Separator-Based Graph Embedding into Higher-Dimensional Grids with Small Congestion

金沢大学理工研究域電子情報学系We study the problem of embedding a guest graph into an optimally-sized grid with minimum edge-congestion. Based on a wellknown notion of graph separator, we prove that any guest graph can be embedded with a smaller edge-congestion as the guest graph has a smaller separator, and as the host grid has a higher dimension. Our results imply the following: An N-node planar graph with maximum node degree Δ can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ2 logN) if d = 2, O(Δ2 log logN) if d = 3, and O(Δ2) otherwise. An N-node graph with maximum node degree Δ and a treewidth O(1), such as a tree, an outerplanar graph, and a series-parallel graph, can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ) for d ≥ 2

Separator-based graph embedding into higher-dimensional grids with small congestion

金沢大学理工研究域電子情報学系We study the problem of embedding a guest graph into an optimally-sized grid with minimum edge-congestion. Based on a wellknown notion of graph separator, we prove that any guest graph can be embedded with a smaller edge-congestion as the guest graph has a smaller separator, and as the host grid has a higher dimension. Our results imply the following: An N-node planar graph with maximum node degree Δ can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ2 logN) if d = 2, O(Δ2 log logN) if d = 3, and O(Δ2) otherwise. An N-node graph with maximum node degree Δ and a treewidth O(1), such as a tree, an outerplanar graph, and a series-parallel graph, can be embedded into an N-node d-dimensional grid with an edge-congestion of O(Δ) for d ≥ 2

Expansion of layouts of complete binary trees into grids

AbstractLet Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if |x1−x2|+|y1−y2|=1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th)=n(T)/(2h+1−1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n×n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1−1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals

Separator-based graph embedding into multidimensional grids with small edge-congestion

We study the problem of embedding a guest graph with minimum edge-congestion into a multidimensional grid with the same size as that of the guest graph. Based on a well-known notion of graph separators, we show that an embedding with a smaller edge-congestion can be obtained if the guest graph has a smaller separator, and if the host grid has a higher but constant dimension. Specifically, we prove that any graph with NN nodes, maximum node degree ΔΔ, and with a node-separator of size ss, where ss is a function such that s(n)=O(nα)s(n)=O(nα) with 0≤α1/(1−α)d>1/(1−α), O(ΔlogN)O(ΔlogN) if d=1/(1−α)d=1/(1−α), and View the MathML sourceO(ΔNα−1+1d) if d1/(1−α)d>1/(1−α), and matches an existential lower bound within a constant factor if d≤1/(1−α)d≤1/(1−α). Our result implies that if the guest graph has an excluded minor of a fixed size, such as a planar graph, then we can obtain an edge-congestion of O(ΔlogN)O(ΔlogN) for d=2d=2 and O(Δ)O(Δ) for any fixed d≥3d≥3. Moreover, if the guest graph has a fixed treewidth, such as a tree, an outerplanar graph, and a series–parallel graph, then we can obtain an edge-congestion of O(Δ)O(Δ) for any fixed d≥2d≥2. To design our embedding algorithm, we introduce edge-separators bounding extension , such that in partitioning a graph into isolated nodes using edge-separators recursively, the number of outgoing edges from a subgraph to be partitioned in a recursive step is bounded. We present an algorithm to construct an edge-separator with extension of O(Δnα)O(Δnα) from a node-separator of size O(nα)O(nα)

Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the successful rst edition of this workshop in 20061 and second edition in New-York in 20082, the third edition was held in Toronto on September 22 20113. Contributions were solicited in Riemannian and group theoretical methods, geometric measurements of the anatomy, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, modeling of growth and longitudinal shape changes. 22 submissions were reviewed by three members of the program committee. To guaranty a high level program, 11 papers only were selected for oral presentation in 4 sessions. Two of these sessions regroups classical themes of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session of 5 papers addresses more application oriented works on computational anatomy

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

Estimating functional performance for use in the aesthetic design process

In engineering fields such as automobile design, optimisation of functional performance properties often conflicts with aesthetic optimisation. Functional performance feedback into the aesthetic design software may therefore improve the convergence of the design process. Unfortunately, many functional performance scores such as aerodynamic drag require intensive computational effort. We consider the use of machine learning approaches to instead provide estimates of these functional performance scores. We study the problems encountered when developing such an estimation function. The use of a historically accumulated data set of STereoLithography-format designs and their performance scores is suggested. We first look at preparing such a data set as training data for a machine learning task. Our first major novel contribution combats this problem in a manner similar to voxelisation. We next look at generating the regression function, seeking to achieve good generalisation across a large space of possible designs and for a problem where dimensionality reduction is challenging. Our second major novel contribution deals with this problem using an ensemble regression framework incorporating multiple data representations. Finally, we look at strategies of combining these two novel systems into a complete system. Upon evaluation, we conclude that our original aims have been met by this complete system