Best Time to Buy and Sell Stock

Every potential selling day can only achieve its best profit against the lowest purchase price seen before it. Walking forward while maintaining that running minimum converts the problem from comparing all pairs into evaluating a single subtraction at each day. The highest subtraction encountered across the entire sequence is the answer.

Full solution

Running-minimum tracking: at each day, price minus the lowest price seen so far is the best profit ending that day. One linear sweep, O(n) time and O(1) space, versus the O(n^2) brute-force pair check.

def max_profit(prices: list[int]) -> int:
    min_price = prices[0]
    max_profit_so_far = 0
    for price in prices[1:]:
        # compute profit before updating the minimum, else buy/sell same day
        max_profit_so_far = max(max_profit_so_far, price - min_price)
        min_price = min(min_price, price)
    return max_profit_so_far

function maxProfit(prices) {
  let minPrice = prices[0];
  let maxProfitSoFar = 0;
  for (let i = 1; i < prices.length; i++) {
    const price = prices[i];
    maxProfitSoFar = Math.max(maxProfitSoFar, price - minPrice);
    minPrice = Math.min(minPrice, price);
  }
  return maxProfitSoFar;
}

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.