Minimum Absolute Difference in an Array

Solutions

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The brute force solution consists in calculating the absolute difference of each pair and reducing to the minimum value. This takes $O(N^2)$ .

We can find better solution, instead, by making a simple observation: the closest two numbers, the lowest their difference. It’s just like calculating the distance between points on the same axis. Indeed, the formula to calculate the distance between two points $x1, x2$ on the same axis is simply $|x1-x2|$ .

Thus, we need to “fill the gaps” between numbers. This means, we need to sort the array. Then, we do the difference between each adjacent pair and calculate the minimum value along the way.

Here is a possible implementation in Python 2:

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def minimumAbsoluteDifference(arr, n):
    arr.sort()
    minVal =  sys.maxint
    for i in range(n - 1):
        minVal = min(minVal, abs(arr[i] - arr[i+1]))
    return minVal

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A pattern brings out from the code above: zip | map | reduce:

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def minimumAbsoluteDifference(arr, n):
    arr.sort()
    return min(map(lambda x: abs(x[0]-x[1]), zip(arr[1:], arr)))

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The same approach in C#, by Claudio Bacchelli:

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static int minimumAbsoluteDifference(int[] arr) 
{
    var lista = arr.ToList.Sort();
    return lista.Zip(lista.Skip(1), (a, b) => Math.Abs(a - b)).Min();
}

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An alternative way of writing the solution above in Python 3 by Eduard “Eddie” Rubio Cholbi and Stefano Ligabue:

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def minimumAbsoluteDifference(arr):
    arr = sorted(arr)
    return min(map(lambda a, b: b - a, arr[:-1], arr[1:]))

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Eddie and Stefano wrote generic way of to zip things in Python and applied it to the problem:

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def zipper(arr, n):
    return (arr[i:len(arr)-n+i+1] for i in range(n))

# Complete the minimumAbsoluteDifference function below.
def minimumAbsoluteDifference(arr):
    arr = sorted(arr)
    return min(map(lambda a, b: b - a, *zipper(arr, 2)))

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zipper zips a sequence with itself shifted n times. For example, on arr = [1,2,3,4,5]:

With n=1 (does nothing):

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unzipper(arr, 1)
=> [1, 2, 3, 4, 5]

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With n=2:

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unzipper(arr, 2)
=> [1, 2, 3, 4]
   [2, 3, 4, 5]

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With n=3:

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unzipper(arr, 2)
=> [1, 2, 3]
   [2, 3, 4]
   [3, 4, 5]

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Etc.

This is an example of “freely expressing oneself” we encourage at Coding Gym. Eddie and Stefano solved the challenge very quickly and started experimenting with their favourite language, getting out of their “state of the art”. Since Coding Gym is not time nor prize driven, anyone sets her own targets.

Just for fun, in C++ zip|map|reduce can be written in terms of inner_product:

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sort(begin(v), end(v));
cout << inner_product(next(begin(v)), end(v), begin(v), numeric_limits<int>::max(), 
              [](int a, int b) { return min(a,b); },
              [](int a, int b) { return abs(a-b); }
);

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In C++20 we can use ranges, left as an exercise.

In Haskell we can write the whole program (input parsing included) in 125 characters.

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import Data.List
f=minimum.(zipWith(\a b->abs(a-b))<*>tail).sort
main=interact(show.f.(fmap read).words.head.(drop 1).lines)

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But it’s better explained by this code:

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import Data.List

minAbsDiff xs = minimum diffs
  where
    absDiff x y = abs (x - y)
    diffs = zipWith absDiff (tail sorted) sorted
    sorted = sort xs

parseInput = (fmap read) . words . head . (drop 1) . lines

main = interact $ show . minAbsDiff . parseInput

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In the compact version above alepez used function composition to create a point-free version of the minAbsDiff function.

Point-free functions are just functions which do not mention the actual arguments they will be applied to.

We can also implement it with foldl:

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f xs = foldl op 10000000000 $ zip sorted (tail sorted)
  where
    sorted = sort xs
    op a (x, y) = min a (abs (x - y))

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We can also implement innerProduct like C++:

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f xs = innerProduct min (\x y -> abs (x - y)) 10000000000 ys (tail ys)
  where
    ys = sort xs

innerProduct op1 op2 init xs ys =
  foldl (\a (x, y) -> op1 a (op2 x y)) init (zip xs ys)

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A solution in Scala by benny1693:

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def minimumAbsoluteDifference(arr: Array[Int]): Int =
    (arr.sorted zip arr.sorted.tail).map({case (a,b) => abs(a-b)}).min

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A solution in Python 3 by Massimo Dalla Cia:

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def minimumAbsoluteDifference(arr):
    arr.sort(key=int)

    sort_array = arr

    delta = 999999999

    for i in range(1, len(sort_array)):
        if delta > abs(sort_array[i] - sort_array[i-1] ):
            delta = abs(sort_array[i] - sort_array[i-1])

    return int(delta)

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A solution in Python 3 by ranzatoweb:

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def minimumAbsoluteDifference(arr):
    s_arr = sorted(arr)
    return min(map(lambda x: abs(x[0] - x[1]), zip(s_arr,s_arr[1:])))

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A solution in Python 3 by Enrico Lovisotto:

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def minimumAbsoluteDifference(arr):
    s_arr = sorted(arr)
    diffs = map(lambda c: abs(c[0] - c[1]), zip(s_arr, s_arr[1:]))
    return sorted(diffs)[0]

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patterns sort functional