Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

Let $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Berlin, Germany, 200

Orderly Spanning Trees with Applications

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Compact Routing on Planar Graphs

This thesis delves into the exploration of shortest path queries in planar graphs, with an emphasis on the utilization of space-efficient data structures. Our investigation primarily targets connected, undirected, static pointer planar graphs, focusing on scenarios where queries predominantly start or end at a select subset of nodes. The shortest path problem, central to our study, boasts a rich historical context and has profound real-world implications in diverse fields such as web mapping, robotics, and VLSI circuit design. Our research is pivoted on the space-efficient representation of planar graphs, a critical consideration in 2D visualizations and city map representations. In this thesis, shortest path queries are delineated into three categories: shortest path, distance oracle, and port queries, each with distinct computational characteristics and storage requirements. A significant portion of our research is focused on center-based configurations in graphs, where a small subset of nodes, designated as ‘centers,’ plays a pivotal role. These centers are crucial, either due to their strategic importance within the graph, which necessitates more prompt responses to queries, or due to their high frequency in the query list. We explore various scenarios within this configuration. Our approach prioritizes handling queries involving these centers more efficiently, aiming to provide rapid responses for strategically important queries and to enhance overall query processing speed. This method is particularly effective, as addressing the queries linked to these relatively few but significant centers can substantially improve the efficiency of the entire system. Such prioritization reflects practical applications like urban navigation, where focusing on key locations can significantly expedite overall navigation and operational efficiency. For shortest path queries in a center-based configuration, we have developed a data structure that efficiently answers queries from other nodes to centers in O(length of the path) time. In the first scenario, where all queries are from or to a center, the space requirement is 3n+2m+2km+o(nk), where n represents the number of nodes, m the number of edges, and k the number of centers. Additionally, our approach supports distributed storage and processing, facilitating parallel computing. For distance oracle queries in unweighted graphs within a center-based configuration, our methods manage responses in O(log^(1+ϵ) n) time with an additional o(nk) space requirement. In general, for unweighted graphs without any specific configuration, the distance oracle requires 2n + 2m + 2nm + o(n) bits of space, offering responses in a similar time frame. The strength of our approach lies in its distributability across multiple servers, which enhances concurrent query processing, a feature particularly beneficial in center-based configurations. Moreover, we introduce a specialized data structure for distributed routing tables, capable of responding to port queries in constant time. This structure efficiently utilizes space, limiting the aggregate bit requirement for all routing tables within graph G to 3.2n^2+o(n^2) bits

On Compact Encoding of Pagenumber kk Graphs

In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;

Small world models and a compact routing scheme

Orientador: Prof. Dr. André Luís VignattiTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 17/10/2018Inclui referências: p.78-82Área de concentração: Ciência da ComputaçãoResumo: Os modelos geradores de redes de mundo pequeno são alternativas aleatorizadas de construção de redes onde os nós estão interligados por caminhos curtos. Nesse contexto, um caminho curto tem o comprimento dado por uma função logarítmica no tamanho da rede. Aplicações em redes peer-to-peer, de sensores sem fio e implementação de redes em chip são alguns exemplos computacionais que utilizam redes geradas por modelos matemáticos de mundo pequeno. Esse trabalho apresenta o modelo de redes de mundo pequeno toroidal não direcionado (UTSW). O modelo gera uma grade bidimensional base para modelar o agrupamento dos nós, e conexões aleatórias para modelar caminhos curtos. A presença de caminhos curtos não implica a existência de um algoritmo de roteamento de mensagens que os encontra. Porém, existe um algoritmo de roteamento guloso na literatura que encontra caminhos curtos em redes UTSW, sendo que um caminho curto tem o comprimento dado por uma função polilogarítmica no tamanho da rede. O modelo de mundo pequeno octaédrico (OSW) também é apresentado. Ele gera um grafo planar base sobre o octaedro, que é provado ser similar à geração sobre esferas. Assim, ele simula pessoas e vizinhanças na superfície do planeta. O modelo OSW também gera conexões aleatórias para modelar caminhos curtos. Um algoritmo de roteamento que encaminha mensagens por caminhos curtos também é definido para esse modelo. Ambos os algoritmos de roteamento utilizam o paradigma guloso e decidem para qual vizinho encaminhar uma mensagem com somente a informação na tabela de roteamento do nó. Esses algoritmos requerem pouca memória por nó, encontram caminhos curtos e executam em tempo constante, nos dois modelos apresentados. Por isso, eles são boas opções em roteamento, além do roteamento por caminhos mínimos. No entanto, ambos necessitam de informações de posicionamento geradas pelo modelo matemático. No caso do modelo UTSW, cada nó possui um par ordenado de números naturais que o posiciona na grade bidimensional. Encontrar essas posições quando elas são desconhecidas é desafiador, pois as conexões geradas aleatoriamente dificultam a identificação da grade. Porém, a conexão aleatória de um nó quebra o padrão de grade em sua vizinhança com baixa probabilidade. Essa propriedade topológica das redes UTSW motivou o projeto de um algoritmo de rotulação que encontra as posições na grade com alta probabilidade. Ele realiza uma busca local em cada nó e remove a conexão aleatória, se ela for identificada. Após a varredura, o algoritmo realiza uma busca em largura global na rede resultante, posicionando cada nó com base nas posições já encontradas de seus vizinhos. Um esquema de roteamento compacto para grafos UTSW é então apresentado. Ele é composto por um algoritmo de pré-processamento que utiliza o algoritmo de rotulação para gerar estruturas de dados de tamanho logarítmico para todos os nós e que executa em tempo linear no tamanho da rede. Com isso, o algoritmo de roteamento guloso encaminha mensagens utilizando as estruturas de dados geradas pelo algoritmo de pré-processamento. Palavras-chave: redes de mundo pequeno. modelos geradores. roteamento guloso. algoritmos de rotulação. esquemas de roteamento compactos.Abstract: Small world networks generative models are randomized alternatives of network building where nodes are interconnected by small paths. In this context, a small path has the length given by a logarithmic function in the size of the network. Applications in peer-to-peer networks, wireless sensors and networks on chip are some computational examples that use networks generated by small world mathematical models. This work presents the undirected toroidal small world networks model (UTSW). The model generates a base two-dimensional grid to model the clustering of the nodes, and random connections to model small paths. The presence of small paths does not imply in the existence of a message routing algorithm that finds them. However, there is a greedy routing algorithm in the literature that finds small paths in UTSW networks, where a small path has the length given by a polylogarithmic function in the size of the network. The octahedral small world model (OSW) is also presented. It generates a base planar graph on the octahedron, which is proved to be similar to the generating on spheres. Then, it simulates people and neighborhoods on the surface of the planet. The OSW model also generates random connections to model small paths. A routing algorithm that routes messages through small paths is also defined for this model. Both routing algorithms use the greedy paradigm and decide to which neighbor to forward a message with only the information in the routing table of the node. These algorithms require small amount of memory per node, find small paths and execute in constant time, in both presented models. So, they are good options in routing, besides the routing thought shortest paths. However, both require positioning information generated by the mathematical model. In the case of UTSW model, each node has an ordered pair of natural numbers that positions it on the two-dimensional lattice. Finding these positions when they are unknown is challenging because the randomly generated connections make the grid identification difficult. However, the random connection of a node breaks the lattice pattern in its neighborhood with small probability. This topological property of UTSW networks motivated the design of a labeling algorithm that finds the positions in the lattice with high probability. It executes a local search on each node and removes the random connection if it is identified. After the sweeping, the algorithm executes a global breadth-first search on the resulting network, positioning each node based on the already found positions of its neighbors. A compact routing scheme for UTSW graphs is then presented. It consists of a preprocessing algorithm that uses the labeling algorithm to generate logarithmic size data structures for all nodes and executes in linear time on the size of the network. Thus, the greedy routing algorithm routes messages using the data structures generated by the preprocessing algorithm. Keywords: small world networks. generative models. greedy routing. labeling algorithms. compact routing schemes

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Hierarchical Routing in Low-Power Wireless Networks

Steen, M.R. van [Promotor