algorithm - Number of 1s in the two's complement binary representations of integers in a range

Question

This problem is from the 2011 Codesprint (http://csfall11.interviewstreet.com/):

One of the basics of Computer Science is knowing how numbers are represented in 2's complement. Imagine that you write down all numbers between A and B inclusive in 2's complement representation using 32 bits. How many 1's will you write down in all ? Input: The first line contains the number of test cases T (<1000). Each of the next T lines contains two integers A and B. Output: Output T lines, one corresponding to each test case. Constraints: -2^31 <= A <= B <= 2^31 - 1

Sample Input: 3 -2 0 -3 4 -1 4 Sample Output: 63 99 37

Explanation: For the first case, -2 contains 31 1's followed by a 0, -1 contains 32 1's and 0 contains 0 1's. Thus the total is 63. For the second case, the answer is 31 + 31 + 32 + 0 + 1 + 1 + 2 + 1 = 99

I realize that you can use the fact that the number of 1s in -X is equal to the number of 0s in the complement of (-X) = X-1 to speed up the search. The solution claims that there is a O(log X) recurrence relation for generating the answer but I do not understand it. The solution code can be viewed here: https://gist.github.com/1285119

I would appreciate it if someone could explain how this relation is derived!

Answer 1

Well, it's not that complicated...

The single-argument solve(int a) function is the key. It is short, so I will cut&paste it here:

long long solve(int a)
{
 if(a == 0) return 0 ;
 if(a % 2 == 0) return solve(a - 1) + __builtin_popcount(a) ;
 return ((long long)a + 1) / 2 + 2 * solve(a / 2) ;
}

It only works for non-negative a, and it counts the number of 1 bits in all integers from 0 to a inclusive.

The function has three cases:

a == 0 -> returns 0. Obviously.

a even -> returns the number of 1 bits in a plus solve(a-1). Also pretty obvious.

The final case is the interesting one. So, how do we count the number of 1 bits from 0 to an odd number a?

Consider all of the integers between 0 and a, and split them into two groups: The evens, and the odds. For example, if a is 5, you have two groups (in binary):

000  (aka. 0)
010  (aka. 2)
100  (aka. 4)

and

001  (aka 1)
011  (aka 3)
101  (aka 5)

Observe that these two groups must have the same size (because a is odd and the range is inclusive). To count how many 1 bits there are in each group, first count all but the last bits, then count the last bits.

All but the last bits looks like this:

00
01
10

...and it looks like this for both groups. The number of 1 bits here is just solve(a/2). (In this example, it is the number of 1 bits from 0 to 2. Also, recall that integer division in C/C++ rounds down.)

The last bit is zero for every number in the first group and one for every number in the second group, so those last bits contribute (a+1)/2 one bits to the total.

So the third case of the recursion is (a+1)/2 + 2*solve(a/2), with appropriate casts to long long to handle the case where a is INT_MAX (and thus a+1 overflows).

This is an O(log N) solution. To generalize it to solve(a,b), you just compute solve(b) - solve(a), plus the appropriate logic for worrying about negative numbers. That is what the two-argument solve(int a, int b) is doing.