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:
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 originvoid 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 Programint 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 originimport 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 originstatic 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 Codepublic 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 Codeif __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 originstatic 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 Codepublic 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 $res;}// Return the Kth smallest way to reach // given coordinate from originfunction Ksmallest($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.?> |
Javascript
<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 originfunction 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 Programvar x = 2, y = 2, k = 2;Ksmallest(x, y, k);// This code is contributed by rutvik_56.</script> |
Output
HVVH
Feeling lost in the world of random DSA topics, wasting time without progress? It’s time for a change! Join our DSA course, where we’ll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 neveropen!
