Google Interview: Find all contiguous subsequence in a given array of integers, whose sum falls in the given range. Can we do better than O(n^2)?

Question

Given an array of Integers, and a range (low, high), find all contiguous subsequence in the array which have sum in the range.

Is there a solution b

Answer 1

If all integers are non-negative, then it can be done in O(max(size-of-input,size-of-output)) time. This is optimal.

Here's the algorithm in C.

void interview_question (int* a, int N, int lo, int hi)
{
  int sum_bottom_low = 0, sum_bottom_high = 0,
      bottom_low = 0, bottom_high = 0,
      top = 0;
  int i;

  if (lo == 0) printf ("[0 0) ");
  while (top < N)
  {
    sum_bottom_low += a[top];
    sum_bottom_high += a[top];
    top++;
    while (sum_bottom_high >= lo && bottom_high <= top)
    {
      sum_bottom_high -= a[bottom_high++];
    }
    while (sum_bottom_low > hi && bottom_low <= bottom_high)
    {
      sum_bottom_low -= a[bottom_low++];
    }
    // print output
    for (i = bottom_low; i < bottom_high; ++i)
      printf ("[%d %d) ", i, top);
  }
  printf("\n");
}

Except for the last loop marked "print output", each operation is executed O(N) times; the last loop is executed once for each interval printed. If we only need to count the intervals and not print them, the entire algorithm becomes O(N).

If negative numbers are allowed, then O(N^2) is hard to beat (might be impossible).