Minimum steps to reach a destination

Given a number line from -infinity to +infinity. You start at 0 and can go either to the left or to the right. The condition is that in i’th move, you take i steps. 

1. Find if you can reach a given number x 
2. Find the most optimal way to reach a given number x, if we can indeed reach it. For example, 3 can be reached in 2 steps, (0, 1) (1, 3) and 4 can be reached in 3 steps (0, -1), (-1, 1) (1, 4).

Source: Flipkart Interview Question

The important thing to note is we can reach any destination as it is always possible to make a move of length 1. At any step i, we can move forward i, then backward i + 1.
Below is a recursive solution suggested by Arpit Thapar here. 

1. Since distance of + 5 and – 5 from 0 is same, hence we find answer for absolute value of destination.
2. We use a recursive function which takes as arguments: 
   1. Source Vertex 
   2. Value of last step taken 
   3. Destination
3.  If at any point source vertex = destination; return number of steps.
4. Otherwise we can go in both of the possible directions. Take the minimum of steps in both cases.
   • From any vertex we can go to : 
     • (current source + last step +1) and 
     • (current source – last step -1)
5. If at any point, absolute value of our position exceeds the absolute value of our destination then it is intuitive that the shortest path is not possible from here. Hence we make the value of steps INT_MAX, so that when i take the minimum of both possibilities, this one gets eliminated. 

If we don’t use this last step, the program enters into an INFINITE recursion and gives RUN TIME ERROR.

Below is the implementation of above idea. Note that the solution only counts steps. 

C++

// C++ program to count number of
// steps to reach a point
#include<bits/stdc++.h>
using namespace std;

// Function to count number of steps
// required to reach a destination

// source -> source vertex
// step -> value of last step taken
// dest -> destination vertex
int steps(int source, int step, int dest)
{
    // base cases
    if (abs(source) > (dest))
         return INT_MAX;
    if (source == dest) return step;

    // at each point we can go either way

    // if we go on positive side
    int pos = steps(source + step + 1,
                      step + 1, dest);

    // if we go on negative side
    int neg = steps(source - step - 1,
                      step + 1, dest);

    // minimum of both cases
    return min(pos, neg);
}

// Driver code
int main()
{
    int dest = 11;
    cout << "No. of steps required to reach "
                            << dest << " is "
                        << steps(0, 0, dest);
    return 0;
}

Java

// Java program to count number of
// steps to reach a point
import java.io.*;

class GFG
{

    // Function to count number of steps
    // required to reach a destination

    // source -> source vertex
    // step -> value of last step taken
    // dest -> destination vertex
    static int steps(int source, int step,
                                int dest)
    {
        // base cases
        if (Math.abs(source) > (dest))
            return Integer.MAX_VALUE;

        if (source == dest)
            return step;

        // at each point we can go either way

        // if we go on positive side
        int pos = steps(source + step + 1,
                        step + 1, dest);

        // if we go on negative side
        int neg = steps(source - step - 1,
                        step + 1, dest);

        // minimum of both cases
        return Math.min(pos, neg);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int dest = 11;
        System.out.println("No. of steps required"+
                                " to reach " + dest +
                       " is " + steps(0, 0, dest));
    }
}

// This code is contributed by Prerna Saini

Python3

# python program to count number of
# steps to reach a point
import sys

# Function to count number of steps
# required to reach a destination

# source -> source vertex
# step -> value of last step taken
# dest -> destination vertex
def steps(source, step, dest):

    #base cases
    if (abs(source) > (dest)) :
        return sys.maxsize

    if (source == dest):
        return step

    # at each point we can go
    # either way

    # if we go on positive side
    pos = steps(source + step + 1,
                    step + 1, dest)

    # if we go on negative side
    neg = steps(source - step - 1,
                    step + 1, dest)

    # minimum of both cases
    return min(pos, neg)


# Driver Code
dest = 11;
print("No. of steps required",
               " to reach " ,dest ,
        " is " , steps(0, 0, dest));


# This code is contributed by Sam007.

C#

// C# program to count number of
// steps to reach a point
using System;

class GFG
{
    // Function to count number of steps
    // required to reach a destination

    // source -> source vertex
    // step -> value of last step taken
    // dest -> destination vertex
    static int steps(int source, int step,
                                int dest)
    {
        // base cases
        if (Math.Abs(source) > (dest))
            return int.MaxValue;

        if (source == dest)
            return step;

        // at each point we can go either way

        // if we go on positive side
        int pos = steps(source + step + 1,
                        step + 1, dest);

        // if we go on negative side
        int neg = steps(source - step - 1,
                        step + 1, dest);

        // minimum of both cases
        return Math.Min(pos, neg);
    }

    // Driver Code
    public static void Main()
    {
        int dest = 11;
        Console.WriteLine("No. of steps required"+
                             " to reach " + dest +
                      " is " + steps(0, 0, dest));
    }
}

// This code is contributed by Sam007

PHP

<?php
// PHP program to count number
// of steps to reach a point

// Function to count number
// of steps required to reach
// a destination

// source -> source vertex
// step -> value of last step taken
// dest -> destination vertex
function steps($source, $step, $dest)
{
    // base cases
    if (abs($source) > ($dest))
        return PHP_INT_MAX;
    if ($source == $dest)
        return $step;

    // at each point we
    // can go either way

    // if we go on positive side
    $pos = steps($source + $step + 1,
                   $step + 1, $dest);

    // if we go on negative side
    $neg = steps($source - $step - 1,
                   $step + 1, $dest);

    // minimum of both cases
    return min($pos, $neg);
}

// Driver code
$dest = 11;
echo "No. of steps required to reach ",
     $dest, " is ", steps(0, 0, $dest);

// This code is contributed by aj_36
?>

Javascript

<script>
// JavaScript program to count number of
// steps to reach a point

// Function to count number of steps
// required to reach a destination

// source -> source vertex
// step -> value of last step taken
// dest -> destination vertex
function steps(source, step, dest)
{

    // base cases
    if (Math.abs(source) > (dest))
        return Number.MAX_SAFE_INTEGER;
    if (source == dest) return step;

    // at each point we can go either way
    // if we go on positive side
    let pos = steps(source + step + 1,
                    step + 1, dest);

    // if we go on negative side
    let neg = steps(source - step - 1,
                    step + 1, dest);

    // minimum of both cases
    return Math.min(pos, neg);
}

// Driver code
    let dest = 11;
    document.write("No. of steps required to reach "
                            + dest + " is "
                        + steps(0, 0, dest));

// This code is contributed by Surbhi Tyagi.
</script>

Output

No. of steps required to reach 11 is 5

Time Complexity : O(2^n)

Auxiliary Space : O(2^n)

Thanks to Arpit Thapar for providing above algorithm and implementation. 

Optimized Solution : Find minimum moves to reach target on an infinite line

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.