Given a tetrahedron(vertex are A, B, C, D), the task is to find the number of different cyclic paths with length n from a vertex.
Note: Considering only a single vertex B i.e. to find the number of different cyclic paths of length N from B to itself.
Input: 2 Output: 3 The paths of length 2 which starts and ends at D are: B-A-B B-D-B B-C-B Input: 3 Output: 6
Approach: Dynamic Programming can be used to keep track of the number of paths for previous values of N. Check for the number of moves which are left and where are we when we are moving in a path. That is 4n states, each with 3 options. Observe that all the vertices A, B, C are equivalent. Let zB be 1 initially and as at 0 steps, we can reach B itself only. Let zACD be 1 as paths for reaching other vertexes A, C and D is 0. Hence the recurrence relation formed will be:
Paths for N steps to reach b is = zADC*3
At every step, zADC gets multiplied by 2 (2 states) and it is added by zB since zB is the number of paths at step n-1 which comprises of the remaining 2 states.
Below is the implementation of the above approach:
- Number of paths with exactly k coins
- Count number of paths with at-most k turns
- Number of palindromic paths in a matrix
- Number of Paths of Weight W in a K-ary tree
- Paths with maximum number of 'a' from (1, 1) to (X, Y) vertically or horizontally
- Total number of decreasing paths in a matrix
- Number of paths from source to destination in a directed acyclic graph
- Number of shortest paths to reach every cell from bottom-left cell in the grid
- Count number of paths whose weight is exactly X and has at-least one edge of weight M
- Maximize the number of segments of length p, q and r
- Number of palindromic subsequences of length k where k <= 3
- Number of binary strings such that there is no substring of length ≥ 3
- Total number of odd length palindrome sub-sequence around each centre
- Number of sub-sequences of non-zero length of a binary string divisible by 3
- Find the number of binary strings of length N with at least 3 consecutive 1s
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