z2 * Math.pow(10,N))  +  (long)((z1-z2-z0) * Math.pow(10,(N/2)))  +  (z0);
    }

以下是一些测试案例:

1)karatsuba(1234,5678)>>> 6952652

*应为7006652

2)唐津(4589,7831)>>> 34649459

*应为35936459

3)唐津(911,482)>>> 44722

*应为472842

最佳答案

您的方法有两个明显的问题。

首先,您应该从最后一位(最低有效数字)开始而不是第一位进行分割。因此,如果您有以下两个数字:

1234
567890

当前,您按如下方式拆分它们:
123   4 (123*1000+4)
567 890 (567*1000+890)

这会给您错误的结果,因为1234 != 123*1000+4
您应该像这样拆分它们:
  1 234  (1*1000+234)
567 890  (567*1000+890)

当您将所有内容重新添加在一起时,我发现了第二个错误。
return  (long)(z2 * Math.pow(10,N))  +  (long)((z1-z2-z0) * Math.pow(10,(N/2)))  +  (z0);

对于N奇数,将返回不正确的结果,因为N/2会四舍五入,因此N != ((N/2)*2)
我结合了两个修复程序,现在得到了正确的结果:
public static long karatsuba(long x, long y){
    //base case:
    if (x < 10 || y < 10) return x * y;

    //length of digits:
    int xSize = String.valueOf(x).length();
    int ySize = String.valueOf(y).length();
    int halfN     = Math.max(xSize, ySize) / 2; // store N/2 instead of N
    int splitX = xSize - halfN;  // count the split point from xSize down
    int splitY = ySize - halfN;  // count the split point from ySize down

    //split each number in half (by length of digits):
    long numX_hi = Long.valueOf((String.valueOf(x).substring(0, splitX)));
    long numX_lo = Long.valueOf((String.valueOf(x).substring(splitX)));
    long numY_hi = Long.valueOf((String.valueOf(y).substring(0, splitY)));
    long numY_lo = Long.valueOf((String.valueOf(y).substring(splitY)));

    //solve multiplications recursively:
    long z0 = karatsuba(numX_lo,numY_lo);
    long z1 = karatsuba((numX_hi+numX_lo),(numY_hi+numY_lo));
    long z2 = karatsuba(numX_hi,numY_hi);

    //answer:
    return  (long)(z2 * Math.pow(10,halfN*2))  +  (long)((z1-z2-z0) * Math.pow(10,halfN))  +  (z0);
}

09-26 22:09