Editorial for Another Contest 3 Problem 2 - Camelot
Remember to use this editorial only when stuck, and not to copy-paste code from it. Please be respectful to the problem author and editorialist.
Submitting an official solution before solving the problem yourself is a bannable offence.
Submitting an official solution before solving the problem yourself is a bannable offence.
If we rotate the plane by 45 degrees, we're solving the same problem of finding a point that minimizes the sum of the distances to that point, except the distance metric is Manhattan distance, where dimensions are independent. In the one-dimensional variant where the distance metric is Manhattan distance, the point that minimizes the distance is the median of the points.
To rotate the points by 45 degrees, map to
. We can compute the median of the sums and differences to get a candidate point in the original space, but note that the given point might not be a lattice point (this happens when the medians have different parities). Nonetheless, we can try all points in the neighborhood of that point, one of which will be optimal.
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