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#### Spanning trees and spanning Eulerian subgraphs with small degrees

, Article Discrete Mathematics ; Volume 338, Issue 8 , August , 2015 , Pages 1317-1321 ; 0012365X (ISSN) ; Sharif University of Technology
Elsevier
2015

Abstract

Liu and Xu (1998) and Ellingham, Nam and Voss (2002) independently showed that every k-edge-connected simple graph G has a spanning tree T such that for each vertex v, dT(v) ≤ ⌈ d(v)/k ⌉ + 2. In this paper we show that every k-edge-connected graph G has a spanning tree T such that for each vertex v, dT(v)≤ ⌈ d(v)-2/k ⌉ + 2; also if G has k edge-disjoint spanning trees, then T can be found such that for each vertex v, dT(v) ≤ ⌈ d(v)-1/k ⌉ + 1. This result implies that every (r-1)-edge-connected r-regular graph (with r ≥ 4) has a spanning Eulerian subgraph whose degrees lie in the set {2,4,6}; also reduces the edge-connectivity needed for some theorems due to Barát and Gerbner (2014) and...

#### Multispanning tree zone-ordered label-based routing algorithms for irregular networks

, Article IEEE Transactions on Parallel and Distributed Systems ; Vol. 22, issue. 5 , 2011 , p. 817-832 ; ISSN: 10459219 ; Moinzadeh, P ; Sarbazi-Azad, H ; Zomaya, A.Y ; Sharif University of Technology
Abstract

In this paper, a diverse range of routing algorithms is classified into a new family of routings called zone-ordered label-based routing algorithms. The proposed classification is based on three common steps (factors) for generating such routings, namely, graph labeling, deadlock-free zones, and zone ordering. The main goal of this classification is to define several new routing concepts and streamline the knowledge on routing algorithms. Following the classification, a novel methodology is proposed to generate routing algorithms for irregular networks. The methodology uses the three mentioned steps to generate deadlock-free routings. Consequently, the methodology-based routings fall into...

#### Multispanning tree zone-ordered label-based routing algorithms for irregular networks

, Article IEEE Transactions on Parallel and Distributed Systems ; Volume 22, Issue 5 , July , 2011 , Pages 817-832 ; 10459219 (ISSN) ; Moinzadeh, P ; Sarbazi Azad, H ; Zomaya, A. Y ; Sharif University of Technology
Abstract

In this paper, a diverse range of routing algorithms is classified into a new family of routings called zone-ordered label-based routing algorithms. The proposed classification is based on three common steps (factors) for generating such routings, namely, graph labeling, deadlock-free zones, and zone ordering. The main goal of this classification is to define several new routing concepts and streamline the knowledge on routing algorithms. Following the classification, a novel methodology is proposed to generate routing algorithms for irregular networks. The methodology uses the three mentioned steps to generate deadlock-free routings. Consequently, the methodology-based routings fall into...

#### Computing Shortest path and Minimum Spanning Tree under Uncertainty

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

Computing shortest paths and minimum spanning trees are basic and well-known problems in computer science and graph theory; also they have been considered as basic subroutines for many practical algorithms. There exists optimal algorithms for solving these problems when information about the basic graph is certain and specified. But, in real applications when the graph is obtained by using measurement tools which have limited computing precision, practically we are facing a graph in which locations of vertices or weights of edges are estimated.In such cases determining upper bounds or lower bounds for solutions of the shortest path and the minimum spanning tree are estimations of the...

#### Special classes of mathematical programming models with fuzzy random variables [electronic resource]

, Article Journal of Intelligent and Fuzzy Systems, Published In: IOS Press ; Volume 19, Number 2, 2008 ; Nematian, Javad ; Sharif University of Technology
Abstract

In this paper, we will discuss two special classes of mathematical programming models with fuzzy random variables. In the first model, a linear programming problem with fuzzy decision variables and fuzzy random coefficients is introduced. Then an algorithm is developed to solve the model based on fuzzy optimization method and fuzzy ranking method. In the second model, a fuzzy random quadratic spanning tree problem is presented. Then the proposed problem is formulated and solved by using the scalar expected value of fuzzy random variables. Furthermore, illustrative numerical examples are also given to clarify the methods discussed in this paper

#### Special classes of mathematical programming models with fuzzy random variables

, Article Journal of Intelligent and Fuzzy Systems ; Volume 19, Issue 2 , 2008 , Pages 131-140 ; 10641246 (ISSN) ; Nematian, J ; Sharif University of Technology
2008

Abstract

In this paper, we will discuss two special classes of mathematical programming models with fuzzy random variables. In the first model, a linear programming problem with fuzzy decision variables and fuzzy random coefficients is introduced. Then an algorithm is developed to solve the model based on fuzzy optimization method and fuzzy ranking method. In the second model, a fuzzy random quadratic spanning tree problem is presented. Then the proposed problem is formulated and solved by using the scalar expected value of fuzzy random variables. Furthermore, illustrative numerical examples are also given to clarify the methods discussed in this paper

#### Integrating module checking and deduction in a formal proof for the perlman Spanning Tree Protocol (STP)

, Article Journal of Universal Computer Science ; Volume 13, Issue 13 , 2007 , Pages 2076-2104 ; 0958695X (ISSN) ; Nakhost, H ; Sirjani, M ; Sharif University of Technology
2007

Abstract

In the IEEE 802.1D standard for the Media Access Control layer (MAC layer) bridges, there is an STP (Spanning Tree Protocol) definition, based on the algorithm that was proposed by Radia Perlman. In this paper, we give a formal proof for correctness of the STP algorithm by showing that finally a single node is selected as the root of the tree and the loops are eliminated correctly. We use formal inductive reasoning to establish these requirements. In order to ensure that the bridges behave correctly regardless of the topology of the surrounding bridges and LANs, the Rebeca, modular verification techniques are applied. These techniques are shown to be efficiently applicable in model checking...

#### Spanning trees with minimum weighted degrees

, Article Information Processing Letters ; Volume 104, Issue 3 , 2007 , Pages 113-116 ; 00200190 (ISSN) ; Mahini, H ; Mirjalali, K ; Oveis Gharan, S ; Sayedi Roshkhar, A. S ; Zadimoghaddam, M ; Sharif University of Technology
2007

Abstract

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2 - ε factor. © 2007 Elsevier B.V. All rights reserved

#### Multicolored parallelisms of isomorphic spanning trees

, Article SIAM Journal on Discrete Mathematics ; Volume 20, Issue 3 , 2006 , Pages 564-567 ; 08954801 (ISSN) ; Alipour, A ; Fu, H. L ; Lo, Y. H ; Sharif University of Technology
2006

Abstract

A subgraph in an edge-colored graph is multicolored if all its edges receive distinct colors. In this paper, we prove that a complete graph on 2m (m ≠ 2) vertices K2m can be properly edge-colored with 2m - 1 colors in such a way that the edges of K2m can De partitioned into m multicolored isomorphic spanning trees. © 2006 Society for Industrial and Applied Mathematics

#### Determinantal Processes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

Determinantal processes are a special family of stochastic processes that arise in physics (fermions), random matrices (eigenvalues), and in combinatorics (random spanning trees and non-intersecting paths). These processes have repelling property (points close to each other are chosen with low probability). Because of this repelling property, determinantal processes are approporiat for modeling some physical quantities (e.g. the position of electrons). Their probabilistic structure is described by operators on complex vector spaces and their eigenvalues. Determinantal processes have interesting properties, e.g. number of points in a region is a sum of independent Bernoulli random variables....

#### Kinetic pie delaunay graph and its applications

, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 7357 LNCS , 2012 , Pages 48-58 ; 03029743 (ISSN) ; 9783642311543 (ISBN) ; Rahmati, Z ; Zarei, A ; Sharif University of Technology
2012

Abstract

We construct a new proximity graph, called the Pie Delaunay graph, on a set of n points which is a super graph of Yao graph and Euclidean minimum spanning tree (EMST). We efficiently maintain the Pie Delaunay graph where the points are moving in the plane. We use the kinetic Pie Delaunay graph to create a kinetic data structure (KDS) for maintenance of the Yao graph and the EMST on a set of n moving points in 2-dimensional space. Assuming x and y coordinates of the points are defined by algebraic functions of at most degree s, the structure uses O(n) space, O(nlogn) preprocessing time, and processes O(n 2 λ 2s∈+∈2(n)β s + 2(n)) events for the Yao graph and O(n 2 λ 2s + 2(n)) events for the...

#### Kinetic Euclidean minimum spanning tree in the plane

, Article Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) ; Volume 7056 LNCS , 2011 , Pages 261-274 ; 03029743 (ISSN) ; 9783642250101 (ISBN) ; Zarei, A ; Sharif University of Technololgy
2011

Abstract

This paper presents the first kinetic data structure (KDS) for maintenance of the Euclidean minimum spanning tree (EMST) on a set of n moving points in 2-dimensional space. We build a KDS of size O(n) in O(nlogn) preprocessing time by which their EMST is maintained efficiently during the motion. In terms of the KDS performance parameters, our KDS is responsive, local, and compact

#### An optimal natural-gas network using minimum spanning tree

, Article 40th International Conference on Computers and Industrial Engineering: Soft Computing Techniques for Advanced Manufacturing and Service Systems, CIE40 2010, 25 July 2010 through 28 July 2010 ; July , 2010 ; 9781424472956 (ISBN) ; Mohajeri, A ; Arabmaghsudi, M ; Yahyanejad, M. H ; Taghipourian, F ; Mahdavi Amiri, N ; Sharif University of Technology
2010

Abstract

We consider the design of an optimal natural-gas network. Our proposed network contains two echelons, Town Broad Stations (TBSs), and consumers (demand zones). Here, our aim is a two-stage cost minimization. We first determine locations of the TBS so that the location-allocation cost is minimized. Then, we show how to distribute the flow of gas among the TBS minimizing the flow cost by using Minimum Spanning Tree (MST). A case study in Mazandaran Gas Company in Iran is made to assess the validity and effectiveness of our proposed model

#### A new approach for sensitivity analysis in network flow problems

, Article International Journal of Industrial Engineering : Theory Applications and Practice ; Volume 27, Issue 1 , 2020 , Pages 72-87 ; Eshghi, K ; Salehipour, A ; Sharif University of Technology
University of Cincinnati
2020

Abstract

This paper proposes a new approach to study the sensitivity analysis in the network flow problems, in particular, the minimum spanning tree and shortest path problems. In a sensitivity analysis, one looks for the amount of changes in the edges’ weights, number of edges or number of vertices such that the optimal solution, i.e., the minimum spanning tree or shortest path does not change. We introduce a novel approach, and develop associated equations and mathematics. We discuss two illustrative examples to show the applicability of the proposed approach. © International Journal of Industrial Engineering

#### Clustering and Embedding Graphs into Trees

, M.Sc. Thesis Sharif University of Technology ; Daneshgar, Amir (Supervisor)
Abstract

In this thesis, we study the following question stating that “how well a tree structure can approximate the clustering structure of a graph”.To do this, we first focus on the DJS algorithm proposed by Daneshgar et.al. and second we consider the minimum distortion tree approximation algorithm proposed by Abraham et.al.We conclude, using some experimental results, that the minimum spanning tree algorithm extracts some geometric aspects of the data set that the Abraham et.al. algorithm can not track

#### Combinatorial changes of euclidean minimum spanning tree of moving points in the plane

, Article Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, CCCG 2010, 9 August 2010 through 11 August 2010, Winnipeg, MB ; 2010 , Pages 43-45 ; Zarei, A ; Sharif University of Technology
2010

Abstract

In this paper, we enumerate the number of combinatorial changes of the the Euclidean minimum spanning tree (EMST) of a set of n moving points in 2- dimensional space. We assume that the motion of the points in the plane, is defined by algebraic functions of maximum degree s of time. We prove an upper bound of O(n3β2s(n2)) for the number of the combinatorial changes of the EMST, where βs(n)= λs(n)/n and λs(n) is the maximum length of Davenport-Schinzel sequences of order s on n symbols which is nearly linear in n. This result is an O(n) improvement over the previously trivial bound of O(n4)

#### Decomposing claw-free subcubic graphs and 4-chordal subcubic graphs

, Article Discrete Applied Mathematics ; Volume 296 , 2021 , Pages 52-55 ; 0166218X (ISSN) ; Ahanjideh, M ; Akbari, S ; Sharif University of Technology
Elsevier B.V
2021

Abstract

Hoffmann–Ostenhof's conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a 2-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and 4-chordal subcubic graphs. © 2021 Elsevier B.V

#### Decomposing claw-free subcubic graphs and 4-chordal subcubic graphs

, Article Discrete Applied Mathematics ; Volume 296 , 2021 , Pages 52-55 ; 0166218X (ISSN) ; Ahanjideh, M ; Akbari, S ; Sharif University of Technology
Elsevier B.V
2021

Abstract

Hoffmann–Ostenhof's conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a 2-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and 4-chordal subcubic graphs. © 2021 Elsevier B.V

#### Minimum Color-spanning Tree

, M.Sc. Thesis Sharif University of Technology ; Abam, Mohammad Ali (Supervisor)
Abstract

In the general case of minimum color-spanning tree which is one of the color-spanning set problems, given a weighted graph with n vertices of k different colors, the goal is to find a subtree of minimum weight such that vertices of this subtree include all the colors in the graph. In the planar case, the input is a complete graph with n colored vertices on the plane and the weight of each edge is the Euclidean distance between its corresponding vertices. In this thesis we consider the problem of minimum color-spanning tree. To this end, first we present various color-spanning set problems and some other related problems like Steiner tree and we study the previous work on these problems. Then...

#### Investigating Geometric Proximity Problems on Moving Points

, M.Sc. Thesis Sharif University of Technology ; Zarei, Alireza (Supervisor)
Abstract

An interesting theoretical and practical set of problems in computer science is concerned with the study of spatial relations among objects in a geometric space. Examples of such problems for a set of points P are finding the closest pair of the points P, partitioning space into regions such that all points of a region have minimum distance to the same point in P, and computing the Euclidean minimum spanning tree on P. Moreover, we need mechanisms to efficiently update these properties when the points P are allowed to move or may be inserted or deleted. This is to avoid re-computation of these properties from scratch. Here, we consider the Euclidean minimum spanning tree (EMST) of a set of...