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Questions about algorithmic solutions of geometric problems, or other algorithms making usage of geometry.

1
vote
If you don't care about time complexity, the following algorithm would do. Go over all pairs of edges. For each pair, we have to find whether the two segments intersect. There are two cases: the two s …
answered Apr 2 '14 by Yuval Filmus
1
vote
Hint: Find some constant $C$ such that if the turtle ever crosses its own path, then it must hit one of the last $C$ segments in the path. (I think that $C=5$ should work.)
answered Oct 3 '15 by Yuval Filmus
3
votes
Suppose that the points are $(x_1,y_1),\ldots,(x_n,y_n)$. You are looking for the smallest (in some sense) bounding box $[X_m,X_M] \times [Y_m,Y_M]$ containing all your points. For every $i$, we must …
answered Apr 11 '18 by Yuval Filmus
5
votes
The problem can be solved in time $\tilde{O}(n^{4/3})$, using several algorithms: Agarwal, Partitioning arrangements of lines II: Applications. Chazelle, Cutting hyperplanes for divide-and-conquer. …
answered Sep 11 by Yuval Filmus
2
votes
The reason we need complexity assumptions like the 3SUM conjecture is that we cannot prove meaningful lower bounds in any non-trivial computational models. Our best lower bounds are in bounded depth m …
answered Apr 22 '16 by Yuval Filmus
2
votes
Garey, Johnson, Preparata and Tarjan came up with a simple $O(n\log n)$ algorithm back in 1978. It is described in many lecture notes, for example these lecture notes of Piotr Indyk.
answered May 13 '14 by Yuval Filmus
2
votes
A sphere centered at the origin with a radius $1$ consists of all points whose distance from the origin is at most $1$. The distance between two points $(x_1,y_1,z_1),(x_2,y_2,z_2)$ is $\sqrt{(x_1-x_2 …
answered Sep 19 '13 by Yuval Filmus
2
votes
For a point $x$, let $d(x)$ be the sum of distances between $x$ and points in $A$. For $x \notin A$, the derivative $d'(x)$ has the nice formula $$ d'(x) = |\{y \in A : y < x\}| - |\{y \in A : y > x\} …
answered Feb 3 '13 by Yuval Filmus
3
votes
This is an active area of research. Fortunately, recently there has been a breakthrough due to Vempala and Cousins [1]. They provide a MATLAB implementation of their algorithm. If you're interested in …
answered Jun 30 '15 by Yuval Filmus
1
vote
We can apply a linear transformation to get into the situation $$ \begin{align*} x' &= \frac{x+C}{Ax+By+1} \\ y' &= \frac{y+D}{Ax+By+1} \end{align*} $$ The Jacobian of the mapping $$ J = \begin{vmatri …
answered Mar 12 '16 by Yuval Filmus
3
votes
For the case $d = 3$, there exist $O(n\log n)$ algorithms due to Clarkson and Shor (randomized) and Ramos (deterministic). For higher dimensions there seem to exist only approximation algorithms, see …
answered Jan 24 '16 by Yuval Filmus
2
votes
Let $p,q$ be two points in your set of points. The point $p$ can be represented as some convex combination $\sum_i \alpha_i p_i$ of points on the convex hull. The triangle inequality gives $$ \|p - q\ …
answered Apr 10 '14 by Yuval Filmus
1
vote
Use a greedy algorithm to cover your points with circular disks of radius 2. In other words, repeatedly choose an uncovered point, and cover it with a circular disk of radius 2 centered at the point. …
answered Jan 9 '18 by Yuval Filmus
6
votes
A similar question was asked on Mathoverflow. The commenters mentioned a paper of Kenyon, which shows that the minimum number of squares required to tile a $w \times (w-1)$ rectangle is $\Theta(\log w …
answered Nov 8 '13 by Yuval Filmus
2
votes
Usually these algorithms can be adapted to the general case, but the details are messy and hard to get right. Another option is perturbation: Move each line segment by some small amount in all direc …
answered Apr 17 '14 by Yuval Filmus

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