What is the time complexity of the two‑sum algorithm that returns the indices of two numbers that add up to a target?

The classic two‑sum problem asks for the indices of two numbers in an array that add up to a given target. A naive solution checks every pair of elements, resulting in O(n²) time. A more efficient approach uses a hash map (dictionary) to store each number’s index as you iterate through the array. For each element x, you compute its complement y = target – x and check whether y is already in the map. If it is, you return the stored index of y and the current index. Each lookup and insertion in the hash map is O(1) on average, so the entire pass over the array takes O(n) time. The space complexity is also O(n) because you may store up to n entries in the map. This hash‑based solution is the standard optimal answer for two‑sum and is often used in interview settings to demonstrate knowledge of hash tables and linear‑time algorithms.