Solve two-sum variants at scale

Base task: Given nums = [3, 1, 2, 2, 4] and target = 4, return the 0-based index pair (i, j) with i < j such that nums[i] + nums[j] = target. If multiple pairs exist, return the lexicographically smallest (i, j). Achieve expected O(n) time and O(n) extra space; justify correctness for duplicates. Follow-ups: 1) If nums is already sorted ascending, provide an O(n) time, O(1) space solution and prove minimal lexicographic pair selection still holds. 2) Streaming variant: nums arrives as an unbounded stream and you must answer membership queries two_sum_exists(x) online with sublinear memory. Propose a data structure, update/query costs, and false-positive/false-negative trade-offs if any. 3) Very large array (hundreds of millions of ints) under a 200MB RAM cap. Describe an external-memory or hashing-with-bucketing approach, handling negatives and overflow. 4) Extend to three-sum for a sorted array; give complexity and deduplication strategy.