Maximize AND on a sequence of XORs

Question

Problem

We are given 2 arrays a and b both of length n. We build a third array c by rearranging the values in b. The goal is to find the optimal c that maximizes

result = (a[0] ^ c[0]) & (a[1] ^ c[1]) & ... & (a[n - 1] ^ c[n - 1])

where ^ is XOR and & is AND.

Is it possible to do this efficiently? It's straightforward to iterate through all possible permutations of b, but this is infeasible for large n.

More details

• The order of the values in a is fixed.
• The order of the values in b may be rearranged to form c. That is, starting with b = [1, 2, 3], it may be that the maximum result is obtained when the values are rearranged to c = [2, 1, 3].
• b may be rearranged in-place if needed.
• Since the optimal c is not necessarily unique, any optimal c may be returned.
• Assume all values are 32-bit unsigned integers.
• 1 <= n <= 10,000.

Test cases

Input:
a = [3, 4, 5]
b = [6, 7, 8]
Output:
c = [8, 7, 6] (result = 3)

Input:
a = [1, 11, 7, 4, 10, 11]
b = [6, 20, 8, 9, 10, 7]
Output:
c = [8, 6, 10, 9, 7, 20] (result = 9)

Input:
a = [0, 1, 2, 4, 8, 16]
b = [512, 256, 128, 64, 32, 16]
Output:
c = [16, 32, 64, 128, 256, 512] (result = 0)

Answer 1

For bit #k to be set in the output, you need to arrange the numbers so that bit #k is either set in a[i] and cleared in c[i] or the other way round. Therefore bit #k must be set in exactly n of your 2n inputs. If that is not the case then bit #k cannot be set in the output. So first you determine all k such that bit #k is set in exactly n of the 2n inputs, and ignore all other ks.

Let k be the highest of these bit positions. Split the elements of b into those where bit #k is set and those where it is cleared, and pick c[i] from the correct one of these sets. That's always possible.

Let j be the second highest of these bit positions. Split the elements of b into four sets, where bit #k and #j are both set, bit #k is set and bit #j is cleared, etc. Depending on which bits in a[i] are set, you know from which subset of b you need to pick c[i]. This only works if the sets have the right sizes! If it doesn't work, ignore bit j and take the next highest bit. If it works you can set bit #k and #j in the result and proceed with the next higher bit.