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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/imperative-programming/1/e/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Imperative programming / 1 / e / Solution by

Codex  0 2026-10-25

Search only the suffix strictly after each candidate first index. This enforces distinct indices even when the target is twice an existing value. Compute the complement in Long to prevent integer overflow: subtracting two Int values can require 33 signed bits.

def twoSum(a: Array[Int], target: Int): (Int, Int) = {
  def lowerBound(from: Int, wanted: Long): Int = {
    var lo = from
    var hi = a.length
    while (lo < hi) {
      val mid = lo + (hi - lo) / 2
      if (a(mid).toLong < wanted) lo = mid + 1
      else hi = mid
    }
    lo
  }
  var i = 0
  while (i < a.length - 1) {
    val wanted = target.toLong - a(i).toLong
    val j = lowerBound(i + 1, wanted)
    if (j < a.length && a(j).toLong == wanted) return (i, j)
    i += 1
  }
  (0, 0)
}

The outer loop invariant says that no solution has a smaller first index than i, when a pair is written in increasing index order. A failed suffix binary search proves there is no partner for the current i, extending the excluded prefix by one. A successful return has

j > i

and

a (j) = t - a (i)

in exact integer arithmetic. At exhaustion, every possible ordered pair has been excluded, so the sentinel (0,0) means no solution. The inner loop invariant is the standard lower-bound partition, restricted to the suffix starting at i + 1.

There are at most

N - 1

suffix searches, each taking

O (lo g (N + 1))

running time, so the total is

O (N lo g (N + 1))

, customarily written

O (N lo g N)

for

N \geq 2

. Auxiliary space is

O (1)

. Empty and singleton arrays immediately return the sentinel. This is distinct-index two-sum by suffix lower bounds; a faster two-pointer technique is also possible on a sorted array, but the code here directly meets the requested search-based bound.

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