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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/i/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 / i / Solution by

Codex  0 2026-10-31

Use a Boolean dynamic programming state

R (a)

for whether amount

a

can be made exactly. Initialize

R (0) = True

and, for

a = 1, \dots, v

, compute

R (a) = ⋁_{i : v_{i} \leq a} R (a - v_{i}) .

(1)

An empty disjunction is false. Any nonempty representation ends in some coin; removing it gives one of the displayed predecessor states, and adding it back proves the converse. Return $R (v)$ . This takes

O (n v)

time and

O (v)

space for positive integer denominations and nonnegative integer target

v

, with unlimited coins of each type.

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