partition a sequence of 2n real numbers so that

Question

I\'m currently reading The Algorithm Design Manual and I\'m stuck on this exercise.

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Take a sequence of 2n real numbers as input. Design an O(n log n) algorithm tha

Answer 1

I think I can prove this for a sequence with no duplicated numbers, and it should be a reasonably simple exercise for the reader to extend the proof to non-unique sequences.

Pair x₀, x_2n together, then pair all other numbers according to an optimal solution.

Now consider the pairing of (x₀, x_2n) against any other pair x_y, x_z from the optimal subset. x_2n + either x_y or x_z will be greater than x_y+x_z and also x_2n+x₀, therefore the pairing of x_2n, x₀ was optimal.

The proof now extends by induction to the pairing of X₁, X_2n-1, and further partitions of the subset, eventually producing the OP's pairing.