Find the subsequence with largest sum of elements in an array

Question

I recently interviewed with a company and they asked me to write an algorithm that finds the subsequence with largest sum of elements in an array. The elements in the array can be negative. Is there a O(n) solution for it? Any good solutions are very much appreciated.

Answer 1

If you want the largest sum of sequential numbers then something like this might work:

$cur = $max = 0;
foreach ($seq as $n)
{
  $cur += $n;
  if ($cur < 0) $cur = 0;
  if ($cur > $max) $max = $cur;
}

That's just off the top of my head, but it seems right. (Ignoring that it assumes 0 is the answer for empty and all negative sets.)

Edit:

If you also want the sequence position:

$cur = $max = 0;
$cur_i = $max_i = 0; 
$max_j = 1;

foreach ($seq as $i => $n)
{
  $cur += $n;
  if ($cur > $max)
  {
    $max = $cur;
    if ($cur_i != $max_i)
    {
      $max_i = $cur_i;
      $max_j = $max_i + 1;
    }
    else
    {
      $max_j = $i + 1;
    }
  }

  if ($cur < 0)
  {
    $cur = 0;
    $cur_i = $i + 1;
  }
}

var_dump(array_slice($seq, $max_i, $max_j - $max_i), $max);

There might be a more concise way to do it. Again, it has the same assumptions (at least one positive integer). Also, it only finds the first biggest sequence.

Edit: changed it to use max_j (exclusive) instead of max_len.