Given a coordinate (x, y) on a 2D plane. We have to reach (x, y) from the current position which is at origin i.e (0, 0). In each step, we can either move vertically or horizontally on the plane. While moving horizontally each step we write ‘H’ and while moving vertically each step we write ‘V’. So, there can be possibly many strings containing ‘H’ and ‘V’ which represents a path from (0, 0) to (x, y). The task is to find the lexicographically Kth smallest string among all the possible strings.
Examples:
Input: x = 2, y = 2, k = 2
Output: HVVH
Explanation: There are 6 ways to reach (2, 2) from (0, 0). The possible list of strings in lexicographically sorted order: [“HHVV”, “HVHV”, “HVVH”, “VHHV”, “VHVH”, “VVHH”]. Hence, the lexicographically 2nd smallest string is HVHV.
Input : x = 2, y = 2, k = 3
Output : VHHV
Prerequisites: Ways to Reach a Point from Origin
Approach: The idea is to use recursion to solve the problem. Number of ways to reach (x, y) from origin is x + yCx.
Now observe, the number of ways to reach (x, y) from (1, 0) will be (x + y – 1, x – 1) because we have already made a step in the horizontal direction, so 1 is subtracted from x. Also, the number of ways to reach (x, y) from (0, 1) will be (x + y – 1, y – 1) because we have already made a step in the vertical direction, so 1 is subtracted from y. Since ‘H’ is lexicographically smaller than ‘V’, so among all strings starting strings will contains ‘H’ in the beginning i.e initial movements will be Horizontal.
So, if K <= x + y – 1Cx – 1, we will take ‘H’ as first step else we will take ‘V’ as first step and solve for number of goings to (x, y) from(1, 0) will be K = K – x + y – 1Cx – 1.
Below is the implementation of this approach:
// CPP Program to find Lexicographically Kth // smallest way to reach given coordinate from origin #include <bits/stdc++.h> using namespace std;
// Return (a+b)!/a!b! int factorial( int a, int b)
{ int res = 1;
// finding (a+b)!
for ( int i = 1; i <= (a + b); i++)
res = res * i;
// finding (a+b)!/a!
for ( int i = 1; i <= a; i++)
res = res / i;
// finding (a+b)!/b!
for ( int i = 1; i <= b; i++)
res = res / i;
return res;
} // Return the Kth smallest way to reach given coordinate from origin void Ksmallest( int x, int y, int k)
{ // if at origin
if (x == 0 && y == 0)
return ;
// if on y-axis
else if (x == 0) {
// decrement y.
y--;
// Move vertical
cout << "V" ;
// recursive call to take next step.
Ksmallest(x, y, k);
}
// If on x-axis
else if (y == 0) {
// decrement x.
x--;
// Move horizontal.
cout << "H" ;
// recursive call to take next step.
Ksmallest(x, y, k);
}
else {
// If x + y C x is greater than K
if (factorial(x - 1, y) > k) {
// Move Horizontal
cout << "H" ;
// recursive call to take next step.
Ksmallest(x - 1, y, k);
}
else {
// Move vertical
cout << "V" ;
// recursive call to take next step.
Ksmallest(x, y - 1, k - factorial(x - 1, y));
}
}
} // Driven Program int main()
{ int x = 2, y = 2, k = 2;
Ksmallest(x, y, k);
return 0;
} |
// Java Program to find // Lexicographically Kth // smallest way to reach // given coordinate from origin import java.io.*;
class GFG
{ // Return (a+b)!/a!b! static int factorial( int a,
int b)
{ int res = 1 ;
// finding (a+b)!
for ( int i = 1 ;
i <= (a + b); i++)
res = res * i;
// finding (a+b)!/a!
for ( int i = 1 ; i <= a; i++)
res = res / i;
// finding (a+b)!/b!
for ( int i = 1 ; i <= b; i++)
res = res / i;
return res;
} // Return the Kth smallest // way to reach given // coordinate from origin static void Ksmallest( int x,
int y, int k)
{ // if at origin
if (x == 0 && y == 0 )
return ;
// if on y-axis
else if (x == 0 )
{
// decrement y.
y--;
// Move vertical
System.out.print( "V" );
// recursive call to
// take next step.
Ksmallest(x, y, k);
}
// If on x-axis
else if (y == 0 )
{
// decrement x.
x--;
// Move horizontal.
System.out.print( "H" );
// recursive call to
// take next step.
Ksmallest(x, y, k);
}
else
{
// If x + y C x is
// greater than K
if (factorial(x - 1 , y) > k)
{
// Move Horizontal
System.out.print( "H" );
// recursive call to
// take next step.
Ksmallest(x - 1 , y, k);
}
else
{
// Move vertical
System.out.print( "V" );
// recursive call to
// take next step.
Ksmallest(x, y - 1 , k -
factorial(x - 1 , y));
}
}
} // Driver Code public static void main (String[] args)
{ int x = 2 , y = 2 , k = 2 ;
Ksmallest(x, y, k);
} } // This code is contributed // by anuj_67. |
# Python3 Program to find Lexicographically Kth # smallest way to reach given coordinate from origin # Return (a+b)!/a!b! def factorial(a, b):
res = 1
# finding (a+b)!
for i in range ( 1 , a + b + 1 ):
res = res * i
# finding (a+b)!/a!
for i in range ( 1 , a + 1 ):
res = res / / i
# finding (a+b)!/b!
for i in range ( 1 , b + 1 ):
res = res / / i
return res
# Return the Kth smallest way to reach # given coordinate from origin def Ksmallest(x, y, k):
# if at origin
if x = = 0 and y = = 0 :
return
# if on y-axis
elif x = = 0 :
# decrement y.
y - = 1
# Move vertical
print ( "V" , end = "")
# recursive call to take next step.
Ksmallest(x, y, k)
# If on x-axis
elif y = = 0 :
# decrement x.
x - = 1
# Move horizontal.
print ( "H" , end = "")
# recursive call to take next step.
Ksmallest(x, y, k)
else :
# If x + y C x is greater than K
if factorial(x - 1 , y) > k:
# Move Horizontal
print ( "H" , end = "")
# recursive call to take next step.
Ksmallest(x - 1 , y, k)
else :
# Move vertical
print ( "V" , end = "")
# recursive call to take next step.
Ksmallest(x, y - 1 , k - factorial(x - 1 , y))
# Driver Code if __name__ = = "__main__" :
x, y, k = 2 , 2 , 2
Ksmallest(x, y, k)
# This code is contributed by Rituraj Jain |
// C# Program to find // Lexicographically Kth // smallest way to reach // given coordinate from origin using System;
class GFG
{ // Return (a+b)!/a!b! static int factorial( int a,
int b)
{ int res = 1;
// finding (a+b)!
for ( int i = 1;
i <= (a + b); i++)
res = res * i;
// finding (a+b)!/a!
for ( int i = 1; i <= a; i++)
res = res / i;
// finding (a+b)!/b!
for ( int i = 1; i <= b; i++)
res = res / i;
return res;
} // Return the Kth smallest // way to reach given // coordinate from origin static void Ksmallest( int x,
int y, int k)
{ // if at origin
if (x == 0 && y == 0)
return ;
// if on y-axis
else if (x == 0)
{
// decrement y.
y--;
// Move vertical
Console.Write( "V" );
// recursive call to
// take next step.
Ksmallest(x, y, k);
}
// If on x-axis
else if (y == 0)
{
// decrement x.
x--;
// Move horizontal.
Console.Write( "H" );
// recursive call to
// take next step.
Ksmallest(x, y, k);
}
else
{
// If x + y C x is
// greater than K
if (factorial(x - 1, y) > k)
{
// Move Horizontal
Console.Write( "H" );
// recursive call to
// take next step.
Ksmallest(x - 1, y, k);
}
else
{
// Move vertical
Console.Write( "V" );
// recursive call to
// take next step.
Ksmallest(x, y - 1, k -
factorial(x - 1, y));
}
}
} // Driver Code public static void Main (String[] args)
{ int x = 2, y = 2, k = 2;
Ksmallest(x, y, k);
} } // This code is contributed by 29AjayKumar |
<?php // PHP Program to find Lexicographically Kth // smallest way to reach given coordinate from origin // Return (a+b)!/a!b! function factorial( $a , $b )
{ $res = 1;
// finding (a+b)!
for ( $i = 1; $i <= ( $a + $b ); $i ++)
$res = $res * $i ;
// finding (a+b)!/a!
for ( $i = 1; $i <= $a ; $i ++)
$res = $res / $i ;
// finding (a+b)!/b!
for ( $i = 1; $i <= $b ; $i ++)
$res = $res / $i ;
return $res ;
} // Return the Kth smallest way to reach // given coordinate from origin function Ksmallest( $x , $y , $k )
{ // if at origin
if ( $x == 0 && $y == 0)
return ;
// if on y-axis
else if ( $x == 0)
{
// decrement y.
$y --;
// Move vertical
echo ( "V" );
// recursive call to
// take next step.
Ksmallest( $x , $y , $k );
}
// If on x-axis
else if ( $y == 0)
{
// decrement x.
$x --;
// Move horizontal.
echo ( "H" );
// recursive call to
// take next step.
Ksmallest( $x , $y , $k );
}
else
{
// If x + y C x is
// greater than K
if (factorial( $x - 1, $y ) > $k )
{
// Move Horizontal
echo ( "H" );
// recursive call to
// take next step.
Ksmallest( $x - 1, $y , $k );
}
else
{
// Move vertical
echo ( "V" );
// recursive call to
// take next step.
Ksmallest( $x , $y - 1, $k -
factorial( $x - 1, $y ));
}
}
} // Driver Code $x = 2; $y = 2; $k = 2;
Ksmallest( $x , $y , $k );
// This code is contributed // by Code_Mech. ?> |
<script> // Javascript Program to find Lexicographically Kth // smallest way to reach given coordinate from origin // Return (a+b)!/a!b! function factorial(a, b)
{ var res = 1;
// finding (a+b)!
for ( var i = 1; i <= (a + b); i++)
res = res * i;
// finding (a+b)!/a!
for ( var i = 1; i <= a; i++)
res = res / i;
// finding (a+b)!/b!
for ( var i = 1; i <= b; i++)
res = res / i;
return res;
} // Return the Kth smallest way to reach given coordinate from origin function Ksmallest(x, y, k)
{ // if at origin
if (x == 0 && y == 0)
return ;
// if on y-axis
else if (x == 0) {
// decrement y.
y--;
// Move vertical
document.write( "V" );
// recursive call to take next step.
Ksmallest(x, y, k);
}
// If on x-axis
else if (y == 0) {
// decrement x.
x--;
// Move horizontal.
document.write( "H" );
// recursive call to take next step.
Ksmallest(x, y, k);
}
else {
// If x + y C x is greater than K
if (factorial(x - 1, y) > k) {
// Move Horizontal
document.write( "H" );
// recursive call to take next step.
Ksmallest(x - 1, y, k);
}
else {
// Move vertical
document.write( "V" );
// recursive call to take next step.
Ksmallest(x, y - 1, k - factorial(x - 1, y));
}
}
} // Driven Program var x = 2, y = 2, k = 2;
Ksmallest(x, y, k); // This code is contributed by rutvik_56. </script> |
Output
HVVH