Given three integers N, M and X, the task is to find the number of ways to form an array, such that all consecutive numbers of the array are distinct, and the value at any index of the array from 2 to N – 1(Considering 1 based indexing) lies between 1 and M, while the value at index 1 is X and the value at index N is 1.
Note: Value of X lies between 1 and M.
Input: N = 4, M = 3, X = 2
The following arrays are possible:
1) 2, 1, 2, 1
2) 2, 1, 3, 1
3) 2, 3, 2, 1
Input: N = 2, M = 3, X = 2
The only possible array is: 2, 1
Approach: The problem can be solved using Dynamic Programming. Let, f(i) represent the number of ways to form the array till the ith index, such that every consecutive element of the array is distinct. Let f(i, One) represent the number of ways to form the array till the i-th index such that every consecutive element of the array is distinct and arri = 1.
Similarly, let f(i, Non-One) represent the number of ways to form the array till the ith index, such that every consecutive element of the array is distinct and arri is not equal to 1.
The following recurrence is formed:
f(i, Non-One) = f(i - 1, One) * (M - 1) + f(i - 1, Non-One) * (M - 2)
which means that the number of ways to form the array till the ith index with arrayi not equal to 1 is formed using two cases:
- If the number at arrayi – 1 is 1, then opt one number out of (M – 1) options to place at the ith index, since arrayi is not equal to 1.
- If the number at arrayi – 1 is not 1, then we need to opt one number out of (M – 2) options, since arrayi is not equal to 1 and arrayi ≠ arrayi – 1.
Similarly, f(i, One) = f(i – 1, Non-One), since the number of ways to form the array till the ith index with arrayi = 1, is same as number of ways to form the array till the (i – 1)th index with arrayi – 1 ≠ 1, thus at the ith index there is only one option. At the end the required answer if f(N, One) since arrayN needs to be equal to 1.
Below is the implementation of the above approach:
Time Complexity: O(N), where N is the size of the array
- Number of ways to form a heap with n distinct integers
- Ways to form an array having integers in given range such that total sum is divisible by 2
- Maximum sum in circular array such that no two elements are adjacent
- Ways to sum to N using array elements with repetition allowed
- Smallest length string with repeated replacement of two distinct adjacent
- Ways to paint stairs with two colors such that two adjacent are not yellow
- Count ways to form minimum product triplets
- Ways to fill N positions using M colors such that there are exactly K pairs of adjacent different colors
- Maximum sum such that no two elements are adjacent
- Ways to arrange Balls such that adjacent balls are of different types
- Ways to paint N paintings such that adjacent paintings don't have same colors
- Maximum sum in a 2 x n grid such that no two elements are adjacent
- Maximum sum of difference of adjacent elements
- Count of arrays in which all adjacent elements are such that one of them divide the another
- Maximum length subsequence with difference between adjacent elements as either 0 or 1
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.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 : vt_m