Lexicographically Kth smallest way to reach given coordinate from origin

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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 + y}C_x.
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 stringsa starting strings will contains ‘H’ in the beginning i.e inital movements will be Horizontal.
So, if K <= ^{x + y – 1}C_{x – 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 – 1}C_{x – 1}.

Below is the implementation of this approach:

C++

// 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

// 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

# 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#

// 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 
// 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. 
?> 

Output

HVVH

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Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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