# Help needed in Yet Another Minimax Problem

Hello Community!
I am trying to solve this problem . Can someone please explain the approach as I didn’t understood from editorial.

1 Like

Try this (hopefully you can view it OK):

https://www.hackerrank.com/challenges/yet-another-minimax-problem/submissions/code/31410889

Edit:

``````#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
using namespace std;

int main() {
int n;
cin >> n;
vector<unsigned long> a(n);
for (int i = 0; i < n; i++)
{
cin >> a[i];
}
// Tricky one :) First of all: pick the highest power of 2 occurring
// in any of the a's (i.e. appearing in their binary representation).
// If *all* of the a's have this power of 2, then that power of 2
// cannot possibly be in the min score (every permutation eliminates it)
// i.e. the min score is less than that power of 2, so we can remove that power of
// 2 from all a's.
// Then look at the new highest power of 2 occuring in any of  the a's: if
// *that* occurs in all a's, etc.
// Keep going until the highest power of 2 occurring in any of the a's does not
// occur in all of the a's (call it P): then P *must* occur in the min score:
// any permutation must place a value with P next to a value
// without P.  So min score >= P.
// Let x be the number of a's with P (so n - x of them do not have that P).
// Now, re-order a so that the first x elements are those with P,
// and the next n - x are those without it.  Note that none of the first x among
// themselves can give rise to a score >= P (they all contain P, so xor'ing any
// pair of them together gives a value without P i.e. < P).  Likewise, none of the latter
// n - x can give rise to a score with P: none of them has  P, so xor'ing any pair
// of them together cannot give a value with P.
// In fact, the highest contributor to the score is at the single place where a value
// with P is placed next to the value without P i.e. it is, in our newly-ordered a, the value
//   a[x - 1] ^ a[x]
// (a[x - 1] has P; a[x] does not; a[x - 1] ^ a[x] >= P).
// We can then re-order a slightly so that the values occurring in a[x - 1] and a[x] have
// the minimum value after xor'ing; this permutation of a will give the minimum score.
// Note that a potentially very high amount of permutations will give the same minimum score
// since further permuting the first x - 1 and latter n - x - 1 once we've chosen the minimal permutation
// described above has no effect on the score.
uint64_t powerOf2 = static_cast<uint64_t>(1) << 63;
while (powerOf2 > 0)
{
int numWithPowerOf2 = 0;
for (auto value : a)
{
if ((value & powerOf2) != 0)
{
numWithPowerOf2++;
}
}
if (numWithPowerOf2 == n)
{
// This power of 2 cannot occur in the min score, so
// remove it from all a's for simplicity.
for (auto& value : a)
{
value -= powerOf2;
}
}
else if (numWithPowerOf2 != 0)
{
// We've found the highest power of 2 occurring in at
// least one of the a's, but not in all of them.
// This is the "P" described above.
break;
}
powerOf2 >>= 1;
}
vector<unsigned int> withPowerOf2;
vector<unsigned int> withoutPowerOf2;
for (auto value : a)
{
if ((value & powerOf2) != 0)
{
withPowerOf2.push_back(value);
}
else
{
withoutPowerOf2.push_back(value);
}
}
unsigned int minScore = std::numeric_limits<unsigned int>::max();
for (auto value1 : withPowerOf2)
{
for (auto value2 : withoutPowerOf2)
{
const unsigned int score = value1 ^ value2;
minScore = min(minScore, score);
}
}
if (withPowerOf2.empty() || withoutPowerOf2.empty())
{
// Original array must have consisted of all equal values;
// any permutation gives score of 0.
minScore = 0;
}
cout << minScore << endl;
return 0;
}

``````
3 Likes

Great Explanation! Thanks!!

1 Like