递归Karatsuba乘法不起作用? [英] Recursive Karatsuba multiplication not working?

问题描述

我正在尝试通过递归调用实现 Karatsuba乘法.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?

I'm trying to implement Karatsuba multiplication through recursive calls. The code below should work, but I keep getting the wrong answer. Any thoughts?

    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

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

*应该为 7006652

2)karatsuba(4589，7831)>>> 34649459

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

*应为 35936459

3)karatsuba(911，482)>>> 44722

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

*应该为 472842

推荐答案

您的方法有两个明显的问题.

There are two distinct problems with your method.

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:

Firstly, you should split starting from the last (least significant) digit, not the first. So if you've got these two numbers:

1234
567890

您当前按如下方式拆分它们:

You currently split them like this:

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

这会为您带来错误的结果，因为1234 != 123*1000+4

This gets you the wrong result because 1234 != 123*1000+4

您应该像这样拆分它们:

You should instead split them like this:

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

---

我发现的第二个错误发生在将事物重新添加在一起时.

---

The second error I discovered happens when you add things back together.

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

对于奇数的N，将返回错误的结果，因为N/2会四舍五入为向上，因此为N != ((N/2)*2)

Will return an incorrect result for odd Ns, as N/2 will be rounded up down and therefore N != ((N/2)*2)

我将这两个修复程序结合在一起，现在我得到了正确的结果:

I've combined the two fixes and now I get the correct results:

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);
}

这篇关于递归Karatsuba乘法不起作用?的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！