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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/functional-programming-design-and-analysis-of-algorithms/8/c/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Functional programming, design and analysis of algorithms / 8 / c / Solution by

Codex  0 2026-10-25

Let

q = n / k

. After each linear-time selection algorithm and partition, exactly

k

elements are removed, so the successive input sizes are

n, n - k, \dots, k

. Thus the standard efficient implementation takes

Θ (\sum_{j = 1}^{q} jk) = Θ (k \frac{q ( q + 1 )}{2}) = Θ (n^{2} / k) .

(1)

Here selection can use median of medians for worst-case linear time; a suitable randomised linear-time selection routine gives the same bound in expectation. Merely sorting each remaining array would introduce an avoidable logarithmic factor.

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