Optimal realisations of two-dimensional, totally-decomposable metrics (original) (raw)
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Journal of Classification, 2007
Let G = (V, E, w) be a graph with vertex and edge sets V and E, respectively, and w : E → IR + a function which assigns a positive weigth or length to each edge of G. G is called a realization of a finite metric space (M, d), with M = {1, ..., n} if and only if {1, ..., n} ⊆ V and d(i, j) is equal to the length of the shortest chain linking i and j in G ∀i, j = 1, ..., n. A realization G of (M, d), is said optimal if the sum of its weights is minimal among all the realizations of (M, d). Consider a partition of M into two nonempty subsets K and L, and let e be an edge in a realization G of (M, d); we say that e is a bridge linking K with L if e belongs to all chains in G linking a vertex of K with a vertex of L. The Metric Bridge Partition Problem is to determine if the elements of a finite metric space (M, d) can be partitioned into two nonempty subsets K and L such that all optimal realizations of (M, d) contain a bridge linking K with L. We prove in this paper that this problem is polynomially solvable. We also describe an algorithm that constructs an optimal realization of (M, d) from optimal realizations of (K, d| K ) and (L, d| L ).
Metric graph theory and geometry: a survey
Surveys on Discrete and Computational Geometry, 2008
The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.
Optimal Embeddings of Finite Metric Spaces Into Graphs
Anadolu Üniversitesi Bilim Ve Teknoloji Dergisi - B Teorik Bilimler, 2015
We consider the embedding of a finite metric space into a weighted graph in such a way that the total weight of the edges is minimal. We discuss metric spaces with = 3,4,5 points in detail and show that the already known classification for these cases can be obtained by simple operations on the associated graph of the given metric space.
Injective optimal realizations of finite metric spaces
Discrete Mathematics, 2012
A realization of a finite metric space (X, d) is a weighted graph (G, w) whose vertex set contains X such that the distances between the elements of X in G correspond to those given by d. Such a realization is called optimal if it has minimal total edge weight. Optimal realizations have applications in fields such as phylogenetics, psychology, compression software and internet tomography. Given an optimal realization (G, w) of (X, d), there always exist certain ''proper'' maps from the vertex set of G into the so-called tight span of d. In [A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984) 321-402], Dress conjectured that any such map must be injective. Although this conjecture was recently disproven, in this paper we show that it is possible to characterize those optimal realizations (G, w) for which certain generalizations of proper maps-that map the geometric realization of (G, w) into the tight span instead of its vertex set-must always be injective. We also prove that these ''injective'' optimal realizations always exist, and show how they may be constructed from non-injective ones. Ultimately it is hoped that these results will contribute towards developing new ways to compute optimal realizations from tight spans.
On the metric dimension of few network sheets
Let M = {v1, v2,…, vn} be an ordered set of vertices in a graph G(V,E). Then (d(u, v1), d(u, v2),...d(u, vn)) is called the M-coordinates of a vertex u of G. The set M is called a resolving set if the vertices of G have distinct M-coordinates. A metric basis is a resolving set M with minimum cardinality. If M is a metric basis then it is clear that for each pair of vertices u and v in the set of vertices V of G not in M, there is a vertex m in M such that the distance between u and m is not equal to the distance between v and m. The cardinality of a metric basis of G is called metric dimension. The members of a metric basis are called landmarks. A metric dimension problem is to find a metric basis. The problem of finding metric dimension is an NP-Complete for general graphs. In this paper we have studied the metric dimension of a new graph called Octo–Nano windows , HDN like networks namely Equilateral Triangular Tetra sheets and Rectangular Tetra Sheet networks.
Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs
Discrete & Computational Geometry, 1998
The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any n-point metric space can be represented by a (weighted) graph with n vertices and n 1+O(1/r) edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g/4 − 3 2. In the special case when |V (G)| = |V (H)| and G has strictly less edges than H , an improved bound of g/3 − 1 is shown. In addition, we discuss the case χ(G) < χ(H) − 1, as well as some interesting higher-dimensional analogues. The proofs employ basic techniques of algebraic topology.
Near-Optimal Compression for the Planar Graph Metric
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, 2018
The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k 2 log n) bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k = √ n, the upper bound is O(n) and their constructions imply an Ω(n 3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric intoÕ(min(k 2 , √ k • n)) bits, which is optimal up to log factors. Our data structure circumvents an Ω(k 2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs withÕ(√ k • n) space, andÕ(n 3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n 1/2) vs. Ω(n 1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.
On the structure of the tight-span of a totally split-decomposable metric
European Journal of Combinatorics, 2006
The tight-span of a finite metric space is a polytopal complex with a structure that reflects properties of the metric. In this paper we consider the tight-span of a totally split-decomposable metric. Such metrics are used in the field of phylogenetic analysis, and a better knowledge of the structure of their tight-spans should ultimately provide improved phylogenetic techniques. Here we prove that a totally split-decomposable metric is cell-decomposable. This allows us to break up the tight-span of a totally split-decomposable metric into smaller, easier to understand tight-spans. As a consequence we prove that the cells in the tight-span of a totally split-decomposable metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra.