Van Phuc Pham Abstract

Transcription

1 Flight Scheduling using Dijstra s Algorithm and Empirical Modelling Van Phuc Pham Abstract Graph theory is one of the most popular subjects in computer science. A mathematician Leonhard Euler first introduced the theory in eighteen century to solve a famous problem called Konigsberg Bridge. The theory is widely used by many modern applications to solve problems in various areas. For example, find the routes for transfer data in a computer network or obtain the quickest path from one city to another. Furthermore, the graph theories are also used as data structures to store data, which can optimise the large data manipulation. This paper will discuss about shortest path on graph algorithms and illustrate the role of EM in modelling this algorithm by explaining the construction of a simple Flight Schedule application. The paper is also aim to achieve further understanding the differences of Empirical Modelling and modern programming concepts. Introduction Graphs provide a discrete structure consisting of vertices and these vertices are connected together by edges. The structure of graph is used to store data or objects, while the algorithms are designed to process these data structures. The problem in identify the shortest paths between two vertices had been intensively research and there are various techniques to solve this problem. The Dijkstra s algorithm is one of the significant algorithms that can able to obtain the shortest paths between two vertices with a low computational cost. The paper will look at the in details this algorithm before deploy the algorithm into the application in the project. Empirical Modelling describes the characteristic of a construal with reference to three key concepts: Observables, dependences and agents. A construal can be seen as embodying patterns of observable, dependencies and agency that are characteristics of the situation that being constructed [1]. Therefore, in this project the concept of observable will be employed to observe the graph- processing algorithm, which lead to the model ling of the application. The paper begins with a brief introduction of different graph algorithms. Follow by the examination of Dijkstra s algorithm to identify the properties of this algorithm. The examination will look at the structure of the algorithm and analyse the conditions for the shortest path to exist on a graph. After that, the paper will describe the application and its construction using EM tools. The details of developing the application help to have further understanding the use of the EM tools such as Scout, Eden or Donal. This part also allows identifying the EM programming concepts and their features. Following previous section, there will be an evaluation about the empirical modelling and its tools. This aim to demonstrates the understanding of the empirical modelling concepts and recommending further improvement for the EM tools. At last, the paper concludes with an overview about the EM concepts and the project. 2. Graphs and Shortest Path Algorithms. 2.1 Graph A graph includes vertices with edges connect these vertices together. A simple undirected graph contains vertices and undirected edges, while the

2 directed graph has edges marked with the directions explicitly. Vertices can have multi- edges in a graph and this graph is called multi- graph. The edge weight is a value assign the edge to represent the distance between to vertices. There are two main methods to representing graphs, which are listing all the edges and using adjacency matrix. Graphs are used widely used in many models. For example: state machine in software engineering project or modelling the influence of a person in a group of people. In this project, graph is presented as the world map where each country is a node and they are linked to each other by the edges. In order to find the shortest path between two nodes on the graph, there are numbers algorithm can be employed. There are three popular algorithms: 2.2 Shortest Path Algorithms Bellman- Ford algorithm computes the shortest path to all the nodes in graph from on single nodes. Most importantly the algorithm accepts the graph, which have the negative edge values. The idea of the algorithm is to examine the entire shortest path on the graph. The algorithm uses nodes relaxation, progressively decreasing an estimate value on the weight of the shortest path from the source to each vertex until the algorithm find the find the actual shortest path on the graph. The second algorithm is called Floyd- Warshall algorithm. This algorithm is able to calculate the shortest path for ever pair of vertex on the graph. Same as the Bellman- Ford algorithm, this algorithm also works on the graph contain negative weight value, but not negative weight cycles. The Floyd- Warshal s algorithm is based upon on the examination, that a path connected any two vertices, which may have zero or more intermediate vertices. The algorithm proceeds by allowing an additional intermediate at each step. The last algorithm called the Dijkstra s algorithm. This algorithm is one of the most popular solutions for shortest path problem. However, this algorithm only works on directed graph without negative edge weights and takes single node as source. The output is all shortest paths from the source to other vertices on the graph. The structure of the algorithm is going to be discussing in the next section, as this algorithm is chosen to employ in the project. 3. Shortest Path Propeties and Dijkstra s Algorithm Model 3.1 Shortest Path Properties. In this section, the Dijkstra s algorithm will be describe in details to show how the shortest path problem can be solved and the properties First we consider a directed graph G with consist of the vertexes V with edges E: G = (V,E). For every edge on the graph, each edge is assigned with a real number R known as weight w: (w): W R. A path p in a graph is denoted as p = v 1 + v 2 + v v n The shortest path from vertex u to v on G is a path of minimum weight from u to v and the minimum weight is defined as: (u,v)=min {w(p): p from u to v} The shortest path may not exist on the on a directed graph if there are some negative weights on the graph. This means that on a graph with some negatives edges, the algorithms such Bellman- Ford or Floyd- Warshal can be applied to solve this graph. However, if there is a circle of vertices exists on the path from u to v, and one of these vertices has a negative weight value, then the shortest path does not exist. This case causes the weight from u to v decreasing. The problem is called

3 negative path cycle. The figure.2 illustrate the case. is considered as settled or solved when the algorithm has found the shortest distance from the source to the vertex. The Dijkstra algorithm is able to identify the shortest path from the source vertex to all other vertices or a single verticex on the graph. Below are steps ho the Dijkstra s algorithm works: 1. Initialise the source vertex with 0 and set the vertex into the settled set. In additional, there is no shortest path between two vertices if these two vertices are not connected together. This also means that there is no path reach to the destination from the source vertex. The graph also has to avoid the triangle inequality problem so that algorithm can able to identify the shortest path. The triangle inequality problem is described as follow: For all vertices u,v,z V w:(u,v) w:(u,z) +w(x,z). This means the sum of two edge weights should be greater than a single edge weight in a graph that formed a triangle. 2. Identify the nodes that linked with the any solved vertex. 3. Calculate the distance between the solved nodes with the linked nodes using the edge weights, which connect between solved nodes and linked nodes. Compare the weight of unsolved nodes with each other. The unsolved nodes with the smallest weight will be move to settled set. 4. The process start again from step 1 but with the solved nodes just moved to settle set in step 4. The pseudo code of the algorithm: If the a directed graph is satisfied the all conditions above then it is possible to find a shortest path on this graph. 3.2 Dijkstra s Algorithm Model The Dijkstra s algorithm is an algorithm that can find a short path on a graph effectively. The algorithm separates the vertices into two separate, the set of unsettle vertices and the set of settled vertices and unsettle vertices. A vertex

4 4. The application and Empirical Modelling. 4.1 The application This empirical modelling project is about building a flight scheduling application, which allows user to find the shortest path to their preference destination, based on an algorithm and data structure model. The users can use the model to get a flight to the select destination at quickest route. In case user want to travel to the destination via a certain country, the application also will able find the shortest path. The graphical features are also applied to the model, so that user can visualise their journey. The focus of the application is to provide user a simple friendly graphical interface. The model will be built by employ number of tools that has been taught through out the course. The Dijkstra s algorithm will be used as the main factor to implement the application. As the application require representing locations on world map, therefore the application can be modelled using the graph theory. Base on the graph theory, the graph can represent the map, the vertices represent the locations, and the edges are the connection between the locations. 4.2 Empirical Modelling One of the main concepts in Empirical Modelling is observable and the slides [2] from Introduction to Empirical Modelling mentioned that EM is also means experience of construction. Therefore, after observing the Dijkstra s algorithm and analyse the algorithm s structure, this algorithm can be modelled with simple steps using EM construction tools. The implementation of the EM model of the Flight schedule application involves modelling the data structure and execution steps for algorithms. The data of the application are vertices and edges. These data can be represent on a table structure, where header of table holds the vertices and the row beneath this header are used to assign the weight of the nodes on headers. The figure below illustrate the structure: The algorithm that works in the application will perform its operation on the table above to manipulate the data. The next step is to define the steps in executing the algorithm to implement the Dijkstra algorithm base on the data structure above. Integrate the empirical modelling techniques, the following procedure are defined to implement the Dijkstra s algorithm to find the shortest path between two locations on the map. 1. Initialise weight of input vertices with Find the vertices that are connected to the source vertices. 3. Adding the weight of the input source to the weight of found connected vertices. 4. Update the new value to the table. However if the weight value can only be updated when it is small than the source vertex. 5. Remove the smallest header with its weight from the table. 6. Find the vertex with the smallest weight on the table again and go back to first step. Feed this value back into the algorithm. The procedures are repeated until the destination is reached. Generally the algorithm can be implemented using the available tools in Empirical Modelling. The algorithm above can be implement with provided EM tools such as EDEN language with its compiler tkeden. Eden provides a simple and efficient data structure, which is ideal for implement

5 application, which requires high computational resources. Simple List type provided by Eden is used to construct the data structure of graph to store the data. Then a major techniques use to process the data structured in this application is loop function. The extracted data from the graph by looping through each node on the graph are compared and analyse to identify the necessary. 5. Evaluate Empirical Modelling and its Before evaluate the Empirical concepts, this section will first discuss two main existing concepts which procedure and object- oriented programming. The object oriented is now has been adopted into many programming language. The main concepts are modelled components within an application as objects. The objects are described by data structure and behaviours are express as the functions. The object- oriented concepts allow to reused codes and data thoroughly. The objects can sometime be a in a form of a data type, therefore it can also be declared in another objects. However in a large- scale project it is difficult to identify the interactions between the objects. This leads to the difficulty in revealing fault in a system. The procedure- programming concept is also known as imperative concept, because the concept explicitly references to the state of the execution environment. The concept is contradicted to the object- oriented, the application is modelled by sequence of function, which execute step by step. Algorithms are easier to constructed by applying the procedure programming. In term of programming concepts, the Empirical Modelling has an abstract concept about modelling software. The application modelling in EM is based on the understanding, experiment and observing the factors that could be contributed into construction of the application. Therefore, EM does not restrict the software modelling into certain style as existing concepts. This leads to freedom- of- thinking in programming in an abstract way. Although, the tools for modelling in EM was built by existing concept and have shared some fundamental structured such as variables or operators, but the tool like Eden provides a much simpler data structure mechanism than existing concepts. The tkeden engine has a remark able computational ability and allows changing the state of the application in real- time. However, the tools such as Eden should have a better mechanism to manage the data within the data as well as extend API. The data management mechanism allows developer to find fault rapidly and monitor the state of the application. Furthermore, with more facility in extended API will help the developer to concentrate on modelling the problems. 6.Conclusion This paper has discussed in details about the graph theory, Dijkstra s algorithm and their implementation in the Flight Scheduling application. This report also described how Empirical Modelling techniques are applied to model the application. At last, a evaluation was given to evaluate compare EM concepts with existing concepts and the evaluate the different aspect of Empirical Modelling. The most valuable experience acquired from this project is that modelling is the most important process before construct the application. Reference: [1] /dcs/research/em/teaching/cs405/lect mon2.pdf [2] s/research/em/teaching/cs405/lectmo n2.pdf

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