scholarly journals An Exact Performance Bound for an ${O (m+n)}$ Time Greedy Matching Procedure

10.37236/1310 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Andrew Shapira
Keyword(s):  
Lower Bound ◽  
Extremal Function ◽  
Hybrid Algorithms ◽  
Maximum Matching ◽  
Performance Bound ◽  
Sparse Graphs ◽  
Augmenting Path ◽  
Dense Graphs ◽  
Matching Procedure

We prove an exact lower bound on $\gamma(G)$, the size of the smallest matching that a certain $O(m+n)$ time greedy matching procedure may find for a given graph $G$ with $n$ vertices and $m$ edges. The bound is precisely Erdős and Gallai's extremal function that gives the size of the smallest maximum matching, over all graphs with $n$ vertices and $m$ edges. Thus the greedy procedure is optimal in the sense that when only $n$ and $m$ are specified, no algorithm can be guaranteed to find a larger matching than the greedy procedure. The greedy procedure and augmenting path algorithms are seen to be complementary: the greedy procedure finds a large matching for dense graphs, while augmenting path algorithms are fast for sparse graphs. Well known hybrid algorithms consisting of the greedy procedure followed by an augmenting path algorithm are shown to be faster than the augmenting path algorithm alone. The lower bound on $\gamma(G)$ is a stronger version of Erdős and Gallai's result, and so the proof of the lower bound is a new way of proving of Erdős and Gallai's result.

scholarly journals A Lower Bound on the Average Degree Forcing a Minor

10.37236/8847 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Sergey Norin ◽  
Bruce Reed ◽  
Andrew Thomason ◽  
David R. Wood
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Factor ◽  
Average Degree ◽  
Complete Graphs ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
A Minor

We show that for sufficiently large $d$ and for $t\geq d+1$,  there is a graph $G$ with average degree $(1-\varepsilon)\lambda  t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.

scholarly journals Algorithms for Finding Shortest Paths in Networks with Vertex Transfer Penalties

Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 269 ◽  
Author(s):  
Rhyd Lewis
Keyword(s):  
Waiting Times ◽  
Shortest Paths ◽  
Dijkstra’S Algorithm ◽  
Weighted Graphs ◽  
Worst Case ◽  
Sparse Graphs ◽  
Graph Expansion ◽  
Advantages And Disadvantages ◽  
Dijkstra's Algorithm ◽  
Dense Graphs

In this paper we review many of the well-known algorithms for solving the shortest path problem in edge-weighted graphs. We then focus on a variant of this problem in which additional penalties are incurred at the vertices. These penalties can be used to model things like waiting times at road junctions and delays due to transfers in public transport. The usual way of handling such penalties is through graph expansion. As an alternative, we propose two variants of Dijkstra’s algorithm that operate on the original, unexpanded graph. Analyses are then presented to gauge the relative advantages and disadvantages of these methods. Asymptotically, compared to using Dijkstra’s algorithm on expanded graphs, our first variant is faster for very sparse graphs but slower with dense graphs. In contrast, the second variant features identical worst-case run times.

scholarly journals Typology of schools operating in the Moscow Electronic School system based on the analysis of network indicators

2021 ◽  
Vol 98 ◽  
pp. 03001
Author(s):  
Svetlana Vachkova ◽  
Elena Petryaeva ◽  
Evgeny Patarakin
Keyword(s):  
Network Analysis ◽  
School System ◽  
Multiple Roles ◽  
Digital Repository ◽  
Sparse Graphs ◽  
Main Method ◽  
Dense Graphs ◽  
Common Activity ◽  
Digital Objects ◽  
Key Participants

Network analysis methods are actively used to research the behavior of digital repository users who utilize and create digital objects. At the same time, the research into the collective behavior of a group of participants who are members of the same school is much less common. The library of the Moscow Electronic School is a rather complicated system with multiple roles offered to users. The actors of the repository are teachers, students, parents, and publishers – anyone performing any actions with the objects. In this study, the school is seen as an actor performing actions with objects – lesson scenarios within the Moscow Electronic School repository of digital objects. Within the study, the authors compare the sociograms of schools that unite teachers and the scenarios created by the teachers and divide schools into factions based on network indicators in sociograms. The main method of presenting and analyzing data is network analysis and sociogram creation. The authors identify two types of networks: the network of single participant’s relationships and the network of relationships of teachers from a single school. The authors not only describe the data structure in the Moscow Electronic School system that records the digital trace of every individual and collective user but also create a digital map that reflects the dynamics of actions in the Moscow Electronic School system and identify the indicators that characterize the common activity of key participants. Moreover, the authors identify graph factions for schools that characterize the degree of interaction between teachers: disconnected groups, sparse graphs, crystallization centers, dense graphs.

Statistical performance analysis of transient-based extended blockage detection in a water supply pipeline

2019 ◽  
Vol 68 (5) ◽  
pp. 346-357 ◽  
Author(s):  
Alireza Keramat ◽  
Roohollah Zanganeh
Keyword(s):  
Lower Bound ◽  
Water Supply ◽  
Fisher Information ◽  
Noise Level ◽  
Minimum Mean Square Error ◽  
Information Matrix ◽  
Physical Parameters ◽  
Resolution Limit ◽  
Performance Bound ◽  
Water Supply Pipeline

Abstract The aim of the present research is to quantify the maximum feasible accuracy (location and size) for the transient-based blockage detection in a water supply pipeline. Owing to the randomness of transient measurements, a performance bound for the extended blockage detection exists which estimated parameters cannot exceed. The Cramér–Rao lower bound (CRLB) theorem is utilized to compute the lower bound variance of noise-induced estimation errors. It gives the minimum mean square error of any estimator according to information obtained from measurements and quantified by Fisher information. The Fisher information matrix is computed using direct differentiation of the compatibility equations obtained by the method of characteristics. The influence of relevant physical parameters including valve closure time, measurement time length and noise level on the best possible localization of blockage is investigated. The connection between the signal bandwidth, noise level and the performance limit is quantified for a typical case study. The results demonstrate trade-off between the size and the location/length of blockage estimates subject to different maneuver times, roughly offering half the wave speed times maneuver duration as the resolution limit.

The Blow-up Lemma

1999 ◽  
Vol 8 (1-2) ◽  
pp. 161-176 ◽  
Author(s):  
JÁNOS KOMLÓS
Keyword(s):  
Graph Theory ◽  
Blow Up ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Regularity Lemma ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
Recent Developments ◽  
Szemeredi's Regularity Lemma

Extremal graph theory has a great number of conjectures concerning the embedding of large sparse graphs into dense graphs. Szemerédi's Regularity Lemma is a valuable tool in finding embeddings of small graphs. The Blow-up Lemma, proved recently by Komlós, Sárközy and Szemerédi, can be applied to obtain approximate versions of many of the embedding conjectures. In this paper we review recent developments in the area.

scholarly journals Tight Localizations of Feedback Sets

2021 ◽  
Vol 26 ◽  
pp. 1-19
Author(s):  
Michael Hecht ◽  
Krzysztof Gonciarz ◽  
Szabolcs Horvát
Keyword(s):  
Lower Bound ◽  
Error Bounds ◽  
Empirical Validation ◽  
Constant Ratio ◽  
Circuit Testing ◽  
Feedback Vertex Set ◽  
Sparse Graphs ◽  
Feedback Arc Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem

The classical NP–hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs ε ⊆ E or vertices ν ⊆ V whose removal G ∖ ε, G ∖ ν makes a given multi–digraph G =( V , E ) acyclic, respectively. Though both problems are known to be APX–hard, constant ratio approximations or proofs of inapproximability are unknown. We propose a new universal O (| V || E | 4 )–heuristic for the directed FASP. While a ratio of r ≈ 1.3606 is known to be a lower bound for the APX–hardness, at least by empirical validation we achieve an approximation of r ≤ 2. Most of the relevant applications, such as circuit testing , ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds with our approach.

scholarly journals Szemerédi's Regularity Lemma for Matrices and Sparse Graphs

2011 ◽  
Vol 20 (3) ◽  
pp. 455-466 ◽  
Author(s):  
ALEXANDER SCOTT
Keyword(s):  
Local Density ◽  
Regularity Lemma ◽  
Density Condition ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
Szemeredi's Regularity Lemma ◽  
Real Matrices

Szemerédi's Regularity Lemma is an important tool for analysing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

scholarly journals On the Staircases of Gyárfás

10.37236/5697 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
János Csányi ◽  
Peter Hajnal ◽  
Gábor V. Nagy
Keyword(s):  
Lower Bound ◽  
Extremal Problem ◽  
Extremal Function ◽  
Upper Bounds ◽  
Ramsey Problem ◽  
Symmetric Version

In a 2011 paper, Gyárfás investigated a geometric Ramsey problem on convex, separated, balanced, geometric $K_{n,n}$. This led to appealing extremal problem on square 0-1 matrices. Gyárfás conjectured that any 0-1 matrix of size $n\times n$ has a staircase of size $n-1$.We introduce the non-symmetric version of Gyárfás' problem. We give upper bounds and in certain range matching lower bound on the corresponding extremal function. In the square/balanced case we improve the $(4/5+\epsilon)n$ lower bound of Cai, Gyárfás et al. to $5n/6-7/12$. We settle the problem when instead of considering maximum staircases we deal with the sum of the size of the longest $0$- and $1$-staircases.

scholarly journals Performance Bound for Joint Multiple Parameter Target Estimation in Sparse Stepped-Frequency Radar: A Comparison Analysis

Sensors ◽  
2019 ◽  
Vol 19 (9) ◽  
pp. 2002
Author(s):  
Qiushi Chen ◽  
Xin Zhang ◽  
Qiang Yang ◽  
Lei Ye ◽  
Mengxiao Zhao
Keyword(s):  
Lower Bound ◽  
Estimation Error ◽  
Specific Effect ◽  
Practical Significance ◽  
Performance Bound ◽  
Multiple Parameter ◽  
Different Types ◽  
Stepped Frequency ◽  
The Relationship ◽  
Selection Of

A performance bound—Cramér-Rao lower bound (CRLB) for target estimation and detection in sparse stepped frequency radars is presented. The vector formulation of this CRLB is used to obtain a lower bound on the estimation error. The estimation performance can be transformed into different types of CRLB structures. Therefore, the expressions of bounds under three equivalent models are derived separately: time delay and Doppler stretch estimator, joint multiple parameter estimator, and sparse-based estimator. The variables to be estimated include the variances of unknown noise, range, velocity, and the real and imaginary parts of the amplitude. A general performance expression is proposed by considering the echo of the target in the line-of-sight. When the relationship between CRLB and various parameters are discussed in detail, the specific effect of waveform parameters on a single CRLB is compared and analyzed. Numerical simulations demonstrated that the resulting CRLB exhibits considerable theoretical and practical significance for the selection of optimal waveform parameters.

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