递归Karatsuba乘法不起作用？

Question

我正试图通过递归调用实现Karatsuba乘法.下面的代码应该可行,但我一直得到错误的答案.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);
    }

以下是一些测试用例:

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

*应为 7006652

2)karatsuba(4589,7831)>>> 34649459

*应为 35936459

3)karatsuba(911,482)>>> 44722

*应为 472842

Answer 1

您的方法存在两个明显的问题.

首先,您应该从最后一个(最不重要)数字开始分割,而不是从第一个数字开始.所以,如果你有这两个数字:

1234
567890

您目前将它们拆分为:

123   4 (123*1000+4)
567 890 (567*1000+890)

这会导致错误的结果,因为 1234 != 123*1000+4

你应该像这样拆分它们:

  1 234  (1*1000+234)
567 890  (567*1000+890)

我发现的第二个错误发生在你重新添加东西时.

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

将返回奇数Ns 的错误结果,因为N/2将向上舍入,因此N != ((N/2)*2)

我结合了两个修复程序,现在我得到了正确的结果:

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);
}