# Count of numbers upto N having absolute difference of at most K between any two adjacent digits

Given an integer N, the task is to count the numbers up to N having an absolute difference of at most K between any two adjacent digits.

Note: Count of integer 0 for any digits is considerable.

Examples:

Input: N = 20, K = 2
Output: 15
Explanation:
The required numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 20. Notice that 14, 15, 16, 17, 18 and 19 all have adjacent digit’s absolute difference greater than K = 2 and thus they are not counted.

Input: N = 30, K = 3
Output: 22
Explanation:
All numbers upto 30 except 15, 16, 17, 18, 19, 26, 27, 28, 29 are accepted.

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

Naive Approach: The idea is to iterate till N and check for all the numbers that the difference of K exists or not. If yes then count it otherwise skip the number and keep iterating.

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

Efficient Approach: This problem can be optimized using Digit Dynamic Programming. Following are the detailed dp states for the given problem.

• In Digit Dp, we consider our number as a sequence of digits so a state position is needed so mark at which state we are currently at. In each recursive call, try to build the sequence from left to right by placing a digit from 0 to 9 and increment the position.
• Previous digit stores only those digits are put which have an absolute difference of atmost K from the previous digit. So, another state needed for the previous digit.
• Tight, state tells whether the number we are trying to build has already become smaller than N, so that in the upcoming recursive calls we can place any digit from 0 to 9. Otherwise, we can place till the digit of N at the current position.
• Initiliaze a boolean variable Start which tells whether the number has started or not. If the number has not yet started, we can start the number by placing digits from 1 to the upper limit with respect to tight in the current position. Otherwise, recur forward without starting the number.
• In each recursive call, set the current digit with respect to the previous digit such that the absolute difference between them never exceeds K. In the base case, return 1 if reached the last position.

Below is the implementation of the above approach:

## C++

 `// C++ program to get the count ` `// of numbers upto N having ` `// absolute difference at most K ` `// between any two adjacent digits ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Table to store solution ` `// of each subproblem ` `long` `long` `dp; ` ` `  `// Function to calculate ` `// all possible numbers ` `long` `long` `possibleNumbers( ` `    ``int` `pos, ``int` `previous, ``bool` `tight, ` `    ``bool` `start, string N, ``int` `K) ` `{ ` `    ``// Check if position reaches end that is ` `    ``// is equal to length of N ` `    ``if` `(pos == N.length()) ` `        ``return` `1; ` ` `  `    ``// Check if the result is ` `    ``// already computed ` `    ``// simply return it ` `    ``if` `(dp[pos][previous][tight][start] != -1) ` `        ``return` `dp[pos][previous][tight][start]; ` ` `  `    ``int` `res = 0; ` ` `  `    ``// Maximum limit upto which we can place ` `    ``// digit. If tight is false, means number has ` `    ``// already become smaller so we can place ` `    ``// any digit, otherwise N[pos] ` `    ``int` `upper_limit ` `        ``= (tight) ` `              ``? (N[pos] - ``'0'``) ` `              ``: 9; ` ` `  `    ``int` `new_tight; ` ` `  `    ``// Chekc if start is false the number ` `    ``// has not started yet ` `    ``if` `(!start) { ` ` `  `        ``// Check if we do not start ` `        ``// the number at pos ` `        ``// then recur forward ` `        ``res = possibleNumbers( ` `            ``pos + 1, previous, ` `            ``false``, ``false``, N, K); ` ` `  `        ``// If we start the number ` `        ``// we can place any digit ` `        ``// from 1 to upper_limit ` `        ``for` `(``int` `i = 1; i <= upper_limit; i++) { ` ` `  `            ``// Finding the new tight ` `            ``new_tight ` `                ``= (tight ` `                   ``&& i == upper_limit) ` `                      ``? 1 ` `                      ``: 0; ` `            ``res += possibleNumbers( ` `                ``pos + 1, i, new_tight, ` `                ``true``, N, K); ` `        ``} ` `    ``} ` ` `  `    ``// Condition if the number ` `    ``// has already started ` `    ``else` `{ ` ` `  `        ``// We can place digit upto ` `        ``// upperbound & absolute difference ` `        ``// with previous digit much ` `        ``// be atmost K ` `        ``for` `(``int` `i = 0; i <= upper_limit; i++) { ` ` `  `            ``new_tight = (tight ` `                         ``&& i == upper_limit) ` `                            ``? 1 ` `                            ``: 0; ` `            ``// Absolute difference atmost K ` `            ``if` `(``abs``(i - previous) <= K) ` `                ``res += possibleNumbers( ` `                    ``pos + 1, i, ` `                    ``new_tight, ``true``, N, K); ` `        ``} ` `    ``} ` ` `  `    ``// Store the solution ` `    ``// to this subproblem ` `    ``dp[pos][previous][tight][start] = res; ` ` `  `    ``return` `dp[pos][previous][tight][start]; ` `} ` ` `  `// Driver code ` `int` `main(``void``) ` `{ ` ` `  `    ``string N = ``"20"``; ` `    ``int` `K = 2; ` ` `  `    ``// Initialising the ` `    ``// table with -1 ` `    ``memset``(dp, -1, ``sizeof` `dp); ` ` `  `    ``// Function call ` `    ``cout << possibleNumbers( ` `                ``0, 0, ``true``, ` `                ``false``, N, K) ` `         ``<< endl; ` `} `

## Python3

 `# Python3 program to get the count of  ` `# numbers upto N having absolute ` `# difference at most K between any  ` `# two adjacent digits ` ` `  `# Table to store solution ` `# of each subproblem ` `dp ``=` `[[[[ ``-``1` `for` `i ``in` `range``(``2``)] ` `             ``for` `j ``in` `range``(``2``)] ` `             ``for` `i ``in` `range``(``10``)] ` `             ``for` `j ``in` `range``(``1002``)] ` ` `  `# Function to calculate ` `# all possible numbers ` `def` `possibleNumber(pos, previous, tight, ` `                   ``start, N, K): ` ` `  `    ``# Check if position reaches end  ` `    ``# that is equal to length of N ` `    ``if``(pos ``=``=` `len``(N)): ` `        ``return` `1` ` `  `    ``# Check if the result is ` `    ``# already computed ` `    ``# simply return it ` `    ``if``(dp[pos][previous][tight][start] !``=` `-``1``): ` `        ``return` `dp[pos][previous][tight][start] ` ` `  `    ``res ``=` `0` ` `  `    ``# Maximum limit upto which we can place ` `    ``# digit. If tight is false, means number has ` `    ``# already become smaller so we can place ` `    ``# any digit, otherwise N[pos] ` `    ``if``(tight): ` `        ``upper_limit ``=` `ord``(N[pos]) ``-` `ord``(``'0'``) ` `    ``else``: ` `        ``upper_limit ``=` `9` ` `  `    ``# Chekc if start is false the number ` `    ``# has not started yet ` `    ``if``(``not` `start): ` ` `  `        ``# Check if we do not start ` `        ``# the number at pos ` `        ``# then recur forward ` `        ``res ``=` `possibleNumber(pos ``+` `1``, previous, ` `                             ``False``, ``False``, N, K) ` ` `  `        ``# If we start the number ` `        ``# we can place any digit ` `        ``# from 1 to upper_limit ` `        ``for` `i ``in` `range``(``1``, upper_limit ``+` `1``): ` ` `  `            ``# Finding the new tight ` `            ``if``(tight ``and` `i ``=``=` `upper_limit): ` `                ``new_tight ``=` `1` `            ``else``: ` `                ``new_tight ``=` `0` ` `  `            ``res ``+``=` `possibleNumber(pos ``+` `1``, i,  ` `                                  ``new_tight, ` `                                  ``True``, N, K) ` `                                   `  `    ``# Condition if the number ` `    ``# has already started ` `    ``else``: ` `         `  `        ``# We can place digit upto ` `        ``# upperbound & absolute  ` `        ``# difference with previous ` `        ``# digit much be atmost K ` `        ``for` `i ``in` `range``(upper_limit ``+` `1``): ` `            ``if``(tight ``and` `i ``=``=` `upper_limit): ` `                ``new_tight ``=` `1` `            ``else``: ` `                ``new_tight ``=` `0` ` `  `            ``# Absolute difference atmost K ` `            ``if``(``abs``(i ``-` `previous) <``=` `K): ` `                ``res ``+``=` `possibleNumber(pos ``+` `1``, i, ` `                                      ``new_tight, ` `                                        ``True``, N, K) ` ` `  `    ``# Store the solution to this subproblem ` `    ``dp[pos][previous][tight][start] ``=` `res ` ` `  `    ``return` `dp[pos][previous][tight][start] ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `     `  `    ``N ``=` `"20"` `    ``K ``=` `2` `     `  `    ``# Function call ` `    ``print``(possibleNumber(``0``, ``0``, ``True``, ` `                         ``False``, N, K)) ` ` `  `# This code is contributed by Shivam Singh `

Output:

```15
```

Time Complexity: O( D * 10 * 2 * 2 * 10), considering N has D digits.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : SHIVAMSINGH67