Intereting Posts

Deducing a $\cos (kx)$ summation from the $e^{ikx}$ summation
Uniform convergence of the series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\sin\frac{1}{nx}$ on $(0,+\infty)$
Is $\{n\>mod\> \pi: n \in \mathbb{N}\}$ dense in $$?
Homotopy/Homology groups of rationals
Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?
What is the difference between a predicate and function?
Limit: $\lim_{n\to \infty} \frac{n^5}{3^n}$
Let $f$ be a bounded measurable function on $E$. Show that there are sequences of simple functions converging uniformly on $E$.
Proving something it NOT an integral domain
Prove: ${n\choose 0}-\frac{1}{3}{n\choose 1}+\frac{1}{5}{n\choose 2}-…(-1)^n\frac{1}{2n+1}{n\choose n}=\frac{n!2^n}{(2n+1)!!}$
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
the knot surgery – from a $6^3_2$ knot to a $3_1$ trefoil knot
do eigenvectors correspond to direction of maximum scaling?
Question about a notation.
Laplace Transform Piecewise Function

I’ve got a specific problem which I’ll try to describe as clearly as possible.

I have a defined rectangular region on a cartesian plane, and within that region there are other given rectangular sub-regions that are described in terms of their 4 vertices, ie {(x1, y1), (x1, y2), (x2, y1), (x2, y2)}, so that these regions form ‘occlusions’ on the plane. These regions don’t overlap, but they can form more complex polygons when different-sized rectangles adjacent to each other appear joined.

here’s an illustration:

- Solving very large matrices in “pieces”
- Computing GCD of all permutations (of the digits) of a given number
- Why is the number of possible subsequences $2^n$?
- Largest prime factor of 600851475143
- gradient descent optimal step size
- Knuth's algorithm for Mastermind question

I am interested in the space between these shapes, and how to define the space in the same way the occlusions are defined, that is as a set of rectangles. In particular I want the definition optimised so that the space is described using the smallest possible number of rectangles. For example, an incomplete rendering might look like this:

Can anyone suggest a way forward with this? How can I have the original set of vertices (describing the black rectangles) generate the ‘complementary’ set of vertices (red rectangles) such that the number of red rectangles is minimal?

I suspect it’s a variation of a ‘packing problem’, but I have a feeling it might be fairly simple…

- Horse Race question: how to find the 3 fastest horses?
- Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$
- Computational complexity of least square regression operation
- Worst case analysis of MAX-HEAPIFY procedure .
- Complexity of counting the number of triangles of a graph
- Is 'every exponential grows faster than every polynomial?' always true?
- Minimum Sum that cannot be obtained from the 1…n with some missing numbers
- Application of computers in higher mathematics
- Knuth's Mastermind Algoritm: “The last step”
- How to accurately calculate the error function erf(x) with a computer?

This problem has been studied in the 1980’s. You can find it named the problem of finding

a minimum rectangle partition of an orthogonal polygon with holes. In the older literature, sometimes “orthogonal” is replaced with “rectilinear.” If you wrap your black rectangles with the minimum bounding box, then you have converted it to an orthogonal polygon with holes.

I believe the first result was by Lipsky in 1979, but I am not finding that

paper. Here is a somewhat later paper:

“Partitioning rectilinear figures into rectangles.” 1988. (ACM link)

Although many problems superficially analogous to this are NP-hard,

this rectangle-partition problem can be solved in

$O(n^{5/2})$ time for a polygon of $n$ vertices in total.

The algorithms depend on finding a maximum independent set of the

intersection graph of certain chords in the polygon.

(Added

a

partition can be found in $O(n \log \log n)$ time.

- Help with convergence in distribution
- Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.
- Is there a problem when defining exponential with negative base?
- Convergence of nested radicals
- Quadratic Polynomial factorization
- How prove this $(abc)^4+abc(a^3c^2+b^3a^2+c^3b^2)\le 4$
- Video Lessons in Complex Analysis
- A question on norm of error vector
- Is a contracted primary ideal the contraction of a primary ideal?
- How to show that $ \sum_{n = 0}^{\infty} \dfrac {1}{n!} = e $
- How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution
- Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
- Parseval's Theorem Proof
- $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_k}$ not an integer
- Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$