Partition of a set into K subsets with equal sum
Given an integer array of N elements, the task is to divide this array into K non-empty subsets such that the sum of elements in every subset is same. All elements of this array should be part of exactly one partition.
Input : arr = [2, 1, 4, 5, 6], K = 3 Output : Yes we can divide above array into 3 parts with equal sum as [[2, 4], [1, 5], ] Input : arr = [2, 1, 5, 5, 6], K = 3 Output : No It is not possible to divide above array into 3 parts with equal sum
We can solve this problem recursively, we keep an array for sum of each partition and a boolean array to check whether an element is already taken into some partition or not.
First we need to check some base cases,
If K is 1, then we already have our answer, complete array is only subset with same sum.
If N < K, then it is not possible to divide array into subsets with equal sum, because we can’t divide the array into more than N parts.
If sum of array is not divisible by K, then it is not possible to divide the array. We will proceed only if k divides sum. Our goal reduces to divide array into K parts where sum of each part should be array_sum/K
In below code a recursive method is written which tries to add array element into some subset. If sum of this subset reaches required sum, we iterate for next part recursively, otherwise we backtrack for different set of elements. If number of subsets whose sum reaches the required sum is (K-1), we flag that it is possible to partition array into K parts with equal sum, because remaining elements already have a sum equal to required sum.
Partitions into equal sum is possible.
Time Complexity: O(2^(N * K)).
Because if we have K trees stacked on top of each other, the new height of the tree is K * n. i.e one subset is not independent from other.
Space Complexity: O(N).
Extra space is required for visited array.
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