Efficient queriable data structure to represent a screen with windows on it

Question

(this is related to my other question, see here)

Imagine a screen, with 3 windows on it:

I'd like to find an efficient data structure to represent this, while supporting these actions:

• return a list of coordinates where a given window can be positioned without overlapping with others
  • for the above example, if we want to insert a window of size 2x2, possible positions will be (8, 6), (8, 7), ..
• resizing a window on the screen without overlapping other windows while maintaining aspect ratio
• insert window at position x, y (assuming it doesn't overlap)

Right now my naive approach is keeping an array of windows and going over all points on the screen, checking for each one if it's in any of the windows. This is $O(n\cdot m\cdot w)$ where $n, m$ are the width, height of the screen and $w$ is the number of windows in it. Note that in general $w$ will be small (say < 10) where each window is taking a lot of space.

Answer 1

An easy optimization to the naive algorithm is to skip some points when you check one covered by a window. Say you scan left-to-right, top-to-bottom. If you encounter an $(x, y)$ in window $w = (l, r, w, h)$, you can jump from $x$ to $l + w + 1$ and continue. If windows are big and don't overlap, off the cuff, I'd wildly conjecture that would give you an $O(p)$ algorithm, where $p$ is the number of points in the set you return.

In fact, it could be beneficial to see whether the windows are generally wider or taller. For tall windows, it would be better to scan top-to-bottom, left-to-right; for wide windows, the method discussed above should win out. You might be able to take the idea and do scan/skip diagonally to get balanced performance across the board.

In fact, scanning left-to-right, top-to-bottom, you can remember for subsequent $h$ rows (values of $y$) that you will need to skip at the same $x$ value to the same $l + w + 1$ value, if you reach the $x$ (you might not if there's another window that overlaps). This would make top-to-bottom, left-to-right and diagonal scanning unnecessary to get equivalently good performance.

More generally, you now have two data structures: one with $O(1)$ preprocessing/construction overhead and $O(w)$ lookups, and one with $O(p)$ (possibly; maybe you can do better, or maybe my optimization doesn't really achieve this) preprocessing/construction and possibly $O(1)$ (hash/lookup table) or $O(\log p)$ (BST). So there are already two alternatives, both of which are pretty good, really...