Two Sum¶

Question¶

Given an array of integers nums and an integer target, return the indices of the two numbers such that they add up to target. You may assume that each input would have exactly one solution, and you may not use the same element twice.

Solution¶

Pattern¶

Complement Search via Hash Map Store elements in a Hash Map as you iterate. For each element, calculate its "complement" (target - current element) and check if it already exists in the map.

How to Identify¶

The problem asks for "pairs" that satisfy a mathematical condition (sum, difference).
You need to optimize an $O(n^2)$ nested-loop brute force search into a linear sequence.
The solution requires returning indices, rendering sorting (which destroys original indices) less viable unless paired with extra space.

Description¶

Step-by-step explanation:

Create an empty Hash Map. This will map the integer values we see to their indices in the array (Value -> Index).
Iterate through the array nums sequentially using a single pointer i.
At each step, calculate the complement, which is target - nums[i]. This is the exact number we need to find to complete our pair.
Check the Hash Map. Does the complement already exist as a key?
If yes, we are done. We retrieve the index of the complement from the map, and return it alongside our current index i.
If no, we haven't seen the needed number yet. We insert our current number nums[i] and its index i into the Hash Map so it can potentially act as a complement for future numbers.
If the loop terminates without returning, no valid pair exists (though the problem guarantees one).

The Intuition¶

Think of this like a "Wanted Poster" mental model. Imagine you are walking down a line of people holding random amounts of money. You need exactly $10 total between two people. When you meet Alice, she has $3. She looks at the "Wanted" board for someone with $7, but the board is empty. So, Alice puts her *own* name and amount ($3) on the board and waits. Next, you meet Bob, who has $7. Bob looks at the board, sees Alice is looking for a partner to make $10. Bob realizes he is the $7 she needs. Match found! By recording what we have, we instantly know when we find what we need.

Complexity¶

Label	Worst	Average
Time Complexity	$O(n^2)$	$O(n)$
Space Complexity	$O(n)$	$O(n)$

Time Complexity¶

On average, inserting and searching in a Hash Map takes $O(1)$ time. Since we iterate through the array of size $n$ exactly once, the average time complexity is $O(n)$. The worst-case is theoretically $O(n^2)$ if every hash insertion results in a collision (though modern Java uses balanced trees for heavy collisions, bringing it to $O(n \log n)$).

Space Complexity¶

We use a Hash Map to store previously seen elements. In the worst-case scenario (the pair is at the very end of the array), we will store $n-1$ key-value pairs in the Hash Map, resulting in $O(n)$ auxiliary space.

Code¶

class Solution {
    public int[] twoSum(int[] nums, int target) {
        // Map stores seen numbers and their original indices: <Number, Index>
        Map<Integer, Integer> numToIndex = new HashMap<>();

        for (int i = 0; i < nums.length; i++) {
            int complement = target - nums[i];

            // Check if we've already processed the required complement
            if (numToIndex.containsKey(complement)) {
                return new int[] {numToIndex.get(complement), i};
            }

            // Add the current number to the "seen" map
            numToIndex.put(nums[i], i);
        }

        throw new IllegalArgumentException("No pair found matching the target.");
    }
}

Caveats¶

Memory Constraints:
If working in an embedded system or processing terabytes of data, an O(n) memory overhead for a Hash Map might trigger Out-Of-Memory errors.
Sorted Data:
If the input array is already sorted, using a Hash Map is a waste of memory. A Two-Pointer approach from both ends of the array achieves O(n) time and O(1) space.

Concepts to Think About¶

Hash Collisions: Understand how your language handles underlying map collisions (e.g., Separate Chaining vs. Open Addressing).
Sorting Trade-offs: If we didn't need to return indices (e.g., returning the values themselves), sorting the array (O(n log n)) + two pointers (O(n)) yields O(1) space.
Handling Duplicates: If the array allows multiple pairs and you need to return all unique pairs, the logic gets substantially more complex.
One-pass vs Two-pass: A two-pass hash map approach is easier to write initially but fails if an element is exactly half the target and appears only once (e.g., target = 6, nums = [3], it might match 3 with itself). One-pass naturally avoids this.
Data Locality: Hash maps have poor CPU cache locality compared to contiguous arrays. For very small n, a brute-force O(n²) double loop might actually benchmark faster than a Hash Map due to caching.
Sorted vs. Unsorted: If the array were sorted, could we do better? (Hint: Two Pointers, O(1) space).
Memory Constraints: What if the array has 10 billion numbers and won't fit in memory? (Hint: Sharding or External Merge Sort).
Collision Handling: How does your language of choice handle Hash Map collisions? (Java uses Linked Lists, then converts to Red-Black Trees for O(logn) worst-case).
Duplicate Values: Does the Hash Map approach handle duplicates like nums = [3, 3], target = 6? (Yes, because the first 3 is stored before the second 3 is checked).

Logical Follow-up¶

Question: What if the input array is sorted in ascending order? Can you solve it using O(1) space? Solution: Yes. We use the Two Pointer technique. Place a left pointer at index 0 and a right pointer at n - 1.

Calculate:

sum = nums[left] + nums[right]

If sum == target, return the indices.
If sum < target, increment left to increase the sum.
If sum > target, decrement right to decrease the sum.

This runs in O(n) time and strictly O(1) space.

Question: What if we are designing a data structure to receive a stream of numbers and answer "Two Sum" queries on demand? (Two Sum III) Solution: We must balance add() and find() operations.

If find() is called frequently:
Store all pairwise sums in a HashSet during add()
Complexity:
- add() → O(n)
- find() → O(1)
If add() is frequent but find() is rare:
Maintain a standard frequency map
During find(), iterate through the keys and search for complements
Complexity:
- add() → O(1)
- find() → O(n)

Question: How would you extend this to find 3 numbers that sum to a target? (3Sum) Solution: First, sort the array:

O(n log n)

Then iterate through the array, treating nums[i] as a fixed value.

The remaining problem becomes a Two Sum search with target:

target - nums[i]

Since the array is sorted, solve this using the O(1) space Two Pointer technique in O(n) time.

Overall complexity:

O(n²)

Label	Worst	Average
Time Complexity	\(O(n^2)\)	\(O(n)\)
Space Complexity	\(O(n)\)	\(O(n)\)