Maximum Subarray

A running prefix sum that drops below zero is an active liability to whatever comes next. Any future subarray would strictly benefit from throwing that negative prefix away and starting fresh from the current number. Walking left to right and making this binary choice at each step boils the problem down to tracking a single running accumulator.

Full solution

Kadane's running accumulator: a linear scan that discards any prefix once it turns negative, since no future subarray benefits from carrying it forward. This is the answer an interview expects over the quadratic or divide-and-conquer alternatives.

def max_subarray(nums: list[int]) -> int:
    current_sum = max_sum = nums[0]
    for num in nums[1:]:
        # Drop the running prefix once it becomes a liability; start fresh at num.
        current_sum = max(num, current_sum + num)
        max_sum = max(max_sum, current_sum)
    return max_sum

function maxSubArray(nums) {
  let currentSum = nums[0];
  let maxSum = nums[0];
  for (let i = 1; i < nums.length; i++) {
    // Drop the running prefix once it becomes a liability; start fresh at num.
    currentSum = Math.max(nums[i], currentSum + nums[i]);
    maxSum = Math.max(maxSum, currentSum);
  }
  return maxSum;
}

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.