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Minimum No of operations required to convert a given number to 1  Integer Replacement Problem.
Objective: Given a number N, write an algorithm to convert that number to 1. Below are the allowed operations.
 If N is even, do N = N/2.
 If N is odd, either do N =N  1 or do N = N + 1
Example:
N = 16 Output: 4 16 → 8 → 4 → 2 → 1 N = 11 Output: 5 11 → 10 → 5 → 4 → 2 → 1 OR 11 → 12 → 6 → 3 → 2 → 1
Approach: Recursion
 If N is even, then divide it by 2 and add 1 to the result and solve the rest of the problem recursively.
 If N is odd, then solve recursively for both options, N + 1 and N  1 and pick the one which requires a minimum and adds that to the final result.
 NOTE: when solving for N + 1, do 1 + (n  1) / 2 and add 2 in result ( to avoid out of range Exception).
Time Complexity: O(2^{n})
Here we are solving many subproblems repeatedly. See the image below
As you can see the for N = 4 and 2, the problem was solved multiple times. We can optimize the solution by using Dynamic Programming.
Dynamic Programming  TopDown Approach
 Every time we solve recursively for any N, save it in HashMap
 Before solving it for any N, check if we have a solution stored for N in HashMap (means we have already solved it earlier).
Complete Code:
Output:
N = 11, Minimum replacements required : 5
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