结构之法 算法之道 面试题 9、编程实现两个正整数的除法 编程实现两个正整数的除法，当然不能用除法操作符。

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编程实现两个正整数的除法

编程实现两个正整数的除法，当然不能用除法操作符。
// return x/y.
int div(const int x, const int y) 
{
  ....
}

答案解析 

类似折半查找

常人会从1，2，3，4，5，6。。。。n。。。。这么一直试下去，发现=效率很低，为什么不学学折半查找呢，查找商值，

我会这么做

（1）先查找出一个商的范围，这个跳转很快的，我个人认为 O（log 正无穷 ）

我设的范围 是 （1，2），（3，6），（7，14），（15，30）。。。。。。。。

直到找到 （m，n）, m <= 商值 <= n )

（2） 折半查找就好了 O（log n）

总结  总的时间复杂度为O( log 正无穷 +log
n)

实验数据

experiment result（文本，这次没有上传照片，太长了，不好截图，请谅解）

请输入正整数 x,y

23421543534

34

x = 1946707054, y = 34,d = 1

x = 1946707054, y = 34,d = 2

x = 1946707054, y = 34,d = 2

x = 1946707054, y = 34,d = 3

x = 1946707054, y = 34,d = 6

x = 1946707054, y = 34,d = 6

x = 1946707054, y = 34,d = 7

x = 1946707054, y = 34,d = 14

x = 1946707054, y = 34,d = 14

x = 1946707054, y = 34,d = 15

x = 1946707054, y = 34,d = 30

x = 1946707054, y = 34,d = 30

x = 1946707054, y = 34,d = 31

x = 1946707054, y = 34,d = 62

x = 1946707054, y = 34,d = 62

x = 1946707054, y = 34,d = 63

x = 1946707054, y = 34,d = 126

x = 1946707054, y = 34,d = 126

x = 1946707054, y = 34,d = 127

x = 1946707054, y = 34,d = 254

x = 1946707054, y = 34,d = 254

x = 1946707054, y = 34,d = 255

x = 1946707054, y = 34,d = 510

x = 1946707054, y = 34,d = 510

x = 1946707054, y = 34,d = 511

x = 1946707054, y = 34,d = 1022

x = 1946707054, y = 34,d = 1022

x = 1946707054, y = 34,d = 1023

x = 1946707054, y = 34,d = 2046

x = 1946707054, y = 34,d = 2046

x = 1946707054, y = 34,d = 2047

x = 1946707054, y = 34,d = 4094

x = 1946707054, y = 34,d = 4094

x = 1946707054, y = 34,d = 4095

x = 1946707054, y = 34,d = 8190

x = 1946707054, y = 34,d = 8190

x = 1946707054, y = 34,d = 8191

x = 1946707054, y = 34,d = 16382

x = 1946707054, y = 34,d = 16382

x = 1946707054, y = 34,d = 16383

x = 1946707054, y = 34,d = 32766

x = 1946707054, y = 34,d = 32766

x = 1946707054, y = 34,d = 32767

x = 1946707054, y = 34,d = 65534

x = 1946707054, y = 34,d = 65534

x = 1946707054, y = 34,d = 65535

x = 1946707054, y = 34,d = 131070

x = 1946707054, y = 34,d = 131070

x = 1946707054, y = 34,d = 131071

x = 1946707054, y = 34,d = 262142

x = 1946707054, y = 34,d = 262142

x = 1946707054, y = 34,d = 262143

x = 1946707054, y = 34,d = 524286

x = 1946707054, y = 34,d = 524286

x = 1946707054, y = 34,d = 524287

x = 1946707054, y = 34,d = 1048574

x = 1946707054, y = 34,d = 1048574

x = 1946707054, y = 34,d = 1048575

x = 1946707054, y = 34,d = 2097150

x = 1946707054, y = 34,d = 2097150

x = 1946707054, y = 34,d = 2097151

x = 1946707054, y = 34,d = 4194302

x = 1946707054, y = 34,d = 4194302

x = 1946707054, y = 34,d = 4194303

x = 1946707054, y = 34,d = 8388606

x = 1946707054, y = 34,d = 8388606

x = 1946707054, y = 34,d = 8388607

x = 1946707054, y = 34,d = 16777214

x = 1946707054, y = 34,d = 16777214

x = 1946707054, y = 34,d = 16777215

x = 1946707054, y = 34,d = 33554430

x = 1946707054, y = 34,d = 33554430

x = 1946707054, y = 34,d = 33554431

x = 1946707054, y = 34,d = 67108862

x = 1946707054, y = 34,d = 67108862

x = 1946707054, y = 34,d = 50331646

x = 1946707054, y = 34,d = 50331646

x = 1946707054, y = 34,d = 50331647

x = 1946707054, y = 34,d = 67108861

x = 1946707054, y = 34,d = 67108861

x = 1946707054, y = 34,d = 58720254

x = 1946707054, y = 34,d = 58720254

x = 1946707054, y = 34,d = 58720254

x = 1946707054, y = 34,d = 50331648

x = 1946707054, y = 34,d = 58720253

x = 1946707054, y = 34,d = 58720253

x = 1946707054, y = 34,d = 54525950

x = 1946707054, y = 34,d = 54525950

x = 1946707054, y = 34,d = 54525951

x = 1946707054, y = 34,d = 58720252

x = 1946707054, y = 34,d = 58720252

x = 1946707054, y = 34,d = 56623101

x = 1946707054, y = 34,d = 56623101

x = 1946707054, y = 34,d = 56623102

x = 1946707054, y = 34,d = 58720251

x = 1946707054, y = 34,d = 58720251

x = 1946707054, y = 34,d = 57671676

x = 1946707054, y = 34,d = 57671676

x = 1946707054, y = 34,d = 57671676

x = 1946707054, y = 34,d = 56623103

x = 1946707054, y = 34,d = 57671675

x = 1946707054, y = 34,d = 57671675

x = 1946707054, y = 34,d = 57147389

x = 1946707054, y = 34,d = 57147389

x = 1946707054, y = 34,d = 57147390

x = 1946707054, y = 34,d = 57671674

x = 1946707054, y = 34,d = 57671674

x = 1946707054, y = 34,d = 57409532

x = 1946707054, y = 34,d = 57409532

x = 1946707054, y = 34,d = 57409532

x = 1946707054, y = 34,d = 57147391

x = 1946707054, y = 34,d = 57409531

x = 1946707054, y = 34,d = 57409531

x = 1946707054, y = 34,d = 57278461

x = 1946707054, y = 34,d = 57278461

x = 1946707054, y = 34,d = 57278461

x = 1946707054, y = 34,d = 57147392

x = 1946707054, y = 34,d = 57278460

x = 1946707054, y = 34,d = 57278460

x = 1946707054, y = 34,d = 57212926

x = 1946707054, y = 34,d = 57212926

x = 1946707054, y = 34,d = 57212927

x = 1946707054, y = 34,d = 57278459

x = 1946707054, y = 34,d = 57278459

x = 1946707054, y = 34,d = 57245693

x = 1946707054, y = 34,d = 57245693

x = 1946707054, y = 34,d = 57245694

x = 1946707054, y = 34,d = 57278458

x = 1946707054, y = 34,d = 57278458

x = 1946707054, y = 34,d = 57262076

x = 1946707054, y = 34,d = 57262076

x = 1946707054, y = 34,d = 57262076

x = 1946707054, y = 34,d = 57245695

x = 1946707054, y = 34,d = 57262075

x = 1946707054, y = 34,d = 57262075

x = 1946707054, y = 34,d = 57253885

x = 1946707054, y = 34,d = 57253885

x = 1946707054, y = 34,d = 57253886

x = 1946707054, y = 34,d = 57262074

x = 1946707054, y = 34,d = 57262074

x = 1946707054, y = 34,d = 57257980

x = 1946707054, y = 34,d = 57257980

x = 1946707054, y = 34,d = 57257980

x = 1946707054, y = 34,d = 57253887

x = 1946707054, y = 34,d = 57257979

x = 1946707054, y = 34,d = 57257979

x = 1946707054, y = 34,d = 57255933

x = 1946707054, y = 34,d = 57255933

x = 1946707054, y = 34,d = 57255934

x = 1946707054, y = 34,d = 57257978

x = 1946707054, y = 34,d = 57257978

x = 1946707054, y = 34,d = 57256956

x = 1946707054, y = 34,d = 57256956

x = 1946707054, y = 34,d = 57256956

x = 1946707054, y = 34,d = 57255935

x = 1946707054, y = 34,d = 57256955

x = 1946707054, y = 34,d = 57256955

x = 1946707054, y = 34,d = 57256445

x = 1946707054, y = 34,d = 57256445

x = 1946707054, y = 34,d = 57256445

x = 1946707054, y = 34,d = 57255936

x = 1946707054, y = 34,d = 57256444

x = 1946707054, y = 34,d = 57256444

x = 1946707054, y = 34,d = 57256190

x = 1946707054, y = 34,d = 57256190

x = 1946707054, y = 34,d = 57256190

x = 1946707054, y = 34,d = 57255937

x = 1946707054, y = 34,d = 57256189

x = 1946707054, y = 34,d = 57256189

x = 1946707054, y = 34,d = 57256063

x = 1946707054, y = 34,d = 57256063

x = 1946707054, y = 34,d = 57256064

x = 1946707054, y = 34,d = 57256188

x = 1946707054, y = 34,d = 57256188

x = 1946707054, y = 34,d = 57256126

x = 1946707054, y = 34,d = 57256126

x = 1946707054, y = 34,d = 57256126

x = 1946707054, y = 34,d = 57256065

x = 1946707054, y = 34,d = 57256125

x = 1946707054, y = 34,d = 57256125

x = 1946707054, y = 34,d = 57256095

x = 1946707054, y = 34,d = 57256095

x = 1946707054, y = 34,d = 57256095

x = 1946707054, y = 34,d = 57256066

x = 1946707054, y = 34,d = 57256094

x = 1946707054, y = 34,d = 57256094

x = 1946707054, y = 34,d = 57256080

x = 1946707054, y = 34,d = 57256080

x = 1946707054, y = 34,d = 57256081

x = 1946707054, y = 34,d = 57256093

x = 1946707054, y = 34,d = 57256093

x = 1946707054, y = 34,d = 57256087

x = 1946707054, y = 34,d = 57256087

x = 1946707054, y = 34,d = 57256088

x = 1946707054, y = 34,d = 57256092

x = 1946707054, y = 34,d = 57256092

x = 1946707054, y = 34,d = 57256090

x = 1946707054, y = 34,d = 57256090

x = 1946707054, y = 34,d = 57256090

x = 1946707054, y = 34,d = 57256089

商=57256089

请按任意键继续. . .

代码实现

test_01.cpp

// test_01.cpp : Defines the entry point for the console application.
//

#include<stdio.h>
#include<stdlib.h>

int main()
{
int half(int m,int n,int x,int y);
int x,y,result;

printf("请输入正整数 x,y \n");
scanf("%d",&x);
scanf("%d",&y);
result=half(1,2,x,y);
printf("商=%d \n",result) ;
system("pause");
return 0;
}

// 类似折半查找商值：running time（log n）
/* functiong check()
input(m，n,x,y)
out result

if( m <=result<=n )
折半查找就ok
else return half( n+1，2*n+2 ,x,y )
*/
int half(int m,int n,int x,int y)
{
int check(int x,int y, int d);

if(check(x,y,m)==0 )
{
return m ;
}
else if( check(x,y,n)==0 )
{
return n;
}
else if( check(x,y,n) == -1 )
{
if( check(x,y,(n+m)/2) == 0 )
return (n+m)/2;
else if( check(x,y,(n+m)/2) == 1 )
return half( (n+m)/2+1,n-1,x,y );
else if( check(x,y,(n+m)/2) == -1 )
return half( m+1,(n+m)/2-1,x,y );
}
else return half( n+1,2*n+2,x,y );

}

// check the relation of the right result and d :running time(1)
/* functiong check()
input(x,y,d)
if( d==result )
return 0;
else if( d>result )
return -1;
else return 1;
*/

int check(int x,int y, int d)
{
printf( "x = %d, y = %d,d = %d\n",x,y,d );
if( x-d*y<y && x-d*y>=0 )
{
return 0 ;
}
else if( x-d*y>=y )
{
return 1;
}
else if( x-d*y< 0 )
{
return -1;
}
}

上面包含不严谨之处，前几天经过有心善良之人提醒，于2017-6-16 16:34 更新

#include<stdio.h>
#include<stdlib.h>
#include<math.h>
#include<limits.h>

//#define INT_MAX 0x7fffffff

unsigned int half(unsigned int m,unsigned int n,unsigned int x,unsigned int y);
unsigned int check(unsigned int x,unsigned int y, unsigned int d);
unsigned int max(unsigned int a, unsigned int b);
unsigned int min(unsigned int a, unsigned int b);

unsigned int main(unsigned int argc, char *argv[])
//unsigned int main()
{
unsigned int x, y, result;
char** left;
printf("正整数 x = %s, y = %s \n", argv[1], argv[2]);
x = atoi(argv[1]);
y = atoi(argv[2]);
result=half(1,y,x,y);
printf("商=%u \n", result);
return 0;
}

unsigned int half(unsigned int m,unsigned int n,unsigned int x,unsigned int y)
{
unsigned int dis;
printf( "m = %u, n = %u, x = %u, y = %u, check m = %u, check n = %u \n", m,n,x,y, check(x, y, m), check(x, y, n));
if(check(x,y,m)==-1){
return 0;
}
else if(check(x,y,m)==0 )
{
return m ;
}
else if( check(x,y,n)==0 )
{
return n;
}
else if( check(x,y,n) == -1 )
{
dis = sqrt(n-m);
dis = dis * sqrt(dis);
return max(half(m,m+dis,x,y), half(m+dis+1,n,x,y));
}
else return half( n+1, min(INT_MAX ,pow(n+1, 2)), x, y );
}

// max : max of a and b

unsigned int max(unsigned int a, unsigned int b)
{
return a > b ? a : b ;
}

// min : min of a and b
unsigned int min(unsigned int a, unsigned int b)
{
return a > b ? b : a ;
}

// check: d 和 正确商的距离
unsigned int check(unsigned int x,unsigned int y, unsigned int d)
{
if( y*d <= x && y*(d+1) > x)
{
return 0 ;
}
else if( y*(d+1) <= x )
{
return 1;
}
else if(y*d > x)
{
return -1;
}
}