ANIMATION OF ALGORITHMS ON GRAPHS

Transcription

1 Master Informatique 1 ère année MASTER 1 ENGLISH REPORT YEAR ANIMATION OF ALGORITHMS ON GRAPHS AUTHORS : TUTOR : MICKAEL PONTON FREDERIC SPADE JEAN MARC NICOD

2 ABSTRACT Among the units studied in our fourth year in Computer Sciences at the University of Franche-Comte, the supervised project is one of the more important practice parts during the second semester. This project's particularity is that we have to write, as well as present it orally, in English. The purpose of this report is to present the work done on our project whose subject was "Animation of algorithms on graphs". This presentation rstly contains some basic denitions on graphs to allow everyone to easily understand the subject; secondly, a description of purpose of application and specications; thirdly, an explanation of our resolution of the matter.

3 Contents 1 Introduction 1 2 BASIC DEFINITION AND MAIN USES OF GRAPHS Denition Main uses OUR PROJECT SPECIFICATION Context Specication OUR ANALYSIS 6 5 SOLUTIONS Tutors Maintenance algorithms Compilation tool Colours of the graph Students REGRETS 9 7 Conclusion 10

4 1 Introduction A graph is the data structure the more general to describe many situations. They are therefore used in our daily lives often to solve dicult problems. For example, nding the best schedule for a review session minimizing its duration is equivalent to a colouring problem on graphs, nding the best duration of a project can be summed up in search of the longest path in the graph of the tasks of the siteâ Algorithms on graphs are numerous. Their understanding is not always easy. However, with an example, the explanations are more convincing. Therefore, Mr Nicod, lecturer at the University of Franche-ComtÃ, would like to make available to its students the animations of the behaviour of algorithms being presented. The aim of our project is to answer this need. 1

5 2 BASIC DEFINITION AND MAIN USES OF GRAPHS The representation of a problem by a plan, a drawing, an outline often contributes to a better and quicker understanding. The theory of graphs is constructed, originally, on this principle. 2.1 Denition A minimum of two points, a link between these two points, here is the minimalist structure of a graph! More seriously, there are several ways to dene a graph. The easiest way may be to see it as a set of vertices or points connected by edges. Indeed, graphs are used to describe a set of objects and their relationships, i.e. links between objects. So the objects are nodes of the graph and links between objects are vertices of the graph. On the scheme above, we can see these two dierent components: The nodes which are the bubbles marked by letters a to h The edges that are linking nodes each other Edges have two particularities: 1. They can be oriented or non-oriented. The direction of an edge matches with the direction allowed to go from one node to another. In case of an oriented edge, the line turns into eche and is also called an arc. 2

6 2. They can be weighted. A value, also named weight, is attributed for the path between both nodes linked by the weighted edge. This value can be for example a time or a distance. This type of structure allows modeling many situations such as a computer network, a network of water or electricity, road maps. Thanks to the graphs and the application of specic algorithms on these graphs, it becomes possible to resolve and even to optimize many complex problems like nding optimal paths and cycles. A path of a graph is a sequence of the edges travelled to go from a node to another node. A cycle of a graph is the chain of nodes covering the entire graph and in which the initial node is also the nal one, and in which every edge is used only one time during the "travel". On the scheme above, we can see in red: a path and in green: a cycle. 2.2 Main uses The range of graph uses is large. Indeed, the pair graph / algorithm on this graph is a resolution tool commonly used in our environment. For example: In the city of Grenoble, the optimization of circuit garbage collection is based on a graph and its algorithm. In the industrial world, this tool is essential in warehouses for an optimal management of the movement of loaded robots that retrieve pallets of products. On the urban side, the choice of a street to become one-way without making impossible city trac is validated by a graph and its algorithm. The pair graph/algorithm is also very useful to solve the problem of construction of highways linking major cities with the lowest total mileage. More surprising, the algorithm for graph 3

7 coloring has determined that only four colors are necessary, on the France map, to full color all departments with no neighbour of the same color. 4

8 3 OUR PROJECT SPECIFICATION Our project focuses on the animation of algorithms on graphs. The aim is to provide to M. Nicod, a powerful educational tool to its students under the teaching of graphs. The animation of the behavior of algorithms presented will be a precious complement in convincing explanation and understanding. 3.1 Context There is existing software written in Java. In fact, two students of professional license have made the codes allowing the graphic seizure of a graph. Then, as part of their supervised project of Master 1 in Computer Science, two others students have added the export of an algorithm on the current graph in a Flash animation. These developments will be our base of work. 3.2 Specication This project involves 2 phases. On the one hand: The rst phase consists in making a reliable mode of production of animations. Indeed, we will have to correct the existing code, provide additional functionality and validate the code. On the other hand: The second part focuses on the programming of several algorithms on graphs. To enable the display of up animation graphs, the program needs a library of algorithms applied on graphs. The aim of this phase is to understand and write algorithms such as nding shortest paths, the computation of a spanning tree of minimum weight of the current graph. The animation of the graphs must operate under dierent operating systems (MS Windows, Linux, Mac OSX). The programmation language is Java. The educational aim of this project is the implementation of all the teachings related to Java programming and graph theory. It is, moreover, a subject with a strong algorithmic connotation. 5

9 4 OUR ANALYSIS We are convinced of the enormous interest of this application in educational level. Be able to broadcast animation graphs with no limit of the environment, without special software, except a browser, is a vital asset. The application could, for example, nd all its value in the context of distance education. Each student would be able to see the animation of all the algorithms described in the fascicles of the course with a single click in his browser. The courses would be easier to begin with, more alive, more interactive and more understandable through these illustrations. This requires that: The application allows the management of a maximum of algorithms (add/modify/delete), simulations are reliable and customizable, export in the ash le is eective Our main goal was to complete and make reliable to the maximum the existing software. To achieve this, we took ownership of the existing code. This has been a long and dicult, but necessary, step in this project. We have been able to purify, sanitize, commenting the code. In addition, improved ergonomics and user-friendly software has been one of our concerns with each change or addition of functionality. Our approach has always been the same for modify the existing as for add functionality: report of a problem or a lack, analysis, integration of our solution. In our view, there are two categories of users of the application: Tutors who are responsible for managing the addition / modication / deletion of graph algorithms to simulate as well as the simulation modes Students must be able to view the animations in any browser in any environment. 6

10 5 SOLUTIONS 5.1 Tutors The application must enable the management of a maximum of algorithms. We have therefore extended the capabilities of existing software to avoid restrictions in the addition of algorithms to animate. Each algorithm requires specic settings that can greatly vary from one algorithm to another: for example, the number of parameters, the weight of edges, the direction of an edge, and many other more technical. We changed the code in order not to restrict the application. Furthermore, tutors need an interface functional and reliable to manage algorithms. We mainly provide solutions to following problems: Dicult maintenance of algorithms Compilation tool inecient and not user-friendly Delicate management of simulation modes Maintenance algorithms After analysis, it appears that maintenance algorithms is dicult because all the algorithms are in a single le. Each public method is an algorithm. Our solution is to separate each algorithm and create as many le as algorithm Compilation tool Compile an algorithm always fails in the editing window, even if the algorithm is correct. In addition, after the compilation of an algorithm, no message appears in the result window. The user has no explanation of errors occurred during compilation. We wanted to add results of the compilation in the result window. Now, after modifying the code, the application displays, in addition to the message of success or failure of the compilation, the output of the compiler. This includes details about the errors and warnings. Before modication: 7

11 After modication: Furthermore, when compiling the project, are now erased binaries les.class, executables, documentation Java.doc and the user les (cong, graphs and algorithms) Colours of the graph Colour management for an algorithm is dicult. Indeed, the user must rst create the dierent colours required for the algorithm; otherwise, the algorithm does not work. This is now xed. In addition, in the original program, all algorithms have the same colours. For purposes of visibility, we decided to propose a personalization of colours for each algorithm. We therefore integrate the separation of modes of simulation algorithm and also changed the code so that the predened modes (default and selection) and user modes are stored in dierent lists. 5.2 Students Students need to view Flash animations of algorithms on graph. Moreover, these animations are subject to the constraint of portability. So, Flash export must be reliable 100To achieve this, we have solved the following problems: 8

12 In some situations, the ash does nothing (when adding the cartridge for example), In the animation exported from some algorithms, the edges do not change colour or there is no animation at all, Export on Windows does not work 6 REGRETS Writing algorithms was approached just at the end of the project. So we could not write a lot of algorithms to simulate. However, we have tried to implement the algorithm of Prim. The aim of this algorithm is to solve a large family of problems involving interactions between dierent components of a system where you want a service or provide communications at low cost. It is important to see that we are here from the perspective of the designer and not the user. Most often that family problems back to physically connect the components by eliminating redundancy for a system as simple as possible: a tree. For example, an operator must connect cities with a new broadband network. A number of direct connections point to point between the cities are technically possible. The distance between two cities in the nal network has little importance in view of the ow but the costs of the network are obviously not the same for the various point-to point. The operator therefore needs to determine how to connect all cities in minimizing the total cost of the network. As we can see in the scheme below, this algorithm selects just any edges (red marqued link). So to connect cities each others, it can be necessary to go through an indirect path. The direct edge is sometimes not marqued by the algorithm. For example to link Caen to Rennes, the path of minimum cost is not directly Caen-Rennes, but Caen-Angers-Rennes. 9

13 7 Conclusion Figure 1: example of the prim algorihm To conclude, this project was very instructive. From a technical point of view, it was for us a real addition to our course on graph theory. We have been able to go a little deeper into this fascinating subject. Indeed, we had to dissect, understand and try to write algorithms on graphs such as Prim or Bellman-Ford. Moreover, the reliability phase of the existing program is also highly instructive. This step was very timeconsuming. The diculty in understanding existing code led us to measure the importance of standard programming, the creation of an intuitive code. Several times, the desire to rewrite many parts of the programs has been very strong. Each developer has its own logic of development and project organization. However, existing sources surely allowed us to more easily identify the parts of code to change, what could fail and has focused on features that seemed interesting to implement in the program. From a human point of view, it is not always easy to work on several on the same project for many reasons. First of all, each person has his own vision of a project and the developments that he wants to give it despite a same specication. Then, each has his own way to plan, organize the tasks it must accomplish. Finally, we must know how to optimize human resources team like as knowledge, skills, experience of each to overcome the various diculties met throughout the project. We therefore measured the importance of working procedures, coordination of development and communication. In our point of view, the main improvement is assuredly expanding to the maximum the library of algorithms. 10