Mathematical Modelling for Transportation With Application to Airline Transportation Network

2021 ◽  
pp. 28-51
Author(s):  
Sadiqah Almarzooq ◽  
Njwd Albishi
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Transportation Networks ◽  
Transportation Network ◽  
Weighted Graph ◽  
Search Method ◽  
Nearest Neighbour ◽  
Water Pipelines ◽  
Regional Airports

Graph theory is a basic tool to solve real-world problems such as communication between people, water pipelines, and transportation networks. A transportation network can be modeled as connected weighted graph. This chapter starts by introducing some fundamental concepts of graph theory to be applied to three main problems: the minimum spanning tree, the shortest path, and the travel salesperson. The authors discuss some appropriated algorithms such as depth first algorithm, Prim's algorithm, Kruskal's algorithm, Dijkstra's algorithm, the nearest neighbour algorithm, the minimum spanning tree depth first search method (MST-DFS) algorithm, and the Christofides' algorithm to solve these problems and apply them the airlines network between international and regional airports in Saudi Arabia.

scholarly journals ALGORITMA “HANCURKAN SEMUA SIKEL” UNTUK MENENTUKAN POHON PERENTANG MINIMUM DARI SUATU GRAF BERBOBOT

2019 ◽  
Vol 21 (2) ◽  
pp. 91-98
Author(s):  
Ricky Aditya
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Alternative Algorithm

The minimum spanning tree is one of the applications of graph theory in various fields. There are several algorithms for determining the minimum spanning tree of a weighted graph, such as Kruskal's algorithm and Prim's algorithm. These two algorithms are not really easy to teach to students in general. Therefore in this paper presented an alternative algorithm called the algorithm "Destroy All Sikel", which is more intuitive and easier to understand. Furthermore, there are also examples of implementation and comparison with two other algorithms.

DEVELOPING PROXIMITY GRAPHS BY PHYSARUM POLYCEPHALUM: DOES THE PLASMODIUM FOLLOW THE TOUSSAINT HIERARCHY?

2009 ◽  
Vol 19 (01) ◽  
pp. 105-127 ◽  
Author(s):  
ANDREW ADAMATZKY
Keyword(s):  
Foraging Behavior ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Physarum Polycephalum ◽  
Minimum Spanning Tree ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Proximity Graphs ◽  
Relative Neighborhood Graph ◽  
Plasmodium Of Physarum Polycephalum

Plasmodium of Physarum polycephalum spans sources of nutrients and constructs varieties of protoplasmic networks during its foraging behavior. When the plasmodium is placed on a substrate populated with sources of nutrients, it spans the sources with protoplasmic network. The plasmodium optimizes the network to deliver efficiently the nutrients to all parts of its body. How exactly does the protoplasmic network unfold during the plasmodium's foraging behavior? What types of proximity graphs are approximated by the network? Does the plasmodium construct a minimal spanning tree first and then add additional protoplasmic veins to increase reliability and through-capacity of the network? We analyze a possibility that the plasmodium constructs a series of proximity graphs: nearest-neighbour graph (NNG), minimum spanning tree (MST), relative neighborhood graph (RNG), Gabriel graph (GG) and Delaunay triangulation (DT). The graphs can be arranged in the inclusion hierarchy (Toussaint hierarchy): NNG ⊆ MST ⊆ RNG ⊆ GG ⊆ DT . We aim to verify if graphs, where nodes are sources of nutrients and edges are protoplasmic tubes, appear in the development of the plasmodium in the order NNG → MST → RNG → GG → DT , corresponding to inclusion of the proximity graphs.

scholarly journals A Constant-Factor Approximation for the Generalized Cable-Trench Problem

2019 ◽  
Author(s):  
Marcelo Benedito ◽  
Lehilton Pedrosa ◽  
Hugo Rosado
Keyword(s):  
Shortest Path ◽  
Steiner Tree ◽  
Spanning Tree ◽  
Optimization Problem ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Constant Factor ◽  
Steiner Tree Problem ◽  
Negative Factor ◽  
Path Lengths

In the Cable-Trench Problem (CTP), the objective is to find a rooted spanning tree of a weighted graph that minimizes the length of the tree, scaled by a non-negative factor , plus the sum of all shortest-path lengths from the root, scaled by another non-negative factor. This is an intermediate optimization problem between the Single-Destination Shortest Path Problem and the Minimum Spanning Tree Problem. In this extended abstract, we consider the Generalized CTP (GCTP), in which some vertices need not be connected to the root, but may serve as cost-saving merging points; this variant also generalizes the Steiner Tree Problem. We present an 8.599-approximation algorithm for GCTP. Before this paper, no constant approximation for the standard CTP was known.

scholarly journals FUZZY SHORTEST ROUTE ALGORITHM FOR TELEPHONE LINE CONNECTION USING THE LC-MST ALGORITHM

2015 ◽  
Vol 2 (2) ◽  
pp. 37-39
Author(s):  
Vijayalakshmi D ◽  
Kalaivani R
Keyword(s):  
Transmission Lines ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Minimum Cost ◽  
Distribution Networks ◽  
Minimum Length ◽  
Fuzzy Graph ◽  
Telephone Line ◽  
Water Pipelines ◽  
Communication Links

In computer science, there are many algorithms that finds a minimum spanning tree for a connected weighted undirected fuzzy graph. The minimum length (or cost) spanning tree problem is one of the nicest and simplest problems in network optimization, and it has a wide variety of applications. The problem is tofind a minimum cost (or length) spanning tree in G. Applications include the design of various types of distribution networks in which the nodes represent cities, centers etc.; and edges represent communication links (fiber glass phone lines, data transmission lines, cable TV lines, etc.), high voltage power transmissionlines, natural gas or crude oil pipelines, water pipelines, highways, etc. The objective is to design a network that connects all the nodes using the minimum length of cable or pipe or other resource in this paper we find the solution to the problem is to minimize the amount of new telephone line connection using matrixalgorithm with fuzzy graph.

scholarly journals IMPLEMENTASI ALGORITMA KRUSKAL DAN ALGORITMA PRIM SUATU GRAPH DENGAN APLIKASI BERBASIS DESKTOP

2020 ◽  
Vol 3 (2) ◽  
pp. 89-93
Author(s):  
Siti Alvi Sholikhatin ◽  
Adi Budi Prasetyo ◽  
Ade Nurhopipah
Keyword(s):  
Spanning Tree ◽  
Research Study ◽  
Minimum Spanning Tree ◽  
Greedy Algorithms ◽  
Relevant Literature ◽  
Weighted Graph ◽  
Self Assessment ◽  
Technological Readiness ◽  
Actual Environment ◽  
Application Design

A graph has several algorithms in its solution, including the Kruskal algorithm and Prim algorithm, both of which are greedy algorithms for determining the minimum spanning tree. Completion of graphs is useful in various fields of life, so an accurate graph calculation is important. Making an application to solve a graph, especially the Kruskal algorithm and Prim algorithm aims to facilitate the work of the graph so as to produce an accurate final result. The flow of research carried out are: a background review of research, study of literature and relevant literature, application design, building desktop-based applications, as well as implementation and application tests. The level of technological readiness or TKT in this research is based on self-assessment which is at level 7, meaning the prototype demonstration system in the actual environment, with details of the TKT indicators as follows: TKT indicator 1 is met, TKT indicator 2 is met, TKT indicator 3 is not met, TKT indicator 4, TKT indicator 5 are met, TKT indicator 6 are met, TKT indicator 7 is met, TKT indicator 8 and 9 are not met. The application that has been built is useful for completing a graph with the Kruskal algorithm and Prim algorithm. This research was conducted to answer the crucial needs of a weighted graph settlement application.

On the Euclidean Minimum Spanning Tree Problem

2005 ◽  
Vol 1 (1) ◽  
pp. 11-14 ◽  
Author(s):  
Sanguthevar Rajasekaran
Keyword(s):  
Spanning Tree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Linear Time ◽  
Randomized Algorithm ◽  
Weighted Graph ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithms ◽  
Tree Algorithms ◽  
Minimum Spanning Tree Problem

Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms

scholarly journals Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

2014 ◽  
Vol XL-8 ◽  
pp. 1105-1114 ◽  
Author(s):  
S. Dutta ◽  
D. Patra ◽  
H. Shankar ◽  
P. Alok Verma
Keyword(s):  
Spanning Tree ◽  
Road Network ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Transportation Network ◽  
Travelling Salesman Problem ◽  
Optimal Path ◽  
Road Transportation ◽  
Gis Technology ◽  
Weighted Network

minimum spanning tree (MST) of a connected, undirected and weighted network is a tree of that network consisting of all its nodes and the sum of weights of all its edges is minimum among all such possible spanning trees of the same network. In this study, we have developed a new GIS tool using most commonly known rudimentary algorithm called Prim’s algorithm to construct the minimum spanning tree of a connected, undirected and weighted road network. This algorithm is based on the weight (adjacency) matrix of a weighted network and helps to solve complex network MST problem easily, efficiently and effectively. The selection of the appropriate algorithm is very essential otherwise it will be very hard to get an optimal result. In case of Road Transportation Network, it is very essential to find the optimal results by considering all the necessary points based on cost factor (time or distance). This paper is based on solving the Minimum Spanning Tree (MST) problem of a road network by finding it’s minimum span by considering all the important network junction point. GIS technology is usually used to solve the network related problems like the optimal path problem, travelling salesman problem, vehicle routing problems, location-allocation problems etc. Therefore, in this study we have developed a customized GIS tool using Python script in ArcGIS software for the solution of MST problem for a Road Transportation Network of Dehradun city by considering distance and time as the impedance (cost) factors. It has a number of advantages like the users do not need a greater knowledge of the subject as the tool is user-friendly and that allows to access information varied and adapted the needs of the users. This GIS tool for MST can be applied for a nationwide plan called Prime Minister Gram Sadak Yojana in India to provide optimal all weather road connectivity to unconnected villages (points). This tool is also useful for constructing highways or railways spanning several cities optimally or connecting all cities with minimum total road length.

scholarly journals A graph theory approach of the study of macroeconomics networks

2017 ◽  
Author(s):  
Γεώργιος-Αντώνιος Σαραντίτης
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dominating Set ◽  
Theory Approach ◽  
Minimum Dominating Set

Πολλά σύγχρονα οικονομικά συστήματα χαρακτηρίζονται από αυξημένο βαθμό πολυπλοκότητας. Οι οντότητες αυτών των συστημάτων αναπτύσσουν διακριτές, αναδυόμενες και μη γραμμικές συμπεριφορές που δεν μπορούν να περιγραφούν πλήρως με οικονομετρικές τεχνικές. Τα τελευταία χρόνια, λόγω της γρήγορης αύξησης της υπολογιστικής ισχύος και της εξέλιξης των αλγορίθμων, η επιστήμη της Ανάλυσης Δικτύων ενσωματώθηκε στην ανάλυση τέτοιων πολύπλοκων οικονομικών συστημάτων, συμπληρώνοντας τη χρήση της οικονομετρίας.Μια κοινώς χρησιμοποιούμενη τεχνική στο πλαίσιο της Ανάλυσης Δικτύων είναι το Minimum Spanning Tree (MST). Το MST παράγει ένα υπο-δίκτυο του αρχικού δικτύου στο οποίο είναι συνδεδεμένοι όλοι οι κόμβοι έτσι ώστε να μην υπάρχουν βρόχοι. Ωστόσο, το MST φέρει κάποια εγγενή μειονεκτήματα που προέρχονται άμεσα από τη διαδικασία αλγοριθμικού προσδιορισμού του και μπορεί να το καταστήσουν ακατάλληλο για τη μελέτη οικονομικών δικτύων. Αυτή η διατριβή αποσκοπεί στο να αναδείξει τα μειονεκτήματα του MST όταν χρησιμοποιείται στα οικονομικά δίκτυα και να επισημάνει τα πλεονεκτήματα μιας νέας τεχνικής βελτιστοποίησης, που ονομάζεται Threshold-Minimum Dominating Set (T-MDS) ως μια καταλληλότερη λύση. Επιπλέον, εισάγεται το Threshold Weighted - Minimum Dominating Set (TW-MDS), το οποίο διατηρεί όλα τα πλεονεκτήματα του T-MDS και, ανάλογα με το δεδομένο σύνολο, μπορεί να είναι πιο κατάλληλο για διαχρονικές αναλύσεις που εκτελούνται στην πάροδο του χρόνου.Η ανωτερότητα των T-MDS και TW-MDS σε σχέση με το κλασικό MST αρχικά επισημαίνεται σε αυτή τη διατριβή με κατάλληλα θεωρητικά παραδείγματα. Στη συνέχεια συνεχίζουμε παρέχοντας ένα ευρύ φάσμα μακροοικονομικών εφαρμογών: τον συγχρονισμό των οικονομικών κύκλων, την εξέλιξη της ανισότητας εισοδήματος και τη μέτρηση του πληθωρισμού πυρήνα. Με αυτόν τον τρόπο τονίζουμε την καταλληλότητα των προτεινόμενων μεθοδολογιών στη μακροοικονομική ανάλυση. Έτσι, αυτή η διατριβή έχει διπλή συμβολή στην ανάλυση των σύνθετων οικονομικών δικτύων: από τη θεωρητική πλευρά επεκτείνει τη σχετική βιβλιογραφία παρέχοντας ένα πιο κατάλληλο εργαλείο από αυτό που χρησιμοποιείται προς το παρόν, ενώ από την εμπειρική πλευρά παρέχει νέα αποτελέσματα από τις διαφορετικές οικονομικές Εφαρμογές του T-MDS.

Clustering Vertices in Weighted Graphs

2011 ◽  
pp. 285-298
Author(s):  
Derry Tanti Wijaya ◽  
Stephane Bressan
Keyword(s):  
Graph Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Graph Clustering ◽  
Weighted Graphs ◽  
Similarity Relations ◽  
Natural Notion ◽  
Cluster Density ◽  
Markov Clustering

Clustering is the unsupervised process of discovering natural clusters so that objects within the same cluster are similar and objects from different clusters are dissimilar. In clustering, if similarity relations between objects are represented as a simple, weighted graph where objects are vertices and similarities between objects are weights of edges; clustering reduces to the problem of graph clustering. A natural notion of graph clustering is the separation of sparsely connected dense sub graphs from each other based on the notion of intra-cluster density vs. inter-cluster sparseness. In this chapter, we overview existing graph algorithms for clustering vertices in weighted graphs: Minimum Spanning Tree (MST) clustering, Markov clustering, and Star clustering. This includes the variants of Star clustering, MST clustering and Ricochet.

Interrelationships Among the Liliatae: a Graph Theory Approach

1980 ◽  
Vol 28 (2) ◽  
pp. 261 ◽  
Author(s):  
HT Clifford ◽  
WT Williams
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Theory Approach ◽  
Nearest Neighbours ◽  
Shared Ancestry ◽  
Evolutionary Convergence

The interrelationships of the families of Liliatae were investigated by using a network generating program (NETS) which, unlike minimum spanning tree programs, considers both first and second nearest neighbours. Within the resultant network the distribution of families reflects many traditionally accepted assemblages, including the Zingiberales and Alismatales. The affinities of the Arecaceae in particular suggests that the observed similarities between some families may be due to evolutionary convergence rather than to a shared ancestry. This fact, plus the tendency of the network to branch rather than to form loops, helps to account for the difficulty in classifying the class.

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