Coursera_Stanford_Algorithms_Karatsuba/Merge sort

Coursera上斯坦福的算法课。为了提升英语水平特地去听的全英版的，可能需要适应一段时间。用的教材是 Algorithms Illuminated。上课讲的东西几乎和教材一致。课上有不懂的东西就直接去逼站搜机翻的视频或者看教材。

第一节课是导入，讲了Karatsuba的大数相乘的算法，和Merge sort（递归排序），以及大欧，西塔，欧米噶这些符号。后三个符号在学校的数据结构课上老师有讲过。但毕竟不是计算机专业，老师讲的没这么深入，只是停留在“感性”认识的阶段。这门课中的定义方法有点数学分析的感觉。很舒服~

相应笔记和python代码实现如下：

1. Karatsuba：
   其实感觉挺鸡肋的。主要是思想。
   因为python内部的算法也非常牛逼。直接把这两个大数相乘瞬间出结果。 甚至用time库计时的结果是0。
   用定义Karatsuba函数进行相乘其实是更慢的…

def str_minus(a,b):     #时间复杂度为n
    tw = 0
    k = len(a) - 1
    outstr = ""
    for i in range(1,len(b)+1):
        if (eval(a[k]) - eval(b[len(b)-i]) - tw) < 0 :
            outstr = str(10 + eval(a[k]) - eval(b[len(b)-i]) - tw) + outstr
            tw = 1
        else:
            outstr = str(eval(a[k]) - eval(b[len(b)-i]) - tw) + outstr
            tw = 0
        k = k - 1
    j = 0
    for i in range(len(b), len(a)): #多余位数的计算。
        k = len(a) - i - 1
        if eval(a[k]) - tw < 0:
            outstr = str(10 + eval(a[k]) - tw) + outstr
            tw = 1
        else:
            outstr = str(eval(a[k]) - tw) + outstr
            tw = 0

    return outstr

def str_plus(a,b):
    if len(a) > len(b):
        for i in range(len(b), len(a)):
            b = "0" + b
    else:
        for i in range(len(a), len(b)):
            a = "0" + a
    jw = 0
    k1 = len(a)
    k2 = len(b)
    outstr = ""

    for i in range(1,k1+1):
        if (eval(b[k2-i]) + eval(a[k1-i]) + jw < 10):
            outstr = str(eval(b[k2-i]) + eval(a[k1-i]) + jw ) + outstr
            jw = 0
        else:
            outstr = str(eval(b[k2 - i]) + eval(a[k1 - i]) + jw)[1] + outstr
            jw = 1

    if jw:
        outstr = "1" + outstr

    if len(outstr) > len(a):
        for i in range(len(outstr),2*len(a)):
            outstr = "0" + outstr
    return outstr


def kara(x,y):
    if len(x) > len(y):
        for i in range(len(y),len(x)):
            y = "0" + y
    else:
        for i in range(len(x),len(y)):
            x = "0" + x

    n = len(x)//2     #默认x，y等长。并且是2的幂级数
    a = x[0:n]
    b = x[n:]
    c = y[0:n]
    d = y[n:]
    if (len(a) != 1) and (len(b) != 1) and (len(c) != 1) and (len(d) != 1):
        result_ac = kara(a,c)
        result_bd = kara(b,d)
        o1 = kara(str_plus(a,b),str_plus(c,d))
        result_mid = str_minus(str_minus(o1,result_bd),result_ac)
        op = str_plus(str_plus(result_ac + 2*n*"0",result_mid + n*"0"),result_bd)
        return op
    else:
        op = str(100 * (eval(a) * eval(c)) + 10 * (eval(a) * eval(d) + eval(b) * eval(c)) + eval(b) * eval(d))
        if len(op) != 4:
            for i in range(4 - len(op)):
                op = "0" + op
        return op


print(kara("3141592653589793238462643383279502884197169399375105820974944592","2718281828459045235360287471352662497757247093699959574966967627"))

print(str(3141592653589793238462643383279502884197169399375105820974944592*2718281828459045235360287471352662497757247093699959574966967627))

结果为：8.539733754338333e+126

Merge sort：
因为python里有自带的sorted方法，，所以其实这个也没啥卵用。但在C里可能会很有用。

def ms(ls):
    n = len(ls)
    if n != 1:
        ls_1 = ms(ls[0:n // 2])
        ls_2 = ms(ls[n // 2:])
        s_ls = []
        ls_1.append(9999)
        ls_2.append(9999)
        i = 0
        j = 0
        for k in range(n):
            if ls_1[i] < ls_2[j]:
                s_ls.append(ls_1[i])
                i = i+1
            else:
                s_ls.append(ls_2[j])
                j = j+1
        return s_ls
    else:
        return ls



ls = [2,5,346,457,5,8,35,2435,7,5,86,5]
print(ms(ls))

---

下面是从别的地方抄过来的C++的递归排序代码。

#include<iostream>
using namespace std;
void merging(int*a,int head,int mid,int tail)
{
    int len=tail-head+1,i=head,j=mid+1,index=0;
    int temp[len]={};//临时数组
    while(i<=mid&&j<=tail)
    {
        if(a[i]<=a[j])
            temp[index++]=a[i++];
        else
            temp[index++]=a[j++];
    }
    while(i<=mid)
        temp[index++]=a[i++];
    while(j<=tail)
        temp[index++]=a[j++];
    for(int i=0;i<len;i++)
        a[head+i]=temp[i];
}
void merge_sort(int*a,int head,int tail)
{
    if(head<tail)//当head<tail说明是多个数据一组,就继续拆分
                 //而当head=tail说明一个数据一组了,就返回开始合并
    {
        int mid=(head+tail)/2;
        merge_sort(a,head,mid);//拆分左半部
        merge_sort(a,mid+1,tail);//拆分左半部
        merging(a,head,mid,tail);//合并两部分
    }
}
int main()
{
    int a[]={9,1,2,8,5,3,7,4,6};
    merge_sort(a,0,8);
    for(int i=0;i<9;i++)
        cout<<a[i]<<" ";
    cout<<endl;
    return 0;
}

来源：
https://blog.youkuaiyun.com/qq_40930559/article/details/104190984?ops_request_misc=%257B%2522request%255Fid%2522%253A%2522160188448719725255563051%2522%252C%2522scm%2522%253A%252220140713.130102334…%2522%257D&request_id=160188448719725255563051&biz_id=0&utm_medium=distribute.pc_search_result.none-task-blog-2_allsobaiduend~default-3-104190984.first_rank_ecpm_v3_pc_rank_v2&utm_term=%E9%80%92%E5%BD%92%E6%8E%92%E5%BA%8Fc&spm=1018.2118.3001.4187