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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/2/e/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 / 2 / e / Solution by

Codex  0 2026-10-25

Define the ordered merge algorithm explicitly:

mergePos :: [Integer] -> [Integer] -> [Integer]
mergePos [] ys = ys
mergePos xs [] = xs
mergePos xs@(x:xt) ys@(y:yt)
  | x <= y = x : mergePos xt ys
  | otherwise = y : mergePos xs yt

ilv :: Seq a -> Seq a -> Seq a
ilv s t v = mergePos (map (2*) (s v))
                    (map (\j -> 2*j+1) (t v))

The first sequence occupies even positions and the second occupies odd positions. Within each transformed list, order is preserved; merging puts all positions for a queried token into increasing order. The two transformed lists have disjoint parities, so equality cannot occur between their heads on valid inputs. The length assumption ensures that these positions describe the intended interleaved sequence without a gap at its finite end.

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