Sum bitdiff between all pairs of numbers in a collection
There are many ways to compare two integers. One of them is to compare each bit of numbers e.g. comparing 2 and 1:
2 -> ...0010
1 -> ...0001
Two last bits are different, so the result is 2. Here is one of many possible implementations:
public static int bitDiff(int a, int b)
{
int count = 0;
for (int i = 0; i < 32; i++)
{
int aBit = (1 << i) & a;
int bBit = (1 << i) & b;
if (aBit != bBit)
{
count++;
}
}
return count;
}
Now let’s have a look at more complex problem. From a collection of integers calculate bitDiff of all pairs and then sum the results.
∑ bitDiff(array[i], array[j])
input: {1,2,3} -> pairs {(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)} -> output: 8
This can be optimized by generating just half of the pairs (bitDiff operation is transitive), but it’s still O(n^2). Can we do O(n)? Scroll for a hint!
What if we just have 0’s and 1’s in the array?
Let’s consider a simple case:
[0,1] -> 1
[1,1] -> 0
[0,0] -> 0
[1,0] -> 1
Clearly, the result is 1 for pairs that are different. We can count 1’s and 0’s and multiply each other to get the number of pairs that result is 1 and then multiply by 2 (permutation of a pair)
[0,1] = 1 * 1 * 2 = 2
[0,1,1] = 1 * 2 * 2 = 4
[0,1,0,1] = 2 * 2 * 2 = 8
How to get the generalization for ints?
We can threat numbers as 32 arrays of 1’s and 0’s
public static bool isBitSet(int number, int index)
{
int bit = (1 << index) & number;
return bit > 0;
}
public static int sumPairBitDiff(int[] numbers)
{
int sum = 0;
for (int i = 0; i < 32; i++)
{
int oneCount = 0;
int zeroCount = 0;
foreach (var num in numbers)
{
if (isBitSet(num, i)) oneCount++;
}
zeroCount = numbers.Length - oneCount;
sum += oneCount * zeroCount * 2;
}
return sum;
}
Awesome!