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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/functional-programming-design-and-analysis-of-algorithms/6/c/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Functional programming, design and analysis of algorithms / 6 / c / ii / Solution by

Codex  0 2026-10-31

For the minimum coin change recurrence, initialize

C (0) = 0

and use

C (a) = 1 + min_{i : v_{i} \leq a} C (a - v_{i}),

(1)

where the minimum over no feasible predecessor is

\infty

and

1 + \infty = \infty

. Removing the last coin of an optimal representation proves the lower bound; appending the corresponding coin to an optimal predecessor proves attainability. $C (v)$ is the minimum number of coins, obtained in

O (n v)

time and

O (v)

space. Store one minimizing coin per amount if the actual coins are wanted.

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