Maximum Product Subarray

Multiplication by a negative number flips signs: a tiny negative product suddenly becomes a huge positive product when struck by another negative. Because an extreme negative is only one step away from becoming the maximum, tracking both the running maximum and the running minimum at every step is necessary to capture sign flips.

Full solution

Dual max-min dynamic tracking: keep the running maximum AND minimum product ending at each index, swapping them when the current number is negative. One pass, constant space, and it is the only linear approach that survives sign flips and zeros.

from typing import List


def max_product(nums: List[int]) -> int:
    best = cur_max = cur_min = nums[0]
    for x in nums[1:]:
        if x < 0:
            cur_max, cur_min = cur_min, cur_max  # a negative flips which extreme is which
        cur_max = max(x, cur_max * x)
        cur_min = min(x, cur_min * x)
        best = max(best, cur_max)
    return best

function maxProduct(nums) {
  let best = nums[0];
  let curMax = nums[0];
  let curMin = nums[0];
  for (let i = 1; i < nums.length; i++) {
    const x = nums[i];
    if (x < 0) [curMax, curMin] = [curMin, curMax]; // a negative flips which extreme is which
    curMax = Math.max(x, curMax * x);
    curMin = Math.min(x, curMin * x);
    best = Math.max(best, curMax);
  }
  return best;
}

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.