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Minimum time to reach a point with +t and -t moves at time t

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  • Difficulty Level : Easy
  • Last Updated : 31 Jul, 2022

Given a positive coordinate ‘X’ and you are at coordinate ‘0’, the task is to find the minimum time required to get to coordinate ‘X’ with the following move : 
At time ‘t’, you can either stay at the same position or take a jump of length exactly ‘t’ either to the left or to the right. In other words, you can be at coordinate ‘x – t’, ‘x’ or ‘x + t’ at time ‘t’ where ‘x’ is the current position.
Examples: 
 

Input: 6
Output: 3
At time 1, jump from x = 0 to x = 1 (x = x + 1)
At time 2, jump from x = 1 to x = 3 (x = x + 2)
At time 3, jump from x = 3 to x = 6 (x = x + 3)
So, minimum required time is 3.

Input: 9
Output: 4
At time 1, do not jump i.e x = 0 
At time 2, jump from x = 0 to x = 2 (x = x + 2)
At time 3, jump from x = 2 to x = 5 (x = x + 3)
At time 4, jump from x = 5 to x = 9 (x = x + 4)
So, minimum required time is 4.

 

Approach: The following greedy strategy works: 
We just find the minimum ‘t’ such that 1 + 2 + 3 + … + t >= X. 
 

  • If (t * (t + 1)) / 2 = X then answer is ‘t’.
  • Else if (t * (t + 1)) / 2 > X, then we find (t * (t + 1)) / 2 – X and remove this number from the sequence [1, 2, 3, …, t]. The resulting sequence sums up to ‘X’.

Below is the implementation of the above approach: 

C++




// C++ implementation of the above approach
 
#include <iostream>
 
using namespace std;
 
   // returns the minimum time
    // required to reach 'X'
    long cal_minimum_time(long X)
    {
 
        // Stores the minimum time
        long t = 0;
        long sum = 0;
 
        while (sum < X) {
 
            // increment 't' by 1
            t++;
 
            // update the sum
            sum = sum + t;
        }
 
        return t;
    }
 
// Driver code
int main()
{
        long n = 6;
        long ans = cal_minimum_time(n);
        cout << "The minimum time required is : " << ans ;
 
   return 0;
    
   // This code is contributed by ANKITRAI1
}

Java




// Java implementation of the above approach
class GFG {
 
    // returns the minimum time
    // required to reach 'X'
    static long cal_minimum_time(long X)
    {
 
        // Stores the minimum time
        long t = 0;
        long sum = 0;
 
        while (sum < X) {
 
            // increment 't' by 1
            t++;
 
            // update the sum
            sum = sum + t;
        }
 
        return t;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        long n = 6;
        long ans = cal_minimum_time(n);
        System.out.println("The minimum time required is : " + ans);
    }
}

Python3




# Python 3 implementation of the
# above approach
 
# returns the minimum time
# required to reach 'X'
def cal_minimum_time(X):
 
    # Stores the minimum time
    t = 0
    sum = 0
 
    while (sum < X):
         
        # increment 't' by 1
        t = t + 1
         
        # update the sum
        sum = sum + t;
     
    return t;
 
# Driver code
if __name__ == '__main__':
    n = 6
    ans = cal_minimum_time(n)
    print("The minimum time required is :", ans)
     
# This code is contributed By
# Surendra_Gangwar

C#




// C#  implementation of the above approach
using System;
 
public class GFG{
     
    // returns the minimum time
    // required to reach 'X'
    static long cal_minimum_time(long X)
    {
 
        // Stores the minimum time
        long t = 0;
        long sum = 0;
 
        while (sum < X) {
 
            // increment 't' by 1
            t++;
 
            // update the sum
            sum = sum + t;
        }
 
        return t;
    }
 
    // Driver code
    static public void Main (){
        long n = 6;
        long ans = cal_minimum_time(n);
        Console.WriteLine("The minimum time required is : " + ans);
    }
}

PHP




<?php
// PHP implementation of the
// above approach
 
// returns the minimum time
// required to reach 'X'
function cal_minimum_time($X)
{
    // Stores the minimum time
    $t = 0;
    $sum = 0;
 
    while ($sum < $X)
    {
 
        // increment 't' by 1
        $t++;
 
        // update the sum
        $sum = $sum + $t;
    }
 
    return $t;
}
 
// Driver code
$n = 6;
$ans = cal_minimum_time($n);
echo "The minimum time required is : " . $ans;
 
// This code is contributed
// by ChitraNayal
?>

Javascript




<script>
 
// JavaScript implementation of the above approach
 
    // returns the minimum time
    // required to reach 'X'
    function cal_minimum_time(X)
    {
   
        // Stores the minimum time
        let t = 0;
        let sum = 0;
   
        while (sum < X) {
   
            // increment 't' by 1
            t++;
   
            // update the sum
            sum = sum + t;
        }
   
        return t;
    }
 
// driver code
 
  let n = 6;
  let ans = cal_minimum_time(n);
  document.write("The minimum time required is : " + ans);
   
</script>

Output: 

The minimum time required is : 3

 

Time Complexity: O(n)

Auxiliary Space: O(1)


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