Problem Link
Author: Noszály Áron
Tester: Misha Chorniy
Editorialist: Bhuvnesh Jain
Difficulty
EASY
Prerequisites
BFS/Floyd Warshall, Observations
Problem
You are to convert number A to number B using the following operation in the minimum number of steps:
- Write A as a binary number with arbitrary number of leading zeros (possibly none)
- Shuffle the binary digits of the A in an arbitrary order. Call it A'
- The output is (A' + 1). Let us call this number as X.
Remember the notation of A' and X as it is used everywhere in the editorial.
Explanation
Subtask 1: A, B ≤ 128
The problem asks to minimise the number of steps to go from A to B. Generally, these type of problems can be easily modelled as a graph and perform BFS to find the answer. Here is a general idea:
Let us model each step as an edge from initial number to final number with an edge of some cost. So, the path with the smallest weight from source to destination is the required answer and also gives us a way to achieve the transition.
In this problem, we can just do what the operations ask us to do. Just all possible shuffling of the digits of A in binary representation and consider only those ones which give end result within the desired range ([0, 128]). Let us call this number as X. Add an edge of weight 1 from A to X. Build the complete graph based on all possible transitions. Make sure that this graph will be directed one as transforming from A to X might not be the same as transforming from X to A due to the last operation which adds 1 to the result. Once, we have the directed graph, we can simply perform any all pair shortest path algorithm like BFS, Floyd Warshall (or Dynamic Programming as a graph is directed) and precalculate the answer for all possible cases. Once, the answers are precalculated, each test case can be answered in constant complexity. For more details, one can refer to the editorialist solution below.
The maximum number of edges from a number A emerging out will be 7! = 5040 in the worst case, but in practice, it will be quite less. The number of vertices in the graph will be 128. Using Floyd Warshall algorithm the precomputation can be achieved in O(128^3), i.e. O(A^3). This is enough to solve this subtask.
Subtask 2: A, B ≤ {10}^{18}
The constraints in this subtask are quite large and the number of test cases also suggest we need a logarithmic approach. This leads us to write A and B in binary representation and finding some observations to proceed with the solution.
Let us first simplify the approach by trying to convert A to (B-1) as in the end 1 would be added as part of the last operation.
We can see that in one step we can shuffle the digits of A in such a manner that an extra 1 is introduced in the binary representation of the newly formed number. This can be done by simply shuffling the digits to contain binary digit 1 from the {2}^{nd} position. For example: Let A = 3. In binary representation A = 011. We can shuffle the digits as A' = 110. On adding 1, we get X = 111, i.e. X = 7. Thus, we can introduce a binary digit 1 in one step.
Also, we can decrease any number of binary digit 1 from A. This can be done by easily placing the required in the number of 1 in towards the end, followed by a 0 and then placing the digits in any order we like. For example: Let A = 13, i.e. A = 1101 in binary representation. Say we want to decrease one binary digit 1, we can arrange the digits as A' = 1011. On adding 1, we get X = 1100. If we wanted to decrease two binary digit 1, we can arrange the digits as A' = 0111. On adding 1, we get X = 1000. Thus, we decrease any number of binary digit 1 in one step.
With the above 2 scenarios, the following is the algorithm-
- Find the number of ones in the binary representation of A and (B - 1). Let us denote then by OA and OB
- If OA > OB, then we can achieve the operation in 2 steps. First decreasing the number of ones in the first step and then rearranging the digits in another step.
- If OA <= OB, then we need (OB - OA) steps to first make the number of ones equal (see decrease operation takes place at one step each). Finally, we can arrange the digits to achieve the desired number.
The only corner case is as follows:
- If B is 0 then, we can’t achieve the desired state whatever the value of A is. Since we are adding 1 in the last step we are guaranteed to have atleast one binary digit 1 in binary representation. Thus the answer for this case is -1.
- If B is 1 then, we can achieve the desired state only if A = 0, in one step. In another case, it is impossible as even though decreasing the number of binary digit 1, the end result would be a number greater than 1. So the answer is 1 if A = 0, else -1.
The number of ones in binary representation can be calculated using the below pseudo-code:
def count_ones(integer x):
ones = 0
while x > 0:
if x % 2 == 1:
ones += 1
x /= 2
return ones
The time complexity of the above pseudo-code will be O(\log{x}) as each iteration of the while loop decreases the value of x by 2.
Feel free to share your approach, if it was somewhat different.
Time Complexity
O(\log{A} + \log{B}) per test case.
Space Complexity
O(1)
AUTHOR’S AND TESTER’S SOLUTIONS:
Author’s solution can be found here. (It will pass subtask 1 only and uses dynammic programming approach)
Tester’s solution can be found here.
Editorialist’s solution for subtask 1 can be found here.
Editorialist’s solution for full problem can be found here.
