Algorithmics, Complexity, Computer Algebra, Computational Geometry

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.

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

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.

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).

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:

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:

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.

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).

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph G0 in the common random graph models ${\cal G}$(n,m) and ${\cal G}$(n,p). Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of G0. This extends an argument given earlier by the second author for G0=K3 with a more restricted range of average degree. For all strictly balanced subgraphs G0, our results give much information on the distribution of the number of copies of G0 that are not in large ‘clusters’ of copies. The probability that a random graph in ${\cal G}$(n,p) has no copies of G0 is shown to be given asymptotically by the exponential of a power series in n and p, over a fairly wide range of p. A corresponding result is also given for ${\cal G}$(n,m), which gives an asymptotic formula for the number of graphs with n vertices, m edges and no copies of G0, for the applicable range of m. An example is given, computing the asymptotic probability that a random graph has no triangles for p=o(n−7/11) in ${\cal G}$(n,p) and for m=o(n15/11) in ${\cal G}$(n,m), extending results of the second author.

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.

This book surveys the mathematical and computational properties of finite sets of points in the plane, covering recent breakthroughs on important problems in discrete geometry, and listing many open problems. It unifies these mathematical and computational views using forbidden configurations, which are patterns that cannot appear in sets with a given property, and explores the implications of this unified view. Written with minimal prerequisites and featuring plenty of figures, this engaging book will be of interest to undergraduate students and researchers in mathematics and computer science. Most topics are introduced with a related puzzle or brain-teaser. The topics range from abstract issues of collinearity, convexity, and general position to more applied areas including robust statistical estimation and network visualization, with connections to related areas of mathematics including number theory, graph theory, and the theory of permutation patterns. Pseudocode is included for many algorithms that compute properties of point sets.

Algorithmics, Complexity, Computer Algebra, Computational Geometry

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7009 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

1 - A Happy Ending

Summary

13 - Integer Realizations

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9 - General Position

11 - Convexity

15 - Configurations from Graphs

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18 - The Big Picture

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4 - Subconfigurations

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10 - General-Position Partitions

2 - Overview

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6 - Computing with Configurations

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

The Probability of Non-Existence of a Subgraph in a Moderately Sparse Random Graph

Triangle-Tilings in Graphs Without Large Independent Sets

Forbidden Configurations in Discrete Geometry

Index

Index of Symbols

Frontmatter

7 - Permutation Invariance and Unitarily Invariant Measures

4 - Unital Channels and Majorization

2 - Basic Notions of Quantum Information

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Forbidden Configurations in Discrete Geometry