Largest N squares that fit in a rectangle

Question

I was working on a project and I needed to display N squares inside a rectangle area and I want them to be as large as possible, no rotations. More formally:

Problem: Given N equal-sized squares and a rectangle with width W and height H, find out the maximum size of the squares L such that the squares fit inside the rectangle in such a way that their sides are orthogonal to the rectangle sides.

So far, I think we have a O(sqrt(N)) solution, in which we assume W>H, start by trying to place all squares in a single row, then in two rows and so on, until we get to O(sqrt(N)). For formally, let nRows be a variable from 1 to sqrt(N), nCols = ceil(N/nRows) and then do L = min(H/nRows, W/nCols).

I'm not completely sure the idea above is optimal. I was also wondering if we can solve this problem in O(1)?

I've tried looking for orthogonal square packing, but the problem seems a bit different (or could we easily reduce on to another?)

Answer 1

Let

\begin{align} C&=\left\lfloor\frac WL\right\rfloor& R&=\left\lfloor\frac HL\right\rfloor \end{align}

be the maximum number of rows and columns respectively. It sounds as though you're looking for the maximum $L^*$ such that, when $L=L^*$, we have $CR\ge N$. Let

\begin{align} L_0&=\sqrt{\frac{WH}N}\\ C_0&=\left\lceil\frac W{L_0}\right\rceil=\left\lceil\sqrt{\frac{WN}H}\right\rceil& R_0&=\left\lceil\frac H{L_0}\right\rceil=\left\lceil\sqrt{\frac{HN}W}\right\rceil. \end{align}

Observe that $L\le\min{\{W/C_0,H/R_0\}}$ is sufficient, and $L\le L_0$ is necessary:

\begin{align} \frac{WH}{L^2}&\ge\left\lfloor\frac WL\right\rfloor\left\lfloor\frac HL\right\rfloor=CR\ge N\\ L&\le\sqrt{\frac{WH}N}=L_0, \end{align}

with equality only if $W/L_0$ and $H/L_0$ both are integers. As a result, the optimal solution has dimension either $C_0\times\lceil N/C_0\rceil$ or $\lceil N/R_0\rceil\times R_0$.

Answer 2

Another possible approach is to look for good rational approximations to the ratio $W/H$, i.e., positive integers $r,s$ such that $r/s \approx W/H$.

Any good solution to your problem immediately yields such a good rational approximation (just let $r = $ the number of rows of squares and $s = $ the number of columns). Similarly, any good rational approximation to $W/H$ suggests a candidate solution to your problem (let $k=\lceil \sqrt{N/rs} \rfloor$, and then consider an arrangement with about $kr$ rows and $ks$ columns). So, a plausible approach might be to generate a bunch of good rational approximations to $W/H$, then test each corresponding candidate layout to see which is the best.

There are various algorithms for generating good rational approximations to a desired number $\alpha \in \mathbb{R}$. For instance, you can use continued fractions to generate a set of candidate approximations; there is a mathematical guarantee that each candidate will be a very good approximation, in some well-defined mathematical sense. The running time to generate and examine all such continued fractions will be poly-logarithmic in $N$. However, I don't know whether this will be better in practice than your simple $O(\sqrt{N})$ time algorithm.