CareerCup Arrange the 2 x 1 boards on the 2 x n board

taoqick

于 2014-02-22 20:40:01 发布

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本文探讨了如何计算使用2x1板砖完全填充2xN大小的板砖的不同方式的数量。通过将问题简化为寻找第N个斐波那契数的方法，提供了一个高效的解决方案，并进一步介绍了如何在O(log N)时间内找到该数值。

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Given a board made of 2 x n squares, and boards made of 2 x 1 squares, write a function that will calculate the number of possible ways to arrange the 2 x 1 boards on the 2 x n board, in a way that will fill it completely.

(Asked to refine the solution to be more efficient)

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At first it seems like a dynamic programming type of problem but looking more closely it reduces to the simple problem of finding the nth Fibonacci number. Denote by f(n) the number of possibilities to tile a 2xn board.

Consider tiling a 2xn board for some n>2. Let us sort all the possibilities according to the last 2 columns. There are f(n-1) possibilities where the last column includes a single tile and there are f(n-2) possibilities where the last 2 columns include 2 horizontally placed tiles. This covers all possibilities and hence: f(n) = f(n-1) + f(n-2) which is exactly the nth Fibonacci number.

AND there is way to get f(n) in log(n) time.