Cut rectangle in minimum number of squares

Question

I\'m trying to solve the following problem:

A rectangular paper sheet of M*N is to be cut down into squares such that:

1. The paper i

Answer 1

I'd write this as a dynamic (recursive) program.

Write a function which tries to split the rectangle at some position. Call the function recursively for both parts. Try all possible splits and take the one with the minimum result.

The base case would be when both sides are equal, i.e. the input is already a square, in which case the result is 1.

function min_squares(m, n):

    // base case:
    if m == n: return 1

    // minimum number of squares if you split vertically:
    min_ver := min { min_squares(m, i) + min_squares(m, n-i)  |  i ∈ [1, n/2] }

    // minimum number of squares if you split horizontally:
    min_hor := min { min_squares(i, n) + min_squares(m-i, n)  |  i ∈ [1, m/2] }

    return min { min_hor, min_ver }

To improve performance, you can cache the recursive results:

function min_squares(m, n):

    // base case:
    if m == n: return 1

    // check if we already cached this
    if cache contains (m, n):
        return cache(m, n)

    // minimum number of squares if you split vertically:
    min_ver := min { min_squares(m, i) + min_squares(m, n-i)  |  i ∈ [1, n/2] }

    // minimum number of squares if you split horizontally:
    min_hor := min { min_squares(i, n) + min_squares(m-i, n)  |  i ∈ [1, m/2] }

    // put in cache and return
    result := min { min_hor, min_ver }
    cache(m, n) := result
    return result

In a concrete C++ implementation, you could use int cache[100][100] for the cache data structure since your input size is limited. Put it as a static local variable, so it will automatically be initialized with zeroes. Then interpret 0 as "not cached" (as it can't be the result of any inputs).

Possible C++ implementation: http://ideone.com/HbiFOH