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Asked: 2026-06-03T04:37:38+00:00

The problem here is to find set of all integer points which gives minimum

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The problem here is to find set of all integer points which gives minimum sum over all Manhattan distances from given set of points!

For example:

lets have a given set of points { P1, P2, P3…Pn }

Basic problem is to find a point say X which would have minimum sum over all distances from points { P1, P2, P3… Pn }.

i.e. |P1-X| + |P2-X| + …. + |Pn-X| = D, where D will be minimum over all X.

Moving a step further, there can be more than one value of X satisfying above condition. i.e. more than one X can be possible which would give the same value D. So, we need to find all such X.

One basic approach that anyone can think of will be to find the median of inputs and then brute force the co-ordinates which is mentioned in this post

But the problem with such approach is: if the median gives two values which are very apart, then we end up brute forcing all points which will never run in given time.

So, is there any other approach which would give the result even when the points are very far apart (where median gives a range which is of the order of 10^9).

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1 Answer

  1. Editorial Team

    If the median gives you an interval of the order of 10^9 then each point in that interval is as good as any other.

    So depending on what you want to do with those points later on you can either return the range or enumerate points in that range. No way around it..

    Obviously in two dimensions you’ll get a bouding rectangle, in 3 dimensions a bounding cuboid etc..

    The result will always be a cartesian product of ranges obtained for each dimension, so you can return a list of those ranges as a result.

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