Minimum bitwise OR after removing at most K elements from given Array

Given an array A[] of length N, the task is to find the minimum possible value of bitwise OR of the elements after removing at most K elements.

Examples: 

Input: A = {1, 10, 9, 4, 5, 16, 8}, K = 3
Output: 11
Explanation: Remove 4, 5, 16 for the given array.
The remaining elements are {1, 10, 9, 5}. 
The OR of these elements are 11 which is the minimum possible.

Input: A = {1, 2, 4, 8}, K = 1
Output: 7
Explanation: Remove 8 form the Array, Minimum OR= 1 | 2 | 4 = 7

Approach: This problem can be solved using bit manipulation on the basis of the following idea:

Try to remove the highest MSB elements first, then the elements having 2nd highest MSB and so on until K elements are removed.

Follow the illustration given below for a better understanding.

Illustration :

Consider A[] = [1, 10, 9, 4, 5, 16, 8 ]

• Binary representation of the elements are:
  1 – 00001
  10 – 01010
  9 – 01001
  4 – 00100
  5 – 00101
  16  – 10000
  8 – 01000
• The positions of the MSB of every elements are:
  1 -> 0, 10 -> 3, 9 -> 3, 4 -> 2, 5 -> 2, 16  -> 4, 8 -> 3
• Therefore count of elements having MSB at ith position are as given below:
  setBitCount = [1, 0, 2, 3, 1 ]
  MSB position    0, 1, 2, 3, 4
• Traverse array and check if current setBitCount is less than k, then remove all elements with current Bit as FirstSetBit.
  For, K = 3, traverse array: 
  =>When i = 4: setBitCount[i] = 1, which is less than K so remove 16, and now K = 2.
  =>When i = 3: K < setBitCount[i] (i.e 3). So don’t remove it.
  =>When i = 2: setBitCount[i] = 2 ≤ K, so remove 4 and 5. Now, K =0.
• Calculate OR for all remaining elements i.e (1, 5, 9, 10) = 11

Follow the steps mentioned below to implement the above observation:

• Create a hash array.
• Traverse the given array and increment the MSB Index of every element into the hash array.
• Now, traverse the hash array from MSB and in each iteration:
  • Check if it is possible to remove all elements having the ith bit as MSB  i.e. setBitCount ≤ K.
  • If setBitCount ≤ K, don’t calculate OR for that elements and, just decrease the K.
  • if not, calculate the OR for elements with ith bit as MSB.
• Return Calculate OR after removing K elements.

Below is the implementation of the above approach:

Time Complexity: O(N* log(N))
Auxiliary Space: O(N)