Find the missing and duplicate elements in an array in linear time and constant space

Question

You’re given an array of N 64 bit integers. N may be very large. You know that every integer 1..N appears once in the array, except there is one integer mis

Answer 1

Using the basic idea from a related interview question you could do:

• Sum up all the numbers (we shall call this S1) and their squares (S2)
• Compute the expected sum of the numbers, without modifications, i.e. E1 = n*(n+1)/2 and E2 = n*(n+1)*(2n+1)/6
• Now you know that E1 - S1 = d - m and E2 - S2 = d^2 - m^2, where d is the duplicated number and m the missing one.

• Solve this system of equations and you'll find that:

  m = 1/2 ((E2 - S2)/(E1 - S1) - (E1 - S1))
  d = 1/2 ((E2 - S2)/(E1 - S1) + (E1 - S1)) // or even simpler: d = m + (E1 - S1)
  

.

$S1 = $S2 = 0;
foreach ($nums as $num) {
    $S1 += $num;
    $S2 += $num * $num;
}

$D1 = $n * ($n + 1) / 2                - $S1;
$D2 = $n * ($n + 1) * (2 * $n + 1) / 6 - $S2;

$m = 1/2 * ($D2/$D1 - $D1);
$d = 1/2 * ($D2/$D1 + $D1);