Minimize steps required to reach the value N

Last Updated : 10 Jun, 2021

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.

Examples:

Input: N = 18
Output: 6
Explanation:
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
Output: 2
Explanation:
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:

C++

 `// C++ program for the above approach`   `#include ` `using` `namespace` `std;`   `// Function to find the minimum steps` `// required to reach N by either moving` `// i steps forward or 1 steps backward` `int` `minimumsteps(``int` `N)` `{`   `    ``// Stores the required count` `    ``int` `steps = 0;`   `    ``// IF total moves required` `    ``// is less than double of N` `    ``while` `(steps * (steps + 1) < 2 * N) {`   `        ``// Update steps` `        ``steps++;` `    ``}`   `    ``// Steps required to reach N` `    ``if` `(steps * (steps + 1) / 2 == N + 1) {`   `        ``// Update steps` `        ``steps++;` `    ``}`   `    ``cout << steps;` `}`   `// Driver Code` `int` `main()` `{`   `    ``// Given value of N` `    ``int` `N = 18;` `    ``minimumsteps(N);` `}`

Java

 `// Java program for the above approach ` `import` `java.util.*;` `class` `GFG{` `      `  `// Function to find the minimum steps` `// required to reach N by either moving` `// i steps forward or 1 steps backward` `static` `void` `minimumsteps(``int` `N)` `{`   `    ``// Stores the required count` `    ``int` `steps = ``0``;`   `    ``// IF total moves required` `    ``// is less than double of N` `    ``while` `(steps * (steps + ``1``) < ``2` `* N)` `    ``{`   `        ``// Update steps` `        ``steps++;` `    ``}`   `    ``// Steps required to reach N` `    ``if` `(steps * (steps + ``1``) / ``2` `== N + ``1``) ` `    ``{`   `        ``// Update steps` `        ``steps++;` `    ``}`   `    ``System.out.println(steps);` `}` `  `  `// Driver code` `public` `static` `void` `main(String[] args)` `{` `  `  `    ``// Given value of N` `    ``int` `N = ``18``;` `    ``minimumsteps(N);` `}` `}`   `// This code is contributed by code_hunt.`

Python3

 `# Function to find the minimum steps` `# required to reach N by either moving` `# i steps forward or 1 steps backward`   `def` `minimumsteps(N) :`   `    ``# Stores the required count` `    ``steps ``=` `0`   `    ``# IF total moves required` `    ``# is less than double of N` `    ``while` `(steps ``*` `(steps ``+` `1``) < ``2` `*` `N) :`   `        ``# Update steps` `        ``steps ``+``=` `1`   `    ``# Steps required to reach N` `    ``if` `(steps ``*` `(steps ``+` `1``) ``/` `2` `=``=` `N ``+` `1``) :`   `        ``# Update steps` `        ``steps ``+``=` `1` `    ``print``(steps)`     `# Driver code` `N ``=` `18``;` `minimumsteps(N)`   `# This code is contributed by Dharanendra L V`

C#

 `// C# program for the above approach ` `using` `System;`   `class` `GFG {`   `    ``// Function to find the minimum steps` `    ``// required to reach N by either moving` `    ``// i steps forward or 1 steps backward` `    ``static` `void` `minimumsteps(``int` `N)` `    ``{`   `        ``// Stores the required count` `        ``int` `steps = 0;`   `        ``// IF total moves required` `        ``// is less than double of N` `        ``while` `(steps * (steps + 1) < 2 * N) {`   `            ``// Update steps` `            ``steps++;` `        ``}`   `        ``// Steps required to reach N` `        ``if` `(steps * (steps + 1) / 2 == N + 1) {`   `            ``// Update steps` `            ``steps++;` `        ``}`   `        ``Console.WriteLine(steps);` `    ``}`   `    ``// Driver code` `    ``static` `public` `void` `Main()` `    ``{`   `        ``// Given value of N` `        ``int` `N = 18;` `        ``minimumsteps(N);` `    ``}` `}`   `// This code is contributed by Dharanendra LV`

Javascript

 ``

Output:

`6`

Time CompleNity: O(sqrt(N))
AuNiliary Space: O(1)