Is Tiling a Rectangle with the Fewest Squares easy or hard?

Tiling a Rectangle with the Fewest Squares is classified as a Hard problem. It requires advanced backtracking, careful pruning strategies, and efficient state representation to avoid exponential explosion. Many candidates struggle with designing the search state and optimization techniques.

Tiling a Rectangle with the Fewest Squares Python/Java solution

Python and Java solutions usually implement recursive backtracking with memoization. The algorithm tracks filled cells, places the largest valid square, and recursively explores the next state. With pruning and caching, the solution handles rectangles up to 13 x 13 efficiently.

How to solve Tiling a Rectangle with the Fewest Squares efficiently?

Use recursive backtracking that always selects the first uncovered cell and tries squares from largest to smallest. Maintain the grid state and recursively fill remaining regions. Add memoization for previously seen configurations and prune branches when the square count exceeds the current best. This strategy finds the optimal solution quickly for the problem constraints.

What is the best approach for Tiling a Rectangle with the Fewest Squares?

Recursive backtracking with memoization is the most effective approach. The algorithm always fills the first empty cell and tries placing the largest possible square, recursively exploring the remaining space. Memoization stores previously visited states and pruning stops paths that exceed the current minimum. This dramatically reduces the exponential search space.

Is Tiling a Rectangle with the Fewest Squares asked at Google/Amazon/Meta?

Rectangle tiling and backtracking optimization problems appear in interviews at companies like Google and Amazon because they test search pruning, recursion, and state representation. Variants of grid covering and minimum tiling are common in advanced algorithm interviews.

What data structure is used in Tiling a Rectangle with the Fewest Squares?

Typical implementations use a 2D grid or a height profile array to represent the current filled state of the rectangle. Hash maps or dictionaries store memoized states for pruning repeated configurations. The algorithm heavily relies on recursion stacks for backtracking.

What is the time complexity of Tiling a Rectangle with the Fewest Squares?

The backtracking solution has exponential worst‑case time complexity because it explores multiple square placement combinations. In practice, pruning and memoization reduce the search significantly for the constraint n, m ≤ 13. The dynamic programming rectangle-splitting approach runs in roughly O(n^2 * m^2) time with O(n * m) space.

Description Hints 3 Solution Video 38:44

1240. Tiling a Rectangle with the Fewest Squares

Hard55.0% acceptance3 companies

Given a rectangle of size n x m, return the minimum number of integer-sided squares that tile the rectangle.

Example 1:

Input: n = 2, m = 3
Output: 3
Explanation: 3 squares are necessary to cover the rectangle.
2 (squares of 1x1)
1 (square of 2x2)

Example 2:

Input: n = 5, m = 8
Output: 5

Example 3:

Input: n = 11, m = 13
Output: 6

Constraints:

1 <= n, m <= 13

Asked at 3 companies

Google Amazon Bloomberg

Topics

Backtracking

Hints open one at a time

Hint 1

Can you use backtracking to solve this problem ?.

Hint 2

Suppose you've placed a bunch of squares. Where is the natural spot to place the next square ?.

Hint 3

The maximum number of squares to be placed will be ≤ max(n,m).

38:44

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