scholarly journals Low-congestion shortcut and graph parameters

2021 ◽  
Author(s):  
Naoki Kitamura ◽  
Hirotaka Kitagawa ◽  
Yota Otachi ◽  
Taisuke Izumi
Keyword(s):  
Lower Bound ◽  
Minimum Spanning Tree ◽  
Hard Core ◽  
Construction Technique ◽  
Graph Problems ◽  
Graph Classes ◽  
Graph Class ◽  
Network Topologies ◽  
Graph Parameters

AbstractDistributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of $${\tilde{\Omega }}(\sqrt{n} + D)$$ Ω ~ ( n + D ) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class $${\mathcal {C}}$$ C , an f-round algorithm for constructing shortcuts of quality q for any instance in $${\mathcal {C}}$$ C results in $${\tilde{O}}(q + f)$$ O ~ ( q + f ) -round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for $${\mathcal {C}}$$ C . The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking $${\tilde{O}}(\sqrt{n}+D)$$ O ~ ( n + D ) -round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

Online Minimum Spanning Tree with Advice

2018 ◽  
Vol 29 (04) ◽  
pp. 505-527
Author(s):  
Maria Paola Bianchi ◽  
Hans-Joachim Böckenhauer ◽  
Tatjana Brülisauer ◽  
Dennis Komm ◽  
Beatrice Palano
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Competitive Ratio ◽  
Spanning Tree ◽  
Online Algorithm ◽  
Minimum Spanning Tree ◽  
Optimal Solution ◽  
Bounded Degree ◽  
Advice Complexity ◽  
Graph Classes

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals Converting MST to TSP Path by Branch Elimination

2020 ◽  
Vol 11 (1) ◽  
pp. 177
Author(s):  
Pasi Fränti ◽  
Teemu Nenonen ◽  
Mingchuan Yuan
Keyword(s):  
Spanning Tree ◽  
Closed Loop ◽  
Minimum Spanning Tree ◽  
Travelling Salesman Problem ◽  
Open Loop ◽  
Travelling Salesman ◽  
Elimination Algorithm ◽  
Number Of Iterations ◽  
Number Of Branches

Travelling salesman problem (TSP) has been widely studied for the classical closed loop variant but less attention has been paid to the open loop variant. Open loop solution has property of being also a spanning tree, although not necessarily the minimum spanning tree (MST). In this paper, we present a simple branch elimination algorithm that removes the branches from MST by cutting one link and then reconnecting the resulting subtrees via selected leaf nodes. The number of iterations equals to the number of branches (b) in the MST. Typically, b << n where n is the number of nodes. With O-Mopsi and Dots datasets, the algorithm reaches gap of 1.69% and 0.61 %, respectively. The algorithm is suitable especially for educational purposes by showing the connection between MST and TSP, but it can also serve as a quick approximation for more complex metaheuristics whose efficiency relies on quality of the initial solution.

scholarly journals The Volatility Forecasting Power of Financial Network Analysis

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Nicolás S. Magner ◽  
Jaime F. Lavin ◽  
Mauricio A. Valle ◽  
Nicolás Hardy
Keyword(s):  
Spanning Tree ◽  
Predictive Power ◽  
Minimum Spanning Tree ◽  
Financial Networks ◽  
Order Effects ◽  
Structural Factors ◽  
Financial Variables ◽  
Road Map ◽  
Daily Returns

This investigation connects two crucial economic and financial fields, financial networks, and forecasting. From the financial network’s perspective, it is possible to enhance forecasting tools, since econometrics does not incorporate into standard economic models, second-order effects, nonlinearities, and systemic structural factors. Using daily returns from July 2001 to September 2019, we used minimum spanning tree and planar maximally filtered graph techniques to forecast the stock market realized volatility of 26 countries. We test the predictive power of our core models versus forecasting benchmarks models in and out of the sample. Our results show that the length of the minimum spanning tree is relevant to forecast volatility in European and Asian stock markets, improving forecasting models’ performance. As a new contribution, the evidence from this work establishes a road map to deepening the understanding of how financial networks can improve the quality of prediction of financial variables, being the latter, a crucial factor during financial shocks, where uncertainty and volatility skyrocket.

scholarly journals Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 615
Author(s):  
Hongzhuan Wang ◽  
Piaoyang Yin
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Resistance Distance ◽  
Harary Index ◽  
H Index ◽  
Electronic Networks ◽  
Minimum Number ◽  
Graph Parameters

Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

scholarly journals Large Linguistic Corpus Reduction with SCP Algorithms

2015 ◽  
Vol 41 (3) ◽  
pp. 355-383 ◽  
Author(s):  
Nelly Barbot ◽  
Olivier Boëffard ◽  
Jonathan Chevelu ◽  
Arnaud Delhay
Keyword(s):  
Lower Bound ◽  
Set Covering ◽  
Covering Problem ◽  
Huge Amount ◽  
Text Corpora ◽  
Corpus Design ◽  
Text Content ◽  
The Cost ◽  
Greedy Solution

Linguistic corpus design is a critical concern for building rich annotated corpora useful in different domains of applications. For example, speech technologies such as ASR (Automatic Speech Recognition) or TTS (Text-to-Speech) need a huge amount of speech data to train data-driven models or to produce synthetic speech. Collecting data is always related to costs (recording speech, verifying annotations, etc.), and as a rule of thumb, the more data you gather, the more costly your application will be. Within this context, we present in this article solutions to reduce the amount of linguistic text content while maintaining a sufficient level of linguistic richness required by a model or an application. This problem can be formalized as a Set Covering Problem (SCP) and we evaluate two algorithmic heuristics applied to design large text corpora in English and French for covering phonological information or POS labels. The first considered algorithm is a standard greedy solution with an agglomerative/spitting strategy and we propose a second algorithm based on Lagrangian relaxation. The latter approach provides a lower bound to the cost of each covering solution. This lower bound can be used as a metric to evaluate the quality of a reduced corpus whatever the algorithm applied. Experiments show that a suboptimal algorithm like a greedy algorithm achieves good results; the cost of its solutions is not so far from the lower bound (about 4.35% for 3-phoneme coverings). Usually, constraints in SCP are binary; we proposed here a generalization where the constraints on each covering feature can be multi-valued.

Evaluating the Quality of the Residential Environment

1970 ◽  
Vol 2 (1) ◽  
pp. 23-32 ◽  
Author(s):  
J.F. Kain ◽  
J.M. Quigley
Keyword(s):  
Lower Bound ◽  
Urban Renewal ◽  
Metropolitan Area ◽  
Explanatory Variables ◽  
Urban Households ◽  
Residential Quality ◽  
Potential Benefits ◽  
Police Protection ◽  
Quality Measurements

This paper attempts to measure systematically the extent of blight in a single metropolitan area. Using a sample of individual dwelling units, the paper first describes a method of quantifying some previously neglected aspects of residential quality and demonstrates that they are highly valued by urban households. Secondly, it illustrates the feasibility of generalizing these quality measurements of sample dwellings to all city blocks using widely available explanatory variables. The analysis finds strong inferential evidence of an important, but elusive, relationship among the level of public services provided to particular dwelling units (police protection and schools), measures of residential quality, and the market's valuation of these units. The models described in this paper relate to an important range of urban renewal questions. For example, they can be used to obtain lower bound estimates of the potential benefits of urban renewal programs.

Research on the Network Topology and Reliability of the Communication System in Smart Substations

2013 ◽  
Vol 765-767 ◽  
pp. 1757-1760
Author(s):  
Zhi Ming Wang ◽  
Xian Li Sun ◽  
Li Liu ◽  
Xiao Rui Zhang
Keyword(s):  
Network Topology ◽  
Network Performance ◽  
Vital Role ◽  
Network System ◽  
Entire System ◽  
Advantages And Disadvantages ◽  
Network Topologies ◽  
The Relationship ◽  
Smart Substation

Communication network plays a vital role in smart substations. Different network topology provides different network performance, while the quality of network performance directly impacts the reliability of the entire system. In view of the network topology of smart substations, this paper compares the advantages and disadvantages of different network topologies as well as different network schemes of substation layer and process layer, and also analyzes the relationship between the reliability of network system and network topology, which play an important role in establishing a reliable and efficient smart substation system.

scholarly journals Note on Nordhaus-Gaddum Problems for Colin de Verdière type Parameters

10.37236/2570 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Wayne Barrett ◽  
Shaun M. Fallat ◽  
H. Tracy Hall ◽  
Leslie Hogben
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minimum Rank ◽  
Maximum Nullity ◽  
Graph Parameters

We establish the bounds $\frac 4 3 \le b_\nu \le b_\xi\le \sqrt 2$, where $b_\nu$ and $b_\xi$ are  the Nordhaus-Gaddum sum upper bound multipliers, i.e., $\nu(G)+\nu(\overline{G})\le b_\nu |G|$ and $\xi(G)+\xi(\overline{G})\le b_\xi | G|$ for all graphs $G$, and $\nu$ and $\xi$ are Colin de Verdiere type graph parameters. The Nordhaus-Gaddum sum lower bound for $\nu$ and $\xi$ is conjectured to be $|G| - 2$, and if these parameters are replaced by the maximum nullity $M(G)$, this bound is called the Graph Complement Conjecture in the study of minimum rank/maximum nullity problems.

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