Structural properties of bichromatic non-crossing matchings
Introduction
Let and be sets of red and blue points in the plane, respectively, with and . Let be a perfect matching of points in to points in using straight line segments, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. If the line segments do not cross, we refer to such a matching as a bichromatic non-crossing matching.
Geometric non-crossing matchings by straight line segments are widely researched. In case there are two groups of objects and the members of one group are to be matched with the members of the other, we naturally arrive to the bichromatic matchings, often also referred to as the red-blue matchings. The examples of real-life problems that fall into this category are numerous, with a whole range of the so-called problems of supply and demand, e.g. matching shoppers and shops, antennas and receivers, etc. A survey by Kaneko and Kano [1] gives an overview of various problems on red and blue points in the plane, including the matching problems. Several papers [2], [3] take a closer look at the algorithms for finding non-crossing planar straight line matchings between red points on one side and various blue objects (in more generality) on the other, devoting particular attention to the special case of blue objects also being points. Note that some geometric versions of the Monge-Kantorovich transportation problem, an optimization problem for matching mines with factories to minimize cost, see [4] for a survey, result in crossing-free straight line matchings of mines and factories, with possible multiplicities depending on the weight distribution.
Several papers [5], [6], [7] study the collection of all possible bichromatic non-crossing straight line red-blue matchings of two given equal-sized planar sets of red and blue points. A number of interesting structural properties of this collection are established, looking into the possibilities to gradually change one matching into the other through a sequence of matchings, such that every pair of consecutive matchings in the sequence has a non-crossing union. A similar problem on monochromatic point sets, where every pair of points is allowed to be matched, has also been looked at, see e.g. [8].
The problem of finding a geometric non-crossing Hamilton path that alternately visits the elements of a given red point set and equally sized blue point set, with certain additional requirements, was studied in [9], with an obvious connection to the red-blue matchings (as we can find one in each such Hamilton path). In [10], the connections of crossing-free red-blue planar matchings and crossing-free red-blue spanning trees are explored. Bounds on the total number of crossing-free red-blue perfect matchings are given in [11].
We take a closer look at the bichromatic non-crossing matchings of points in convex position. In this case, it is straightforward to see that two line segments of a matching cross if and only if their two pairs of endpoints are interleaved in the cyclic order around the convex hull of the given point set. Therefore, the collection of all valid matchings is fully determined by the sequence of red and blue points around the convex hull.
In order to efficiently deal with bichromatic non-crossing matchings on points in convex position we introduce a structure that we refer to as orbits, which turn out to capture well the properties of such matchings. As we will show, the points naturally partition into sets, i.e., orbits, in such a way that two points of different colors can be connected by a segment in a non-crossing perfect matching if and only if they belong to the same orbit. We go on to study the structure of individual orbits, their properties, as well as the relationship of different orbits of the same point set.
This apparatus enables us to get a grip on the bichromatic non-crossing matchings of points in convex position and work with them in a more efficient manner, with a potential to apply our machinery on the whole range of problems dealing with these matchings. We will present one such application in this article. It is worth noting that another application to several matching optimisation problems recently appeared in [12].
We will illustrate the applicability of our theory of orbits on the problem of efficiently finding the so-called bottleneck bichromatic non-crossing matching of points in convex position.
Denote the length of a longest line segment in a straight segment geometric matching with , which we also call the value of . We aim to find a perfect matching under given constraints that minimizes . Any such matching is called a bottleneck matching of .
The monochromatic variant of the problem is the case where points are not assigned colors, and any two points are allowed to be matched.
In [13], Chang, Tang and Lee gave an -time algorithm for computing a bottleneck matching of a point set, but allowing crossings. This result was extended by Efrat and Katz [14] to higher-dimensional Euclidean spaces.
The problem of computing bottleneck monochromatic non-crossing matching of a point set is shown to be NP-complete by Abu-Affash et al. [15]. They also proved that it does not allow a PTAS, gave a factor approximation algorithm, and showed that the case where all points are in convex position can be solved exactly in time. We improved this result in [16] by constructing an -time algorithm.
The problem of finding a bottleneck bichromatic non-crossing matching was proved to be NP-complete by Carlsson et al. [17]. But for the version where crossings are allowed, Efrat et al. [18] that a bottleneck matching between two point sets can be found in time.
Biniaz et al. [19] studied special cases of bottleneck bichromatic non-crossing matchings. They showed that the case where all points are in convex position can be solved in time, utilizing an algorithm similar to the one for monochromatic case presented in [15]. They also considered the case where the points of one color lie on a line and all points of the other color are on the same side of that line, providing an algorithm to solve it. The same results for these special cases are independently obtained in [17]. An even more restricted problem is studied in [19], a case where all points lie on a circle, for which an -time algorithm is given.
A variant of the bichromatic case is the so-called bicolored (or multicolored, when there are arbitrary many colors) case, where only the points of the same color are allowed to be matched. Abu-Affash et al. [20] examined bicolored matchings that minimize the number of crossings between edges matching different color sets. They presented an algorithm to compute a bottleneck matching of points in convex position among all matchings that have no crossings of this kind.
Using the orbit theory we solve the problem of finding a bottleneck bichromatic non-crossing matching of points in convex position in time, improving upon the best previously known algorithm of -time complexity. Also, combining the same tool set with a geometric analysis we design an optimal algorithm for the same problem when the points lie on a circle, where the best previously known algorithm has -time complexity.
As we deal with bichromatic perfect matchings without crossings, from now on, when we talk about matchings, it is understood that we refer to bichromatic matchings that are both perfect and crossing-free.
Also, we assume that the given points in are in convex position, i.e., they are the vertices of a convex polygon . Let us label the points of by in the positive (counterclockwise) direction. To simplify the notation, we will often use only indices when referring to points. We write to represent the set . Arithmetic operations on indices are done modulo . Note that is not necessarily less than , and that is not the same as . Definition 1 Balanced, Blue-heavy, Red-heavy A bichromatic set of points is balanced if it contains the same number of red and blue points. If the set has more red points than blue, we say that it is red-heavy, and if there are more blue points than red, we call it blue-heavy.
As we already mentioned, we assume that consists of red and blue points, i.e., it is balanced.
The following lemma is a well-known result that ensures the existence of a balanced matching on a point set. A couple of proofs, along with an algorithm that computes one such matching in time, can be found in [21]. Lemma 1 Every balanced set of points admits a matching. Definition 2 Feasible pair We say that is a feasible pair if there exists a matching containing .
We will make good use of the following characterization of feasible pairs. Lemma 2 A pair is feasible if and only if and have different colors and is balanced. Proof If is feasible, then and have different colors. Also, there is a matching that contains the pair , and at the same time the set , containing all points on one side of the line , is matched. Then must be balanced, so is balanced as well. On the other hand, if and are of different colors and is balanced, then both and are also balanced. Thus we can match with , and Lemma 1 ensures that each of the sets and can be matched. Clearly, the obtained matching remains crossing-free. □
The statement of Lemma 2 is quite simple, and we will apply it on many occasions. To avoid its numerous mentions that could make some of our proofs unnecessarily cumbersome, from now on we will use it without explicitly stating it.
The rest of the paper is organized as follows. In Section 2 we formally define orbits and derive numerous properties that hold for them. We note the existence of a structured relationship between orbits. This leads us to the definition of orbit graphs for which we show certain properties. In Section 3 we construct an efficient algorithm for finding a bottleneck matching of points in convex position. For this we follow the general idea from [16], but now we use orbits and their properties for the proofs. In Section 4 we again use properties of orbits and orbit graph to solve the problem of finding a bottleneck matching for points on a circle in time.
Section snippets
Orbits and their properties
Definition 3 (Functions and ) By we denote functions, such that is the first point starting from in the positive direction with being feasible, and is the first point starting from in the negative direction with being feasible.
As is balanced, Lemma 1 guarantees that both and are well-defined. Proposition 3 If a set is such that the number of points in of the same color as is not larger than the number of points of the other color, then . If a set
Finding bottleneck matchings
For the problem of finding a bottleneck bichromatic matching of points in convex position, we will utilize the theory that is developed for orbits and the orbit graph, combining it with the approach used in [16] to tackle the monochromatic case.
For the special configuration where colors alternate, i.e., two points are colored the same if and only if the parity of their indices is the same, we note that every pair where and are of different parity is feasible. This is also the case
Points on a circle
It this section we consider the case where all points lie on a circle. Obviously, the algorithm for the convex case can be applied here, but utilizing the geometry of a circle we can do better.
Employing the properties of orbits that we developed, we construct an time algorithm for the problem of finding a bottleneck matching.
We will make use of the following lemma. Lemma 33 [19] If all the points of lie on the circle, then there is a bottleneck matching in which each point is connected either to
Acknowledgments
We are grateful to the anonymous referees, whose useful and detailed comments improved our paper.
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Partly supported by Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-9/2021-14/200125).Partly supported by Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina (Grant No. 142-451-3227/2020-01).