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Spaceview: a Visualization Tool for Matroids of Rank at Most 3 The Pennsylvania State University The Graduate School Capital College SpaceView: A visualization tool for matroids of rank at most 3 A Master's Paper in Computer Science by Padma Gunturi °c 2002 Padma Gunturi Submitted in Partial Ful¯llment of the Requirements for the Degree of Master of Science April 2002 Abstract This paper presents some algorithms and a program called SpaceView for visualizing and manipulating matroids with rank at most 3. Simple matroids with rank at most 3 are also known as combinatorial geometries or linear spaces. The points and lines diagram for such a matroid is called its geometric representation. SpaceView takes as input a matroid of rank at most 3 and outputs its geometric representation. The user can move the points and lines, thereby animating the entire con¯guration. The user can perform matroid operations such as deletion, contraction and relaxation and can also obtain the bipartite graph corresponding to the inclusion relation between the points and lines. i Table of Contents Abstract i Acknowledgement iii List of Figures iv 1 Introduction 1 2 SpaceView 8 3 Design and Algorithms 13 4 Conclusion 19 References 20 ii Acknowledgement I would like to take this opportunity to express my deepest gratitude to my project advisor, Dr. S. R. Kingan. Her guidance and inspiration have been of tremendous help to me during the course of this project. Her patience and experience have helped me go through some of the most di±cult times in the project. I would like to extend my hearty thanks to Dr. T. N. Bui, not only for serving on my advisory committee, but also for introducing me to Matroid Theory. His suggestions and comments during the several group meetings were extremely helpful. I also would like to thank my academic advisor, Dr. L. Null and Prof. H. Royer for their immense guidance and support during my stay at Penn State, Harrisburg. I wish to thank Dr. R. J. Kingan for helping us build this visualization tool and continually giving us a helpful hand whenever we needed it. I would like to thank him also for sharing some of his experiences with us during this project. Thanks to Dr. Haidong Wu for recommending the book The Theory of Finite Linear Spaces. The book introduced me to the embeddability of linear spaces in projective planes and this knowledge has greatly helped us in developing SpaceView. Thanks to Karishma, Kate and Zhaoxia for making my stay at PSH both enjoyable and memorable. My special thanks to K. Richwine for all the help and encouragement during my master's program. Finally, I would like to thank my husband for his loving a®ection and constant encouragement during my study. iii List of Figures 1 Representation of K4 . 3 2 Well-known linear spaces . 5 3 Projective planes of order 3 and 5 . 7 4 Some other interesting linear spaces . 9 5 Fano representations after performing the matroid operations . 11 iv 1 INTRODUCTION 1 1 Introduction Computer science professor Andrew J. Hanson describes the importance of visualization in mathematics and physics as follows [8]: \Visualization in general embodies a transformation between a body of knowledge and a picture, or perhaps an interactive animation, capable of representing features of the data to the viewer. In general, the hope is that the displayed features will stimulate associations in the mind of the user that will lead to further insights, suggest new hypotheses to test, and thus advance the progress of science more rapidly than without this methodology. Mathematics and mathematical physics (like most other visualization domains, in fact) have potentially direct relations between the con- cepts in question and the choice of representation: it seems obvious to try to represent a geometric object by a faithful picture of the ge- ometry. Several situations typically arise when we ask whether such a picture can assist mathematical research. There is nothing con- ceptually new about the picture, but it was di±cult or impossible to construct without either computer assistance, or a clever technique for producing an easy computer representation of the geometry, or both. A new way of transforming the data to an image exposes aspects of the object not evident from the geometry alone. The ability to inter- actively alter, deform, optimize, re-parameterize, or manipulate the object permits us to examine limits and features that were unavail- able without a computer graphics representation and manipulation system. In general, the idea of simply seeing global, holistic features of an object that has never been depicted before can potentially sug- gest associations that algebraic analysis alone might miss." We designed and implemented a Java program called SpaceView for vi- sualizing and manipulating matroids with rank at most 3. We begin by reviewing background material on matroids with rank at most 3. In Chapter 2, we describe how SpaceView can be used as a research and pedagogical 1 INTRODUCTION 2 tool. In Chapter 3, we give the design and algorithm details. We conclude in Chapter 4 with suggestions for future work in this area. Matroids are a generalization of a number of combinatorial objects. Among them are graphs, matrices, linear spaces, a±ne spaces and projective spaces. The matroid terminology used here follows Oxley [13]. Portions of this in- troduction are taken from [5]. A matroid M is de¯ned as an ordered pair (E; I) consisting of a ¯nite set E and a collection I of subsets of E called independent sets such that: I1 ; 2 I I2 If I1 2 I and I2 ⊆ I1, then I2 2 I I3 If I1; I2 2 I and jI1j < jI2j then there exists an element e 2 I2 ¡ I1 such that, I1 [ e 2 I. A dependent set is a subset of E not in I. A circuit is a minimal dependent set. A matroid is called simple if it has no 1- or 2-point circuits. A basis is a maximal independent set. The rank of M is the size of a basis set. For a subset X of E, the rank of X is the size of a maximal independent subset in X. The closure of X is the union of X and all points e not in X, such that r(X [ e) = r(X). We say X is closed if r(X [ e) = r(X) + 1 for all points e not in X. A flat is a closed set. A hyperplane is a flat of rank r(M) ¡ 1. Matroids of rank at most 3 can be represented as a geometry in which the points, lines, and planes correspond to the rank-1, rank-2, and rank-3 flats, respectively. To simplify the diagram, lines with fewer than three points on them are not shown in the representation. Figure 1 and Figure 2 give representations of some popular matroids. A geometric representation for a matroid of rank at most 3 is a diagram of points and lines governed by the following rules: (1) All loops are marked in a single inset. (2) Parallel points (2-point circuits) are represented by touching points. 1 INTRODUCTION 3 1 4 2 5 5 6 1 3 ¡ ¢¡¢ 2 4 3 6 graphic geometric Figure 1: Representation of K4 (3) Any two distinct points belong to exactly one line and any line contains at least 2 distinct points. It follows from rule (3) that any two distinct lines meet in at most one point. For a graph G, let E be the set of edges and C be the set of cycles of G. Then (E; C) is a matroid called the cycle matroid of G. A matroid is called graphic if it can be represented as the cycle matroid of a graph. For a matrix A over a ¯eld F , let E be the set of columns and I be the set of independent sets of columns. Then (E; I) is a matroid called the vector matroid of A over the ¯eld F . A matroid is called representable if it can be represented as a matrix over a ¯eld. Since we are focusing only on rank 3 matroids, the only simple graphic matroids are subgraphs of the complete graph on four vertices, K4. A graphic and a geometric representation for K4 are shown in Figure 1. It should be noted that this paper is not really about graphs and the reader is encouraged not to think of these diagrams in terms of vertices and edges but rather as a geometry consisting of points and lines where the lines can be curved. We will now view rank 3 matroids from the perspective of linear spaces. Simple rank 3 matroids are also called linear spaces. There is much literature on linear spaces and an entire book called The Theory of Finite Linear Spaces which makes no mention of the word matroid. In fact, many matroid theorists prefer to call simple matroids combinatorial geometries. The linear space 1 INTRODUCTION 4 terminology used here will follow Batten and Beutelspacher [4]. This portion is taken from a survey on linear spaces by S. R. Kingan [11]. A linear space is a pair S = (P; L) consisting of a set P of points called points and a set L of distinguished subsets of points called lines satisfying the following: (S1) Any two distinct points of S belong to exactly one line of S; and (S2) Any line of S has at least two distinct points of S. A linear space is called non-trivial if there are three points not on a com- mon line. The number of points and lines are denoted by v and b, respectively.
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