Need some help understanding python solutions of leetcode 371. "Sum of Two Integers". I found https://discuss.leetcode.com/topic/49900/python-solution/2 is the most voted python solution, but I am having problem understand it.
class Solution(object): def getSum(self, a, b): """ :type a: int :type b: int :rtype: int """ MAX_INT = 0x7FFFFFFF MIN_INT = 0x80000000 MASK = 0x100000000 while b: a, b = (a ^ b) % MASK, ((a & b) << 1) % MASK return a if a <= MAX_INT else ~((a % MIN_INT) ^ MAX_INT)
Let's disregard the
MIN_INT for a second.
Why does this black magic bitwise stuff work?
The reason why the calculation works is because
(a ^ b) is "summing" the bits of
b. Recall that bitwise xor is
1 when the bits differ, and
0 when the bits are the same. For example (where D is decimal and B is binary), 20D == 10100B, and 9D = 1001B:
10100 1001 ----- 11101
and 11101B == 29D.
But, if you have a case with a carry, it doesn't work so well. For example, consider adding (bitwise xor) 20D and 20D.
10100 10100 ----- 00000
Oops. 20 + 20 certainly doesn't equal 0. Enter the
(a & b) << 1 term. This term represents the "carry" for each position. On the next iteration of the while loop, we add in the carry from the previous loop. So, if we go with the example we had before, we get:
# First iteration (a is 20, b is 20) 10100 ^ 10100 == 00000 # makes a 0 (10100 & 10100) << 1 == 101000 # makes b 40 # Second iteration: 000000 ^ 101000 == 101000 # Makes a 40 (000000 & 101000) << 1 == 0000000 # Makes b 0
b is 0, we are done, so return
a. This algorithm works in general, not just for the specific cases I've outlined. Proof of correctness is left to the reader as an exercise ;)
What do the masks do?
All the masks are doing is ensuring that the value is an integer, because your code even has comments stating that
b, and the return type are of type
int. Thus, since the maximum possible
int (32 bits) is 2147483647. So, if you add 2 to this value, like you did in your example, the
int overflows and you get a negative value. You have to force this in Python, because it doesn't respect this
int boundary that other strongly typed languages like Java and C++ have defined. Consider the following:
def get_sum(a, b): while b: a, b = (a ^ b), (a & b) << 1 return a
This is the version of
getSum without the masks.
print get_sum(2147483647, 2)
print Solution().getSum(2147483647, 2)
due to the overflow.
The moral of the story is the implementation is correct if you define the
int type to only represent 32 bits.
Here is solution works in every case
cases - - - + + - + +
python default int size is not 32bit, it is very large number, so to prevent overflow and stop running into infinite loop, we use 32bit mask to limit int size to 32bit (0xffffffff)
a,b=-1,-1 mask=0xffffffff while (b & mask): carry=a & b a=a^b b=carray <<1 print( (a&Mask) if b>0 else a)
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