Low-Distortion Embeddings of Trees

Babilon, Robert; Matoušek, Jiřı́; Maxová, Jana; Valtr, Pável

doi:10.1007/3-540-45848-4_27

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…Some work has been devoted to establish a theoretical framework and to propose generic embeddings with good properties concerning the distortion introduced by the embedding (see [11] for a good review). Some of these ideas have been applied to trees [12] and graphs [13] in the context of image categorization. The application of the generic framework of embedding to graphs permits to convert graphs into points in any vector space.…”

Section: Introductionmentioning

confidence: 99%

A generic framework for median graph computation based on a recursive embedding approach

Ferrer

Karatzas

Valveny

et al. 2011

Computer Vision and Image Understanding

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The median graph has been shown to be a good choice to obtain a representative of a set of graphs. However, its computation is a complex problem. Recently, graph embedding into vector spaces has been proposed to obtain approximations of the median graph. The problem with such an approach is how to go from a point in the vector space back to a graph in the graph space. The main contribution of this paper is the generalization of this previous method, proposing a generic recursive procedure that permits to recover the graph corresponding to a point in the vector space, introducing only the amount of approximation inherent to the use of graph matching algorithms. In order to evaluate the proposed method, we compare it with the set median and with the other state-of-the-art embedding-based methods for the median graph computation. The experiments are carried out using four different databases (one semi-artificial and three containing real-world data). Results show that with the proposed approach we can obtain better medians, in terms of the sum of distances to the training graphs, than with the previous existing methods.

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Section: Introductionmentioning

confidence: 99%

A generic framework for median graph computation based on a recursive embedding approach

Ferrer

Karatzas

Valveny

et al. 2011

Computer Vision and Image Understanding

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“…Let ε ∈ (0, 1) be given, let n be sufficiently large, and let [6]. Unweighted trees [2] and, more generally, unweighted outerplanar graphs [3] embed in R 2 with distortion O(n 1/2 ). On the other hand, there exist unweighted planar graphs for which c R 2 is Ω(n 2/3 ) [3].…”

Section: Introductionmentioning

confidence: 99%

Inapproximability for metric embeddings into $\mathbb{R}^{d}$

Matoušek

Sidiropoulos

2010

Trans. Amer. Math. Soc.

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Abstract. We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into R d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1/12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1/(22d−10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Väisälä).For d ≥ 3, we obtain a stronger inapproximability result by a different reduction: assuming P =NP, no polynomial-time algorithm can distinguish between spaces embeddable in R d with constant distortion from spaces requiring distortion at least n c/d , for a constant c > 0. The exponent c/d has the correct order of magnitude, since every n-point metric space can be embedded in R d with distortion O(n 2/d log 3/2 n) and such an embedding can be constructed in polynomial time by random projection.For d = 2, we give an example of a metric space that requires a large distortion for embedding in R 2 , while all not too large subspaces of it embed almost isometrically.

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Low-Distortion Embeddings of Trees

Cited by 2 publications

References 6 publications

A generic framework for median graph computation based on a recursive embedding approach

A generic framework for median graph computation based on a recursive embedding approach

Inapproximability for metric embeddings into $\mathbb{R}^{d}$

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