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0
votes
1answer
71 views

The random densification technique-JL lemma

In Ailon's paper (p.3): How $1/(20nd)$ is obtained?
10
votes
4answers
279 views

Which properties of planar graphs generalize to higher dimension / hypergraphs?

A planar graph is a graph which can be embedded in the plane, without having crossing edges. Let $G=(X,E)$ be a $k$-uniform-hypergraph, i.e. an hypergraph such that all its hyperedges have size k. ...
6
votes
1answer
197 views

Proof Haar matrices satisfy JL lemma

The Johnson-Lindenstrauss lemma says roughly that for any collection $S$ of $n$ points in $\mathbb{R}^d$, there exists a linear map $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ where $k = O(\log ...
2
votes
0answers
48 views

Quality measure for clusters of a metric space embedding of a graph?

When evaluating clustering algorithms for networks, we have well-established metrics like Modularity and Surprise for evaluating the quality of the resulting partition. If we then embed our graph ...
4
votes
1answer
176 views

Adjacency-Preserving 2D Grid Embedding

Consider a 2D grid, and a given planar graph $G$ with $\Delta<4$ (max node degree) and without odd cycles. What conditions should $G$ satisfy so that when it is mapped (or embedded) into the 2D ...
7
votes
1answer
224 views

Fast deletion / contraction in combinatorial embedding

I wonder if there is a sublinear algorithm to make deletion or contraction of an edge in a combinatorial embedding of, lets say, planar graph? Since in combinatorial embedding we have to maintain ...
3
votes
2answers
121 views

Is there any result on approximating an arbitrary tree metric by an HST metric?

Is there any known result on approximating an arbitrary tree metric by an HST metric (or an Ultrametric)? What is the distortion? Thanks.
1
vote
1answer
173 views

2D grid placement problem

Data for the problem: 2D grid(lattice) of size NxN n nodes placed on the grid:node_1,node_2,…node_n Each of nodes contain some data: a. node_i is presented by 3 parameters (x_i,y_i,t_i) b. ...
26
votes
1answer
809 views

Isometric embedding of L2 into L1

It is known that given an $n$-point subset of $\ell_2^d$ (that is, given $n$ points in ${\mathbb R}^d$ with Euclidean distance) it is possible to embed them isometrically in $\ell^{n\choose 2}_1$. ...
5
votes
2answers
320 views

Graph planar drawing, with each edge's length is known

Assuming I have a graph $G$, with edges $E$ and vertices $V$, and the length of each edge is known, but the coordinates of vertices are not. Further assume that this is a graph that can be embedded ...
11
votes
2answers
346 views

Average distortion embeddings

Consider two metric spaces $(X, d)$ and $(Y, f)$, and an embedding $\mu : X \rightarrow Y$. Traditional metric space embeddings measure the quality of $\mu$ as the worst-case ratio of original to ...
11
votes
4answers
415 views

Dimensionality reduction with slack?

The Johnson-Lindenstrauss lemma says roughly that for any collection $S$ of $n$ points in $\mathbb{R}^d$, there exists a map $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ where $k = O(\log n/\epsilon^2)$ ...