HackWithInfy Round 2 - The Number Game

I recently attended HackWithInfy Round 2 and this was one of the questions they asked.
Two numbers will be given A and B. Constraints: 1<=A,B<=10^18.

Four operations are allowed:

  1. If a number is odd, you can subtract 1 from it
  2. If a number is even, you can add 1 to it
  3. If a number is even or odd, you can multiply 2 to it
  4. If a number is even, you can divide it by 2

What is the minimum number of operations to reach B from A. If not possible, just print 1.

I tried this with recursion and it gave TLE for large numbers. I couldn’t do using memorization also. If anyone came up with a better approach, please share it here.

My approach:
The Number Game

You could use bfs which would have time complexity O(n)

@shravan660 Please Sir,can you explain how should I approach this problem??

Sir,@coderag, did you find any impossible test cases???

No, I didn’t find any.

If possible, can you implement and share?

see this solution


Can be done in O(log(n)) using bit manipulation. Look at the binary form of the two numbers.
It doesn’t matter whether you go from A to B or B to A. The process is reversible. So, I am gonna focus on going from the smaller number to the bigger number.
The operations on the binary numbers can be looked as follows-

  1. If the number has a 1 bit at the end, you can make it 0
  2. If a number has a 0 bit at the end, you can make it 1
  3. You can add a 0 at the end of the number whenever you want
  4. If there is a zero at the end, you can remove it

Notice, we can build any number we want with the given type of operations.

Let A < B

  1. Look at how much of the prefix of A and B is same.
  2. Then perform the operations to reduce A to just the common prefix.
  3. Then perform the operations to build A same as B.

@harisisbatman @nehra_1997 Thanks for the approach. I really didn’t think of bit manipulation during the contest.

Yeap… Happens.

I would like to share my approach.What I did was try to bring A and B to some common term.The common term that we can bring A and B to provided A and B are both>=2 is 2 with the help of given operations.So
1)list all the numbers obtained in the process of converting B to 2.
2)this conversion is carried out in the following way
1->if B is odd, decrease it.
2->ifB is even divide by 2.This iteration here i think will also be taking log(B) at most.
3)try to convert A to 2 in the same way.If in the process ,if we encounter some number that is common with the numbers in the process of converting B to 2, we stop the iteration.
4)ans will be sum of (steps taken to convert A to that common number)+(steps taken to convert B to that common number).

Although Sample test cases passed, I cannot prove optimality.@harisisbatman, would you like to comment about this solution?

Thanks for your solution also…

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Yes, the time complexity is O(log(n)) and you are doing exactly the same thing I described with a little different way of thinking. According to my understanding, your solution is correct and efficient. I hope you did not forget to take care of cases where A or B is 1. I think you should have just gone to the number 1 rather than 2 and it would have automatically taken care of it.

damn, how much is this question worth in terms of points?

@harisisbatman thanks so much for taking time to go through my solution and comment.
You really have been a tremendous help to beginners like us.
However what I really miss is how are you going to prove that your solution or mine will give the minimum number of steps?

It was 75 points.But i think you will be angry to know that the question included a statement of impossibility even though that impossible case shall never come. I never seen such type of misguiding statements ever in my life in CP. These people take CP contests but violate certain basic courtsies.

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We can’t use bfs?

I don’t know whether people who did this question noticed or not. I was checking my solution for some custom test cases during the contest. For many outputs, which can be obtained only by multiplying it by 2, the expected output was 1 less than the actual answer.
For example:
2 8
For this example, the output was 1, but we can never reach 8 by making only one operation. It should be two. Similarly I checked for all cases in which result can be obtained only by multiplying the number by 2 ‘n’ times. The output was always ‘n-1’. Anyway I submitted.

These people…I don’t know…why do they host contest if they can’t follow some basic norms of CP?

Yeah I faced the same problem. I ignored the checker. Most likely it was designed to confuse us. Though I hate the fact that there were no pretests and the solution was just saved. It’s not like you can run a stress test on an online compiler.

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Sir,if they did so, believe me it’s cheating.Total misguidance.