Combination Sum II

Posted: 28 Jan, 2021
Difficulty: Moderate

PROBLEM STATEMENT

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You are given an array ‘Arr’ of ‘N’ positive integers. You are also given a positive integer ‘target’.

Your task is to find all unique combinations of the array ‘Arr’ whose sum is equal to ‘target’. Each number in ‘Arr’ may only be used once in the combination.

Elements in each combination must be in non-decreasing order and you need to print all unique combinations in lexicographical order.

Note:
In lexicographical order, combination/array  ‘A’  comes before array ‘B’ if ‘A’ is the prefix of array ‘B’, or if none of them is a prefix of the other and at the first position where they differ integer in ‘A’ is smaller than the integer in ‘B’.
Example:
Let the array ‘Arr’ be [1, 2, 3, 1] and ‘target’ = 5. Then all possible valid combinations in lexicographical order are -:
(1, 1, 3)
(2, 3)
Input Format:
The first line of input contains an integer ‘T’ denoting the number of test cases.

Then the first line of each test case contains two space-separated integers  ‘N’ and ‘target’ denoting the number of elements in ‘Arr’ and the ‘target'

The second line of each test case contains N space-separated integers the elements of array ‘Arr’.
Output Format:
For each test case, print all possible valid combinations in a separate line in the lexicographical order. Elements in each combination must be in non-decreasing order. Print a new line after each test case.
Note:
You do not need to print anything, it has already been taken care of. Just implement the given function.
Constraints:
1 <= T <= 10 
1 <= N <= 20
1 <= Arr[i] <= 30
1 <= target <= 30

Time Limit: 1 sec
Approach 1

First, sort the given array in non-decreasing order, it will help to generate combinations in non-decreasing order.  There will be 2 ^ N possible combinations of the given array, We create a vector ‘result’ and then we one by one check for all possible combinations, whether the sum of its elements is equal to ‘target’ or not. If the sum of the combination is equal to ‘target’, we will append it in vector ‘result’.

 

We finally sort vector ‘result’ in lexicographical order and remove all duplicates from it.

 

We can find all combinations of array both iteratively or recursively, Here, we will be using the iterative approach only.

 

Algorithm:

 

  • Sort the given array ‘arr’.
  • Create a vector ‘result’ It will keep all combinations having sum equal to ‘target’.
  • Run a loop where ‘i’ ranges from 0 to 2^N-1, and in each iteration do the following -:
    • Create an empty vector ‘comb’
    • Run a loop where ‘j’ ranges from 0 to N - 1 and if ‘jth’ bit is set in ‘i’ then append element arr[j] in ‘comb’.
    • If the sum of all elements of ‘comb’ is equal to ‘target’ then add it in vector ‘result’
    • Sort the vector ‘result’
    • Remove all duplicates from vector ‘result’.  This can be done easily by either using built-in library functions or using the fact that duplicate entries are grouped together after sorting.
  • Return ‘result’.
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