# Tiling Problem using Divide and Conquer algorithm

Given a n by n board where n is of form 2^{k} where k >= 1 (Basically n is a power of 2 with minimum value as 2). The board has one missing cell (of size 1 x 1). Fill the board using L shaped tiles. A L shaped tile is a 2 x 2 square with one cell of size 1×1 missing.

This problem can be solved using Divide and Conquer. Below is the recursive algorithm.

// n is size of given square, p is location of missing cell Tile(int n, Point p) 1) Base case: n = 2, A 2 x 2 square with one cell missing is nothing but a tile and can be filled with a single tile. 2) Place a L shaped tile at the center such that it does not cover the n/2 * n/2 subsquare that has a missing square.Now all four subsquares of size n/2 x n/2 have a missing cell(a cell that doesn't need to be filled). See figure 2 below. 3) Solve the problem recursively for following four. Let p1, p2, p3 and p4 be positions of the 4 missing cells in 4 squares. a) Tile(n/2, p1) b) Tile(n/2, p2) c) Tile(n/2, p3) d) Tile(n/2, p3)

The below diagrams show working of above algorithm

Figure 2: After placing first tile

Figure 3: Recurring for first subsquare.

Figure 4: Shows first step in all four subsquares.

**Time Complexity:**

Recurrence relation for above recursive algorithm can be written as below. C is a constant.

T(n) = 4T(n/2) + C

The above recursion can be solved using Master Method and time complexity is O(n^{2})

**How does this work?**

The working of Divide and Conquer algorithm can be proved using Mathematical Induction. Let the input square be of size 2^{k} x 2^{k} where k >=1.

Base Case: We know that the problem can be solved for k = 1. We have a 2 x 2 square with one cell missing.

Induction Hypothesis: Let the problem can be solved for k-1.

Now we need to prove to prove that the problem can be solved for k if it can be solved for k-1. For k, we put a L shaped tile in middle and we have four subsqures with dimension 2^{k-1} x 2^{k-1} as shown in figure 2 above. So if we can solve 4 subsquares, we can solve the complete square.

**References:**

http://www.comp.nus.edu.sg/~sanjay/cs3230/dandc.pdf

This article is contributed by **Abhay Rathi**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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