Sort Colors

Because only three distinct keys exist, sorting is equivalent to partitioning the array into three contiguous sections: 0s on the left, 1s in the middle, and 2s on the right. Marching a mid pointer forward while swapping 0s toward the low boundary and 2s toward the high boundary places every value into its permanent bucket in a single pass.

Full solution

Dutch National Flag three-pointer partition: low/mid/high sweep the array once, swapping 0s left and 2s right in place. It is the one to write because counting sort needs two passes and a library sort is exactly what the problem forbids.

def sort_colors(nums: list[int]) -> None:
    """Dutch National Flag: one pass, in place."""
    low, mid, high = 0, 0, len(nums) - 1
    while mid <= high:
        if nums[mid] == 0:
            nums[low], nums[mid] = nums[mid], nums[low]
            low += 1
            mid += 1
        elif nums[mid] == 1:
            mid += 1
        else:
            # do not advance mid: the value swapped in from high is unexamined
            nums[mid], nums[high] = nums[high], nums[mid]
            high -= 1

function sortColors(nums) {
  // Dutch National Flag: one pass, in place.
  let low = 0;
  let mid = 0;
  let high = nums.length - 1;
  while (mid <= high) {
    if (nums[mid] === 0) {
      [nums[low], nums[mid]] = [nums[mid], nums[low]];
      low += 1;
      mid += 1;
    } else if (nums[mid] === 1) {
      mid += 1;
    } else {
      // do not advance mid: the value swapped in from high is unexamined
      [nums[mid], nums[high]] = [nums[high], nums[mid]];
      high -= 1;
    }
  }
}

The theory behind it

Arrays — the ground this problem stands on. All Arrays problems

What Arrays is

An array is a row of fixed boxes laid side by side in computer memory, like numbered lockers in a hallway. Because each box occupies identical space and sits directly next to its neighbors, jumping to locker zero or locker ten thousand takes the exact same tiny fraction of time. Every box holds an item of the same type, addressed by an offset number called an index.

When to reach for it

Reach for an array when items arrive in a known sequence and need immediate retrieval by position number. Problems asking for running totals, prefix accumulations, cyclic rotations, or in-place rearrangements signal array mechanics. Whenever constraints require constant-time random lookups or contiguous cache scans across fixed collections, a flat sequence is the default container.

How the pattern works

Visualize a tape with zero-indexed slots stretching from start to end. Keep track of write and read cursors when modifying contents without allocating helper buffers. For running computations, maintain an invariant such as having processed all elements left of the current index while pending elements wait to the right. When modifying entries in place, consider scanning backwards from the end so unread data is not overwritten.