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Count of N-digit numbers whose absolute difference between adjacent digits is non-increasing

  • Last Updated : 30 Jun, 2021

Given a positive integer N, the task is to count the number of N-digit numbers having absolute difference between consecutive digits in non-increasing order.

Examples:

Input: N = 1
Output: 10
Explanation:
All numbers from 0 to 9 satisfy the given condition as there is only one digit.

Input: N = 3
Output: 495

Naive Approach: The simplest approach to solve the given problem is to iterate over all possible N-digit numbers and count those numbers whose digits are in non-increasing order. After checking for all the numbers, print the value of count as the result. 



Time Complexity: O(N * 10N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized by using Dynamic Programming because the above problem has Overlapping subproblems and Optimal substructure. The subproblems can be stored in dp[][][] table using memoization where dp[digit][prev1][prev2] stores the answer from the digitth position till the end, when the previous digit selected, is prev1 and the second previous digit selected is prev2. Follow the steps below to solve the problem:

  • Define a recursive function, say countOfNumbers(digit, prev1, prev2) by performing the following steps.
    • If the value of digit is equal to N + 1 then return 1 as a valid N-digit number is formed.
    • If the result of the state dp[digit][prev1][prev2] is already computed, return this state dp[digit][prev1][prev2].
    • If the current digit is 1, then any digit from [1, 9] can be placed. If N = 1, then 0 can be placed as well.
    • If the current digit is 2, then any digit from [0, 9] can be placed.
    • Otherwise iterate through all the numbers from i = 0 to i = 9, and check if the condition (abs(prev1 – i) <= abs(prev1 – prev2) ) holds valid or not and accordingly place satisfying ‘i’ values in the current position.
    • After making a valid placement, recursively call the countOfNumbers function for index (digit + 1).
    • Return the sum of all possible valid placements of digits as the answer.
  • Print the value returned by the function countOfNumbers(1, 0, 0, N) as the result.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
int dp[100][10][10];
 
// Function to count N-digit numbers
// having absolute difference between
// adjacent digits in non-increasing order
int countOfNumbers(int digit, int prev1,
                   int prev2, int n)
{
    // If digit = n + 1, a valid
    // n-digit number has been formed
    if (digit == n + 1) {
        return 1;
    }
 
    // If the state has
    // already been computed
    int& val = dp[digit][prev1][prev2];
 
    if (val != -1) {
        return val;
    }
    val = 0;
 
    // If the current digit is 1,
    // then any digit from [1-9]
    // can be placed
    if (digit == 1) {
 
        for (int i = (n == 1 ? 0 : 1);
             i <= 9; ++i) {
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // If the current digit is 2, any
    // digit from [0-9] can be placed
    else if (digit == 2) {
 
        for (int i = 0; i <= 9; ++i) {
 
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // For other digits, any digit i
    // can be placed which satisfies
    // abs(prev1 - i) <= abs(prev1 - prev2)
    else {
        int diff = abs(prev2 - prev1);
 
        for (int i = 0; i <= 9; ++i) {
 
            // If absolute difference is
            // less than or equal to diff
            if (abs(prev1 - i) <= diff) {
 
                val += countOfNumbers(
                    digit + 1, i,
                    prev1, n);
            }
        }
    }
    return val;
}
 
// Function to count N-digit numbers with
// absolute difference between adjacent
// digits in non increasing order
int countNumbersUtil(int N)
{
    // Initialize dp table with -1
    memset(dp, -1, sizeof dp);
 
    // Function Call
    cout << countOfNumbers(1, 0, 0, N);
}
 
// Driver code
int main()
{
    int N = 3;
    countNumbersUtil(N);
 
    return 0;
}

Java




// Java program for the above approach
import java.io.*;
 
class GFG{
   
static int dp[][][] = new int[100][10][10];
 
// Function to count N-digit numbers
// having absolute difference between
// adjacent digits in non-increasing order
static int countOfNumbers(int digit, int prev1,
                          int prev2, int n)
{
     
    // If digit = n + 1, a valid
    // n-digit number has been formed
    if (digit == n + 1)
    {
        return 1;
    }
 
    // If the state has
    // already been computed
    int val = dp[digit][prev1][prev2];
 
    if (val != -1)
    {
        return val;
    }
    val = 0;
 
    // If the current digit is 1,
    // then any digit from [1-9]
    // can be placed
    if (digit == 1)
    {
        for(int i = (n == 1 ? 0 : 1);
                i <= 9; ++i)
        {
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // If the current digit is 2, any
    // digit from [0-9] can be placed
    else if (digit == 2)
    {
        for(int i = 0; i <= 9; ++i)
        {
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // For other digits, any digit i
    // can be placed which satisfies
    // abs(prev1 - i) <= abs(prev1 - prev2)
    else
    {
        int diff = Math.abs(prev2 - prev1);
 
        for(int i = 0; i <= 9; ++i)
        {
             
            // If absolute difference is
            // less than or equal to diff
            if (Math.abs(prev1 - i) <= diff)
            {
                val += countOfNumbers(
                    digit + 1, i,
                    prev1, n);
            }
        }
    }
    return val;
}
 
// Function to count N-digit numbers with
// absolute difference between adjacent
// digits in non increasing order
static void countNumbersUtil(int N)
{
     
    // Initialize dp table with -1
    for(int i = 0; i < 100; i++)
    {
        for(int j = 0; j < 10; j++)
        {
            for(int k = 0; k < 10; k++)
            {
                dp[i][j][k] = -1;
            }
        }
    }
     
    // Function Call
    System.out.println(countOfNumbers(1, 0, 0, N));
}
 
// Driver code
public static void main(String[] args)
{
    int N = 3;
     
    countNumbersUtil(N);
}
}
 
// This code is contributed by Dharanendra L V.

Python3




# Python3 program for the above approach
dp = [[[0 for i in range(10)]
          for col in range(10)]
          for row in range(100)]
 
# Function to count N-digit numbers
# having absolute difference between
# adjacent digits in non-increasing order
def countOfNumbers(digit, prev1, prev2, n):
     
    # If digit = n + 1, a valid
    # n-digit number has been formed
    if (digit == n + 1):
        return 1
 
    # If the state has
    # already been computed
    val = dp[digit][prev1][prev2]
 
    if (val != -1):
        return val
         
    val = 0
 
    # If the current digit is 1,
    # then any digit from [1-9]
    # can be placed
    if (digit == 1):
        i = 1
        if n == 1:
            i = 0
             
        for j in range(i, 10):
            val += countOfNumbers(digit + 1, j, prev1, n)
 
    # If the current digit is 2, any
    # digit from [0-9] can be placed
    elif (digit == 2):
        for i in range(0, 10):
            val += countOfNumbers(digit + 1, i, prev1, n)
 
    # For other digits, any digit i
    # can be placed which satisfies
    # abs(prev1 - i) <= abs(prev1 - prev2)
    else:
        diff = abs(prev2 - prev1)
        for i in range(0, 10):
             
            # If absolute difference is
            # less than or equal to diff
            if (abs(prev1 - i) <= diff):
                val += countOfNumbers(digit + 1, i, prev1, n)
    return val
 
# Function to count N-digit numbers with
# absolute difference between adjacent
# digits in non increasing order
def countNumbersUtil(N):
     
    # Initialize dp table with -1
    for i in range(0, 100):
        for j in range(0, 10):
            for k in range(0, 10):
                dp[i][j][k] = -1
 
    # Function Call
    print(countOfNumbers(1, 0, 0, N))
 
# Driver code
N = 3
 
countNumbersUtil(N)
 
# This code is contributed by amreshkumar3

C#




// C# program for the above approach
using System;
 
class GFG{
     
static int[,,] dp = new int[100, 10, 10];
 
// Function to count N-digit numbers
// having absolute difference between
// adjacent digits in non-increasing order
static int countOfNumbers(int digit, int prev1,
                          int prev2, int n)
{
     
    // If digit = n + 1, a valid
    // n-digit number has been formed
    if (digit == n + 1)
    {
        return 1;
    }
 
    // If the state has
    // already been computed
    int val = dp[digit, prev1, prev2];
 
    if (val != -1)
    {
        return val;
    }
    val = 0;
 
    // If the current digit is 1,
    // then any digit from [1-9]
    // can be placed
    if (digit == 1)
    {
        for(int i = (n == 1 ? 0 : 1);
                i <= 9; ++i)
        {
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // If the current digit is 2, any
    // digit from [0-9] can be placed
    else if (digit == 2)
    {
        for(int i = 0; i <= 9; ++i)
        {
            val += countOfNumbers(
                digit + 1, i, prev1, n);
        }
    }
 
    // For other digits, any digit i
    // can be placed which satisfies
    // abs(prev1 - i) <= abs(prev1 - prev2)
    else
    {
        int diff = Math.Abs(prev2 - prev1);
 
        for(int i = 0; i <= 9; ++i)
        {
             
            // If absolute difference is
            // less than or equal to diff
            if (Math.Abs(prev1 - i) <= diff)
            {
                val += countOfNumbers(
                    digit + 1, i,
                    prev1, n);
            }
        }
    }
    return val;
}
 
// Function to count N-digit numbers with
// absolute difference between adjacent
// digits in non increasing order
static void countNumbersUtil(int N)
{
     
    // Initialize dp table with -1
    for(int i = 0; i < 100; i++)
    {
        for(int j = 0; j < 10; j++)
        {
            for(int k = 0; k < 10; k++)
            {
                dp[i, j, k] = -1;
            }
        }
    }
     
    // Function Call
    Console.WriteLine(countOfNumbers(1, 0, 0, N));
}
 
// Driver code
static public void Main()
{
    int N = 3;
     
    countNumbersUtil(N);
}
}
 
// This code is contributed by splevel62
Output
495

Time Complexity: O(N * 103)
Auxiliary Space: O(N * 102

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