From da2e1b2d7756a3ce5d3c3782cdd4c5d168a9e0e8 Mon Sep 17 00:00:00 2001 From: Stefan Dresselhaus Date: Thu, 31 Aug 2017 11:43:57 +0200 Subject: [PATCH] added incomplete proof for generating valid random grids --- dokumentation/generating_random_grids.md | 55 ++++++++++++++++++++++ dokumentation/generating_random_grids.pdf | Bin 0 -> 139858 bytes 2 files changed, 55 insertions(+) create mode 100644 dokumentation/generating_random_grids.md create mode 100644 dokumentation/generating_random_grids.pdf diff --git a/dokumentation/generating_random_grids.md b/dokumentation/generating_random_grids.md new file mode 100644 index 0000000..cb796b4 --- /dev/null +++ b/dokumentation/generating_random_grids.md @@ -0,0 +1,55 @@ +# Generating random grids + +## The Task + +Given an amount $p = p_1 \cdot p_2\cdot \dots \cdot p_n$ of random points in an $n$-dimensional unit-cube ($[0..1]^n$), +find a regular grid^[i.e. no intersections of cells, each point inside the grid is connected to $2n$ points] +with grid-dimensions of $p_1 \times \dots \times p_n$. + +## The Algorithm + +This Algorithm is a simple greedy-algorithm recursing over the dimensionality to construct one valid solution. + +Choose two dimensions $i$ and $j$. Sort the points by dimension $i$, divide them into chunks of $p_j$ points and sort the +points inside these chunks along $x_j-x_i$. + +Now you connect every $k$-th ($0 \leq k < p_j$) point of each chunk to generate lines along the $i$-th dimension. +By definition these paths cannot intersect each other in an projection onto the $i$/$j$-plane (see [Proof](#proof)). + +Let the "starting point" (i.e. the smallest in dimension $i$ on the line) be the representative for the line. + +Recurse over the other dimensions without choosing dimension $i$ again in a similar way. In the recursive call the "1:1 point merging" +will become "1:1 line-merging" (and "1:1 grid-merging") where you connect the $k$-th component of each line/grid. + +The resulting grid will be regular because no cell will overlap with another one due to careful construction^[TODO: Proof the same argument as above holds for arbitrary grids using only properties of the representing point] +and each point inside gets 2 new neighbors in each of the $n$ calls of the function resulting in $2n$ neighbors. + +## Application + +For our thesis we only need the cases of $n=2$ and $n=3$, but having a higher-dimensional solution around may come in handy in the future. + +## Proof + +It suffices to show that given 2 points of chunk $i$ and 1 point of chunk $i+1$ there exists no point s.t. the line-segments created by these four points intersect. + +Keep in mind that the intersection is in the $i$/$j$-projection and we are working in a unit-square. + +Let $a,b$ be the points in the $i$th-chunk and $x,y$ the points in the $i+1$-th chunk. Further shall a,b and x be fixed. + +We know (w.l.o.g., by construction), that + +- $\lbrace a_i, b_i \rbrace < \lbrace x_i, y_i \rbrace$ +- $a_j - a_i < b_j - b_i$ + +Assume $y$ is placed in a way that the generated lines intersect. We have two cases to consider: + +1. $x_j - x_i < y_j - y_i$ + +In this case the algorithm says that we have to connect $x$ to $a$ and $y$ to $b$. As we can only choose $y$, this point +has to have a greater distance to $b$ than the $x/a$-line. + +$\lambda a + (1-\lambda) x$ for $\lambda \in [0..1]$ are all points on the line. + + + + diff --git a/dokumentation/generating_random_grids.pdf b/dokumentation/generating_random_grids.pdf new file mode 100644 index 0000000000000000000000000000000000000000..70e4c5789f8eaa5bd0930bf87756211159c75ee0 GIT binary patch literal 139858 zcma&NQ?M{htgg9i`&+hc+qP}nwr$(CZQHhOule`X?5i^uCza|>s&CU(PrXS<E7a&G8voSlRIZUxT6(wXk+Jam1$+wKi}z z5jHWhGd6+Z<%M!`b~G`tfpXtW(U@~PYKQGTtGie5z;SxlTI{|xD7$&J(bH?|%G$cd zW^OgK8co%nkY)Yz0Sru($Z%y$6E1{60O9inIEaIHyYox*CSX?DoE#ZC5+omS)S|AQ z$WS(YsMqzL{MBf4(Zk7db?Q^@nE0&Jar5%reV*}b|NFkDTO?Muh+Y~-m(pnLMkzir z&`Rnb@~LRDd>3ULq?J_FwAeX}mWl^vG2?1C6xxwR|dl?1?D-8RV3pzDu7|G-Ervm6~LyYfIsDkr&*uq{Qokx;71TH=#Xqt$0|f zb7P}UZmU6HR)3BQ364yC@BxsDs<8$@8+pyXO~-Pk;RH7O!n8a{Z=~EkOl4|8k*>g$ zaniZ5b~Ep6(<1a>5rVi?iXts8(IjW%q+Khkm%W2QDH)&#D~GR)E-h#K!TZ_j3|ZQcqEi%{LFM?pAIi$K`sc`Y>1BV% 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