Counting numbers of n digits that are monotone

Call decimal number a monotone if: D[\, i]\, \leqslant D[\, i+1]\, 0 \leqslant i \leqslant |D|            . Write a program that takes the positive number n on input and returns a number of decimal numbers of length n that are monotone. Numbers can't start with 0.

Examples :

Input : 1
Output : 9
Numbers are 1, 2, 3, ... 9
Input : 2
Output : 45
Numbers are 11, 12, 13, .... 22, 23
...29, 33, 34, ... 39.
Count is 9 + 8 + 7 ... + 1 = 45

Explanation: Let's start by example of monotone numbers:\{111\}, \{123\}, \{12223333444\}            All those numbers are monotone as each digit on higher place is \geq            than the one before it. What are the monotone numbers are of length 1 and digits 1 or 2? It is question to ask yourself at the very beginning. We can see that possible numbers are: \{1\}, \{2\}            That was easy, now lets expand the question to digits 1, 2 and 3: \{1\}, \{2\}, \{3\}            Now different question, what are the different monotone numbers consisting of only 1 and length 3 are there? \{111\}            Lets try now draw this very simple observation in 2 dimensional array for number of length 3, where first column is the length of string and first row is possible digits: \begin{array}{c c c c c c c c c c} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &\\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &\\ 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\\ 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\\ \end{array}            Let's try to fill 3rd row 3rd column(number of monotone numbers consisting from numbers 1 or 2 with length 2). This should be: \{11\}, \{12\}, \{22\}            If we will look closer we already have subsets of this set i.e: \{11\}, \{12\}            - Monotone numbers that has length 2 and consist of 1 or 2 \{22\}            - Monotone numbers of length 2 and consisting of number 2 We just need to add previous values to get the longer one. Final matrix should look like this: \begin{array}{c c c c c c c c c c} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &\\ 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &\\ 2 & 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & 47 &\\ 3 & 1 & 4 & 10 & 20 & 35 & 56 & 84 & 120 & 167 &\\ \end{array} 

// CPP program to count numbers of n digits
// that are  monotone.
#include <cstring>
#include <iostream>

// Considering all possible digits as {1, 2, 3, ..9}
int static const DP_s = 9;

int getNumMonotone(int len)
{
    // DP[i][j] is going to store monotone numbers of length
    // i+1 considering j+1 digits.
    int DP[len][DP_s];
    memset(DP, 0, sizeof(DP));
    // Unit length numbers
    for (int i = 0; i < DP_s; ++i)
        DP[0][i] = i + 1;
    // Single digit numbers
    for (int i = 0; i < len; ++i)
        DP[i][0] = 1;
    // Filling rest of the entries in bottom
    // up manner.
    for (int i = 1; i < len; ++i)
        for (int j = 1; j < DP_s; ++j)
            DP[i][j] = DP[i - 1][j] + DP[i][j - 1];
    return DP[len - 1][DP_s - 1];
}

// Driver code.
int main()
{
    std::cout << getNumMonotone(10);
    return 0;
}

// This code is contributed by Sania Kumari Gupta

// C program to count numbers of n digits
// that are monotone.
#include <stdio.h>
#include <string.h>

// Considering all possible digits as
// {1, 2, 3, ..9}
int static const DP_s = 9;

int getNumMonotone(int len)
{

    // DP[i][j] is going to store monotone numbers of length
    // i+1 considering j+1 digits.
    int DP[len][DP_s];
    memset(DP, 0, sizeof(DP));

    // Unit length numbers
    for (int i = 0; i < DP_s; ++i)
        DP[0][i] = i + 1;

    // Single digit numbers
    for (int i = 0; i < len; ++i)
        DP[i][0] = 1;

    // Filling rest of the entries in bottom up manner.
    for (int i = 1; i < len; ++i)
        for (int j = 1; j < DP_s; ++j)
            DP[i][j] = DP[i - 1][j] + DP[i][j - 1];

    return DP[len - 1][DP_s - 1];
}

// Driver code.
int main()
{
    printf("%d", getNumMonotone(10));
    return 0;
}

// This code is contributed by Sania Kumari Gupta

// Java program to count numbers 
// of n digits that are monotone.

class GFG 
{
    // Considering all possible 
    // digits as {1, 2, 3, ..9}
    static final int DP_s = 9;
    
    static int getNumMonotone(int len)
    {
    
        // DP[i][j] is going to store 
        // monotone numbers of length 
        // i+1 considering j+1 digits.
        int[][] DP = new int[len][DP_s];
    
        // Unit length numbers
        for (int i = 0; i < DP_s; ++i)
            DP[0][i] = i + 1;
    
        // Single digit numbers
        for (int i = 0; i < len; ++i)
            DP[i][0] = 1;
    
        // Filling rest of the entries 
        // in bottom up manner.
        for (int i = 1; i < len; ++i)
            for (int j = 1; j < DP_s; ++j)
                DP[i][j] = DP[i - 1][j] 
                           + DP[i][j - 1];
    
        return DP[len - 1][DP_s - 1];
    }
    
    // Driver code.
    public static void main (String[] args) 
    {
        System.out.println(getNumMonotone(10));
    }
}

// This code is contributed by Ansu Kumari.

# Python3 program to count numbers of n 
# digits that are monotone.

# Considering all possible digits as
# {1, 2, 3, ..9}
DP_s = 9

def getNumMonotone(ln):

    # DP[i][j] is going to store monotone
    # numbers of length i+1 considering
    # j+1 digits.
    DP = [[0]*DP_s for i in range(ln)]

    # Unit length numbers
    for i in range(DP_s):
        DP[0][i] = i + 1

    # Single digit numbers
    for i in range(ln):
        DP[i][0] = 1

    # Filling rest of the entries  
    # in bottom up manner.
    for i in range(1, ln):

        for j in range(1, DP_s):
            DP[i][j] = DP[i - 1][j] + DP[i][j - 1]

    return DP[ln - 1][DP_s - 1]


# Driver code
print(getNumMonotone(10))


# This code is contributed by Ansu Kumari

// C# program to count numbers 
// of n digits that are monotone.
using System;

class GFG 
{
    // Considering all possible 
    // digits as {1, 2, 3, ..9}
    static int DP_s = 9;
    
    static int getNumMonotone(int len)
    {
    
        // DP[i][j] is going to store 
        // monotone numbers of length 
        // i+1 considering j+1 digits.
        int[,] DP = new int[len,DP_s];
    
        // Unit length numbers
        for (int i = 0; i < DP_s; ++i)
            DP[0,i] = i + 1;
    
        // Single digit numbers
        for (int i = 0; i < len; ++i)
            DP[i,0] = 1;
    
        // Filling rest of the entries 
        // in bottom up manner.
        for (int i = 1; i < len; ++i)
            for (int j = 1; j < DP_s; ++j)
                DP[i,j] = DP[i - 1,j] 
                        + DP[i,j - 1];
    
        return DP[len - 1,DP_s - 1];
    }
    
    // Driver code.
    public static void Main () 
    {
        Console.WriteLine(getNumMonotone(10));
    }
}

// This code is contributed by vt_m.

<script>

// JavaScript program to count numbers of n
// digits that are monotone.

// Considering all possible digits as
// {1, 2, 3, ..9}
let DP_s = 9

function getNumMonotone(ln){

    // DP[i][j] is going to store monotone
    // numbers of length i+1 considering
    // j+1 digits.
    let DP = new Array(ln).fill(0).map(()=>new Array(DP_s).fill(0))

    // Unit length numbers
    for(let i=0;i<DP_s;i++){
        DP[0][i] = i + 1
    }

    // Single digit numbers
    for(let i=0;i<ln;i++)
        DP[i][0] = 1

    // Filling rest of the entries
    // in bottom up manner.
    for(let i=1;i<ln;i++){

        for(let j=1;j<DP_s;j++){
            DP[i][j] = DP[i - 1][j] + DP[i][j - 1]
        }
    }

    return DP[ln - 1][DP_s - 1]
}


// Driver code
document.write(getNumMonotone(10),"</br>")


// This code is contributed by shinjanpatra

</script>

<?php
// PHP program to count numbers 
// of n digits that are monotone.
function getNumMonotone($len)
{
    // Considering all possible
    // digits as {1, 2, 3, ..9}
    $DP_s = 9;


    // DP[i][j] is going to store 
    // monotone numbers of length 
    // i+1 considering j+1 digits.
    $DP = array(array_fill(0, $len, 0),
                array_fill(0, $len, 0));

    // Unit length numbers
    for ($i = 0; $i < $DP_s; ++$i)
        $DP[0][$i] = $i + 1;

    // Single digit numbers
    for ($i = 0; $i < $len; ++$i)
        $DP[$i][0] = 1;

    // Filling rest of the entries 
    // in bottom up manner.
    for ($i = 1; $i < $len; ++$i)
        for ($j = 1; $j < $DP_s; ++$j)
            $DP[$i][$j] = $DP[$i - 1][$j] + 
                          $DP[$i][$j - 1];

    return $DP[$len - 1][$DP_s - 1];
}

// Driver code
echo getNumMonotone(10);

// This code is contributed by mits
?>

43758

Time complexity: O(n*DP_s)
Auxiliary space: O(n*DP_s) 

Efficient approach : Space optimization

In previous approach the current value dp[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.

Implementation steps:

• Create a 1D vector dp of size DP_s.
• Set a base case by initializing the values of DP .
• Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
• At last return and print the final answer stored dp[Dp_s-1] .

Implementation:

// CPP program to count numbers of n digits
// that are  monotone.

#include <cstring>
#include <iostream>

// Considering all possible digits as {1, 2, 3, ..9}
int static const DP_s = 9;

// funtion  to count numbers of n digits
// that are  monotone.
int getNumMonotone(int len)
{
    int DP[DP_s];
    memset(DP, 0, sizeof(DP));
    for (int i = 0; i < DP_s; ++i)
        DP[i] = i + 1;
    
    // iterate over subprobelms
    for (int i = 1; i < len; ++i)
        for (int j = 1; j < DP_s; ++j)
            DP[j] += DP[j - 1];

    // return answer
    return DP[DP_s - 1];
}

// Driver code
int main()
{   
    // function call
    std::cout << getNumMonotone(10);
    return 0;
}

public class MonotoneNumbers {

    // Define a constant for the number of possible digits (1 to 9)
    static final int DP_SIZE = 9;

    /**
     * Function to count the number of n-digit monotone numbers.
     *
     * @param len The number of digits in the monotone numbers.
     * @return The count of monotone numbers with n digits.
     */
    public static int getNumMonotone(int len) {
        // Create an array to store intermediate results for dynamic programming
        int[] DP = new int[DP_SIZE];

        // Initialize DP array with values from 1 to 9
        for (int i = 0; i < DP_SIZE; i++) {
            DP[i] = i + 1;
        }

        // Iterate over subproblems to compute the count of monotone numbers
        for (int i = 1; i < len; i++) {
            for (int j = 1; j < DP_SIZE; j++) {
                DP[j] += DP[j - 1];
            }
        }

        // Return the final count of n-digit monotone numbers
        return DP[DP_SIZE - 1];
    }

    public static void main(String[] args) {
        // Call the function and print the result
        int n = 10; // Change this value to count n-digit 
        // monotone numbers for a different n
        int result = getNumMonotone(n);
        System.out.println(result);
    }
}

def get_num_monotone(length):
    # Initialize a list to store the dynamic programming values
    dp = [0] * 9
    
    # Initialize the values for the one-digit numbers (1 to 9)
    for i in range(9):
        dp[i] = i + 1

    # Iterate to calculate the number of n-digit monotone numbers
    for i in range(1, length):
        for j in range(1, 9):
            # Update the dp values based on the previous row
            dp[j] += dp[j - 1]

    # The final result is stored in dp[8] for an n-digit number
    return dp[8]

if __name__ == "__main__":
    # Function call to get the number of 10-digit monotone numbers
    result = get_num_monotone(10)
    
    # Print the result
    print(result)

// C# program to count numbers of n digits
// that are monotone.
using System;

class GFG
{
  
// Considering all possible digits as {1, 2, 3, ..9}
const int DP_s = 9;
  
  // function to count numbers of n digits
// that are monotone.
static int GetNumMonotone(int len)
{
    int[] DP = new int[DP_s];
    for (int i = 0; i < DP_s; ++i)
        DP[i] = i + 1;

    // iterate over subproblems
    for (int i = 1; i < len; ++i)
        for (int j = 1; j < DP_s; ++j)
            DP[j] += DP[j - 1];

    // return answer
    return DP[DP_s - 1];
}

// Driver code
static void Main()
{
    // function call
    Console.WriteLine(GetNumMonotone(10));
}
}

// JavaScript program to count numbers of n digits
// that are monotone.

// Considering all possible digits as {1, 2, 3, ..9}
const DP_s = 9;

// Function to count numbers of n digits
// that are monotone.
function getNumMonotone(len) {
    let DP = new Array(DP_s).fill(0);

    for (let i = 0; i < DP_s; ++i)
        DP[i] = i + 1;

    // Iterate over subproblems
    for (let i = 1; i < len; ++i)
        for (let j = 1; j < DP_s; ++j)
            DP[j] += DP[j - 1];

    // Return answer
    return DP[DP_s - 1];
}

// Driver code
// Function call
console.log(getNumMonotone(10));

Output

43758

Time complexity: O(n*DP_s)
Auxiliary space: O(DP_s)