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Minimum sum of absolute differences of pairs in a triplet from three arrays
• Last Updated : 15 Apr, 2021

Given three arrays a[], b[] and c[] of sizes A, B and C respectively, the task is to find the minimum possible value of abs(a[i] – b[j]) + abs(b[j] – c[k]) where 0 ≤ i ≤ A, 0 ≤ j ≤ B and 0 ≤ k ≤ C.

Examples:

Input: A = 3, B = 2, C = 2, a[] = {1, 8, 5}, b[] = {2, 9}, c[] = {5, 4}
Output: 3
Explanation:
The triplet (a, b, c), i.e. (1, 2, 4) has minimum sum of absolute difference of pairs, i.e. abs(1 – 2) + abs(2 – 4) = 1 + 2 = 3.

Input: A = 4, B = 3, C = 3, a[] = {4, 5, 1, 7}, b[] = {8, 5, 6}, c[] = {2, 7, 12}
Output: 2
Explanation:
The triplet (a, b, c), i.e. (1, 5, 7) has minimum sum of absolute difference of pairs, i.e. abs(5 – 5) + abs(5 – 7) = 0 + 2 = 2.

Approach: The idea to solve this problem is to sort the arrays a[] and c[] and then traverse the array b[] and find the elements which satisfy the given condition.
Follow the steps below to solve the problem:

• Initialize the variable, say min, as INT_MAX, to store the minimum possible value.
• Sort the arrays a[] and c[] in increasing order.
• Traverse the array b[] and for each element, say b[i], find the closest element to b[i] from the arrays a[] and c[] as arr_close and crr_close and do the following:
• To find the closest element, firstly find the lower_bound of the target element b[i].
• If the lower bound is found, check if it is the first element of the array or not. If it is not, then compare the lower bound and its previous element with the target element and find which is closest to the target element.
• If the lower bound is not found, then the closest element will be the last element of the array.
• Update min as the minimum of abs(b[i] – arr_close) + abs(b[i] – crr_close).
• After completing the above steps, print the value of min as the result.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach`   `#include ` `#include ` `using` `namespace` `std;`   `// Function to find the value` `// closest to K in the array A[]` `int` `closestValue(vector<``int``> A, ``int` `k)` `{` `    ``// Initialize close value as the` `    ``// end element` `    ``int` `close = A.back();`   `    ``// Find lower bound of the array` `    ``auto` `it = lower_bound(A.begin(),` `                          ``A.end(), k);`   `    ``// If lower_bound is found` `    ``if` `(it != A.end()) {`   `        ``close = *it;`   `        ``// If lower_bound is not` `        ``// the first array element` `        ``if` `(it != A.begin()) {`   `            ``// If *(it - 1) is closer to k` `            ``if` `((k - *(it - 1))` `                ``< (close - k)) {` `                ``close = *(it - 1);` `            ``}` `        ``}` `    ``}`   `    ``// Return closest value of k` `    ``return` `close;` `}`   `// Function to find the minimum sum of` `// abs(arr[i] - brr[j]) and abs(brr[j]-crr[k])` `void` `minPossible(vector<``int``> arr,` `                 ``vector<``int``> brr,` `                 ``vector<``int``> crr)` `{` `    ``// Sort the vectors arr and crr` `    ``sort(arr.begin(), arr.end());` `    ``sort(crr.begin(), crr.end());`   `    ``// Initialize minimum as INT_MAX` `    ``int` `minimum = INT_MAX;`   `    ``// Traverse the array brr[]` `    ``for` `(``int` `val : brr) {`   `        ``// Stores the element closest` `        ``// to val from the array arr[]` `        ``int` `arr_close = closestValue(arr, val);`   `        ``// Stores the element closest` `        ``// to val from the array crr[]` `        ``int` `crr_close = closestValue(crr, val);`   `        ``// If sum of differences is minimum` `        ``if` `(``abs``(val - arr_close)` `                ``+ ``abs``(val - crr_close)` `            ``< minimum)`   `            ``// Update the minimum` `            ``minimum = ``abs``(val - arr_close)` `                      ``+ ``abs``(val - crr_close);` `    ``}`   `    ``// Print the minimum absolute` `    ``// difference possible` `    ``cout << minimum;` `}`   `// Driver Code` `int` `main()` `{` `    ``vector<``int``> a = { 1, 8, 5 };` `    ``vector<``int``> b = { 2, 9 };` `    ``vector<``int``> c = { 5, 4 };`   `    ``// Function Call` `    ``minPossible(a, b, c);`   `    ``return` `0;` `}`

## Java

 `// Java program for the above approach` `import` `java.io.*;` `import` `java.lang.*;` `import` `java.util.*;`   `class` `GFG{`   `// Lower_bound function ` `public` `static` `int` `lower_bound(``int` `arr[], ``int` `key)` `{` `    ``int` `low = ``0``;` `    ``int` `high = arr.length - ``1``;` `    `  `    ``while` `(low < high)` `    ``{` `        ``int` `mid = low + (high - low) / ``2``;` `        ``if` `(arr[mid] >= key) ` `        ``{` `            ``high = mid;` `        ``}` `        ``else` `        ``{` `            ``low = mid + ``1``;` `        ``}` `    ``}` `    ``return` `low;` `}`   `// Function to find the value` `// closest to K in the array A[]` `static` `int` `closestValue(``int` `A[], ``int` `k)` `{` `    `  `    ``// Initialize close value as the` `    ``// end element` `    ``int` `close = A[A.length - ``1``];`   `    ``// Find lower bound of the array` `    ``int` `it = lower_bound(A, k);`   `    ``// If lower_bound is found` `    ``if` `(it != A.length) ` `    ``{` `        ``close = A[it];`   `        ``// If lower_bound is not` `        ``// the first array element` `        ``if` `(it != ``0``) ` `        ``{`   `            ``// If *(it - 1) is closer to k` `            ``if` `((k - A[it - ``1``]) < (close - k))` `            ``{` `                ``close = A[it - ``1``];` `            ``}` `        ``}` `    ``}`   `    ``// Return closest value of k` `    ``return` `close;` `}`   `// Function to find the minimum sum of` `// abs(arr[i] - brr[j]) and abs(brr[j]-crr[k])` `static` `void` `minPossible(``int` `arr[], ``int` `brr[], ` `                        ``int` `crr[])` `{` `    `  `    ``// Sort the vectors arr and crr` `    ``Arrays.sort(arr);` `    ``Arrays.sort(crr);`   `    ``// Initialize minimum as INT_MAX` `    ``int` `minimum = Integer.MAX_VALUE;`   `    ``// Traverse the array brr[]` `    ``for``(``int` `val : brr) ` `    ``{` `        `  `        ``// Stores the element closest` `        ``// to val from the array arr[]` `        ``int` `arr_close = closestValue(arr, val);`   `        ``// Stores the element closest` `        ``// to val from the array crr[]` `        ``int` `crr_close = closestValue(crr, val);`   `        ``// If sum of differences is minimum` `        ``if` `(Math.abs(val - arr_close) + ` `            ``Math.abs(val - crr_close) < minimum)`   `            ``// Update the minimum` `            ``minimum = Math.abs(val - arr_close) + ` `                      ``Math.abs(val - crr_close);` `    ``}`   `    ``// Print the minimum absolute` `    ``// difference possible` `    ``System.out.println(minimum);` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `a[] = { ``1``, ``8``, ``5` `};` `    ``int` `b[] = { ``2``, ``9` `};` `    ``int` `c[] = { ``5``, ``4` `};`   `    ``// Function Call` `    ``minPossible(a, b, c);` `}` `}`   `// This code is contributed by Kingash`

Output:

`3`

Time Complexity: O(A*log A + C*log C + B)
Auxiliary Space: O(A + B + C)

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