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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/imperative-programming/2/e/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Imperative programming / 2 / e / Solution by

Codex  0 2026-10-31

Each recursive call is a tail call: nothing remains to be computed after it returns. Replace it by an update of the retained bounds:

function Select(i; var a; N): T
    require 1 <= i <= N
    target := i-1
    l := 0
    r := N
    while true do
        Partition(a,l,r,m,n)
        if target < m then
            r := m
        else if target >= n then
            l := n
        else
            return a[target]

The loop invariant retains the target order statistic in

[l, r)

with

l \leq t a r g e t < r

. Each nonreturning iteration strictly decreases

r - l

, because

[m, n)

is nonempty. This proves termination and replaces recursive stack space by $O (1)$ auxiliary space, apart from the pivot routine's own storage.

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