Given an array of even number of elements, form groups of 2 using these array elements such that the difference between the group with highest sum and the one with lowest sum is maximum.
Note: An element can be a part of one group only and it has to be a part of at least 1 group.
Examples:
Input : arr[] = {1, 4, 9, 6}
Output : 10
Groups formed will be (1, 4) and (6, 9),
the difference between highest sum group
(6, 9) i.e 15 and lowest sum group (1, 4)
i.e 5 is 10.
Input : arr[] = {6, 7, 1, 11}
Output : 11
Groups formed will be (1, 6) and (7, 11),
the difference between highest sum group
(7, 11) i.e 18 and lowest sum group (1, 6)
i.e 7 is 11.
Simple Approach: We can solve this problem by making all possible combinations and checking each set of combination differences between the group with the highest sum and with the lowest sum. A total of n*(n-1)/2 such groups would be formed (nC2).
Time Complexity: O(n^3), because it will take O(n^2) to generate groups and to check against each group n iterations will be needed thus overall it takes O(n^3) time.
Efficient Approach: We can use the greedy approach. Sort the whole array and our result is sum of last two elements minus sum of first two elements.
PHP
<?php// PHP program to find minimum // difference between groups of // highest and lowest sums.function CalculateMax($arr, $n){ // Sorting the whole array. sort($arr); $min_sum = $arr[0] + $arr[1]; $max_sum = $arr[$n - 1] + $arr[$n - 2]; return abs($max_sum - $min_sum);}// Driver code$arr = array (6, 7, 1, 11 );$n = sizeof($arr);echo CalculateMax($arr, $n), "" ;// This code is contributed by ajit?> |
Output:
11
Time Complexity: O (n * log n)
Space Complexity: O(1) as no extra space has been taken.
Further Optimization :
Instead of sorting, we can find a maximum two and minimum of two in linear time and reduce the time complexity to O(n).
Below is the code for the above approach.
PHP
<?php // Php program to // find minimum difference // between groups of highest and lowest // sums. function CalculateMax($arr, $n) { $first_min = min($arr); $second_min = PHP_INT_MAX; for($i = 0; $i < $n ; $i ++) { // If arr[i] is not equal to first min if ($arr[$i] != $first_min) $second_min = min($arr[$i],$second_min); } $first_max = max($arr); $second_max = PHP_INT_MIN; for($i = 0; $i < $n ; $i ++) { // If arr[i] is not equal to first max if ($arr[$i] != $first_max) $second_max = max($arr[$i],$second_max); } return abs($first_max+$second_max-$first_min-$second_min); } $arr = array(6, 7, 1, 11); $n = sizeof($arr); echo CalculateMax($arr, $n);// This code is contributed by Aman Kumar?> |
11
Time Complexity: O(N)
Auxiliary Space: O(1)
Please refer complete article on Maximum difference between groups of size two for more details!
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