SDE - 1

You are present at point ‘A’ which is the top-left cell of an M X N matrix, your destination is point ‘B’, which is the bottom-right cell of the same matrix. Your task is to find the total number of unique paths from point ‘A’ to point ‘B’.In other words, you will be given the dimensions of the matrix as integers ‘M’ and ‘N’, your task is to find the total number of unique paths from the cell MATRIX[0][0] to MATRIX['M' - 1]['N' - 1].

To traverse in the matrix, you can either move Right or Down at each step. For example in a given point MATRIX[i] [j], you can move to either MATRIX[i + 1][j] or MATRIX[i][j + 1].

You are given a connected directed acyclic graph of ‘N’ nodes and ‘N’ - 1 edges, such that there is only one edge between any two nodes. You can perform the below operation on the given graph zero or more times:

1) Choose two nodes, ‘X’ and ‘Y’, such that there exists an edge between ‘X’ and ‘Y’.

2) Change the direction of this edge, i.e., if this edge is directed from ‘X’ to ‘Y’, change the direction of this edge to be directed from ‘Y’ to ‘X’ and vice versa.

Your task is to reorder the edges of the given graph in such a way that there exists a directed path from each node to node 0, using the minimum number of operations.

You have been given a Snake and Ladder Board with 'N' rows and 'N' columns with the numbers written from 1 to (N*N) starting from the bottom left of the board, and alternating direction each row.

For example

For a (6 x 6) board, the numbers are written as follows:

You start from square 1 of the board (which is always in the last row and first column). On each square say 'X', you can throw a dice which can have six outcomes and you have total control over the outcome of dice throw and you want to find out the minimum number of throws required to reach the last cell.
Some of the squares contain Snakes and Ladders, and these are possibilities of a throw at square 'X':
You choose a destination square 'S' with number 'X+1', 'X+2', 'X+3', 'X+4', 'X+5', or 'X+6', provided this number is <= N*N.
If 'S' has a snake or ladder, you move to the destination of that snake or ladder.  Otherwise, you move to S.
A board square on row 'i' and column 'j' has a "Snake or Ladder" if board[i][j] != -1. The destination of that snake or ladder is board[i][j].

Note :

You can only take a snake or ladder at most once per move: if the destination to a snake or ladder is the start of another snake or ladder, you do not continue moving - you have to ignore the snake or ladder present on that square.

For example, if the board is:
-1 1 -1
-1 -1 9
-1 4 -1
Let's say on the first move your destination square is 2  [at row 2, column 1], then you finish your first move at 4 [at row 1, column 2] because you do not continue moving to 9 [at row 0, column 0] by taking the ladder from 4.

A square can also have a Snake or Ladder which will end at the same cell.
For example, if the board is:
-1 3 -1
-1 5 -1
-1 -1 9
Here we can see Snake/Ladder on square 5 [at row 1, column 1] will end on the same square 5.