Count of N-digit numbers having each digit as the mean of its adjacent digits
Given a positive integer N, the task is to count the number of N-digit numbers where each digit in the number is the mean of its two adjacent digits.
Input: N = 1
Explanation: All numbers from 0 to 9 satisfy the given condition as there is only one digit.
Input: N = 2
Naive Approach: The simplest approach to solve the given problem is to iterate over all possible N-digit numbers and count such numbers where each digit is the mean of the two adjacent digits. 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 an 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, consider the three numbers prev1, prev2, and the current digit to be placed which is not yet decided. Among these three numbers, prev1 must be the mean of prev2 and current digit. Hence, current digit = (2*prev1) – prev2. If current >= 0 and current <= 9 then we can place it in the given position. Else return 0.
- Since mean involves integer division, (current + 1) can also be placed at the current position if (current + 1) >= 0 and (current + 1) ≤ 9.
- 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:
Time Complexity: O(N × 102)
Auxiliary Space: O(N × 102)