Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries

Naoki Katoh; Shin-ichi Tanigawa

doi:10.1137/110846944

Correspondence between Multilevel Graph Partitions and Tree Decompositions

Algorithms ◽

10.3390/a12090198 ◽

2019 ◽

Vol 12 (9) ◽

pp. 198

Author(s):

Michael Hamann ◽

Ben Strasser

Keyword(s):

Knowledge Transfer ◽

Rooted Tree ◽

Graph Partitions ◽

Significant Research ◽

Tree Decompositions

We present a mapping between rooted tree decompositions and node separator based multilevel graph partitions. Significant research into both tree decompositions and graph partitions exists. We hope that our result allows for an easier knowledge transfer between the two research avenues.

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HTPC: heterogeneous traffic-aware partition coding for random packet spraying in data center networks

Journal of Cloud Computing Advances Systems and Applications ◽

10.1186/s13677-021-00248-4 ◽

2021 ◽

Vol 10 (1) ◽

Author(s):

Jiawei Huang ◽

Shiqi Wang ◽

Shuping Li ◽

Shaojun Zou ◽

Jinbin Hu ◽

...

Keyword(s):

Data Center ◽

Large Scale ◽

Network Performance ◽

Rooted Tree ◽

Heterogeneous Traffic ◽

Data Center Networks ◽

Packet Reordering ◽

Traffic Characteristics ◽

Network Utilization ◽

The Impact

AbstractModern data center networks typically adopt multi-rooted tree topologies such leaf-spine and fat-tree to provide high bisection bandwidth. Load balancing is critical to achieve low latency and high throughput. Although the per-packet schemes such as Random Packet Spraying (RPS) can achieve high network utilization and near-optimal tail latency in symmetric topologies, they are prone to cause significant packet reordering and degrade the network performance. Moreover, some coding-based schemes are proposed to alleviate the problem of packet reordering and loss. Unfortunately, these schemes ignore the traffic characteristics of data center network and cannot achieve good network performance. In this paper, we propose a Heterogeneous Traffic-aware Partition Coding named HTPC to eliminate the impact of packet reordering and improve the performance of short and long flows. HTPC smoothly adjusts the number of redundant packets based on the multi-path congestion information and the traffic characteristics so that the tailing probability of short flows and the timeout probability of long flows can be reduced. Through a series of large-scale NS2 simulations, we demonstrate that HTPC reduces average flow completion time by up to 60% compared with the state-of-the-art mechanisms.

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Hierarchical Analysis of Nucleotide Diversity in Geographically Structured Populations

Genetics ◽

10.1093/genetics/142.2.629 ◽

1996 ◽

Vol 142 (2) ◽

pp. 629-639 ◽

Cited By ~ 4

Author(s):

Kent E Holsinger ◽

Roberta J Mason-Gamer

Keyword(s):

Nucleotide Diversity ◽

Restriction Site ◽

Rooted Tree ◽

Structured Populations ◽

Hierarchical Analysis ◽

Restriction Site Variation ◽

Hierarchical Levels ◽

Site Variation ◽

The Mean ◽

Mean Time

Abstract Existing methods for analyzing nucleotide diversity require investigators to identify relevant hierarchical levels before beginning the analysis. We describe a method that partitions diversity into hierarchical components while allowing any structure present in the data to emerge naturally. We present an unbiased version of Nei's nucleotide diversity statistics and show that our modification has the same properties as Wright's F ST. We compare its statistical properties with several other F ST estimators, and we describe how to use these statistics to produce a rooted tree of relationships among the sampled populations in which the mean time to coalescence of haplotypes drawn from populations belonging to the same node is smaller than the mean time to coalescence of haplotypes drawn from populations belonging to different nodes. We illustrate the method by applying it to data from a recent survey of restriction site variation in the chloroplast genome of Coreopsis grandiflora.

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AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

International Journal of Foundations of Computer Science ◽

10.1142/s0129054100000247 ◽

2000 ◽

Vol 11 (03) ◽

pp. 365-371 ◽

Cited By ~ 33

Author(s):

LJUBOMIR PERKOVIĆ ◽

BRUCE REED

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Tree Decomposition ◽

Linear Time Algorithm ◽

Input Graph ◽

Primary Motivation ◽

Small Width ◽

Tree Decompositions ◽

Improved Algorithm

We present a modification of Bodlaender's linear time algorithm that, for constant k, determine whether an input graph G has treewidth k and, if so, constructs a tree decomposition of G of width at most k. Our algorithm has the following additional feature: if G has treewidth greater than k then a subgraph G′ of G of treewidth greater than k is returned along with a tree decomposition of G′ of width at most 2k. A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n2) time. This is the primary motivation of this paper.

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Tree decompositions with small cost

Discrete Applied Mathematics ◽

10.1016/j.dam.2004.01.008 ◽

2005 ◽

Vol 145 (2) ◽

pp. 143-154 ◽

Cited By ~ 12

Author(s):

Hans L. Bodlaender ◽

Fedor V. Fomin

Keyword(s):

Tree Decompositions

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Amenability and non-uniform growth of some directed automorphism groups of a rooted tree

Mathematische Zeitschrift ◽

10.1007/s00209-008-0417-3 ◽

2008 ◽

Vol 263 (2) ◽

pp. 265-293 ◽

Cited By ~ 8

Author(s):

Jérémie Brieussel

Keyword(s):

Automorphism Groups ◽

Rooted Tree ◽

Uniform Growth

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Two Short Proofs Concerning Tree-Decompositions

Combinatorics Probability Computing ◽

10.1017/s0963548302005369 ◽

2002 ◽

Vol 11 (6) ◽

pp. 541-547 ◽

Cited By ~ 22

Author(s):

PATRICK BELLENBAUM ◽

REINHARD DIESTEL

Keyword(s):

Duality Theorem ◽

Finite Graph ◽

Tree Decomposition ◽

Tree Width ◽

Tree Decompositions

We give short proofs of the following two results: Thomas's theorem that every finite graph has a linked tree-decomposition of width no greater than its tree-width; and the ‘tree-width duality theorem’ of Seymour and Thomas, that the tree-width of a finite graph is exactly one less than the largest order of its brambles.

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Optimal Flow Control and Single Split Architecture Exploration for Fluid-Based Thermal Management

Volume 2A: 44th Design Automation Conference ◽

10.1115/detc2018-86148 ◽

2018 ◽

Cited By ~ 2

Author(s):

Satya R. T. Peddada ◽

Daniel R. Herber ◽

Herschel C. Pangborn ◽

Andrew G. Alleyne ◽

James T. Allison

Keyword(s):

Flow Control ◽

Thermal Management ◽

High Performance ◽

Dynamic Models ◽

Cooling System ◽

Rooted Tree ◽

Modeling Framework ◽

System Architectures ◽

Tree Graphs ◽

Heat Loads

High-performance cooling is often necessary for thermal management of high power density systems. Both human intuition and vast experience may not be adequate to identify optimal thermal management designs as systems increase in size and complexity. This paper presents a design framework supporting comprehensive exploration of a class of single phase fluid-based cooling architectures. The candidate cooling system architectures are represented using labeled rooted tree graphs. Dynamic models are automatically generated from these trees using a graph-based thermal modeling framework. Optimal performance is determined by solving an appropriate fluid flow control problem, handling temperature constraints in the presence of exogenous heat loads. Rigorous case studies are performed in simulation, with components having variable sets of heat loads and temperature constraints. Results include optimization of thermal endurance for an enumerated set of 4,051 architectures. In addition, cooling system architectures capable of steady-state operation under a given loading are identified.

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