Minimize steps required to reach the value N
Given an infinite number line from the range [-INFINITY, +INFINITY] and an integer N, the task is to find the minimum count of moves required to reach the N, starting from 0, by either moving i steps forward or 1 steps backward in every ith move.
Input: N = 18
To reach to the given value of N, perform the operations in the following sequence: 1 – 1 + 3 + 4 + 5 + 6 = 18
Therefore, a total of 6 operations are required.
Input: N = 3
To reach to the given value of N, perform the operations in the following sequence: 1 + 2 = 3
Therefore, a total of 2 operations are required.
Approach: The idea is to initially, keep adding 1, 2, 3 . . . . K, until it is greater than or equal to the required value N. Then, calculate the required number to be subtracted from the current sum. Follow the steps below to solve the problem:
- Initially, increment by K until N is greater than the current value. Now, stop at some position
pos = 1 + 2 + …………. + steps = steps ∗ (steps + 1) / 2 ≥ N.
Note: 0 ≤ pos – N < steps. Otherwise, the last step wasn’t possible.
- Case 1: If pos = N then, ‘steps’ is the required answer.
- Case 2: If pos ≠ N, then replace any iteration of K with -1.
- By replacing any K with -1, the modified value of pos = pos – (K + 1). Since K ∈ [1, steps], then pos ∈ [pos – steps – 1, pos – 2].
- It is clear that pos – step < N. If N < pos – 1, then choose the corresponding K = pos – N – 1 and replace K with -1 and get straight to the point N.
- If N + 1 = pos, only one -1 operation is required.
Below is the implementation of the above approach:
Time CompleNity: O(sqrt(N))
AuNiliary Space: O(1)