Longest set of Palindrome Numbers from the range [L, R] with at most K difference between its maximum and minimum

Given three positive integers L, R, and K, the task is to find the largest group of palindromic numbers from the range [L, R] such that the difference between the maximum and the minimum element present in the group is less than K.

Examples:

Input: L = 50, R = 78, K = 12
Output: 2
Explanation:
All palindromic numbers from the range [50, 78] are {55, 66, 77}.
The group of palindromic numbers {55, 66} have the difference between the maximum and minimum element as 11, which is less than K( = 12).
Therefore, the size of the group is 2. 

Input: L = 98, R = 112, K = 13
Output: 3

Approach: The given problem can be solved by using Binary Search. Follow the steps below to solve the problem:

• Initialize an auxiliary array, say arr[], and store all the palindrome numbers present in the range [L, R].
• Define a function, say search(arr, X), to find the rightmost index with value less than X:
  • Initialize three variables, say low as 0, high as (arr.size() – 1), and ans as -1.
  • Iterate until low ≤ high and perform the following operations:
    • Calculate mid as low+ (high – low)/2.
    • If the value at index mid is at most X, then update the value of ans as mid and low as (mid + 1).
    • Otherwise, update the value of high as (mid – 1).
  • After completing the above steps, return the value of ans as the result.
• Traverse the array arr[] and perform the following steps:
  • Find the rightmost index, which is less than or equal to (arr[i] + K – 1) using function search() and store it in a variable, say rightIndex.
  • If the value of rightIndex is not equal to -1, then update the value of count as the maximum of count and (rightIndex – i  + 1).
• After completing the above steps, print the value of count as the resultant.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include<bits/stdc++.h>
using namespace std;

// Function to search the
// rightmost index of given number
static int search(vector<int> list, int num)
{
    int low = 0, high = list.size() - 1;

    // Store the rightmost index
    int ans = -1;

    while (low <= high)
    {

        // Calculate the mid
        int mid = low + (high - low) / 2;

        // If given number <= num
        if (list[mid] <= num)
        {

            // Assign ans = mid
            ans = mid;

            // Update low
            low = mid + 1;
        }
        else

            // Update high
            high = mid - 1;
    }

    // return ans
    return ans;
}

// Function to check if the given
// number is palindrome or not
bool isPalindrome(int n)
{
    int rev = 0;
    int temp = n;

    // Generate reverse
    // of the given number
    while (n > 0)
    {
        rev = rev * 10 + n % 10;
        n /= 10;
    }

    // If n is a palindrome
    return rev == temp;
}

// Function to find the maximum size
// of group of palindrome numbers
// having difference between maximum
// and minimum element at most K
int countNumbers(int L, int R, int K)
{

    // Stores the all the palindromic
    // numbers in the range [L, R]
    vector<int> list;

    // Traverse over the range [L, R]
    for(int i = L; i <= R; i++)
    {

        // If i is a palindrome
        if (isPalindrome(i))
        {

            // Append the number
            // in the list
            list.push_back(i);
        }
    }

    // Stores count of maximum
    // palindromic numbers
    int count = 0;

    // Iterate each element in the list
    for(int i = 0; i < list.size(); i++)
    {

        // Calculate rightmost index in
        // the list < current element + K
        int right_index = search(list,
                                 list[i] + K - 1);

        // Check if there is rightmost
        // index from the current index
        if (right_index != -1)
            count = max(count, right_index - i + 1);
    }

    // Return the count
    return count;
}

// Driver Code
int main()
{
    int L = 98, R = 112;
    int K = 13;

    cout << countNumbers(L, R, K);
}

// This code is contributed by ipg2016107

Java

// Java program for the above approach
import java.util.*;

public class Main {

    // Function to find the maximum size
    // of group of palindrome numbers
    // having difference between maximum
    // and minimum element at most K
    static int countNumbers(int L, int R, int K)
    {
        // Stores the all the palindromic
        // numbers in the range [L, R]
        ArrayList<Integer> list
            = new ArrayList<>();

        // Traverse over the range [L, R]
        for (int i = L; i <= R; i++) {

            // If i is a palindrome
            if (isPalindrome(i)) {

                // Append the number
                // in the list
                list.add(i);
            }
        }

        // Stores count of maximum
        // palindromic numbers
        int count = 0;

        // Iterate each element in the list
        for (int i = 0; i < list.size(); i++) {

            // Calculate rightmost index in
            // the list < current element + K
            int right_index
                = search(list, list.get(i) + K - 1);

            // Check if there is rightmost
            // index from the current index
            if (right_index != -1)
                count = Math.max(count,
                                 right_index - i + 1);
        }

        // Return the count
        return count;
    }

    // Function to search the
    // rightmost index of given number
    static int search(
        ArrayList<Integer> list, int num)
    {
        int low = 0, high = list.size() - 1;

        // Store the rightmost index
        int ans = -1;

        while (low <= high) {

            // Calculate the mid
            int mid = low + (high - low) / 2;

            // If given number <= num
            if (list.get(mid) <= num) {

                // Assign ans = mid
                ans = mid;

                // Update low
                low = mid + 1;
            }
            else

                // Update high
                high = mid - 1;
        }

        // return ans
        return ans;
    }

    // Function to check if the given
    // number is palindrome or not
    static boolean isPalindrome(int n)
    {
        int rev = 0;
        int temp = n;

        // Generate reverse
        // of the given number
        while (n > 0) {
            rev = rev * 10 + n % 10;
            n /= 10;
        }

        // If n is a palindrome
        return rev == temp;
    }

    // Driver Code
    public static void main(String args[])
    {
        int L = 98, R = 112;
        int K = 13;

        System.out.print(
            countNumbers(L, R, K));
    }
}

Python3

# Python3 program for the above approach

# Function to find the maximum size
# of group of palindrome numbers
# having difference between maximum
# and minimum element at most K
def countNumbers(L, R, K):

    # Stores the all the palindromic
    # numbers in the range [L, R]
    list = []

    # Traverse over the range [L, R]
    for i in range(L, R + 1):

        # If i is a palindrome
        if (isPalindrome(i)):

            # Append the number
            # in the list
            list.append(i)

    # Stores count of maximum
    # palindromic numbers
    count = 0

    # Iterate each element in the list
    for i in range(len(list)):

        # Calculate rightmost index in
        # the list < current element + K
        right_index = search(list, list[i] + K - 1)

        # Check if there is rightmost
        # index from the current index
        if (right_index != -1):
            count = max(count, right_index - i + 1)

    # Return the count
    return count

# Function to search the
# rightmost index of given number
def search(list, num):
    low, high = 0, len(list) - 1

    # Store the rightmost index
    ans = -1

    while (low <= high):

        # Calculate the mid
        mid = low + (high - low) // 2

        # If given number <= num
        if (list[mid] <= num):

            # Assign ans = mid
            ans = mid

            # Update low
            low = mid + 1
        else:
            # Update high
            high = mid - 1

    # return ans
    return ans

# Function to check if the given
# number is palindrome or not
def isPalindrome(n):
    rev = 0
    temp = n

    # Generate reverse
    # of the given number
    while (n > 0):
        rev = rev * 10 + n % 10
        n //= 10

    # If n is a palindrome
    return rev == temp

# Driver Code
if __name__ == '__main__':
    L, R = 98, 112
    K = 13

    print(countNumbers(L, R, K))

# This code is contributed by mohit kumar 29.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find the maximum size
// of group of palindrome numbers
// having difference between maximum
// and minimum element at most K
static int countNumbers(int L, int R, int K)
{

    // Stores the all the palindromic
    // numbers in the range [L, R]
    List<int> list = new List<int>();

    // Traverse over the range [L, R]
    for(int i = L; i <= R; i++)
    {

        // If i is a palindrome
        if (isPalindrome(i))
        {

            // Append the number
            // in the list
            list.Add(i);
        }
    }

    // Stores count of maximum
    // palindromic numbers
    int count = 0;

    // Iterate each element in the list
    for(int i = 0; i < list.Count; i++)
    {

        // Calculate rightmost index in
        // the list < current element + K
        int right_index = search(list,
                                 list[i] + K - 1);

        // Check if there is rightmost
        // index from the current index
        if (right_index != -1)
            count = Math.Max(count,
                             right_index - i + 1);
    }

    // Return the count
    return count;
}

// Function to search the
// rightmost index of given number
static int search(List<int> list, int num)
{
    int low = 0, high = list.Count - 1;

    // Store the rightmost index
    int ans = -1;

    while (low <= high)
    {

        // Calculate the mid
        int mid = low + (high - low) / 2;

        // If given number <= num
        if (list[mid] <= num)
        {

            // Assign ans = mid
            ans = mid;

            // Update low
            low = mid + 1;
        }
        else

            // Update high
            high = mid - 1;
    }

    // return ans
    return ans;
}

// Function to check if the given
// number is palindrome or not
static bool isPalindrome(int n)
{
    int rev = 0;
    int temp = n;

    // Generate reverse
    // of the given number
    while (n > 0)
    {
        rev = rev * 10 + n % 10;
        n /= 10;
    }

    // If n is a palindrome
    return rev == temp;
}

// Driver Code
public static void Main(string[] args)
{
    int L = 98, R = 112;
    int K = 13;

    Console.WriteLine(countNumbers(L, R, K));
}
}

// This code is contributed by avijitmondal1998

Javascript

<script>

// Javascript program for the above approach

// Function to search the
// rightmost index of given number
function search(list, num)
{
    var low = 0, high = list.length - 1;

    // Store the rightmost index
    var ans = -1;

    while (low <= high)
    {

        // Calculate the mid
        var mid = low + (high - low) / 2;

        // If given number <= num
        if (list[mid] <= num)
        {

            // Assign ans = mid
            ans = mid;

            // Update low
            low = mid + 1;
        }
        else

            // Update high
            high = mid - 1;
    }

    // return ans
    return ans;
}

// Function to check if the given
// number is palindrome or not
function isPalindrome(n)
{
    var rev = 0;
    var temp = n;

    // Generate reverse
    // of the given number
    while (n > 0)
    {
        rev = rev * 10 + n % 10;
        n = parseInt(n/10);
    }

    // If n is a palindrome
    return rev == temp;
}

// Function to find the maximum size
// of group of palindrome numbers
// having difference between maximum
// and minimum element at most K
function countNumbers(L, R, K)
{

    // Stores the all the palindromic
    // numbers in the range [L, R]
    var list = [];

    // Traverse over the range [L, R]
    for(var i = L; i <= R; i++)
    {

        // If i is a palindrome
        if (isPalindrome(i))
        {

            // Append the number
            // in the list
            list.push(i);
        }
    }

    // Stores count of maximum
    // palindromic numbers
    var count = 0;

    // Iterate each element in the list
    for(var i = 0; i < list.length; i++)
    {

        // Calculate rightmost index in
        // the list < current element + K
        var right_index = search(list,
                                 list[i] + K - 1);

        // Check if there is rightmost
        // index from the current index
        if (right_index != -1)
            count = Math.max(count, right_index - i + 1);
    }

    // Return the count
    return count;
}

// Driver Code
var L = 98, R = 112;
var K = 13;
document.write( countNumbers(L, R, K));

// This code is contributed by noob2000.
</script>

Output:

3

Time Complexity: O((R – L) * log(R – L))
Auxiliary Space: O(R – L)