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算法题：求两个大数的差

03-18 面试题/算法&算力分析 308

本文概述

给定两个数字作为字符串。这些数字可能非常大(可能不适合int long int int), 任务是找到这两个数字的差。

例子：

Input : str1 = "11443333311111111100", str2 =     "1144422222221111"
Output : 11442188888888889989

Input :str1 = "122387876566565674", str2 =     "31435454654554"
Output : 122356441111911120

这是基于学校数学的简单方法。我们从头开始遍历两个字符串, 一个接一个的减数位。

  1. 颠倒两个字符串。
  2. 继续从第0个索引开始(以反向字符串)一一减一到小字符串的末尾, 如果对结果的末尾是正数, 则追加差异。如果difference(diff)为负, 则加10, 如果正值为正, 则将进位保持为1, 则进位为0。
  3. 最后, 反转结果。

以下是上述想法的实现：

C++

//C++ program to find difference of two large numbers.
#include <bits/stdc++.h>
using namespace std;
 
//Returns true if str1 is smaller than str2.
bool isSmaller(string str1, string str2)
{
     //Calculate lengths of both string
     int n1 = str1.length(), n2 = str2.length();
 
     if (n1 <n2)
         return true ;
     if (n2 <n1)
         return false ;
 
     for ( int i = 0; i <n1; i++)
         if (str1[i] <str2[i])
             return true ;
         else if (str1[i]> str2[i])
             return false ;
 
     return false ;
}
 
//Function for find difference of larger numbers
string findDiff(string str1, string str2)
{
     //Before proceeding further, make sure str1
     //is not smaller
     if (isSmaller(str1, str2))
         swap(str1, str2);
 
     //Take an empty string for storing result
     string str = "" ;
 
     //Calculate length of both string
     int n1 = str1.length(), n2 = str2.length();
 
     //Reverse both of strings
     reverse(str1.begin(), str1.end());
     reverse(str2.begin(), str2.end());
 
     int carry = 0;
 
     //Run loop till small string length
     //and subtract digit of str1 to str2
     for ( int i = 0; i <n2; i++) {
         //Do school mathematics, compute difference of
         //current digits
 
         int sub
             = ((str1[i] - '0' ) - (str2[i] - '0' ) - carry);
 
         //If subtraction is less then zero
         //we add then we add 10 into sub and
         //take carry as 1 for calculating next step
         if (sub <0) {
             sub = sub + 10;
             carry = 1;
         }
         else
             carry = 0;
 
         str.push_back(sub + '0' );
     }
 
     //subtract remaining digits of larger number
     for ( int i = n2; i <n1; i++) {
         int sub = ((str1[i] - '0' ) - carry);
 
         //if the sub value is -ve, then make it positive
         if (sub <0) {
             sub = sub + 10;
             carry = 1;
         }
         else
             carry = 0;
 
         str.push_back(sub + '0' );
     }
 
     //reverse resultant string
     reverse(str.begin(), str.end());
 
     return str;
}
 
//Driver code
int main()
{
     string str1 = "978" ;
     string str2 = "12977" ;
 
     //Function call
     cout <<findDiff(str1, str2) <<endl;
 
     string s1 = "100" ;
     string s2 = "1000000" ;
   
     //Function call
     cout <<findDiff(s1, s2);
 
     return 0;
}

Java

//Java program to find difference of two large numbers.
import java.util.*;
 
class GFG {
 
     //Returns true if str1 is smaller than str2.
     static boolean isSmaller(String str1, String str2)
     {
         //Calculate lengths of both string
         int n1 = str1.length(), n2 = str2.length();
         if (n1 <n2)
             return true ;
         if (n2 <n1)
             return false ;
 
         for ( int i = 0 ; i <n1; i++)
             if (str1.charAt(i) <str2.charAt(i))
                 return true ;
             else if (str1.charAt(i)> str2.charAt(i))
                 return false ;
 
         return false ;
     }
 
     //Function for find difference of larger numbers
     static String findDiff(String str1, String str2)
     {
         //Before proceeding further, make sure str1
         //is not smaller
         if (isSmaller(str1, str2)) {
             String t = str1;
             str1 = str2;
             str2 = t;
         }
 
         //Take an empty string for storing result
         String str = "" ;
 
         //Calculate length of both string
         int n1 = str1.length(), n2 = str2.length();
 
         //Reverse both of strings
         str1 = new StringBuilder(str1).reverse().toString();
         str2 = new StringBuilder(str2).reverse().toString();
 
         int carry = 0 ;
 
         //Run loop till small string length
         //and subtract digit of str1 to str2
         for ( int i = 0 ; i <n2; i++) {
             //Do school mathematics, compute difference of
             //current digits
             int sub
                 = (( int )(str1.charAt(i) - '0' )
                    - ( int )(str2.charAt(i) - '0' ) - carry);
 
             //If subtraction is less then zero
             //we add then we add 10 into sub and
             //take carry as 1 for calculating next step
             if (sub <0 ) {
                 sub = sub + 10 ;
                 carry = 1 ;
             }
             else
                 carry = 0 ;
 
             str += ( char )(sub + '0' );
         }
 
         //subtract remaining digits of larger number
         for ( int i = n2; i <n1; i++) {
             int sub = (( int )(str1.charAt(i) - '0' ) - carry);
 
             //if the sub value is -ve, then make it
             //positive
             if (sub <0 ) {
                 sub = sub + 10 ;
                 carry = 1 ;
             }
             else
                 carry = 0 ;
 
             str += ( char )(sub + '0' );
         }
 
         //reverse resultant string
         return new StringBuilder(str).reverse().toString();
     }
 
     //Driver code
     public static void main(String[] args)
     {
         String str1 = "978" ;
         String str2 = "12977" ;
 
         //Function call
         System.out.println(findDiff(str1, str2));
 
         String s1 = "100" ;
         String s2 = "1000000" ;
       
         //Function call
         System.out.println(findDiff(s1, s2));
     }
}
 
//This code is contributed by mits

Python3

# Python 3 program to find difference of two large numbers.
 
# Returns true if str1 is smaller than str2.
 
 
def isSmaller(str1, str2):
 
     # Calculate lengths of both string
     n1 = len (str1)
     n2 = len (str2)
 
     if (n1 <n2):
         return True
     if (n2 <n1):
         return False
 
     for i in range (n1):
         if (str1[i] <str2[i]):
             return True
         elif (str1[i]> str2[i]):
             return False
 
     return False
 
# Function for find difference of larger numbers
 
 
def findDiff(str1, str2):
 
     # Before proceeding further, make sure str1
     # is not smaller
     if (isSmaller(str1, str2)):
         temp = str1
         str1 = str2
         str2 = temp
 
     # Take an empty string for storing result
     str3 = ""
 
     # Calculate length of both string
     n1 = len (str1)
     n2 = len (str2)
 
     # Reverse both of strings
     str1 = str1[:: - 1 ]
     str2 = str2[:: - 1 ]
 
     carry = 0
 
     # Run loop till small string length
     # and subtract digit of str1 to str2
     for i in range (n2):
 
         # Do school mathematics, compute difference of
         # current digits
 
         sub = (( ord (str1[i]) - ord ( '0' )) - ( ord (str2[i]) - ord ( '0' )) - carry)
 
         # If subtraction is less then zero
         # we add then we add 10 into sub and
         # take carry as 1 for calculating next step
         if (sub <0 ):
 
             sub = sub + 10
             carry = 1
 
         else :
             carry = 0
 
         str3 = str3 + str (sub)
 
     # subtract remaining digits of larger number
     for i in range (n2, n1):
 
         sub = (( ord (str1[i]) - ord ( '0' )) - carry)
 
         # if the sub value is -ve, then make it positive
         if (sub <0 ):
 
             sub = sub + 10
             carry = 1
 
         else :
             carry = 0
 
         str3 = str3 + str (sub)
 
     # reverse resultant string
     str3 = str3[:: - 1 ]
 
     return str3
 
 
# Driver code
if __name__ = = "__main__" :
     str1 = "978"
     str2 = "12977"
     
     # Function call
     print (findDiff(str1, str2))
 
     s1 = "100"
     s2 = "1000000"
     
     # Function call
     print (findDiff(s1, s2))
 
# This code is contributed by ChitraNayal

C#

//C# program to find difference of two large numbers.
using System;
using System.Collections;
 
class GFG {
 
     //Returns true if str1 is smaller than str2.
     static bool isSmaller( string str1, string str2)
     {
         //Calculate lengths of both string
         int n1 = str1.Length, n2 = str2.Length;
         if (n1 <n2)
             return true ;
         if (n2 <n1)
             return false ;
 
         for ( int i = 0; i <n1; i++)
             if (str1[i] <str2[i])
                 return true ;
             else if (str1[i]> str2[i])
                 return false ;
 
         return false ;
     }
 
     //Function for find difference of larger numbers
     static string findDiff( string str1, string str2)
     {
         //Before proceeding further, make sure str1
         //is not smaller
         if (isSmaller(str1, str2)) {
             string t = str1;
             str1 = str2;
             str2 = t;
         }
 
         //Take an empty string for storing result
         string str = "" ;
 
         //Calculate length of both string
         int n1 = str1.Length, n2 = str2.Length;
 
         //Reverse both of strings
         char [] ch1 = str1.ToCharArray();
         Array.Reverse(ch1);
         str1 = new string (ch1);
         char [] ch2 = str2.ToCharArray();
         Array.Reverse(ch2);
         str2 = new string (ch2);
 
         int carry = 0;
 
         //Run loop till small string length
         //and subtract digit of str1 to str2
         for ( int i = 0; i <n2; i++) {
             //Do school mathematics, compute difference of
             //current digits
             int sub = (( int )(str1[i] - '0' )
                        - ( int )(str2[i] - '0' ) - carry);
 
             //If subtraction is less then zero
             //we add then we add 10 into sub and
             //take carry as 1 for calculating next step
             if (sub <0) {
                 sub = sub + 10;
                 carry = 1;
             }
             else
                 carry = 0;
 
             str += ( char )(sub + '0' );
         }
 
         //subtract remaining digits of larger number
         for ( int i = n2; i <n1; i++) {
             int sub = (( int )(str1[i] - '0' ) - carry);
 
             //if the sub value is -ve, then make it
             //positive
             if (sub <0) {
                 sub = sub + 10;
                 carry = 1;
             }
             else
                 carry = 0;
 
             str += ( char )(sub + '0' );
         }
 
         //reverse resultant string
         char [] ch3 = str.ToCharArray();
         Array.Reverse(ch3);
         return new string (ch3);
     }
 
     //Driver code
     public static void Main()
     {
         string str1 = "978" ;
         string str2 = "12977" ;
 
         //Function call
         Console.WriteLine(findDiff(str1, str2));
 
         string s1 = "100" ;
         string s2 = "1000000" ;
       
         //Function call
         Console.WriteLine(findDiff(s1, s2));
     }
}
 
//This code is contributed by mits

PHP

<?php
//PHP program to find difference of two large numbers.
 
//Returns true if str1 is smaller than str2.
function isSmaller( $str1 , $str2 )
{
     //Calculate lengths of both string
     $n1 = strlen ( $str1 );
     $n2 = strlen ( $str2 );
 
     if ( $n1 <$n2 )
     return true;
     if ( $n2 <$n1 )
     return false;
 
     for ( $i =0; $i <$n1 ; $i ++)
     if ( $str1 [ $i ] <$str2 [ $i ])
         return true;
     else if ( $str1 [ $i ]> $str2 [ $i ])
         return false;
 
     return false;
}
 
//Function for find difference of larger numbers
function findDiff( $str1 , $str2 )
{
     //Before proceeding further, make sure str1
     //is not smaller
     if (isSmaller( $str1 , $str2 )){
         $t = $str1 ;
         $str1 = $str2 ;
         $str2 = $t ;
     }
 
     //Take an empty string for storing result
     $str = "" ;
 
     //Calculate length of both string
     $n1 = strlen ( $str1 );
     $n2 = strlen ( $str2 );
 
     //Reverse both of strings
     $str1 = strrev ( $str1 );
     $str2 = strrev ( $str2 );
     
     $carry = 0;
 
     //Run loop till small string length
     //and subtract digit of str1 to str2
     for ( $i =0; $i <$n2 ; $i ++)
     {
         //Do school mathematics, compute difference of
         //current digits
         
         $sub =((ord( $str1 [ $i ])-ord( '0' ))-(ord( $str2 [ $i ])-ord( '0' ))- $carry );
         
         //If subtraction is less then zero
         //we add then we add 10 into sub and
         //take carry as 1 for calculating next step
         if ( $sub <0)
         {
             $sub = $sub + 10;
             $carry = 1;
         }
         else
             $carry = 0;
 
         $str .= chr ( $sub +48);
     }
 
     //subtract remaining digits of larger number
     for ( $i = $n2 ; $i <$n1 ; $i ++)
     {
         $sub = ((ord( $str1 [ $i ])-ord( '0' )) - $carry );
         
         //if the sub value is -ve, then make it positive
         if ( $sub <0)
         {
             $sub = $sub + 10;
             $carry = 1;
         }
         else
             $carry = 0;
             
         $str .= chr ( $sub +48);
     }
 
     //reverse resultant string
     $str = strrev ( $str );
 
     return $str ;
}
 
//Driver code
 
     $str1 = "978" ;
     $str2 = "12977" ;
 
     //Function call
     echo findDiff( $str1 , $str2 ). "\n" ;
     
     $s1 = "100" ;
     $s2 = "1000000" ;
 
     //Function call
     echo findDiff( $s1 , $s2 );
     
//This code is contributed by mits
?>

输出如下

11999
0999900

优化的解决方案：

通过从末尾遍历它们, 可以避免前两个字符串反向操作。

以下是优化的解决方案。

C++

//C++ program to find difference of two large numbers.
#include <bits/stdc++.h>
using namespace std;
 
//Returns true if str1 is smaller than str2, //else false.
bool isSmaller(string str1, string str2)
{
     //Calculate lengths of both string
     int n1 = str1.length(), n2 = str2.length();
 
     if (n1 <n2)
         return true ;
     if (n2 <n1)
         return false ;
 
     for ( int i = 0; i <n1; i++) {
         if (str1[i] <str2[i])
             return true ;
         else if (str1[i]> str2[i])
             return false ;
     }
     return false ;
}
 
//Function for finding difference of larger numbers
string findDiff(string str1, string str2)
{
     //Before proceeding further, make sure str1
     //is not smaller
     if (isSmaller(str1, str2))
         swap(str1, str2);
 
     //Take an empty string for storing result
     string str = "" ;
 
     //Calculate lengths of both string
     int n1 = str1.length(), n2 = str2.length();
     int diff = n1 - n2;
 
     //Initially take carry zero
     int carry = 0;
 
     //Traverse from end of both strings
     for ( int i = n2 - 1; i>= 0; i--) {
         //Do school mathematics, compute difference of
         //current digits and carry
         int sub = ((str1[i + diff] - '0' ) - (str2[i] - '0' )
                    - carry);
         if (sub <0) {
             sub = sub + 10;
             carry = 1;
         }
         else
             carry = 0;
 
         str.push_back(sub + '0' );
     }
 
     //subtract remaining digits of str1[]
     for ( int i = n1 - n2 - 1; i>= 0; i--) {
         if (str1[i] == '0' && carry) {
             str.push_back( '9' );
             continue ;
         }
         int sub = ((str1[i] - '0' ) - carry);
         if (i> 0 || sub> 0) //remove preceding 0's
             str.push_back(sub + '0' );
         carry = 0;
     }
 
     //reverse resultant string
     reverse(str.begin(), str.end());
 
     return str;
}
 
//Driver code
int main()
{
     string str1 = "88" ;
     string str2 = "1079" ;
   
     //Function call
     cout <<findDiff(str1, str2);
     return 0;
}

Java

//Java program to find difference of two large numbers.
 
class GFG {
 
     //Returns true if str1 is smaller than str2, //else false.
     static boolean isSmaller(String str1, String str2)
     {
         //Calculate lengths of both string
         int n1 = str1.length(), n2 = str2.length();
 
         if (n1 <n2)
             return true ;
         if (n2 <n1)
             return false ;
 
         for ( int i = 0 ; i <n1; i++) {
             if (str1.charAt(i) <str2.charAt(i))
                 return true ;
             else if (str1.charAt(i)> str2.charAt(i))
                 return false ;
         }
         return false ;
     }
 
     //Function for finding difference of larger numbers
     static String findDiff(String str1, String str2)
     {
         //Before proceeding further, make sure str1
         //is not smaller
         if (isSmaller(str1, str2)) {
             String t = str1;
             str1 = str2;
             str2 = t;
         }
 
         //Take an empty string for storing result
         String str = "" ;
 
         //Calculate lengths of both string
         int n1 = str1.length(), n2 = str2.length();
         int diff = n1 - n2;
 
         //Initially take carry zero
         int carry = 0 ;
 
         //Traverse from end of both strings
         for ( int i = n2 - 1 ; i>= 0 ; i--) {
             //Do school mathematics, compute difference of
             //current digits and carry
             int sub
                 = ((( int )str1.charAt(i + diff) - ( int ) '0' )
                    - (( int )str2.charAt(i) - ( int ) '0' )
                    - carry);
             if (sub <0 ) {
                 sub = sub + 10 ;
                 carry = 1 ;
             }
             else
                 carry = 0 ;
 
             str += String.valueOf(sub);
         }
 
         //subtract remaining digits of str1[]
         for ( int i = n1 - n2 - 1 ; i>= 0 ; i--) {
             if (str1.charAt(i) == '0' && carry> 0 ) {
                 str += "9" ;
                 continue ;
             }
             int sub = ((( int )str1.charAt(i) - ( int ) '0' )
                        - carry);
             if (i> 0 || sub> 0 ) //remove preceding 0's
                 str += String.valueOf(sub);
             carry = 0 ;
         }
 
         //reverse resultant string
         return new StringBuilder(str).reverse().toString();
     }
 
     //Driver code
     public static void main(String[] args)
     {
         String str1 = "88" ;
         String str2 = "1079" ;
       
         //Function call
         System.out.println(findDiff(str1, str2));
     }
}
 
//This code is contributed by mits

Python3

# Python3 program to find difference
# of two large numbers.
 
# Returns true if str1 is smaller than
# str2, else false.
def isSmaller(str1, str2):
     
     # Calculate lengths of both string
     n1 = len (str1)
     n2 = len (str2)
 
     if (n1 <n2):
         return True
     if (n2 <n1):
         return False
 
     for i in range (n1):
         if (str1[i] <str2[i]):
             return True
         elif (str1[i]> str2[i]):
             return False
             
     return False
 
# Function for finding difference
# of larger numbers
def findDiff(str1, str2):
     
     # Before proceeding further, # make sure str1 is not smaller
     if (isSmaller(str1, str2)):
         str1, str2 = str2, str1
 
     # Take an empty string for
     # storing result
     Str = ""
     
     # Calculate lengths of both string
     n1 = len (str1)
     n2 = len (str2)
     diff = n1 - n2
     
     # Initially take carry zero
     carry = 0
     
     # Traverse from end of both strings
     for i in range (n2 - 1 , - 1 , - 1 ):
         
         # Do school mathematics, compute
         # difference of current digits
         # and carry
         sub = (( ord (str1[i + diff]) - ord ( '0' )) -
                ( ord (str2[i]) - ord ( '0' )) - carry)
 
         if (sub <0 ):
             sub + = 10
             carry = 1
         else :
             carry = 0
 
         Str + = chr (sub + ord ( '0' ))
     
     # Subtract remaining digits of str1[]
     for i in range (n1 - n2 - 1 , - 1 , - 1 ):
         if (str1[i] = = '0' and carry):
             Str + = '9'
             continue
             
         sub = ( ord (str1[i]) - ord ( '0' )) - carry
         
         # Remove preceding 0's
         if (i> 0 or sub> 0 ):
             Str + = chr (sub + ord ( '0' ))
             
         carry = 0
 
     # Reverse resultant string
     Str = Str [:: - 1 ]
     
     return Str
 
# Driver code
str1 = "88"
str2 = "1079"
 
# Function call
print (findDiff(str1, str2))
 
# This code is contributed by avanitrachhadiya2155

C#

//C# program to find difference of
//two large numbers.
using System;
 
class GFG {
 
     //Returns true if str1 is smaller
     //than str2, else false.
     static bool isSmaller( string str1, string str2)
     {
         //Calculate lengths of both string
         int n1 = str1.Length, n2 = str2.Length;
 
         if (n1 <n2)
             return true ;
         if (n2 <n1)
             return false ;
 
         for ( int i = 0; i <n1; i++) {
             if (str1[i] <str2[i])
                 return true ;
             else if (str1[i]> str2[i])
                 return false ;
         }
         return false ;
     }
 
     //Function for finding difference of
     //larger numbers
     static string findDiff( string str1, string str2)
     {
         //Before proceeding further, //make sure str1 is not smaller
         if (isSmaller(str1, str2)) {
             string t = str1;
             str1 = str2;
             str2 = t;
         }
 
         //Take an empty string for
         //storing result
         String str = "" ;
 
         //Calculate lengths of both string
         int n1 = str1.Length, n2 = str2.Length;
         int diff = n1 - n2;
 
         //Initially take carry zero
         int carry = 0;
 
         //Traverse from end of both strings
         for ( int i = n2 - 1; i>= 0; i--) {
             //Do school mathematics, compute
             //difference of current digits and carry
             int sub = ((( int )str1[i + diff] - ( int ) '0' )
                        - (( int )str2[i] - ( int ) '0' ) - carry);
             if (sub <0) {
                 sub = sub + 10;
                 carry = 1;
             }
             else
                 carry = 0;
 
             str += sub.ToString();
         }
 
         //subtract remaining digits of str1[]
         for ( int i = n1 - n2 - 1; i>= 0; i--) {
             if (str1[i] == '0' && carry> 0) {
                 str += "9" ;
                 continue ;
             }
             int sub = ((( int )str1[i] - ( int ) '0' ) - carry);
             if (i> 0 || sub> 0) //remove preceding 0's
                 str += sub.ToString();
             carry = 0;
         }
 
         //reverse resultant string
         char [] aa = str.ToCharArray();
         Array.Reverse(aa);
         return new string (aa);
     }
 
     //Driver code
     public static void Main()
     {
         String str1 = "88" ;
         String str2 = "1079" ;
 
         //Function call
         Console.WriteLine(findDiff(str1, str2));
     }
}
 
//This code is contributed by mits

PHP

<?php
//PHP program to find difference of two
//large numbers.
 
//Returns true if str1 is smaller than
//str2, else false.
function isSmaller( $str1 , $str2 )
{
     //Calculate lengths of both string
     $n1 = strlen ( $str1 );
     $n2 = strlen ( $str2 );
 
     if ( $n1 <$n2 )
         return true;
     if ( $n2 <$n1 )
         return false;
 
     for ( $i = 0; $i <$n1 ; $i ++)
     {
         if ( $str1 [ $i ] <$str2 [ $i ])
             return true;
         else if ( $str1 [ $i ]> $str2 [ $i ])
             return false;
     }
     return false;
}
 
//Function for finding difference
//of larger numbers
function findDiff( $str1 , $str2 )
{
     //Before proceeding further, make
     //sure str1 is not smaller
     if (isSmaller( $str1 , $str2 ))
     {
         $t = $str1 ;
         $str1 = $str2 ;
         $str2 = $t ;
     }
 
     //Take an empty string for storing result
     $str = "" ;
 
     //Calculate lengths of both string
     $n1 = strlen ( $str1 );
     $n2 = strlen ( $str2 );
     $diff = $n1 - $n2 ;
 
     //Initially take carry zero
     $carry = 0;
 
     //Traverse from end of both strings
     for ( $i = $n2 - 1; $i>= 0; $i --)
     {
         //Do school mathematics, compute
         //difference of current digits and carry
         $sub = ((ord( $str1 [ $i + $diff ]) - ord( '0' )) -
                 (ord( $str2 [ $i ]) - ord( '0' )) - $carry );
         if ( $sub <0)
         {
             $sub = $sub + 10;
             $carry = 1;
         }
         else
             $carry = 0;
 
         $str .= chr ( $sub + ord( "0" ));
     }
 
     //subtract remaining digits of str1[]
     for ( $i = $n1 - $n2 - 1; $i>= 0; $i --)
     {
         if ( $str1 [ $i ] == '0' && $carry> 0)
         {
             $str .= "9" ;
             continue ;
         }
         $sub = (ord( $str1 [ $i ]) - ord( '0' ) - $carry );
         if ( $i> 0 || $sub> 0) //remove preceding 0's
             $str .= chr ( $sub + ord( "0" ));
         $carry = 0;
 
     }
 
     //reverse resultant string
     return strrev ( $str );
 
}
 
//Driver code
$str1 = "88" ;
$str2 = "1079" ;
 
//Function call
print (findDiff( $str1 , $str2 ));
 
//This code is contributed by chandan_jnu
?>

输出如下

991

时间复杂度：O(n1 + n2)

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