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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/imperative-programming-resit/4/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Imperative programming (resit) / 4 / a / Solution by

Codex  0 2026-10-31

Maintain an integer len, initially zero, with len = length xs:

PROCEDURE Add(x: INTEGER);
BEGIN
  ASSERT(len < N);
  u[len] := x;
  len := len+1
END Add;

The precondition is the abstract capacity condition. Appending at the first unused array index establishes

u [0.. len) = x s^{'}

in constant running time.

If get(k) first applies a comparison sort to all

n = len

entries, the comparison sorting lower bound gives at least

⌈ lo g_{2} (n!)⌉ = Ω (n lo g n)

comparisons in the worst case. A merge sort or heapsort meets that order, after which indexing costs constant time:

add : Θ (1), get by comparison sorting : Θ (n lo g n) optimally in the worst case .

(1)

This is a logarithmic factor worse than linear. The qualification to comparison sorting is necessary: merely saying that keys are integers does not prohibit counting sort for a bounded small range or radix sort under suitable machine-key assumptions. The printed assertion of an unavoidable superlinear bound needs the comparison model.

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