Tiling Problem

Posted: 25 Jul, 2020
Difficulty: Hard

PROBLEM STATEMENT

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You have been given a board where there are '2' rows and 'N' columns. You have an infinite supply of 2x1 tiles, and you can place a tile in the following ways:

1. Horizontally as 1x2 tile
2. Vertically as 2x1 tile

Count the number of ways to tile the given board using the available tiles.

Note :
The number of ways might be large so output your answer modulo 10^9 + 7.

Here an example of tile and board for 'N' = 4 :

Tiling Example

Input format :
The first and only line of each test case contains an Integer 'N' which denotes the size of the board, i.e. '2' rows and 'N' columns.
Output format :
For each test case, print the number of ways to tile the board modulo 10^9 + 7.
Note:
You are not required to print the output explicitly, it has already been taken care of. Just implement the function.
Constraints :
1 <= N <= 10^18

Where 'N' is the number of columns in the board. 

Time limit: 1 sec
Approach 1

Try to place the tile to fill the unit column and calculate the number of ways from smaller sub-problems. Then use memoization to convert O(2^N) solution to an O(N) solution.

  1. At any point we are at ‘idx’ column then we can place our tile in two ways to fill this column.
    1. Option 1 -  1 Horizontal Tile

We can place in this way where we have ‘idx-1’ column filled.

          2.   Option 2 - 2 Vertical Tiles

We can place in this way where we have ‘idx-2’ column filled.   

     2. So, numberOfWays(n) = numberOfWays(n-1) + numberOfWays(n-2)

     3. Base cases are:

  1. When n = 1 there is only 1 way - Placing 1 Vertical Tile
  2. When n = 2 there are two ways - Placing 2 Vertical Tile and Placing 2 Horizontal Tiles.
  3. Also, take care of overflow using modulo 10^9 + 7.
  4. Lastly, use a DP Array of size N for memoization to save time over repetitive calls.
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