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8660 results in Discrete Mathematics, Information Theory and Coding

Page 70 of 433

  • 12 - More on Convexity

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 125-140

  • Summary

    A puzzle from the Russian mathematical olympiads asks for a proof that, for any convex pentagon of points in a grid, there is another grid point on or inside the smaller convex pentagon formed by the diagonals of the given pentagon (Figure 12.1). Thus, the grid has no empty pentagons. The existence or nonexistence of empty polygons is not a monotone property, but it suggests the study of other properties, beyond the basic ones studied in Chapter 11, based on what the convex polygons of a configuration contain and not just which points formtheir vertices.

    Weak Convexity

    A configuration S is in weakly convex position if none of the points is interior to the convex hull of S: each point either is a vertex of the convex hull (as in configurations that are in convex position) or lies on a convex hull edge. For instance, grid(2, n) is weakly convex for every n, but grid(3, 3) is not. It is tempting to guess that the weakly convex configurations are defined by the property FORBIDDEN(TETRAD), by analogy to the fact that convex = FORBIDDEN (TETRAD, LINE(3)). The three-point line is allowed in weakly convex positions, so it should be dropped fromthe list of obstacles.However, this guess is not quite right. There is one configuration that is not in weakly convex position but has no tetrad: the quincunx (Figure 12.2), a configuration formed from the four points of a convex quadrilateral together with one more point where the diagonals of the quadrilateral cross.

    The two obstacles LINE(3) and TETRAD for convex position were derived from Carathéodory's theorem, that every point in the convex hull of a set of points belongs to the convex hull of two or three points from the set. The corresponding result for weakly convex position is Steinitz's theorem that every point interior to the convex hull is interior to the convex hull of three or four points from the set. Therefore, the two forbidden patterns for weakly convex position are the tetrad and the quincunx.

  • 3 - Configurations

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 8-18

  • Summary

    We begin our study with an examination of the different ways we can place small numbers of points in the plane, and what it means for two sets of points to be different.

    Small Configurations

    In how many different ways can we place n points in the plane? With a list of all of the possible placements, we could prove statements such as Klein's observation about convex quadrilaterals in five-point sets, automatically, merely by checking all the cases. When n is small enough, we can provide an explicit answer.

    Example 3.1

    There are two different ways of placing three points in the plane: they may either lie on a line, or theymay forma triangle.

    Four points may be arranged in four different ways:

    • a four-point line

    • three points on a line and one off the line

    • a triangle containing one point, or

    • a convex quadrilateral.

    So far, we have used only an intuitive notion for what it means for two sets of points to be the same or different. But when we get to five points, we already need to define more precisely what we mean. Figure 3.2 depicts 13 sets of five points. Are they all different from each other? Two of these are mirror images: should that count as two different sets of points or as two different views of a single way of placing five points?

    We will give a more precise definition of what it means for sets of points to be the same or different in the next section. For the definition we use, mirror images are not (in general) considered to be the same, so there are indeed 13 different ways of arranging five points. Open Problem 3.5, later in this chapter, formalizes the problem of counting configurations of different sizes.

    The numbers of sets of n points grow very quickly, as cn log n for a constant c > 1. Aichholzer et al. (2002) used computer searches to establish a database of small sets of points. To help control the size of the database, they exclude sets that have three points on a line and count mirror-image sets of points as equivalent. Despite these restrictions, the sets in their database still grow as cn log n, but with a smaller c.

  • 16 - Universality

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 177-187

  • Summary

    A planar graph is a graph that can be drawn in the plane with its vertices as points and its edges as noncrossing curves (often drawn as line segments).Universal point sets are point sets that can form the vertex set of any sufficiently small planar graph. Before defining this concept more carefully as the monotone parameter universal (Definition 16.1), we look at a computer puzzle that has popularized some of these concepts.

    Planarity

    Planarity is a computer puzzle game based on planar graph drawing, originally devised by John Tantalo and Mary Radcliffe. In it, one is presented with a tangled drawing of a graph (Figure 16.1). The goal is to untangle it, by moving the vertices one by one, until the result has no crossings. As the vertices move, the edges move with them as straight line segments. Each level of the puzzle presents a more complicated graph to be untangled. Although there exist computer algorithms that can solve these puzzles efficiently, their visual complexity nonetheless makes them a challenge to human puzzle-solvers.

    The original version of this puzzle generates its graphs from arrangements of randomly generated lines (Figure 16.2, left). However, the sharp angles of these arrangements make it difficult to reconstruct them while solving the puzzle. Indeed, even finding an arrangement that generates a given graph is a hard problem, much harder than finding an untangled drawing for the graph.

    Therefore, it can be easier to search for a solution to the puzzle that places the vertices on a grid (or a rough human approximation to a grid), such as the one in Figure 16.2, right. But when we're just starting the puzzle, we don't know what the whole grid drawing will look like, if it even exists. How big should we make the grid squares? If they're too big, the drawing won't fit in the window, and we'll have to waste time and moves shrinking it. But if they're too small, it will be more difficult to precisely place each of the graph vertices into its grid position.

  • 1 - A Happy Ending

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 1-3

  • Summary

    In the early 1930s, Hungarian mathematician Esther Klein made a discovery that, despite its apparent simplicity, would kick off two major lines of research in mathematics. Klein observed that every set of five points in the plane has either three points in a line or four points in a convex quadrilateral. This became one of the first results in the two fields of discrete geometry (the study of combinatorial properties of geometric objects such as points in the Euclidean plane, and the subject of this book) and Ramsey theory (the study of the phenomenon that unstructured mathematical systems often contain highly structured subsystems).

    Klein's observation can be proven by a simple case analysis that considers how many of the points belong to their convex hull. The convex hull is a convex polygon, having some of the given points as its vertices and containing the others. It can be defined mathematically in many ways, for instance as the smallest-area convex polygon that contains all of the given points or as the largest-area simple polygon whose vertices all belong to the given points. The convex hull of points that are not all on a line always has at least three vertices (for otherwise it could not enclose a nonzero area) and, for five given points, at most five vertices. If it has five vertices, any four of them forma convex quadrilateral, and if it has four vertices then it is a convex quadrilateral. The remaining possibility for the convex hull is a triangle, with the other two points either part of a line of three points or inside the triangle.When both points are inside, and the line through themmisses the triangle vertices, it alsomisses one side of the triangle. In this case the two interior points and the two points on the missed side forma convex quadrilateral (Figure 1.1).

    The challenge of extending and generalizing this observation was taken up by two of Klein's friends, Paul Erdʺos and George Szekeres. They proved that, for every k, a convex k-gon can be found in all large enough sets of points, as long as no three of the points lie on a line. Klein later married Szekeres, and their marriage is commemorated in the name of Erdʺos and Szekeres's result: the happy ending theorem.

  • 13 - Integer Realizations

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 141-150

  • Summary

    Every general-position configuration can be realized by points with integer coordinates. However, realizing a configuration that has three or more points in a line may require the use of noninteger coordinates. The existence of an integer-coordinate realization is a monotone property, as is the existence of a realization for which all distances are integers.

    Definition 13.1

    We define the property INTEGER-COORDINATES(S) to be true when S can be realized by points all of whose Cartesian coordinates are integers. We define the property INTEGER-COORDINATES(S) to be true when S can be realized by points all of whose pairwise Euclidean distances are integers.

    Another way of defining integer-coordinates(S) is that it is the property of being a subconfiguration of grid(n) for some n. On the other hand, it is not known whether all grids can be realized with integer distances. Figure 13.1 shows an integer-distance realization of grid(2, 3).

    Open Problem 13.2

    Which grids can be realized with integer distances?

    We could equivalently substitute the rational numbers for the integers. Any set of points whose coordinates or distances are rational can be scaled tomake them integers, without changing its order type. Klee and Wagon (1991) trace the history of the integer distance problem (in its equivalent formulation with rational numbers) to the seventh-century Indianmathematician Brahmagupta, who already asked about rational-distance realizations of POLYGON(4).

    The Perles Configuration

    The realizability of configurations using rational coordinates was considered by Micha Perles in the 1960s. Perles found an unrealizable configuration now called the Perles configuration, consisting of nine points at the corners and center of a regular pentagram (Figure 13.2).

    The properties of this configuration depend on the projective transformations of the plane and on a number defined fromquadruples of collinear points called their cross-ratio. A projective transformation is a transformation of the plane that preserves collinearities among all of the (infinitely many) triples of points in the plane. The cross-ratio of four distinct points A, B, C, and D appearing in that order along a line is defined as

  • 15 - Configurations from Graphs

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 166-176

  • Summary

    Many results on the hardness of subconfiguration-related problems can be based on graph drawing algorithms that create configurations fromthe vertices and edges of a graph. The value of a monotone parameter of configurations, on a configuration generated from a graph, can often provide useful information about the graph that the configuration came from. If this information would allow us to solve a graph problem that is known to be hard, then computing the parametermust be hard as well.

    Definitions

    In this sectionwe briefly summarize the terminology fromgraph theory thatwe need here.

    Definition 15.1 (graphs, vertices, and edges)

    A graph, for our purposes, consists of a pair (V, E) where V is a finite set and E is a set of unordered pairs of elements of V. The elements of V are called vertices (a single one is a vertex) and the elements of E are called edges.1 The two vertices of each edge are called its endpoints. A vertex and edge are incident if the vertex is an endpoint of the edge. Two vertices are adjacent if they are both the endpoints of the same edge.

    Definition 15.2 (graph drawings and planar graphs)

    A drawing of a graph is a representation of the vertices by points in the plane and the edges by curves from one endpoint to the other that avoid the other vertices. A crossing in a drawing is a point where two edges intersect, other than at a shared endpoint of the edges. A drawing is planar if it has no crossings, and a graph is a planar graph if it can be drawn without crossings. Although, as mathematical objects, the vertices of a drawing are points, we will represent them in figures by small disks, as we do for other kinds of points (Figure 15.1).11

    Definition 15.3 (paths and connectivity)

    A path in a graph is an alternating sequence of vertices and edges, starting and ending with vertices, and with each consecutive pair of a vertex and an edge incident to each other. A graph is connected if every two vertices can be the first and last vertex of at least one path. An isolated vertex is a vertex that has no incident edges.

  • 18 - The Big Picture

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 207-210

  • Summary

    As we have seen, many long-known problems from discrete geometry can be brought into the picture of monotone properties and parameters of configurations. Along the way, we have found obstacles and algorithms for many of these parameters.We have also shown thatmany pairs of these parameters are either equivalent for the purposes of fixed-parameter tractability (when each is bounded by a function of the other) or related by bounds in only one direction. Figure 18.1 shows some of these parameters and their relations.

    In the next two sections, we summarize the key results about these properties and parameters.

    Properties

    COLLINEAR is the property of having all points on a single line (Definition 8.2). It may be tested in linear time (Section 8.4). There is only one collinear configuration of each size n, the configuration line(n). It has one obstacle, polygon(3).

    CONVEX is the property of forming the vertices of a convex polygon (Definition 11.1). It may be tested in polynomial time by computing the convex hull (Section 11.4). There is only one convex configuration of each size n, the configuration polygon(n). It has a finite set of obstacles.

    GENERAL-POSITION is the property of being in general position, true of point sets with no three in a line (Definition 9.1). Thus, it has one obstacle, line(3). It may be tested in quadratic time (Section 9.2).

    INTEGER-COORDINATES(S) is the property of being realizable with integer coordinates, true of subconfigurations of grids (Definition 13.1). It includes all configurations in general position and all weakly convex configurations. We do not know its obstacles or computational complexity (Open Problem 13.4).

    INTEGER-DISTANCES(S) is the property of being realizable with all pairwise distances equal to integers (Definition 13.1). It includes all configurations in convex position (Section 13.2). We do not know its obstacles or computational complexity (Open Problem 13.4).

    STRETCHED is a property that characterizes the configurations generated by the stretching transformation (Definition 14.5), used to represent permutations by point sets (Observation 14.7). It may be tested in polynomial time (Theorem 14.23). Finding subconfigurations of stretched configurations is fixedparameter tractable (Theorem 14.26).

    WEAKLY-CONVEX is the property of being in weakly convex position (Definition 12.1). It has a finite set of obstacles andmay be computed in polynomial time (Section 12.2). The weakly convex configurations are not well-quasiordered (Theorem 12.3).

  • 4 - Subconfigurations

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 19-23

  • Summary

    How many sides can a convex polygon with vertices in a 5 × 5 grid have? To avoid spoiling this puzzle we defer the answer to Figure 11.3. Just as one set of points may be a subset of another, one configuration may be a subconfiguration of another. The puzzle seeks a polygon subconfiguration of grid(5, 5). We define subconfigurations more carefully in the rest of this chapter. In Chapter 5, we will use them to define monotone properties and parameters and to find obstacles, the subconfigurations that prevent a given configuration from having a property.

    Definitions

    Definition 4.1

    A subconfiguration of a configuration S is any configuration that can be obtained from a realization of S by removing zero or more points. If T is a subconfiguration of S, an instance of T in S is a one-to-one matching of points in T to a subset of points in S so the matched points have the same orientations in T and in S. Each configuration is a subconfiguration of itself; T is a proper subconfiguration of S if it is a subconfiguration but unequal to S.

    Example 4.2

    Figure 4.1 shows all of the configurations on one to four points, with lines indicating pairs of subconfigurations that differ from each other by the addition or removal of a single point. In this diagram, one configuration is a subconfiguration of another if the two configurations are connected by a monotone path. For instance, the one-point configuration at the bottom of the diagram is a subconfiguration of all of the other configurations.

    Note that the subconfigurations of a configuration T are not subsets of T, because T is not a set of points (it is an equivalence class of sets of points).

    Partial Orders and Quasi-Orders

    The subconfiguration relation between configurations forms an example of a partial order. This means that, if we write S→ T when S is a subconfiguration of T, then this relation →obeys the following three axioms:

  • • Reflexivity: For all configurations X, X → X.

  • • Anti-symmetry: For all configurations X and Y, if X _→Y and Y → X, then X = Y.

  • • Transitivity: For all configurations X, Y, and Z, if X →Y and Y → Z, then X → Z.

  • 2 - Overview

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 4-7

  • Summary

    Many algorithmic and combinatorial problems concerning finite sets of points have been studied in discrete and computational geometry. Often, the answers to these problems depend only on knowing, for each three points, whether they are in clockwise order, counterclockwise order, or lie on a single line. It is safe, for these problems, to throw away the coordinates of the points and retain only their configuration, which tells us this ordering information for each triple of points. In many cases, in addition, the property or quantity to be studied behaves predictably under the removal of points. If removing a point can never cause a quantity of interest to increase, we call that quantity monotonic. Our goal in this work is to provide a systematic study of the monotonic properties of configurations.

    Several old and colorfully named puzzles and games fit this pattern:

  • • The happy ending theorem was famously given its name after its proof led to the marriage of two of the mathematicians who discovered it, Esther Klein and George Szekeres. It is about how many points are needed (no three in a line) to ensure the existence of a convex polygon with a given number of corners.We described it already in Chapter 1.

  • • The orchard-planting problem, which we describe in Section 8.1, dates back to the early nineteenth century. It asks howmany rows of three trees one can formby planting an orchard with a given number of trees.

  • • We describe the no-three-in-line problem in Section 9.1. It was first posed in terms of placing 16 pawns on a chessboard, so that no three of them line up with each other. For this problem, it is important to consider all directions of lines, not just the horizontal, vertical, and diagonal directions of the chessboard.

    • Other topics of more serious past research also concern monotonic properties of configurations.

  • 6 - Computing with Configurations

  • David Eppstein, University of California, Irvine
  • Book:
    Forbidden Configurations in Discrete Geometry
    Published online:
    04 May 2018
    Print publication:
    17 May 2018, pp 33-42

  • Summary

    Howmany triangles are there in the configuration of Figure 6.1? To answer this, one needs either some mathematical insight or tedious calculation. But performing tedious calculations is something computers do well. So, how could a computer be used to solve this puzzle?

    Input Representation

    Before we ask how to design a program to find triangles or other subconfigurations in larger configurations, we need to consider how a configuration can be described as the input to a program. In the case of Figure 6.1, the configuration is already shown with grid lines, making it easy to represent its points using integer Cartesian coordinates. But not every configuration can be represented in this way (see Chapter 13 for details). Even the configurations that do have integer coordinate representations sometimes need huge numbers for those coordinates, forcing algorithms with this kind of input to use multiprecision arithmetic and complicating their analysis.

    Instead, for algorithms on configurations, we will generally assume only that our algorithms have some way to identify the points of a configuration and to determine the orientations of triples of points. Algorithms that access their input in this limited way are more versatile. They can handle inputs using explicit coordinates for the points, as long aswe can compute orientations from coordinates. But they can also handle a three-dimensional matrix of orientations as input. It would even be possible for such an algorithm to take as its input a number of points and a pointer to a subroutine that maps triples of indexes to the orientation of the points with those indexes. When analyzing these algorithms,we will assume that each orientation test takes constant time, but if not, the time bounds we state should be multiplied by the time per orientation test.

    Despite being restricted in this way, it is still possible for an algorithm to carry out many important geometric tasks, such as the construction of the convex hull of the input points. Recall from Chapter 1 that the convex hull is a convex polygon, the smallest convex polygon that contains the set. Figure 6.2 shows an example. For points that are not all on a single line, the convex hull may be determined from the order type, as follows.

  • A (5,5)-Colouring of Kn with Few Colours

  • Part of
  • ALEX CAMERON, EMILY HEATH
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 6 / November 2018
    Published online by Cambridge University Press:
    09 May 2018, pp. 892-912

  • For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

Page 70 of 433