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Extrinsic calibration of large-scale ad hoc networks of cameras is posed as the following problem: Calculate the locations of N mobile, rotationally aligned cameras distributed over an urban region, subsets of which view some common environmental features. We show that this leads to a novel class of graph embedding problems that admit closed-form solutions in linear time via partial spectral decomposition of a quadratic form. The minimum squared error (mse) solution determines locations of cameras and/or features in any number of dimensions. The spectrum also indicates insufficiently constrained problems, which can be decomposed into well-constrained rigid subproblems and analyzed to determine useful new views for missing constraints. We demonstrate the method with large networks of mobile cameras distributed over an urban environment, using directional constraints that have been extracted automatically from commonly viewed features. Spectral solutions yield layouts that are consistent in some cases to a fraction of a millimeter, substantially improving the state of the art. Global layout of large camera networks can be computed in a fraction of a second.

We investigate the computational complexity of determining if a graph G on v vertices embeds in a surface S. Robertson and Seymour have given an O(v 3) decision algorithm for this embedding problem. We show here how the use the yes/no output from their algorithm to construct the embedding, that is, we self-reduce the search algorithm to the decision algorithm. We conclude that for each fixed surface S there exists an O(v 10) algorithm for constructing an embedding or answering that no embedding exists.

It is proved that any n-vertex, k-valent undirected simple graph, G, contains a spanning tree with at least $$\frac{{{\text{(}}k - {\text{2)}}n}}{{{\text{(5}}k - {\text{8}}{\text{.5)}}}}$$ leaves. As a result of this it is shown that any such graph contains an independent set, of size at least n/5k, with the property that deleting the vertices in this set and their incident edges does not disconnect G. This latter result is applied by giving an improved upper bound on the area required to embed arbitrary graphs into grid graphs.

Wireless sensor location estimation is an important area and attracts considerable research interests. In this paper, we present a novel graph embedding method for the localization problem by using signal strengths. We view the wireless sensor nodes as a group of distributed devices, and employ an appropriate kernel function to measure the similarity between sensors. The kernel function can be naturally defined according to the signal strength matrix. Then we formulate the localization problem as a graph embedding problem. Finally, we use the kernel locality preserving projection (KLPP) technique to estimate the relative locations of all sensor nodes. Given sufficient number of anchors, the relative locations can be transformed into physical locations. The main advantage of formulating the localization problem as graph embedding problem is that it allows us to construct a graph to preserve the topological structure of the sensor networks. We evaluate our method based on various network topologies, and analyze its performance. We also compare our method with several existing methods, and demonstrate the high efficiency of our proposed method.

An important feature of an interconnection network is its ability to efficiently simulate one architecture by another. Such a simulation problem can be mathematically formulated as a graph embedding problem. Although the definition of an embedding is an into mapping from Guest Graph to Host Graph, so far in the literature, the embedding has been considered as a mapping from G onto H. In other words, the number of processors in G and H are considered to be the same. In this paper, we increase the number of processors in H by 1. The question is to find the processor in H which does not have the pre-image under the embedding mapping, so that the wirelength of the embedding is minimum. We relate this problem to finding transmission of vertices in the host graph.

Recently, learning embedding of nodes in graphs has attracted increasing research attention. There are two main kinds of graph embedding methods, i.e., the transductive embedding methods and the inductive embedding methods. The former focuses on directly optimizing the embedding vectors, and the latter tries to learn a mapping function for the given nodes and features. However, few works focus on applying the learned model from one graph to another, which is a pervasive idea in Computer Version or Natural Language Processing. Although some of the graph neural networks (GNNs) present similar motivation, none of them considers the graph bias among graphs. In this paper, we present an interesting graph embedding problem called Adaptive Task (AT), and propose a unified framework for this adaptive task, which introduces two types of alignment to learn adaptive node embedding across graphs. Then, based on the proposed framework, a novel graph adaptive embedding network is designed to address the adaptive task. Extensive experimental results demonstrate that our model significantly outperforms the state-of-the-art methods.

Programmable software switches have become crucial building blocks in a wide range of applications, from (virtual) data center networking to telco clouds. Modern use-cases, such as 5G, require very low latency, large throughput, and high availability. Software switches generally realize the packet processing pipeline as a dataflow graph: graph nodes correspond to simple packet processing operations and graph edges represent the control flow. The efficiency and dependability of the software switch critically depends on the way the dataflow graph is mapped to the underlying hardware resources. In this paper, we focus on dataflow graph embedding in a software switching context. We present an embedding approach which minimizes performance loss on inter-CPU communication across packet-processing control flow chains, and is resilient against a single CPU failure. The embedding is easy to generalize to $N$ CPU failures. We formulate the dataflow graph embedding problem as a mathematical program, characterize the computational complexity, and we propose optimal and heuristic algorithms to solve it. The viability of our approach is confirmed in comprehensive numerical analysis on a 5G packet processing pipeline, taken from an industrial 5G NFV benchmark suite.

Metric learning with neural networks has exhibited promising improvements in representation learning. Yet cross-modal retrieval poses a unique challenge to metric learning: how to compute the distance across different modalities such as image and text. Existing neural network based methods tend to establish two branches for images and texts respectively to bridge the modal gap. Also, most of them cannot fully exploit the structure embedded in the multimodal data. This paper introduces embedding layer to provide cross-modal shared representation with non-linearity and reformulates the cross-modal retrieval problem as a graph embedding problem by constructing a multimodal graph. To learn the graph embedding, training pairs and triplets are uniformly generated from random walk sequences on the graph. Then graph pair and triplet constraints are imposed on the embedding layer for structure preservation. Meanwhile, a classifier is trained with labeled data to ensure the learned embedding is coupled with semantic information. For optimization, graph pair and triplet constraints are integrated into a unified multi-task learning with the supervised classifier. Experimental results on the Wiki and NUS-WIDE datasets demonstrate the effectiveness and superiority of the learned embedding for cross-modal retrieval.

Recently, one has seen a surge of interest in developing such methods including ones for learning such representations for (undirected) graphs (while preserving important properties). However, most of the work to date on embedding graphs has targeted undirected networks and very little has focused on the thorny issue of embedding directed networks. In this paper, we instead propose to solve the directed graph embedding problem via a two-stage approach: in the first stage, the graph is symmetrized in one of several possible ways, and in the second stage, the so-obtained symmetrized graph is embedded using any state-of-the-art (undirected) graph embedding algorithm. Note that it is not the objective of this paper to propose a new (undirected) graph embedding algorithm or discuss the strengths and weaknesses of existing ones; all we are saying is that whichever be the suitable graph embedding algorithm, it will fit in the above proposed symmetrization framework., Comment: has been accepted to The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI 2019) Student Abstract and Poster Program

Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful applications such as node classification, node recommendation, link prediction, etc. However, most graph analytics methods suffer the high computation and space cost. Graph embedding is an effective yet efficient way to solve the graph analytics problem. It converts the graph data into a low dimensional space in which the graph structural information and graph properties are maximally preserved. In this survey, we conduct a comprehensive review of the literature in graph embedding. We first introduce the formal definition of graph embedding as well as the related concepts. After that, we propose two taxonomies of graph embedding which correspond to what challenges exist in different graph embedding problem settings and how the existing work address these challenges in their solutions. Finally, we summarize the applications that graph embedding enables and suggest four promising future research directions in terms of computation efficiency, problem settings, techniques and application scenarios., A 20-page comprehensive survey of graph/network embedding for over 150+ papers till year 2018. It provides systematic categorization of problems, techniques and applications. Accepted by IEEE Transactions on Knowledge and Data Engineering (TKDE). Comments and suggestions are welcomed for continuously improving this survey

Virtualized datacenters offer great flexibilities in terms of resource allocation. In particular, by decoupling applications from the constraints of the underlying infrastructure, virtualization supports an optimized mapping of virtual machines as well as their interconnecting network (the so-called virtual cluster ) to their physical counterparts: a graph embedding problem. However, existing virtual cluster embedding algorithms often ignore a crucial dimension of the problem, namely data locality : the input to a cloud application such as MapReduce is typically stored in a distributed, and sometimes redundant, file system. Since moving data is costly, an embedding algorithm should be data locality aware, and allocate computational resources close to the data; in case of redundant storage, the algorithm should also optimize the replica selection . This paper initiates the algorithmic study of data locality aware virtual cluster embeddings on datacenter topologies. We show that despite the multiple degrees of freedom in terms of embedding, replica selection and assignment, many problems can be solved efficiently. We also highlight the limitations of such optimizations, by presenting several NP-hardness proofs; interestingly, our hardness results also hold in uncapacitated networks of small diameter.

Virtualized datacenters offer great flexibilities in terms of resource allocation. In particular, by decoupling applications from the constraints of the underlying infrastructure, virtualization supports an optimized mapping of virtual machines as well as their interconnecting network to their physical counterparts: essentially a graph embedding problem. However, existing embedding algorithms such as Oktopus and Proteus often ignore a crucial dimension of the embedding problem, namely data locality: the input to a cloud application such as MapReduce is typically stored in a distributed, and sometimes redundant, file system. Since moving data is costly, an embedding algorithm should be data locality aware, and allocate computational resources close to the data, in case of redundant storage, the algorithm should also optimize the replica selection. This paper initiates the algorithmic study of data locality aware virtual cluster embeddings on datacenter topologies. We show that despite the multiple degrees of freedom in terms of embedding, replica selection and assignment, many problems can be solved efficiently. We also highlight the limitations of such optimizations, by presenting several NP-hardness proofs, interestingly, our hardness results also hold in uncapacitated networks of small diameter.

Research in VLSI placement, an NP-hard problem, has branched in two different directions. The first one employs iterative heuristics with many tunable parameters to produce a near-optimal solution but without theoretical guarantee on its quality. The other one considers placement as a graph-embedding problem and designs approximation algorithms with provable bounds on the quality of the solution. In this article, we aim at unifying the above two directions. First, we extend the existing approximation algorithms for graph embedding in 1D and 2D grid to those for hypergraphs, which typically model circuits to be placed on a FPGA. We prove an approximation bound of O ( d √log n log log n ) for 1D, that is, linear arrangement and O ( d log n log log n ) for the 2D grid, where d is the maximum degree of hyperedges and n , the number of vertices in the hypergraph. Next, we propose an efficient method based on linear arrangement of the CLBs and the notion of space-filling curves for placing the configurable logic blocks (CLBs) of a netlist on island-style FPGAs with an approximation guarantee of O ( 4 √log n √ kd log log n ), where k is the number of nets. For the set of FPGA placement benchmarks, the running time is near linear in the number of CLBs thus allowing for scalability towards large circuits. We obtained a 33× speed-up, on average, with only 1.31× degradation in the quality of the solution compared to that produced by the popular FPGA tool VPR, thereby demonstrating the suitability of this very fast method for FPGA placement, with a provable performance guarantee.

A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a geometric buildup approach to the problem. Central to this approach is the idea that the coordinates of the atoms in a protein can be determined one atom at a time, with the distances from the determined atoms to the undetermined ones. It can determine a structure more efficiently than other conventional approaches, yet without requiring more distance constraints than necessary.

The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the extension graph of $\Gamma$. They show that each finite induced subgraph $\Delta$ of $\Gamma^e$ gives rise to an embedding between the partially commutative groups $G(\Delta)$ and $G({\Gamma})$. Furthermore, it is proven that in many instances the converse also holds. Our first result is the decidability of the Extension Graph Embedding Problem: there is an algorithm that given two finite simplicial graphs {\Delta} and {\Gamma} decides whether or not $\Delta$ is an induced subgraph of $\Gamma^e$. As a corollary we obtain the decidability of the Embedding Problem for 2-dimensional partially commutative groups. In the second part of the paper, we relate the Embedding Problem between partially commutative groups to the model-theoretic question of classification up to universal equivalence. We use our characterisation to transfer algebraic and algorithmic results on embeddability to model-theoretic ones and obtain some rigidity results on the elementary theory of atomic pc groups as well as to deduce the existence of an algorithm to decide if an arbitrary pc group is universally equivalent to a 2-dimensional one., Comment: 40 pages in International Mathematical Research Notices 2015

In this paper, we consider a graph-theoretic problem,, the so-called "graph embedding problem" that maps the vertices and edges of the given guest graph model into the corresponding vertices and paths of the host graph under the condition of maintaining better performance parameters such as dilation, congestion, and expansion. We firstly propose a new mapping function which can embed the pyramid model with height N into the 3-dimensional mesh massively parallel processor system with the height and the regular 2-dimensional mesh of one side , and analyze the performance of the embedding in terms of the dilation parameter that reflects the number of communication steps between two adjacent vertices under the embedding. We prove that the dilation of the embedding is . This is superior to the previous result of under the same condition.condition.

This paper addresses the problem of state assignment for finite-state machines (FSM). This is an important problem in digital system design where added functionality often comes at the expense of a larger (and slower) FSM to control the system. We present a new scheme to solve the graph embedding problem which is the main step in the state assignment process. This new approach places the graph in a two-dimensional array (grid) while minimizing the total edge length, and then maps this two-dimensional array into an n-dimensional hypercube with dilation of at most 2. Experimental results are presented and compared against those of NOVA. These results demonstrate the effectiveness of the proposed approach.

International audience; Defining similarities or distances between graphs is one of the bases of the structural pattern recognition field. An important trend within this field consists in going beyond the simple formulation of simi- larity measures by studying properties of graph's spaces induced by such distance or similarity measures . Such a problematic is closely related to the graph embedding problem. In this article, we investigate two types of similarity measures. The first one is based on the notion of graph edit distance which aims to catch a global dissimilarity between graphs. The second family is based on comparisons of bags of patterns extracted from graphs to be compared. Both approaches are detailed and their performances are evaluated on different chemoinformatics problems.

Recently discriminative based appearance model has achieved great success in modeling the object. It considers the tracking as the binary classification problem to separate the object from the background. However it neglects the object appearance and the object properties, it will encounter problems when the object lacks the features to separate it from the background. In this paper, we compensate the discriminative model with the object appearance through a unified graph-based constraint embedding framework. Volterra kernels approximation and log-Riemannian transformation are introduced to transform the discriminative model and object subspace learning problem into a graph embedding problem. Through the unified graph-based constraint embedding framework, the discriminative model is enhanced with the help of the object appearance. The effectiveness of our proposed approach is demonstrated in various experiments and quantitative evaluation using several challenging sequences.

This paper outlines a general method for allocating a document collection over a distributed storage system. Documents are organized to make up a corporate semantic web featured by a graph G 1 , each of whose nodes represents a set of documents having a common range of semantic indices. There exists a second graph G 2 modelling the distributed storage system. Our approach consists of embedding G 1 into G 2 , under size restrictions. We use a meta-heuristic called "Ant Colony Optimization", to solve the corresponding instances of the graph embedding problem, which is known to be a NP problem. Our solution provides an efficient mechanism for information storage and retrieval.

LPI is optimal in the sense of local manifold structure. However, LPI is not efficient in time and memory which makes it difficult to be applied to very large data set. In this paper, we propose a optimal algorithm called FLPI. FLPI decomposes the LPI problem as a graph embedding problem plus a regularized least squares problem. Such modification avoids eigen decomposition of dense matrices and can significantly reduce both time and memory cost in computation. Moreover, with a specifically designed graph in supervised situation, LPI only needs to solve the regularized least squares problem which is a further saving of time and memory. Real data experimental results show that FLPI obtains similar or better results comparing to LPI and it is significantly faster.

LPI is not efficient in time and memory which makes it difficult to be applied to very large data set. We propose a optimal algorithm called FLPI. FLPI decomposes the LPI problem as a graph embedding problem plus a regularized least squares problem. Such modification avoids eigen decomposition of dense matrices and can significantly reduce both time and memory cost in computation. Moreover, with a specifically designed graph in supervised situation, LPI only needs to solve the regularized least squares problem which is a further saving of time and memory. Real data experimental results show that FLPI obtains similar or better results comparing to LPI.

LPI is not efficient in time and memory which makes it difficult to be applied to very large data set. In this paper, we propose a optimal algorithm called FLPI which decomposes the LPI problem as a graph embedding problem plus a regularized least squares problem. Such modification avoids eigen decomposition of dense matrices and can significantly reduce both time and memory cost in computation. Moreover, with a specifically designed graph in supervised situation, LPI only needs to solve the regularized least squares problem which is a further saving of time and memory. Real and synthetic data experimental results show that FLPI obtains similar or better results comparing to LPI and it is significantly faster.

We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a long-standing open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE.

In this paper, we study routing and wavelength assignment (RWA) for realizing hypercube communications on a family of WDM optical double-loop networks, chordal rings of degree 4. We design embedding schemes and derive the numbers of wavelengths required for different chord lengths. We also provide the analysis of chord lengths with minimum number of wavelengths to realize hypercube communications on chordal ring of degree 4. Results show that the wavelength requirement for realizing hypercube communications on optical networks has been further reduced on optical double-loop networks compared with some topologies discussed before. Our embedding approaches also take a new view on graph embedding problem for hypercube graphs on double-loop networks.

Mapping virtual processes onto physical processos is one of the most important issues in parallel computing. The problem of mapping of processes/tasks onto processors is equivalent to the graph embedding problem which has been studied extensively. Although many techniques have been proposed for embeddings of two-dimensional grids, hypercubes, etc., there are few efforts on embeddings of three-dimensional grids and tori. Motivated for better support of task mapping for Blue Gene/L supercomputer, in this paper, we present embedding and integration techniques for the embeddings of three-dimensional grids and tori. The topology mapping library that based on such techniques generates high-quality embeddings of two/three-dimensional grids/tori. In addition, the library is used in BG/L MPI library for scalable support of MPI topology functions. With extensive empirical studies on large scale systems against popular benchmarks and real applications, we demonstrate that the library can significantly improve the communication performance and the scalability of applications.

Analyses the graph embedding problem. The major difficulty in dealing with the problem is that the embedding of a biconnected graph is not unique (the order of vertices in the ordered adjacency list is not unique and the vertex sets constituting face boundaries are not unique) and there can be many ways to embed graphs especially those arising in IC layout. The embedding is a graph model of a layout where some elements and connecting lines should be placed apart on the plane. The author presents an algorithm for obtaining an embedding which satisfies the condition that no two vertices in the given vertex subset U appear on the same face boundary. This algorithm is based on the vertex addition algorithm for planarity testing and uses a data structure called a PQ-tree. >

We consider the problem of how to allocate the tasks of a parallel program to the processor elements of a multicomputer system such that the communication overhead of the program is minimized. This problem basically amounts to a graph embedding problem since both the program and the multicomputer can be modeled as graphs. The embedding problem can be characterized as the search for a topology conserving mapping of the source graph to the target graph.

A string p = p0p1...pn-1 of non-negative integers is a Euclidean string if the string (p0 + 1)p1...(pn-1 - 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + ... + pn-1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stem-Brocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.

We have found that a triple helix configuration of points in E3 yields the best value of the Steiner ratio for the Euclidean Steiner Minimal Tree (ESMT) problem. In this paper we explore the properties, configurations, and implications of this topology which yields this best Steiner ratio and its relationship to the Euclidean Graph embedding problem (EGEP) for weighted graphs in E3. The unique equivalence between these problems is also explored in their application for identification and modelling of minimum energy configurations (MECs) such as the biochemical protein structures of Collagen.