Given an array arr of N integer elements, the task is to find sum of average of all subsets of this array.
Input : arr = [2, 3, 5] Output : 23.33 Explanation : Subsets with their average are,  average = 2/1 = 2  average = 3/1 = 3  average = 5/1 = 5 [2, 3] average = (2+3)/2 = 2.5 [2, 5] average = (2+5)/2 = 3.5 [3, 5] average = (3+5)/2 = 4 [2, 3, 5] average = (2+3+5)/3 = 3.33 Sum of average of all subset is, 2 + 3 + 5 + 2.5 + 3.5 + 4 + 3.33 = 23.33
A naive solution is to iterate through all possible subsets, get average of all of them and then add them one by one, but this will take exponential time and will be infeasible for bigger arrays.
We can get a pattern by taking an example,
arr = [a0, a1, a2, a3] sum of average = a0/1 + a1/1 + a2/2 + a3/1 + (a0+a1)/2 + (a0+a2)/2 + (a0+a3)/2 + (a1+a2)/2 + (a1+a3)/2 + (a2+a3)/2 + (a0+a1+a2)/3 + (a0+a2+a3)/3 + (a0+a1+a3)/3 + (a1+a2+a3)/3 + (a0+a1+a2+a3)/4 If S = (a0+a1+a2+a3), then above expression can be rearranged as below, sum of average = (S)/1 + (3*S)/2 + (3*S)/3 + (S)/4
The coefficient with numerators can be explained as follows, suppose we are iterating over subsets with K elements then denominator will be K and numerator will be r*S, where ‘r’ denotes number of times a particular array element will be added while iterating over subsets of same size. By inspection we can see that r will be nCr(N – 1, n – 1) because after placing one element in summation, we need to choose (n – 1) elements from (N – 1) elements so each element will have a frequency of nCr(N – 1, n – 1) while considering subsets of same size, as all elements are taking part in summation equal number of times, this will the frequency of S also and will be the numerator in final expression.
In below code nCr is implemented using dynamic programming method, you can read more about that here,
This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Sum of bitwise OR of all possible subsets of given set
- Backtracking to find all subsets
- Number of distinct subsets of a set
- Number of subsets with product less than k
- Finding all subsets of a given set in Java
- Count number of subsets having a particular XOR value
- Number of subsets with sum divisible by m
- Count no. of ordered subsets having a particular XOR value
- Average of a stream of numbers
- Path with maximum average value
- Average of first n odd naturals numbers
- Average of first n even natural numbers
- Find the subarray with least average
- Count subsets having distinct even numbers
- Minimum difference between max and min of all K-size subsets